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## April 21, 2010

### On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

#### Posted by Simon Willerton

Back in my recent post on intrinsic volumes I said that I was just finishing a paper on the magnitude of certain Riemannian manifolds. It is now essentially finished.

I will put it on the arxiv next week: comments and suggestions are welcome.

There are two main results in the paper. The first is a calculation of the magnitude of the $n$-sphere and the second is the asymptotic behaviour of the magnitude for an arbitrary homogeneous Riemannian manifold in terms of volume and total scalar curvature, a special case of the latter is the asymptotic behaviour of the magnitude of a homogeneous surface. I will explain these below, but first remind you a little about magnitude.

### On the Magnitude…

Magnitude of finite metric spaces (under the name cardinality) was introduced by Tom Leinster at this very café, by analogy with his notion of Euler characteristic of a category. In two posts, Entropy, Diversity and Cardinality (Part 1) and Part 2, he explained how this was related to ideas in ecology and went on to prove a result on how magnitude helps you maximize biological diversity. In another direction Tom and I defined the magnitude of various infinite subsets of Euclidean space by approximating these with finite metric spaces. In the case of a finite metric space, as the space is scaled up, the magnitude gets closer and closer to the number of points: in the case of some nice subsets of Euclidean space we observed that as the subset is scaled up, the magnitude seems to tend to some linear combination of classical ‘intrinsic volumes’ of the subset.

[If you want some idea of the intuition I use to think of the magnitude then you should look at the post More Magnitude of Metric Spaces and Problems with Penguins.]

In this paper I take a different tack in defining the magnitude of infinite metric spaces, one that was suggested to me by Tom Leinster and by David Speyer. This involves using a (signed) measure on a metric space, and it clearly generalizes the idea of having a weight associated to each point in a finite metric space. Using this approach gives the same answer as we got in examples we’d previously calculated.

A key observation, essentially made by David Speyer here, is that for a homogeneous space $X$ (i.e., one that looks the same at every point) the magnitude can be calculated using any measure $\mu$ by the formula

$\left|X\right|= \frac{\int_{x\in X} d\mu}{\int_{x\in X}e^{-d(x,y)} d\mu},\qquad\text{for any} y\in X$

provided that the denominator is finite and non-zero. Aside from working with subsets of the line, this is currently the only way I know of calculating the magnitude of infinite metric spaces.

### …of Spheres,…

There are two obvious metrics to consider on the $n$-sphere: there’s the ‘intrinsic’ metric, coming from thinking of it as an abstract Riemannian manifold and there’s the ‘Euclidean subspace’ metric, coming from thinking of it as a subset of $\mathbb{R}^{n+1}$. On the Earth, the intrinsic distance between Sheffield and Riverside is how far I would have to travel over the surface of the Earth to get from one to the other; whereas the Euclidean subspace distance is how far I would have to travel through the Earth if I dug a straight-line tunnel from one to the other. In previous work, Tom and I concentrated on spaces with the subspace metric, but here I am concerned with the intrinsic metric.

It is a very fun exercise to use Speyer’s formula to explicitly calculate the magnitude for $S_R^n$, the $n$-sphere of radius $R$ with its intrinsic metric (or at least I found it fun). This is done in the paper and gives the following answer.

$\left|S^n_R\right|=\begin{cases} \frac{2 \Big(\Big(\frac{R}{n-1}\Big)^2 + 1\Big)\Big(\Big(\frac{R}{n-3}\Big)^2 + 1\Big) \dots \Bigl(\Bigl(\frac{R}{1}\Bigr)^2 + 1\Bigr) }{1+e^{-\pi R}}& \text{for n even}\\ \frac{\pi R \Bigl(\Bigl(\frac{R}{n-1}\Bigr)^2 + 1\Bigr)\Bigl(\Bigl(\frac{R}{n-3}\Bigr)^2 + 1\Bigr) \dots \Bigl(\Bigl(\frac{R}{2}\Bigr)^2 + 1\Bigr) }{1-e^{-\pi R}}\quad& \text{for n odd}. \end{cases}$

The denominator rapidly tends to one as $R$ increases. So the large, or asymptotic, behaviour of the magnitude is governed by the numerator which, for fixed dimension $n$, is a polynomial in $R$ with some nice properties. For instance the leading order and constant terms can be identified in terms of classical invariants, namely the volume and the Euler characteristic, so, writing $\omega_n$ for the volume of the unit $n$-ball, the numerator is the following form:

$\tfrac{1}{n!\omega_n}vol(S^n_1)R^n +\dots+\chi(S^n_1).$

This is the kind of thing that Tom and I had been expecting for spaces with the Euclidean subspace metric. However, we will see below that the other term we can identify differs from what we naively expect in the Euclidean metric case.

One obvious question to ask here is the following.

• What is the magnitude of the $n$-sphere with the subspace metric?

### …Surfaces…

All closed, compact surfaces (spheres, tori, etc.) can be equipped with a homogeneous Riemannian metric. For the sphere this should be clear as it is naturally thought of as a homogeneous space. Also the torus is commonly thought of as the quotient of the plane by an integer lattice, making it homogeneous. Similarly every higher genus surface can be obtained by quotienting the hyperbolic plane with an appropriate group of isometries, analogous to $\mathbb{Z}^2$.

For a surface with a homogeneous metric we can use Speyer’s formula from above to calculate the magnitude. I haven’t been able to find any nice closed form for the magnitude of higher genus surfaces as in the case of spheres, but there’s a nice computation giving the asymptotic behaviour. For a metric space $X$ and $t\gt0$ we can write $t X$ for $X$ with the metric scaled up by a factor of $t$. The for a surface $\Sigma$ with a homogeneous metric, as $t\to \infty$

\begin{aligned}\left|t\Sigma\right|&= \tfrac{1}{2\pi}Area(t\Sigma)+\chi(t\Sigma) +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}). \end{aligned}

This is precisely the kind of behaviour Tom and I had conjectured for subspaces of Euclidean spaces.

There are four main ingredients in my proof of this result: one is Speyer’s formula for the magnitude of a homogeneous metric space mentioned above, another is Watson’s Lemma, a staple of asymptotic analyis, and the other two are properties of Gauss curvature mentioned in my post on intrinsic volumes, namely that the Gauss curvature determines the circumference of small circles (at least to low order), and that the integral of the Gauss curvature is proportional to the Euler characteristic.

If we are calculating $\left|t\Sigma\right|$ using Speyer’s formula, then the key thing is to understand the denominator; so for any fixed $x_0\in \Sigma$ we want to consider

$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x.$

If $t$ is very large then because of the negative exponential, parts of $\Sigma$ which are not very close to $x_0$ will only give exponentially small contributions. This rigorous way to state this is Watson’s lemma, but the point is that for large $t$ the dominant contributions will come from local information, i.e., curvature type information.

Writing $F_r$ for the subset of $\Sigma$ at a distance of $r$ from $x_0$ we can rewrite the the above integral as follows

$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=\int_{r\ge 0}e^{-t r} Length(F_r) d r.$

But we know that for small $r$ the length, or circumference, of $F_r$ is determined at low order by the Gauss curvature $K(x_0)$.

$Length(F_r)=2\pi r \big(1 - \tfrac{1}{6}K(x_0)r^{2}+O(r^{4})\big)\qquad{as} r\to 0.$

Using this, together with Watson’s Lemma we get that the integral has the form

$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big) \qquad{as} t\to\infty.$

This means that as $t\to \infty$ we find, as we wanted, that

\begin{aligned} \left|t\Sigma\right| &=\frac{\int_{x\in\Sigma} d x}{\int_{x\in\Sigma} e^{-t d(x,x_0)}d x} =\frac{\int_{x\in\Sigma} d x}{2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big)}\\ &=\tfrac{1}{2\pi}t^2\int_{x\in\Sigma} d x + \tfrac{1}{2\pi}\int_{x\in\Sigma} K(x_0)d x +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}), \end{aligned}

by the fact that the Gauss curvature is constant on a homogeneous space and that the integral of the Gauss curvature is proportional to the Euler characteristic.

### …and Other Homogeneous Spaces

For an arbitrary homogeneous Riemannian manifold $X$ of dimension $n$, you can use the same technique as for surfaces to calculate the leading order terms of the asymptotics of the magnitude. The key point is to identify the relevant analogues of the Gauss curvature and the Euler characteristic. These turn out to be the scalar curvature and the total scalar curvature.

At every point $x$ of a Riemannian manifold $X$ there is a scalar curvature $\tau(x)$ which specializes to twice the Gauss curvature in the case of a surface. This has the property of the Gauss curvature that at a point $x$ the volume of the set of points $F_r$ a distance $r$ from $x$ is, for small $r$, determined to leading terms by the scalar curvature:

$vol_{n-1}(F_r)= r^{n-1}\sigma_{n-1}\left(1-\tfrac{\tau(x)r^2}{6n}+\dots\right)\quad \text{as} r\to 0.$

The total scalar curvature, $tsc(X)$, is defined to be the integral of the scalar curvature;

$tsc(X)\coloneqq\int_X \tau(x) d x,$

The total scalar curvature seems to crop up in various parts of mathematics and physics, but I don’t know much about that. The connection with intrinsic volumes is that the total scalar curvature is, for closed manifolds, the next non-trivial intrinsic volume after the volume:

$\mu_{n-2}(X) = \frac{1}{4\pi}tsc(X).$

In the case that $X$ is a surface (so $n=2$) we recover the relationship between the total Gauss curvature and the Euler characteristic

Running through through the same analysis as for the case of surfaces above, you find that for $X$ a homogeneous Riemannian manifold the magnitude looks asymptotically as follows:

$\left|t X\right| = \tfrac{1}{n!\omega_n}vol(X)t^n+\tfrac{n+1}{6n!\omega_n}tsc(X)t^{n-2}+O(t^{n-4}).$

There are a few questions that jump out at this point.

• Can these methods be used to calculate further terms in the asymptotic expansion for homogeneous manifolds?
• Are these leading terms the same for the asymptotic behaviour of non-homogeneous Riemannian manifolds?

I should probably note here that the asymptotic behaviour is only one part of the picture and it would also be good to know what is going on for ‘small’ spaces: what is the magnitude measuring?

Posted at April 21, 2010 7:33 AM UTC

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### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Thanks for the post and paper. I’m a big fan of this magnitude of metric spaces stuff.

Once every couple of years I pick up Street’s paper “Algebra of oriented simplexes” hoping my math kung fu has progressed enough for me to finally understand it. Coincidentally, the last time I picked it up recently made me think of this magnitude of metric spaces stuff. Probably for no good reason other than lack of sleep though. Let me throw some thoughts out there and see if anything makes sense…

I liked the idea of relating the 1-cocycle condition to a composite triangle in a category, i.e.

$\partial[012] = [12]-[02]+[01] = 0$

rearranges to

$[02] = [01]+[12]$

which is interpreted as a commuting triangle.

Similarly

$\partial[0123] = [123]-[023]+[013]-[012] = 0$

rearranges to

$[123]+[013] = [023]+[012].$

The pictures that go with this are great.

Then I had a whacky idea. What if we turn the square brackets into magnitudes instead, i.e.

$|02| = |01|+|12|.$

This says the magnitude of [02] equals the sum of the magnitudes of [01] and [12]. But if magnitude is “volume” then we are talking about lengths of the sides of a triangle. When is the sum of two lengths of a triangle equal to the length of the third side? When the triangle is degenerate, i.e. has no area. Area is zero? Boundary is zero? Could they be related?

This idea bumps up trivially in dimension.

The 2-cocycle condition gives

$|123|+|013| = |023|+|012|$

which says the sum of areas of two triangles of a tetrahedron are equal to the sum of the areas of the two remaining triangles of the tetrahedron. When does that happen? It happens when the 3-simplex is degenerate, i.e. it has no volume.

The picture is kind of pretty. You partition the boundary of an $n$-simplex into two $(n-1)$-complexes. The $(n-1)$-cocycle condition says the magnitude of these two complexes are equal.

We have magnitudes of metric spaces and magnitudes of categories, is there a such thing as a magnitude of a diagram in a category?

Apologies if this makes absolutely no sense and is completely irrelevant to this magnitude business :)

Posted by: Eric Forgy on April 21, 2010 4:14 PM | Permalink | Reply to this

### First reactions

Because a diagram is a functor, it has a domain category, with its own magnitude; but the image is a subcategory, too (unless that notion is evil?), which again has its own magnitude. Maybe you want some combination of these two?

Posted by: some guy on the street on April 22, 2010 12:05 AM | Permalink | Reply to this

### Re: First reactions

Hmm…

If you start with the category $\mathbf{1}$ with two objects 0,1 and one non-identity morphism $\to$ and a functor $F:\mathbf{1}\to C$ with $F(0) = F(1)$, then the image of $F$ is not a subcategory of $C$ unless $F(\to) = 1_{F(0)}$. Is that what you meant by evil?

I spent about a week on the n-Forum (starting around here) making these guys pull their hair out because I was making a mistake thinking the image of a functor was a category.

After all that, please do not now tell me the image of a functor is a category after all :)

We have magnitudes of metric spaces and magnitudes of categories, is there a such thing as a magnitude of a diagram in a category?

I had the magnitude of the image of a functor in mind (which need not be a category). More generally, I’d be interested if there was a magnitude for a general directed graph.

Now the admittedly weak link to magnitudes is that I was wondering if $\partial$ might be considered a functor mapping categories (or diagrams or graphs) to magnitudes.

Posted by: Eric on April 22, 2010 1:54 AM | Permalink | Reply to this

### Re: First reactions

Oops. I guess you guys call that category $\mathbf{2}$ :)

Posted by: Eric on April 22, 2010 4:54 AM | Permalink | Reply to this

### Re: First reactions

oh! hmm… it’s weirder than I took care to think about, this image-of-a-functor business. Of course, the identity morphism will be in there, but unless the image of $\to$ is an involution… I really should try to remember these issues more carefully!

More carefully, we should probably take the right factorization of the diagram $F:\mathcal{D}\rightarrow\mathcal{C}$ as $\mathbf{2}\longrightarrow \Big\langle Hom(0,1)\mathllap{:}{\,=}\mathcal{C}(F(0),F(1))\Big\rangle \longrightarrow \mathcal{C}|_{\{F(0),F(1)\}} \longrightarrow \mathcal{C} ;$ being the factor functors forgetting only structure, only stuff, and only property, respectively — and wherein none of the intermediate categories are really the image of the functor where you wanted to define a magnitude. So… take or leave what you will.

Posted by: some guy on the street on April 22, 2010 7:21 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

One thing that immediately strikes me when looking at this is the similarity of all the definitions here to things that come up when using the energy method for bounding Hausdorff dimensions.

For instance the formula

(1)$\int_X \exp(-d(x,y)) d\mu(y) = 1$

is precisely the expression (in the language of the energy method, which of course is in the language of physics) that the potential felt by the point $x$ with mass distribution $\mu$ and forces that decay exponentially is one.

Similarly the expression

(2)$|X| = \int_X d\mu(x) = \int_X\int_X \exp(-d(x,y)) d\mu(x)d\mu(y)$

is precisely the statement that the magnitude of the space is the total mass required, or equivalently the total potential energy of the system.

This leads to a nice physical interpretation: that the magnitude of a metric space is exactly the amount of mass needed to be spread out over the metric space so that every point feel a total potential energy of one (when the potentials/forces decay exponentially).

I don’t think there’ll be any direct connection to Hausdorff measures though because everything there relies upon a potential which is singular at small distances, whereas this is “regularized” near zero distance and instead cares about large distances. Still the terminology might make what you are saying familiar to more people, which is always good!

A resource, probably not the best one, for this sort of stuff is the excellent book on Brownian Motion by Morters and Peres, a draft of which can be posted here: www.stat.berkeley.edu/~peres/bmbook.pdf. The pertinent information is in chapter 4 about page 109 (a section about Hausdorff measures and dimensions).

Posted by: Brent Werness on April 21, 2010 5:10 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Brent said:

This leads to a nice physical interpretation: that the magnitude of a metric space is exactly the amount of mass needed to be spread out over the metric space so that every point feel a total potential energy of one (when the potentials/forces decay exponentially).

I prefer to think in terms of something like ‘heat generation’ rather than mass because we actually allow signed measures (which I didn’t emphasize in the post before) and I don’t have a good intuition for negative mass. I described this point of view in the post More Magnitude of Metric Spaces and Problems with Penguins and I’ve now linked to this in the main text above.

The similarity with the Energy Method is intriguing. I wasn’t aware of it. The main difference, however, is that the Energy Method uses a power of the distance where we use the exponential. I would really like to find some connection with this fractal dimension ideas though, as the growth of the magnitude does seem connected with ‘dimension’ in some sense. So for instance in On the asymptotic magnitude of subsets of Euclidean space Tom and I show that the magnitude of the Cantor set asymptotically grows like the Hausdorff dimension $\log_3 2$; asymptotically, the magnitude of the length $t$ Cantor set looks like

$f(t)t^{\log_3 2}$

for some ‘almost constant’, multiplicatively periodic function $f$.

Posted by: Simon Willerton on April 22, 2010 1:58 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Typo? The denominator rapidly tends to zero as R increases … zero -> one

Posted by: Charlie C on April 21, 2010 6:07 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Fixed, thanks.

Posted by: Simon Willerton on April 21, 2010 7:13 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

In the Introduction you credit the knowledge that magnitude is always defined for finite subsets of Euclidean space partly to me. Actually I believe that was a combination of observations of Tom, David Corfield, Yemon Choi, and David Speyer. My contributions started with generalizing that away from Euclidean space.

Posted by: Mark Meckes on April 21, 2010 9:04 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I was basing that on my interpretation of Tom’s comments in the post An Adventure in Analysis. I will figure out how to rewrite this before putting it on the arxiv. Thanks.

I see you’re giving some seminars on magnitude. I hope they’re going well!

Posted by: Simon Willerton on April 21, 2010 11:45 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

They seem to be going well. You’ve released this draft just in time for me to talk about it in my last talk tomorrow!

Posted by: Mark Meckes on April 22, 2010 1:24 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Posted by: Tom Leinster on April 21, 2010 11:48 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

It’s interesting to me that you say the measure-based approach to defining magnitude was suggested to you by Tom and David S., whereas your own thoughts had been more along the lines of currents. To me the measure approach is the obvious first thing to look at, but prompted by some things Tom has said to me I now think something like currents might be the way to go.

In any case, I think it’s clear that whenever a weight measure exists, the magnitude should be exactly as you define it. I hope that finding the right class of measure-like things to consider will provide some more versatile tools for proving that weight measures exist for many spaces.

Posted by: Mark Meckes on April 22, 2010 3:08 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark said:

prompted by some things Tom has said to me I now think something like currents might be the way to go.

Do you want to expand on this? My reasons were mainly that on the one hand I had never done any measure theory – I certainly never took a course on it – and on the other hand I was used to using currents, indeed had used them in my thesis. When I did the calculations for the weightings on squares and discs and the like, it was the idea of currents that was brought to mind by the pictures.

Posted by: Simon Willerton on April 22, 2010 7:14 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Do you want to expand on this?

Sure. To begin with, to someone who works with measures a lot, Tom’s original definition of a weighting vector $w$ for a finite metric space $X$ brings measures to mind immediately. After all, how does $w$ show up? The weighted sums $\sum_{x\in X} e^{-d(y,x)} w_x=1$ can be interpreted as integrals of the functions $f_y(x) = e^{-d(y,x)}$ and the sum $\sum_{x\in X} w_x = \vert X \vert$ is then the total (signed) measure of $X$. Even the name “weighting” brings to mind common terminology for discrete measures. If $X$ is no longer finite, no problem, just forget that we ever thought of $w$ as a vector: now it’s a signed measure on $X$.

But we don’t want to fall into the trap of thinking everything is a nail just because we happen to have a hammer handy. Is there some reason that measures arise naturally from the mathematics here? Well, another way to think about the finite sums above is to identify $\mathbb{R}^X$ with its dual space and consider $w \in (\mathbb{R}^X)^*$, so what we have is $w(f_y) = 1$ for every $y\in X$ and then $\vert X \vert = w(1)$. When $X$ is infinite, we would like to replace $\mathbb{R}^X$ with some function space on $X$ which contains all the functions $f_y$ and $1$; this will be an infinite dimensional space, most likely not identifiable with its dual in a reasonable way, and so $w$ will become a different kind of creature.

So which function space should we work with? One that comes to mind is $C_b(X)$, the space of bounded continuous real-valued functions on $X$. (Keep in mind we don’t have a lot of structure to work with. If $X$ is nothing but a metric space with no canonical choice of a measure we can’t talk about $L^p$ spaces, for example.) $C_b(X)$ contains the functions $f_y$ and $1$, and is a Banach space with the sup norm. If $X$ is moreover compact (so we can drop that $b$), then $C(X)^*$ is isometrically isomorphic to $M(X)$, the space of signed Borel measures on $X$; a signed measure acts on a continuous function by integration. So measures really do show up naturally.

I said some abbreviated version of this to Tom in email. He said that he didn’t find this argument completely satisfactory in part because the space $C(X)$ only depends on the topology of $X$, and doesn’t see anything else about the metric structure. (Tom, please correct me if I’m misrepresenting or leaving out something important.) He suggested that it might make more sense to use $Lip(X)$, the space of Lipschitz functions on $X$, or something like it. $Lip(X)$ is also a Banach space with an appropriate norm, which contains $f_y$ and $1$. Furthermore $Lip(X)^*$ (is isometrically isomorphic to a space which) contains $M(X)$, but as a proper subspace when $X$ is infinite. Heuristically, since Lipschitz functions are within spitting distance of being differentiable (when “differentiable” even makes sense), the elements of $Lip(X)^*$ are something like functionals of differentiable functions, so they are something like some class of distributions, or currents. (I’d never actually heard of currents until recently, when I was talking to a differential geometer colleague about this.) So maybe that’s the sort of thing $w$ should be.

I should point out one possible objection to Tom’s hesitation about using $C(X)$ and $M(X)$: although those Banach spaces don’t know much about the metric on $X$, the functions $f_y$ do, and they stay in the picture in any case.

Of course this is not just an aesthetic or philosophical issue. We want definitions of weighting and magnitude that let us prove good theorems. In particular, I’d like a definition of magnitude which is equivalent to yours whenever a weighting measure exists. In the framework I’ve described, this means that whatever function space we work with, we want its dual to contain $M(X)$. It would also be nice to have a definition such that the definitions of magnitude in your paper with Tom, via approximations by finite sets, become theorems (this is what Tom said to me that set me to thinking about all of the above in the first place). To prove such theorems we need to know something about how measures with finite support sit inside our dual space. For both of the function spaces $C(X)$ and $Lip(X)$ I do know some interesting things about this question, but I haven’t managed to use them prove the desired theorems yet.

[Off-topic question for those who understand the tex rendering here: I didn’t bother to make the Lip in $Lip(X)$ Roman, although I normally would, out of laziness. But it appears in Roman anyway, on my browser at least. Why is that?]

Posted by: Mark Meckes on April 23, 2010 3:30 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark’s comments here just suggested something which might be of passing interest if not of real relevance. Since I’m currently a bit sleep-deprived and not really up to cogent mathematical thought today, I’d just like to throw out some disjointed remarks and half-remembered tidbits. Apologies if this derails the discussion a bit; I am really just commenting here while the iron is hot.

- The use of lip(X) or Lip(X) as a kind of coordinate ring (broadly speaking) for compact metric spaces has been championed in some papers and in a book of Nik Weaver, this POV probably goes back to work of Mark Rieffel but I forget the precise referencs.

- Lip(X) is the bidual Banach space of lip(X), and the predual of Lip(X) is I think a named widget, called the Arens-Eells space of X – it is like some linear combination of “molecules” or “atoms” - hopefully someone who’s seen some modern-era Hardy space theory can correct or expand on this bit? I don’t know about Lip(X)* though.

- I seem to remember that the forgetful functor from (Banach spaces, linear maps of norm $\leq 1$) to (metric spaces, distance-non-increasing maps) has a left adjoint. I first saw this in a talk of Gilles Godefroy on some work he did with Nigel Kalton on “Lipschitz-free spaces” - although he didn’t once use the words functor or adjunction if I recall correctly.

Posted by: Yemon Choi on April 23, 2010 10:22 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Let me make some general comments as to why Lipschitz functions might be appropriate in this context, or at least let me speculate on why Tom might think so.

The main point is that the idea of magnitude of metric spaces arose from thinking of metric space as enriched categories (following Lawvere). In that situation, the obvious notion of function (though probably not the only one) is that of Lipschitz function. Lipschitz functions play the role of ‘enriched presheaves’. I’ll try to explain what kind of properties can be deduced from that point of view.

Firstly, I should say what I assume you mean by Lipschitz functions, or rather what I would like you to mean. I will use $[0,\infty]$ to denote the extended non-negative reals $\mathbb{R}_{\ge 0}\cup\infty$. In the present context I would think of a Lipschitz function as being a function $h: X\to [0,\infty]$ such that $d(x,y)\ge h(x) - h(y)$. A good class of examples, indeed a key class of examples, of such functions is given in the following way. Take any $y\in X$ and define the function $h_y:X\to [0,\infty]$ by $h_y(x)\coloneqq d(x,y)$. The triangle inequality ensures that this satisfies the Lipschitz condition. More generally, for any subset $Y\subset X$ we can define $h_Y:X\to [0,\infty]$ by $h_Y(x)\coloneqq inf_{y\in Y} d(x,y)$. I’ll come back to the relevance of these in a second, but the sharp-eyed category theorists should be shouting “Yoneda!” at their screen by now.

Now this wasn’t quite what you were talking about. You were talking about functions like $f_y:X\to [0,1]$ where $f_y=e^{h_y}$, so I would expect that your functions $f:X\to [0,1]$ will satisfy a ‘multiplicative Lipschitz condition’ $f(x)/f(y)\ge e^{-d(x,y)}$ for all $x$ and $y$. So you might want to translate what I’m going to say over to your kind of functions.

Now $Lip(X)$ the set of Lipschitz functions (in my sense) carries a ‘generalized metric’ defined on function $h'$ and $h''$ as follows:

$d(h', h'')\coloneqq sup_{x\in X} [h''(x)- h'(x)]$

where my ad-hoc notation $[a]\coloneqq max(0, a)$ is the ‘non-negative part’. This generalized metric is not symmetric, might take the value $\infty$, and $d(h',h'')=0$ does not imply that $h'=h''$; however it does satisfy the triangle equality and $d(h,h)=0$.

The key point about the above generalized metric is that the function

$X\to Lip(X); \quad y\to h_y$

is an isometric embedding – it’s a nice exercise to prove that. [For the cognoscetti this is an enriched version of the Yoneda embedding, $Lip(X)$ is the $[0,\infty]$-category of $[0,\infty]$-presheaves on $X$.] As John B keeps telling us in a more general setting, we should think of $Lip(X)$ as some kind of cocompletion of $X$. (Okay, yes, when people bandy about words like cocompletion it causes some glazed expression to pass across my eyes.)

That the set of Lipschitz functions $Lip(X)$ is a cocompletion means that it will have all colimits (whatever that means), in particular it will have coproducts, so we will be able to add things together. In $X$ we can’t add two points together, but we can in $Lip(X)$: it makes sense in many circumstances to take the distance from a point $x$ to the union of two points $y$ and $y'$ to be the minimum of distances from $x$ to $y$ and from $x$ to $y'$. In this way we can think of the union of two points as being a Lipschitz function.

This leads to thinking of any subset of points in $X$ as giving an element of $Lip(X)$ as we did above. For $Y\subset X$ define the function $h_Y\in Lip(X)$ by $h_Y(x)\coloneqq inf_{y\in Y} d(x,y)$.

Things get even nicer when we restrict to $Comp(X)$, the set of compact subsets of $X$. Firstly, if $Y$ is compact then we can recover $Y$ from the function $h_Y$ by just taking the zero set of the function. Moreover, we can define a generalized metric on $Comp(X)$ by saying $d(Y',Y'')=\epsilon$ if $\epsilon$ is the least number such that $Y'$ is contained in a closed $\epsilon$-neighbourhood of $Y''$. In particular $d(Y',Y'')=0$ if and only if $Y'\subset Y''$. We obtain the Haussdorf metric (a genuine metric) $D$ on $Comp(X)$ by symmetrizing $d$ in the following way:

$D(Y',Y'')\coloneqq max\bigl( d(Y',Y''),d(Y'',Y')\bigr).$

Although I guess the Haussdorf metric has less information in it. None-the-less, we get an isometric embedding of generalized metric spaces:

$(Comp(X),d)\to (Lip(X),d);\quad Y\to h_Y.$

This can be made into an isometric embedding of genuine metric spaces by using the Haussdorf metric on the set of compact subsets and similarly symmetrizing the generalized metric on the set of Lipschitz functions.

So $Lip(X)$ contains not only the points of $X$ but also the unions of points of $X$. I guess there’s lots more stuff in $Lip(X)$, but I’m not imaginative enough at this point in time to think of anything else. What other colimits would be in there other than coproducts?

Anyway, this is all very natural from the point of view of enriched category theory and everything I’ve said has analogues in other enriched situations. As magnitude of metric spaces arose by thinking of metric spaces as being enriched categories, it makes sense to hope/think/guess that these sorts of functions are the right ones to consider in this context.

Posted by: Simon Willerton on April 24, 2010 3:38 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Thanks for the comments, Simon (and thanks also to Yemon - I’ve only recently started learning about $Lip(X)$ and $lip(X)$ and all that, so pointers to the literature are good to have [1]).

To me your comments actually support working with $C(X)$ instead of $Lip(X)$. Let me explain why. First of all, I’m going to work only with symmetric metrics, not because I disagree with Lawvere but because I don’t know how much classical stuff I know will still work without symmetry [2]. For me, a function $f:X\to \mathbb{R}$ is $L$-Lipschitz for $L \ge 0$ if $sup_{x,y\in X}|f(x)-f(y)| \le L$, so what you call Lipschitz is what I call $1$-Lipschitz; $f$ is said to be Lipschitz if it is $L$-Lipschitz for some (finite) $L$. The functions $f_y(x)=e^{-d(x,y)}$ are $1$-Lipschitz, since $t \mapsto e^{-t}$ is $1$-Lipschitz on $[0,\infty)$.

Since I hope to use Banach space methods here, for me $Lip(X)$ is the set of all Lipschitz functions $f:X\to \R$, since $1$-Lipschitz functions alone don’t form a vector space. Next I need a norm. The symmetric analogue to the distance you put on $1$-Lipschitz functions is the sup norm: $\|f\|_\infty = \sup_{x\in X} |f(x)|$. Now $Lip(X)$ is not complete in this norm. It’s a dense subspace of $C(X)$, and so $(Lip(X), \| \cdot \|_\infty)^* = M(X)$. So your comments seem to support what I meant by “working with $C(X)$”. By “working with $Lip(X)$” I mean instead equipping $Lip(X)$ with a different norm (the bounded Lipschitz norm) which makes it a Banach space, whose dual (and in fact predual) contain $M(X)$ as a proper subspace.

[1] One thing I will point out related to Yemon’s comments is that finitely supported signed measures are norm-dense in the predual of $Lip(X)$ (with the bounded Lipschitz norm). This is one thing I think could be useful in this context. Note that finitely supported signed measures are not norm-dense in $M(X)$, although they are dense in the weak-* topology.

[2] In case you ever meet a skeptical analyst or geometer who might think Lawvere lacks the metric-space-credibility to advocate dropping such a fundamental axiom, you can tell them that Gromov has also argued against the symmetry axiom as too restrictive for applications.

Posted by: Mark Meckes on April 24, 2010 6:48 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

To answer your parenthetical question about typesetting. In itex (which is what is used here) consecutive, abutting Roman letters in mathmode are interpreted as being a word, so are typeset upright.

• $L i p$ gives $L i p$
• $Lip$ gives $Lip$
Posted by: Simon Willerton on April 24, 2010 4:00 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

On page 3, you write “It does seem that the subspace distance metric and the intrinsic metric are infinitessimally the same. I would hope that the asymptotics of the magnitude only depend on metric infinitessimally.” I though I found a counter-example to this here (and you did as well, I think?)

I would not say that I hoped something was true if I knew a counter-example.

Also, on page 6, “we will note distinguish” should read “we will not distinguish”.

Posted by: David Speyer on April 22, 2010 3:37 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I’m glad you’re there to keep me on my toes David.

I guess I had in my memory that there was some problem with your calculation for the subspace metric on $S^2$ (I don’t think I could quite figure it out or something, I don’t remember). But I hadn’t got round to going back to look at it. Anyway, I just sat down and calculated the thing on the nose today. It’s very nice. As you said, the thing to calculate is

$2\pi\int^\pi_{\theta=0} e^{-R 2\sin(\theta/2)} \sin(\theta) d \theta.$

The cunning observation is that

$\sin(\theta) d \theta = 2\sin(\theta/2)\cos(\theta/2)d\theta =2\sin(\theta/2)d(2\sin(\theta/2)).$

So substituting $s=2\sin(\theta/2)$ we get that the integral equals $2\pi\int_{s=0}^2 e^{-R s} s d s$ which evaluates to

$\frac{2\pi(1-e^{-2R}(2R+1))}{R^2}.$

The magnitude of the radius $R$ sphere with the subspace metric is the volume of $S^2$ divided by that integral, whence

$\left|S^2_{R,sub}\right| =\frac{2R^2}{1-e^{-2R}(2R+1)}= 2R^2+exponentially decaying terms.$

However, for the intrinsic metric you we have the magnitude is of the form $2R^2+2 +O(R^{-2})$. So asymptotically we get different answers for the magnitude.

You can do a similar calculation for higher dimensional spheres.

I’m sure I only put those questions there about the subspace metric so that you would goad me into actually doing the calculations!

This then leads to the question: in what sense are the subspace metric and the intrinsic metric infinitesimally the same? I still haven’t got my head round this.

Posted by: Simon Willerton on April 23, 2010 7:10 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

in what sense are the subspace metric and the intrinsic metric infinitesimally the same?

One sense in which this is true is that an $\varepsilon$-ball w.r.t. the intrinsic metric is the same as a $2R \sin (\varepsilon / 2R)$-ball w.r.t. the subspace metric, and $2R\sin (\varepsilon / 2R) \approx \varepsilon$ as $\varepsilon \to 0$.

Posted by: Mark Meckes on April 23, 2010 12:34 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

You say you haven’t calculated the magnitudes of spheres with the subspace metric yet. If you haven’t even checked for $S^1$ yet, I just did and it’s easy to see (the hardest part is using a trig identity) that it matches the expression derived in your paper with Tom.

Also, a typo in Theorem 1: $\mu$ is an invariant measure.

Posted by: Mark Meckes on April 22, 2010 3:43 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Oh, I see the calculation for $S^1$ is also in the discussion David just linked to.
Posted by: Mark Meckes on April 22, 2010 3:48 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark said

You say you haven’t calculated the magnitudes of spheres with the subspace metric yet. If you haven’t even checked for $S^1$ yet, I just did and it’s easy to see (the hardest part is using a trig identity) that it matches the expression derived in your paper with Tom.

I’m not sure which bit you mean you’ve checked!

I should say certainly say a little more in the paper about $S^1$, as we do know a reasonable amount about it. Of course what I meant to say was “I haven’t calculated it for the general $n$-sphere, but we know a fair bit about the circle, though I don’t really want to generalize from that.” However, that isn’t what I said. It was one of the bits of the paper where I needed to fill in the blank before finishing it and I rushed it. Thanks.

In the paper with Tom, it was shown that that the magnitude of a circle has the expected asymptotic behaviour – namely half the length – when it is thought of a subspace of any constant curvature surface. In particular this covers both the Euclidean metric and the intrinsic metric. The non-asymptotic magnitude will in general be different metrics on the circle.

Posted by: Simon Willerton on April 22, 2010 7:05 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

What I checked is that the magnitude of a circle of arbitary fixed circumference, as a subspace of $\mathbb{R}^2$, has the same magnitude whether defined as in the present paper, or as in section 4.1 of your paper with Tom.

Actually I think this has to be the case from measure-theoretic “first principles”, but there are some fiddly functional-analytic issues I want to think about more before I say I’m sure of that.

Posted by: Mark Meckes on April 22, 2010 7:14 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I tried to post another comment on this earlier but it vanished. In fact the reason they have to be the same appeared in your paper with Tom: the sums for the finite approximations of the circle are just Riemann sums for the integral corresponding to the whole circle.

To make it sound fancier, the measures $\mu_n$, which give mass $1/n$ to each of $n$ equally spaced points on a circle, converge in the weak-* topology of $M(S^1)$ (induced by the action as the dual of $C(S^1)$) to the invariant probability measure on $S^1$.

Posted by: Mark Meckes on April 23, 2010 12:58 AM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

It seems I should have waited a while to collect my comments before posting, but it’s too late for that now.

Regarding Theorem 1 (Speyer’s Homogeneous Magnitude Theorem), it’s worth pointing out that if $X$ is a compact homogeneous metric space then (up to normalization) there exists a unique invariant measure $\mu$ on $X$.

Posted by: Mark Meckes on April 22, 2010 4:30 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Furthermore this is a positive measure, so $\int_{x\in X} e^{-d(x,y)} d\mu$ is automatically nonzero and so the magnitude of $X$ is indeed defined.

Folklore has it that a nonzero invariant measure on a noncompact space is (typically at least) infinite. Does anyone know an actual theorem to that effect?

Posted by: Mark Meckes on April 22, 2010 7:04 PM | Permalink | Reply to this

### Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Is http://arxiv.org/abs/math-ph/0210033 “Volumes of Compact Manifolds” of some use, here?

Posted by: Alejandro Rivero on April 27, 2010 1:57 AM | Permalink | Reply to this
Read the post Magnitude of Metric Spaces: A Roundup
Weblog: The n-Category Café
Excerpt: Resources on magnitude of metric spaces.
Tracked: January 10, 2011 4:02 PM

Post a New Comment


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# QlikView Scripting

Discussion Board for collaboration on QlikView Scripting.

Announcements

Contributor

## Grouping Dimensions in a Table

Hello my QlikView friends   I am hoping this is very easy to understand and create.  I have an example table pictured below.  I want to know how to create the grouped dimension section, with the arrow pointing to it.  It is my current understanding that I would have Expressions for Apples and Oranges, summing each.  But how do I group those two fields under another calculated dimension?  Thank you

4 Replies
Valued Contributor

## Re: Grouping Dimensions in a Table

Create a table, like

Category, Item

Fruits, Apples

Fruits, Oranges

Vegetables, Tomato

Vegetables, Potato

];

No, you can use the Category as a dimension and pivot it to columns

Eduardo

Valued Contributor

## Re: Grouping Dimensions in a Table

I forgot something: you can also pivot the Item Dimension to a column, under category, and have what you want.

Eduardo

MVP

## Re: Grouping Dimensions in a Table

=If(Match(FruitType,'Apples','Oranges'),'Fruit',

if(Match(FruitType,'Tomoto','Potato'),'Vegetables'))

Note:- I assume FruitType contains(Apples,Oranges,Tomoto,Potato)

Contributor III

format: plain


text:
Pentachoron

562pages on
this wiki

A pentachoron is a 4-dimensional simplex. It is also called the pyrochoron under the elemental naming scheme. Its Bowers acronym is "pen".

Properties Edit

StructureEdit

The pentachoron is the pyramid on the tetrahedron, with 4 tetrahedra on each vertex.

Hypervolumes Edit

• vertex count = $5$
• edge length = $10l$
• surface area = $\frac { 5\sqrt { 3 } }{ 2 } { l }^{ 2 }$
• surcell volume = $\frac { \sqrt { 2 } }{ 3 } { l }^{ 3 }$
• surteron bulk = $\frac { \sqrt { 5 } }{ 96 } { l }^{ 4 }$


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# Fundamental inequality of entropy in random walks

I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,

$\textbf{Question}$: Under which conditions (on the measure $P$), do the following inequalities hold: $$l_P^2 \leq h_P \leq v l_P.$$

The second inequality (called "fundamental", due to Vershik and/or Guivarc'h, I have unfortunately no access to any of these paper) is valid for a large class of measures, but my concern is about the first one.

The above quantities are defined by: $$H_{n,P} = -\sum_{g \in G} P^{(n)}(g) \ln P^{(n)}(g), \qquad \text{and } \qquad L_{n,P} = \sum_{g \in G} |g| P^{(n)}(g)$$ where $P^{(n)}$ is the convolution of $P$ with itself $n$ times [EDIT:and $|g|$ is the word length of $g$ for the generating set of the Cayley graph]. Then define the [asympotic] entropy, the rate of escape (or drift, or Green speed) and the exponential volume growth to be $$h_P = \lim_{n \to \infty} \tfrac{1}{n} H_{n,P}, \qquad l_P = \lim_{n \to \infty} \tfrac{1}{n} L_{n,P}, \qquad \text{and } v = \lim_{n \to \infty} \tfrac{1}{n} \log |B_n|$$ where $|B_n|$ is the cardinality of the ball of radius $n$ around the identity.

PS: If I understood correctly, a paper of Blachère, Haïssinsky and Mathieu shows that (if $\sum_{g \in G} P^{(1)}(g) \ln P^{(1)}(g)$ is finite) $l_P=0$ is equivalent to $h_P=0$. This would be a consequence of the above inequalities, so I assume they are not too trivial in this generality.

-
In the formula for $L_{n,P}$, what is $|g|$? –  Vaughn Climenhaga Apr 29 at 16:54
$|g|$=word length, right. So that $L_{n,P}$ is the expected distance from the origin in terms of the generators after $n$ steps. Since it grows sublinearly, $l_P$ exists. –  Anthony Quas Apr 29 at 17:38
Now, in what concerns the first inequality, first of all its scope is more restricted than that of the second one. Indeed, the second one makes sense for any measure with a finite first moment (which implies finiteness of entropy - not the other way round as you claim), whereas any random walk with a non-trivial drift on $\mathbb Z$ provides an obvious counterexample to the first one. As it was proved by Karlsson and Ledrappier in 2006 or 2007, this is actually the only obstacle to equivalence of $h=0$ and $l=0$. In particular, for symmetric measures with a finite first moment $h=0 \iff l=0$ (it is completely wrong to attribute this result to Blachère, Haïssinsky and Mathieu).
In the form you formulate the first inequality ($h\ge l^2$), as far as I know it was established only for Brownian motion on cocompact manifolds (by Kaimanovich and Ledrappier). For symmetric random walks on groups with an additional multiplicative constant it was recently proved by Erschler and Karlsson.


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# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function y = 3x^2 - 12x - 2?

Minimum value $f \left(2\right) = - 14$ at the lowest point $\left(2 , - 14\right)$
$x = 2$ the axis of symmetry

#### Explanation:

$y = 3 {x}^{2} - 12 x - 2$
by completing the square method
$y = 3 \left({x}^{2} - 4 x\right) - 2$
$y = 3 \left({x}^{2} - 4 x + 4 - 4\right) - 2$
$y = 3 {\left(x - 2\right)}^{2} - 12 - 2$
$y = 3 {\left(x - 2\right)}^{2} - 14$
$\frac{1}{3} \left(y + 14\right) = {\left(x - 2\right)}^{2}$

$\frac{1}{3} \left(y - - 14\right) = {\left(x - 2\right)}^{2}$

graph{y=3x^2-12x-2[-5,35,-15,5]}

Have a nice day ! from the Philippines..


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# Why is 3 pion decay of eta (η) strong forbidden?

by Smithf
Tags: η, decay, forbidden, pion, strong
 P: 5 I have read that 3 pion decay of the eta is not allowed as a strong reaction, but proceeds as an electromagnetic interaction. I do not see why it is strong forbidden.
 Sci Advisor P: 1,233 That is easiest to see using G parity, which is $$C e^{i\pi I_y}$$. One or three pions have negative G, while the eta has positive G.
 P: 5 How do you find the result of the I2 rotation operation?


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# Browsing Physics by Title

Sort by: Order: Results:

• (American Physical Society, 2012)
Core-collapse supernovae may depend sensitively on charged-current neutrino interactions in warm, low-density, neutron-rich matter. A proton in neutron-rich matter is more tightly bound than is a neutron. This energy shift ...
• (American Physical Society, 2012)
We present results on the chiral and deconfinement properties of the QCD transition at finite temperature. Calculations are performed with 2+1 flavors of quarks using the p4, asqtad, and HISQ/tree actions. Lattices with ...
• (Elsevier, 2010)
Classical point-particle relativistic lagrangians are constructed that generate the momentum–velocity and dispersion relations for quantum wave packets in Lorentz-violating effective field theory.
• (American Physical Society, 2002)
Clock-comparision experiments on satellite and other spacecarft were used to determine unmeasured coefficients for Lorentz and CPT violation. The presence of Lorentz and CPT violation caused frequency shifts in certain ...
• (American Physical Society, 2012)
We derive an effective field-theoretical model for the one-dimensional collective mode associated with a domain wall in a quantum Hall ferromagnetic state, as realized in confined graphene systems at zero filling. To this ...
• (American Physical Society, 1990)
A second-quantized analysis is performed to examine many-body phenomena in closed bosonic strings. The covariant nonpolynomial closed-string field theory is developed in terms of particle fields and shown to contain ...
• (American Physical Society, 2012)
We report the combination of recent measurements of the helicity of the W boson from top quark decay by the CDF and D0 collaborations, based on data samples corresponding to integrated luminosities of $2.7-5.4\text{ ... • (American Physical Society, 2012) We present the results of the combination of searches for the standard model Higgs boson produced in association with a$W$or$Z$boson and decaying into$b\bar{b}$using the data sample collected with the D0 detector in ... • (American Association of Physics Teachers, 2012) We survey various numerical methods for finding solutions of quantum confined states. We especially consider states in two-dimensional (2D) tubes, or 2D surfaces that are confined in the transverse direction but are ... • (American Physical Society, 1999) Constraints from clock-comparison experiments on violations of Lorentz and CPT symmetry are investigated in the context of a general Lorentz-violating extension of the standard model. The experimental signals are shown to ... • (Physics Letters B, 2015) Limits on gravitational Čerenkov radiation by cosmic rays are obtained and used to constrain coefficients for Lorentz violation in the gravity sector associated with operators of even mass dimensions, including ... • (American Physical Society, 2013) Tiny violations of the Lorentz symmetry of relativity and the associated discrete CPT symmetry could emerge in a consistent theory of quantum gravity such as string theory. Recent evidence for linear polarization in gamma-ray ... • (American Physical Society, 2012) The symmetry energy contribution to the nuclear equation of state impacts various phenomena in nuclear astrophysics, nuclear structure, and nuclear reactions. Its determination is a key objective of contemporary nuclear ... • (American Physical Society, 2008) Exceptional sensitivity to spacetime torsion can be achieved by searching for its couplings to fermions. Recent experimental searches for Lorentz violation are exploited to extract new constraints involving 19 of the 24 ... • (American Physical Society, 2001) Infrared, optical, and ultraviolet spectropolarimetry of cosmological sources is used to constrain the pure electromagnetic sector of a general Lorentz-violating standard-model extension. The coefficients for Lorentz ... • (American Physical Society, 1989) For supersymmetric extensions of the standard model, we obtain necessary conditions for$CP$invariance expressed in terms of weak-basis invariants. Among other applications, these conditions allow a simple identification ... • (American Physical Society, 1999) Signals for$CPT$and Lorentz violation at the Planck scale may arise in hydrogen and antihydrogen spectroscopy. We show that certain$1S−2S$and hyperfine transitions can exhibit theoretically detectable effects unsuppressed ... • (American Physical Society, 1998) A theoretical analysis is performed of Penning-trap experiments testing$CPT$and Lorentz symmetry through measurements of anomalous magnetic moments and charge-to-mass ratios. Possible$CPT$and Lorentz violations arising ... • (American Physical Society, 2000) Precision experiments with muons are sensitive to Planck-scale$CPT$and Lorentz violation that is undetectable in other tests. Existing data on the muonium ground-state hyperfine structure and on the muon anomalous magnetic ... • (American Physical Society, 2010) Recent evidence for anomalous$CP$violation in$B$-meson oscillations can be interpreted as resulting from$CPT$violation. This yields the first sensitivity to$CPT$violation in the$B^{0}_{s}\$ system, with the relevant ...


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# Math Help - Compound Angle Formulas - Confused over evaluating

1. ## Compound Angle Formulas - Confused over evaluating

I have done questions like these before, but the ones I did were only in quadrant I. I really wish I knew how to draw the triangles on here for you, so you see everything that I am working with.

Question:
If x is in the interval $(\frac{\pi}{2}, \pi)$ and y is in the interval $(\pi, \frac {3\pi}{2})$ and cosx = $- \frac {5}{13}$ and tany = $\frac {4} {3}$, evaluate the following.

*I will put the thing to evaluate, then the answer (according to the sheet) under it, then put my attempt after*

a) sin (x + y)
Answer: $- \frac{16}{65}$

b) cos (x - y)
Answer: $- \frac {33}{65}$

c) tan (x - y)
Answer: $\frac {56}{33}$

Okay, now I will show my attempts (for some reason I got the answer for b, as the sheet answer for a. And the answer for b as the sheet answer for a.)

a) $\sin (x + y)$
= $\sin x \cos y + \cos x \sin y$
= $((\frac {12}{13}) (\frac {-4}{-5})) + ((\frac {-5}{13}) (\frac {-3}{-5})$
= $(\frac {48}{65}) + (\frac{-3}{13})$
= $\frac {33}{65}$

b) $cos (x - y)$
= $cosx cosy + sinx siny$
= $((\frac{-5}{13})(\frac{-4}{5}) + ((\frac {12}{13})(\frac{-3}{5}))$
= $\frac {4}{13} + \frac {-36}{65}$
= $\frac {-16}{65}$

c) $tan (x - y)$
= $\frac {tanx - tany} {1 + tanxtany}$
= $\frac {(\frac {12}{13}) - \frac {-3}{-4}} {1 + (\frac {12}{13}) (\frac {-3}{-4}) }$
= $\frac { \frac {9}{52} } {1+ \frac {9}{13} \frac {22}{13}}$
= $\frac {9} {88}$

2. Hello, Xenophobe!

If $x$ is in the interval $\left(\tfrac{\pi}{2}, \pi\right)$ and $y$ is in the interval $\left(\pi, \tfrac{3\pi}{2}\right)$

and $\cos x = -\tfrac{5}{13}$ and $\tan y = \tfrac{4}{3}$, evaluate the following:

$(a)\;\sin(x + y) \qquad (b)\;\cos(x - y) \qquad (c)\;\tan (x - y)$

$x$ is in Quadrant 2, and $\cos x \:=\: -\frac{5}{13} \:=\:\frac{adj}{hyp}$
We have: . $adj = -5,\;hyp = 13$
. . Using Pythagorus: . $opp \:=\:12$

$y$ is in Quadrant 3, and $\tan y \:=\:\frac{4}{3} \:=\:\frac{opp}{adj}$

In Quadrant 3: . $opp = -4,\; adj = -3,\; hyp = 5$

We have these values: . $\begin{array}{cccccccc}
\sin x &=& \dfrac{12}{13} & & \sin y &=& \text{-}\dfrac{4}{5} \\ \\[-3mm]
\cos x &=& \text{-}\dfrac{5}{13} & & \cos y &=& \text{-}\dfrac{3}{5} \\ \\[-3mm]
\tan x &=& \text{-}\dfrac{12}{5} & & \tan y &=& \dfrac{4}{3} \end{array}$

Try those in your compound angle formulas . . .


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# TikZ: draw lines connecting border of shapes

Being new to TikZ, I am trying to draw several (labeled) circles which are connected by lines. In its simplest form, I am doing this via

\documentclass{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\draw (0,0) circle [radius=0.5] node (A) {A};
\draw (2,2) circle [radius=0.5] node (B) {B};
\draw (A) -- (B);
\end{tikzpicture}

\end{document}

However, the connecting line reaches inside the circles which I don't want it to do:

How do I make the line reach only from border to border?

• you can make those circles as nodes \node[draw,circle,minimum size=1cm] at (2,2) {B}; etc. – percusse Nov 13 '14 at 11:55

As percusse commented you can draw circular nodes.

\documentclass{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\node[circle,draw, minimum size=1cm] (A) at  (0,0) {A};
\node[circle,draw, minimum size=1cm] (B) at  (2,2)  {B};
\draw (A) -- (B);
\end{tikzpicture}

\end{document}

Your code also works, but the problem is with how you've defined the nodes. I've added draw,red style to better see what happens:

\documentclass{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\draw (0,0) circle [radius=0.5] node[draw,red] (A) {A};
\draw (2,2) circle [radius=0.5] node[draw,red] (B) {B};
\draw (A) -- (B);
\end{tikzpicture}

\end{document}

As you can see, you are drawing two circles with rectangular nodes inside that are not drawn. And command \draw (A)--(B) works correctly, it stops drawing on node boundaries but not on circles boundaries as you expected.

• +1 for explaining why the original code gives an undesired result. – Matthew Leingang Nov 13 '14 at 20:26
• Is\path (0,0) node [circle,draw,minimum size=1cm] (A) {A} ; identical to \node[circle,draw,minimum size=1cm] (A) at (0,0){A}; ? – Money Oriented Programmer Aug 17 '17 at 14:33
• @SinglePackAbs Yes, it is. You can even omit (0,0) because is the default starting coordinate: \node[circle,...] (A) {A}; – Ignasi Aug 18 '17 at 9:11

A PSTricks solution using the pst-node package:

\documentclass{article}

\usepackage{pst-node}

\begin{document}

\begin{pspicture}(-0.35,-0.35)(3,3) % adjust the bounding box manually according to the contents of the pspicture environment
\cnodeput(0.5,0.5){A}{Mammal}
\cnodeput(2.5,2.5){B}{Fish}
\ncline{A}{B}
\end{pspicture}

\end{document}


text:
# James McKeown

## University of Miami Graduate Student

Fall 2017 I am teaching MATH162 (Calculus II).

Fall 2016 I taught MATH161 (Calculus I).

Spring 2016 I taught MATH107 (Pre-Calculus).

Here are some slides from some talks I've given and some papers I'm working on.

Here is a copy of my CV.

Feel free to also check out my research statement and teaching statement.

I haven't put my thesis on arXiv yet, but it is in an acceptable form.

Last summer my brothers, Michael and Jack and I got together with Dalibor Froncek and worked on a graph labeling problem.

Play around with Meinrenken Tilings for affine Weyl groups: $\tilde{A_2}$, $\tilde{B_2}$, $\tilde{C_2}$, $\tilde{G_2}$, $\tilde{A_3}$, $\tilde{B_3}$ or , $\tilde{C_3}$!

James McKeown
Department of Mathematics 528B
University of Miami
Coral Gables, FL 33124
mckeown@math.miami.edu
+1 (305)284-1956


text:
# How to install the ML.NET Command-Line Interface (CLI) tool

The ML.NET CLI (command-line interface) is a tool you can run on any command-prompt (Windows, Mac, or Linux) for generating good quality ML.NET models and source code based on training datasets you provide.

Note

This topic refers to ML.NET CLI and ML.NET AutoML, which are currently in Preview, and material may be subject to change.

## Pre-requisites

You can either run the generated C# code projects with Visual Studio F5 or with dotnet run (.NET Core CLI).

Note: If after installing .NET Core 2.2 SDK the dotnet tool command is not working, sign out from Windows and sign in again.

## Install

The ML.NET CLI is installed like any other dotnet Global Tool. You use the dotnet tool install .NET Core CLI command.

The following example shows how to install the ML.NET CLI in the default NuGet feed location:

dotnet tool install -g mlnet

If the tool can't be installed (that is, if it is not available at the default NuGet feed), error messages are displayed. Check that the feeds you expected are being checked.

If installation is successful, a message is displayed showing the command used to call the tool and the version installed, similar to the following example:

You can invoke the tool using the following command: mlnet
Tool 'mlnet' (version 'X.X.X') was successfully installed.

You can confirm the installation was successful by typing the following command:

mlnet

You should see the help for available commands for the mlnet tool such as the 'auto-train' command.

## Install a specific release version

If you're trying to install a pre-release version or a specific version of the tool, you can specify the framework using the following format:

dotnet tool install -g mlnet --framework <FRAMEWORK>

You can also check if the package is properly installed by typing the following command:

dotnet tool list -g

## Uninstall the CLI package

Type the following command to uninstall the package from your local machine:

dotnet tool uninstall mlnet -g

## Update the CLI package

Type the following command to update the package from your local machine:

dotnet tool update -g mlnet

## Set up CLI suggestions (tab-based auto-completion)

Since the ML.NET CLI is based on System.CommandLine, it has built-in support for tab completion.

An example of how tab auto completion works is shown in the following animation:

'Tab-based auto-completion' (parameter suggestions) works on Windows PowerShell and macOS/Linux bash but it won't work on Windows CMD.

To enable it, in the current preview version, the end user has to take a few steps once per shell, outlined below. Once this is done, completions will work for all apps written using System.CommandLine such as the ML.NET CLI.

On the machine where you'd like to enable completion, you'll need to do two things.

1. Install the dotnet-suggest global tool by running the following command:

dotnet tool install dotnet-suggest -g

2. Add the appropriate shim script to your shell profile. You may have to create a shell profile file. The shim script will forward completion requests from your shell to the dotnet-suggest tool, which delegates to the appropriate System.CommandLine-based app.

• For bash, add the contents of dotnet-suggest-shim.bash to ~/.bash_profile.

• For PowerShell, add the contents of dotnet-suggest-shim.ps1 to your PowerShell profile. You can find the expected path to your PowerShell profile by running the following command in your console:

echo $profile  (For other shells, look for or open an issue.) ## Installation directory The ML.NET CLI can be installed in the default directory or in a specific location. The default directories are: OS Path Linux/macOS $HOME/.dotnet/tools
Windows %USERPROFILE%\.dotnet\tools

These locations are added to the user's path when the SDK is first run, so Global Tools installed there can be called directly.

Note: the Global Tools are user-specific, not machine global. Being user-specific means you cannot install a Global Tool that is available to all users of the machine. The tool is only available for each user profile where the tool was installed.

Global Tools can also be installed in a specific directory. When installed in a specific directory, the user must ensure the command is available, by including that directory in the path, by calling the command with the directory specified, or calling the tool from within the specified directory. In this case, the .NET Core CLI doesn't add this location automatically to the PATH environment variable.


text: 3. Question 3: Round the following numbers to 1 signi+icant +igure (a) eight million, six hundred thousand (b) the product of 19 and 351 Question 4: Tom has been asked to round the number on the calculator to 2 signi+icant igures. What others think . stream The stem and leaf diagram shows this information. "(a) Write 4486 to the nearest thousand. 2K likes. Please let me know your favourites and I will add them to this list. Grade Fifth Math Worksheets Angles New 5th Printable Common Core ... #205100. Interactive Prisms Tool. Model answers & video solution for Rearranging Formulae. Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. stream GCSE Maths - AQA 8300 Here is a detailed explanation of the entire syllabus. Question 1: (a) 40 (b) 20 (c) 80 (d) 70 (e) 100 (f) 100 (g) 500 (h) 300 (i) 700 (j) 900 (k) 400 (l) 700 (m) 100 (n) 1000 A handy explanation as to how rounding errors can occur and a real life analogy as to why the misconception occurs. %PDF-1.3 The Corbettmaths Textbook Exercise on Rounding to Decimal Places. Plano de Gerenciamento do Escopo – Projeto (Exemplo) 6 anos atrás corbett maths quadratic sequences Rounding . Question 3: Daniel has moved the decimal place instead of rounding. Example: The formula for the force, F on a moving object is F = ma, where m is the mass and a is the acceleration. The Corbettmaths Practice Questions on Rounding - Significant Figures. Please also note that the layout in terms of fonts, answer lines and space given to each question does not reflect the actual papers to save space. So far there are answers to all the KS3 topics, and more answers will be added over time. These questions have been collated by me as the basis for a GCSE working party set up by the GLOW maths hub - if you want to get involved please get in touch. Then you look at the digit to the right of the number. Question 4: Any 10 values from 9.15 ≤ x < 9.25 such as 9.18 or 9.218 etc. KS3 Maths Curriculum Area. Quizizz-an alternative to Kahoot! Contents. Matching the answers to the letters in the decoding grid will produce anagrams of maths … Always show your workings Information 1. Identifying where a number lies on a … Multiplication Facts 9 times table. This applet allows you to generate questions on rounding, and to reveal step-by-step solutions using the slider. << /Length 9 0 R /Type /XObject /Subtype /Image /Width 1192 /Height 1684 /ColorSpace Mathster; Corbett Maths Rounding Textbook Answers nearest, whole number, integer, ten, hundred, thousand. Primary 5-a-day, videos, worksheets and more. The dynamic view of each prism can be altered with the following modes: Cross-section The cross-section (in red) can be seen passing through the prism, illustrating that it remains constant through the length of the shape. Subtract 80 64 output (2) input Multiply by 6 The output is equal to the input. Question 5: 0.47g. Here are some other Maths Kahoots! Notice that the number of significant figures in the question is the maximum number of non-zero digits in your answer. Contents. Menu Skip to content Open link MathsPad Resources. The number of passengers on 10 buses was recorded. Answers to the worksheet on rounding. << /Length 5 0 R /Filter /FlateDecode >> Round 0.005089 to 1 significant figure, then 2 significant figures. Interactive tools and activites to aid the teaching of mathematics. Round this number to 2 signi+icant +igures. 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. His answer is 172.5 Explain Daniel’s mistake. Question 2: Walter has worked out a calculation on a calculator Shown on the calculator is the answer. Rounding: to nearest 10 Video 277a Practice Questions Textbook Exercise. endobj You should work your way down this list of tasks. Practice Questions with Answers on the topic of Rounding to the nearest 10, 100 or 1000. If it is 4 or smaller you round it down. ... Decimal Worksheets For Class 6 With Answers Multiplying And ... #205108. come with answers. %��������� Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; September 19, 2018 October 22, 2019 corbettmaths. PiAcademy THANK YOU!, These 11+ Maths Papers and the detailed answers are invaluable! Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. >> Which subject do you need support in? All Answers For Hegarty Maths - localexam.com. 3. come with answers. !Round her answer to four signiﬁcant ﬁgures Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. This resource contains 15 worksheets, with answers, aimed at Key Stage 3, Year 7 and 8, maths topics including HCF and LCM, rounding and one-step equations. Rounding: to whole numbers Video 276 Practice Questions Textbook Exercise. Practice Questions with Answers on the topic of Rounding to the nearest 10, 100 or 1000. come with answers. Please also note that the layout in terms of fonts, answer lines and space given to each question does not reflect the actual papers to save space. The Corbettmaths Practice Questions on Rounding. arrow_back Back to Rounding to Significant Figures Rounding to Significant Figures: Worksheets with Answers. Corbettmaths is a free mathematics website with 1000's of Video Tutorials, Worksheets and much more Explain how you worked out your answer. Rounding Practice Questions ... Rounding Textbook Answers. Even though maths is an unbelievably creative subject often with numerous answers to solutions, some elegant and even beautiful, maths in school classrooms can be (perceived to be) a rather narrow subject with a right or wrong answer All answers for hegarty maths. Posts about Worksheet Answers written by corbettmaths. 5. "���{%�.Z��l�Y(D�����߲6=8FVP^�X'�d��@D��6ZV�����������;��5(����Dc�w�d�. !Holly has worked out the answer to a calculation. Primary 5-a-day, videos, worksheets and more. stream %��������� Contents. It really is one of the very best websites around. Ticktock Maths. Corbett Maths. Question 6: Here is a possible solution – A = 8.33 and B = 5.13 – the difference is 3.2. Before you try the MyMaths homework you have been set, watch the lesson that goes with it. Loc as) fin 38. bd b deccecoe -quaaer in decrecoe (3) The table gives information about the number of people voting for each party at an election. Round this Nigure to one decimal place. Mathster; Corbett Maths And best of all they all (well, most!) 6 0 obj endobj !Her teacher has told her to write all her answers to four signiﬁcant ﬁgures. Workout. Integers & Decimals. Year 7 Route 1. C = 8.3 and B = 5.1 – the difference is 3.2 Upper and lower bounds also covered along with standard form calculations. Contents. To help you round, you need to know the 'place' that you are rounding to eg. Hundreds of randomly generated questions and answers. ... Bank of Maths Double your money in 15 The average annual growth for your investment is 4.5% Great maths teaching ideas. I'm not featuring the well-known websites here - we are very lucky that there are absolutely loads for maths (Hegarty Maths, Dr Frost, Corbett Maths, MyMaths, Mathigon, MathsWatch, MathsPad, White Rose, Times Table Rockstars, Desmos and so on). The worksheet involves some questions rounding to the nearest power of 10, eg thousand. endobj Comprehensive planning that covers the statutory objectives of the National Curriculum for England lie at the heart of Hamilton maths. (a) Round the answer to one decimal place (b) Round the answer to two decimal places Question 3: Daniel has been asked to round 1.725 to one decimal place. /XObject << /Im1 8 0 R >> >> Rounding Textbook Exercise. << /Type /Page /Parent 3 0 R /Resources 6 0 R /Contents 4 0 R /MediaBox [0 0 612 792] 8 0 obj Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; Search Results for: rounding . It contains estimating the count, estimating time, estimating sum, difference, product and quotient, estimating fractions and decimals, rounding whole numbers, fractions and decimals and rounding … Videos, worksheets, 5-a-day and much more The Corbettmaths channel has over 800 mathematical videos on a wide range of topics, for KS2, KS3, GCSE and A-Level. Round Decimals Worksheets Multiplying Grade 6 Dividing And Pdf – pular #205101. These popular activities are great as starters or as part of a homework task. Party Number of Votes Gold Party 12598 Pink Party 9112 Brown Party 20059 Purple Party 4466 L} … © CORBETTMATHS 2019 726 (2) << /ProcSet [ /PDF /ImageB /ImageC /ImageI ] /ColorSpace << /Cs1 7 0 R >> Type 2: Estimating with Equations Estimating with equations is a little bit more difficult, since we also have to interpret the question. The steps involved are: Finding relevant consecutive multiples. 1050 (a) Work out the size of the angle marked x. IOS Give a reason for your answer. 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Primary 5-a-day, videos, worksheets and more PROBLEM SOLVING - free samples 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. If you want some questions on any topic, click on the Corbett maths link, then on "videos on worksheets". Factors, Multiples & Primes. Multiplication Madness. The maximum mark for this paper is 80. We have a new interactive tool to help students to visualise various types of prisms. These questions have been collated by me as the basis for a GCSE working party set up by the GLOW maths hub - if you want to get involved please get in touch. If you want some questions on any topic, click on the Corbett maths link, then on "videos on worksheets". %PDF-1.3 Estimation worksheets encompass a huge collection of printable sheets on rounding and estimation. Notice that the number of significant figures in the question is the maximum number of non-zero digits in your answer. Corbett Maths Videos Remember your 5 a day. Week Beginning. Mathster; Corbett Maths And best of all they all (well, most!) Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. Using … August 29, 2019 corbettmaths. And best of all they all (well, most!) endobj Class 9H2, 9H3 and 9S1. 650 AB is parallel to CD. Estimation worksheets encompass a huge collection of printable sheets on rounding and estimation. Rounding Whole Numbers . 66 ÷ 6 = 11 72 ÷ 6 = 12 11 or 12 200,000 150,000 153,000. 7 0 R /SMask 10 0 R /BitsPerComponent 8 /Filter /FlateDecode >> << /Length 5 0 R /Filter /FlateDecode >> He adds the numbers on the two cards together He then rounds the result to the nearest 10 His answer is 40. "At a football match between City and Rovers, there were 4486 fans. Bounds of Error: Worksheets with Answers. Maths English Biology Chemistry Physics History Geography Economics Psychology (2) (b) Work out the size of the angle marked y. Any value which was equal to 5.435 or higher and it would’ve rounded up to 5.44. Identify factors and multiples of a number. Justin chooses two of these cards. And best of all they all (well, most!) to perform quick calculations. 4 0 obj Videos, worksheets, 5-a-day and much more Estimate the force on an object which has mass 5.87 kg and acceleration 24.02 m/s ^2.. xM�1�0{^1 >;��4t��F�����% h���.L,�C���h�����ç�(���p�?o��N���Tj.���r�޴�&Dه䴓�z}$Ϙ�l���� � � x� p\Wy��$��@(���P��kA�[�P*p���0oË���H��i�mi+��vZ�ʋE�Qi����:u�֣�UqP�,ᢺ�A��v��r';�����{�=��{Www�J���q��w�=�y>����{ϛR� @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � @ � �����Օ�R��X*--���+-ਔWW�$+�UZ^Y��W�������ʕj��V7ʵ\���u�7�����-*��*���j�f�z� ��U7��5�k��u�į�|F�U�����E��3�OB@ � �8������Ў�khdz~�1�P-�MbY��1X�Y��j�41�=R�1�bey��ojyê���p�Ʌ�lq�.shbn�J�Temq|x�N60>��n���O� �O-�VV�Gó;F�*��2㿟Z��Z��-����隭C�� õ�1ǖ��٥��bHcp��7{Ѫ�8R\��(TT�6:?��[K;�&���3��� @ �ؘ+��ɋ��Ƨffggg&��(*�X&�Ixf���d_6���)R�ј�K��鉒�WJ�F�.3��&C��F(����η�. endstream 5 0 obj Using … This takes you from basic rounding to whole numbers up through decimal places, significant figures and beyond! Answers for all worksheets on Corbett Maths are on the website. Rounding: to nearest 100 Video 277b Practice Questions Textbook Exercise In order to round down to 5.43, the capacity of the jug must have been strictly less than 5.435. Rounding BODMAS Percentages of Amounts Ratio Frequency Trees Area of Rectangles/Triangles Venn Diagrams Ratio Negative Indices Equations (letters both sides) Y = rnx + c Cubic Graphs Reciprocal Graph Density Pressure Volume of Spheres/Cones Video number 276, 222 205 to 209 386 256 16 320 318 160 155, 156 253 169, 170 277a, 277b, 278 211 234, 235 GCSE Maths - AQA 8300 Here is a detailed explanation of the entire syllabus. Here is a function machine. (If there is a particular topic for which you would like answers, please get in touch - my email is on the home page.) And best of all they all (well, most!) Identify prime and composite numbers. Videos, worksheets, 5-a-day and much more These questions have been collated by me as the basis for a GCSE working party set up by the GLOW maths hub - if you want to get involved please get in touch. Topic based resource to print or project with 3 sets of differentiated questions and answers. come with answers. come with answers. When each number is rounded to the nearest hundred, the difference between them is 100. and 10. We live outside of UK at an international school so not tutors or help available for 11+, but we found this very useful as my daughter and i could manage on our own easily with PiAcademy's detailed stepwise answers. so 343 150 Circle all the cube numbers. Contents. 112 Check your answers seem right. ... Use rounding to estimate the answer to calculations. 1.!A sign says there are 230 people that live in a village, correct to the nearest 10.!! ; Net The prism is 'unfolded' into its net. Visit the post for more. Answers to worksheets (where available) are included at the bottom of each topic. Estimation: Worksheets with Answers. Find the input. (a) What is the lowest possible number of people that live in the village? Finding the midpoint of the consecutive multiples. if you are rounding to nearest 10, you first look at the digit in the 10s column. If it is 5 or bigger, you round the digit in the 10s place up. The difference between two numbers is 4. Our lower bound is 5.425 and our upper bound is 5.435, so we express our answer as, 5.425 \leq C < 5.435 2 0 obj Key: 1 14 means 14 passengers 0 1 2 7 4 1 Written by highly experienced maths specialists, each day's planning includes a starter, whole class teaching, group work, practice activities and a plenary. October 7, 2019 corbettmaths. Weekly worksheets and selected old syllabus exam questions. e����+�E\h���L�X\����U�Y�-M��l���b Topic. Mental Math addition. Rounding numbers code breaker - mathspad.co.uk; Rounding lessons - Boss Maths; Variationtheory.com rounding tasks; Simple rounding worksheets including number lines - Math Salamanders; Rounding Pixel Puzzle 1 and Puzzle 2 - MissBrookesMaths (corrections here) Rounding slides and Rounding to 2dp Question Trade - Teachit Maths Tom says the answer is 516.16 Can you explain Tom’s mistake? Time: 1 hour 30 minutes 2. Spring Week 2 – Number: Multiplication & Division; Spring Week 1 – Number: Multiplication & Division; Activity Week; Autumn Week 12 – Number: Multiplication & Division Corbett Maths | Indices Answers. I thought it might be helpful to share some of that advice here for my readers who aren't on Twitter. "In the match report, 4486 was rounded to the nearest thousand. Party Number of Votes Gold Party 12598 Pink Party 9112 Brown Party 20059 Purple Party … Past paper exam questions organised by topic and difficulty for Edexcel GCSE Maths. Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. You may use tracing paper. 5,000,000 5,200,000 5,250,000 5,246,000 0.3 3.6 8.1. The Math Hatter It contains estimating the count, estimating time, estimating sum, difference, product and quotient, estimating fractions and decimals, rounding whole numbers, fractions and decimals and rounding … Estimation: Worksheets with Answers. bid —k fin 38. b dccecoe - quaâer in decreae (3) The table gives information about the number of people voting for each party at an election. I have linked to Corbett maths videos where possible, and added some extra notes. ... Corbett Maths Rounding Estimating 4 4 Fractions. Primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 ... 29 x 20 = 580 31 x 20 = 620 both round … 25,500 26,499. Primary 5-a-day, videos, worksheets and more QUICK COVER - free samples A collection of lessons, each lesson has a set of 40 questions, an examples/hints sheet and the soltuions. Corbettmaths. Please answer the following questions so that a customer service representative can start your search for perfect tutor. I have linked to Corbett maths videos where possible, and added some extra notes. 4 0 obj Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. Corbettmaths Primary Round 0.005089 to 1 significant figure, then 2 significant figures. Please also note that the layout in terms of fonts, answer lines and space given to each question does not reflect the actual papers to save space. answers; Post navigation. Mathster; Corbett Maths 18.09: Induction, British values, ground rules, BIDMAS, symbols including <, >, ≤, ≥, √ and negative number arithmetic.Some of this was in the written assessment test you did before joining the course. Skill isolation: rounding. Mathster; Corbett Maths Explain how you worked out your answer. Rounding BODMAS Percentages of Amounts Ratio Frequency Trees Area of Rectangles/Triangles Venn Diagrams Ratio Negative Indices Equations (letters both sides) Y = rnx + c Cubic Graphs Reciprocal Graph Density Pressure Volume of Spheres/Cones Video number 276, 222 205 to 209 386 256 16 320 318 160 155, 156 253 169, 170 277a, 277b, 278 211 234, 235 Write all the numbers between 12 and 50 that are factors of 90 30 © CORBETTMATHS 2019. Rearranging Formulae: Worksheets with Answers. �1��9y�xCg��]��j^�~SLR����ղ�N�n���'dE���@�:�ɞ[�~f��$w�"��(�d@Q�覻��*ߕ�|? A worksheet on Rounding. ! a sign says there are 230 people that live in the question the. 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