Tag: measure

gt.geometric topology – Thurston measure under finite covers

Let $S= S_{g,n}$ be a finite type orientable surface of genus $g$ and $n$ punctures and let $mathcal{ML}(S)$ denote the corresponding space of measured laminations. The Thurston measure, $mu^{Th},$ is a mapping class group invariant and locally finite Borel measure on $mathcal{ML}(S)$ which is obtained as a weak-$star$ limit of (appropriately weighted and rescaled) sums of Dirac measures supported on the set of integral multi-curves.

The Thurston measure arises in Mirzakhani’s curve counting framework. Concretely, given a hyperbolic metric $rho$ on $S_{g,n}$, let $B_{rho} subset mathcal{ML}(S)$ denote the set of measured laminations with $rho$-length at most $1$. Then $mu^{Th}(B_{rho})$ controls the top coefficient of the polynomial that counts multi-curves up to a certain $rho$-length and living in a given mapping class group orbit.

Question: Fix a hyperbolic metric $rho$ on $S$ and a finite (not necessarily regular) cover $p: Y rightarrow S$. Let $rho_{p}$ denote the hyperbolic metric on $Y$ obtained by pulling $rho$ back to $Y$ via $p$. Is there a straightforward relationship between $mu^{Th}(B_{rho_{p}})$ and $mu^{Th}(B_{rho})$? For example, is the ratio
$$ frac{mu^{Th}(B_{rho})}{mu^{Th}(B_{rho_{p}})} $$
uniformly bounded away from $0$ and $infty$? Does it equal a fixed value, independent of $rho$? If so, can it be easily related to the degree of the cover $Y rightarrow S$?

It seems hard to approach the above by thinking about counting curves on $Y$ versus $S$, because “most” simple closed curves on $Y$ project to non-simple curves on $S$. But, maybe the generalizations of curve counting for non-simple curves due to Mirzakhani (https://arxiv.org/pdf/1601.03342.pdf) or Erlandsson-Souto (https://arxiv.org/pdf/1904.05091.pdf) could be useful. Of course, both apply to counting curves in a fixed mapping class group orbit, so it’s not clear (to me) how to apply these results either since multi-curves on $Y$ can project to curves on $S$ with arbitrarily large self intersection.

Thanks for reading! I appreciate any ideas or reading suggestions.

real analysis – Proving exterior measure of a cube is equal to its volume

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This is from Stein and Shakarchi’s Measure Theory. I don’t understand why we must use this whole $epsilon$-proof (below the black line). Wouldn’t the following much simpler proof show $|Q| leq sum_{j=1}^infty |Q_j|$?

We’ve previously proven a lemma that states if $R, R_1, …, R_N$ are rectangles, and $R subseteq cup_{j=1}^infty R_j$, then
$$
|R| leq sum_{j=1}^N |R_j|.
$$

This is for finite $N$, but since each $|R_j|$ is positive, this must hold for an infinite collection of rectangles as well. Therefore, the claim that
$$
|Q| leq sum_{j=1}^infty |Q_j|
$$

follows immediately.

measure theory – Inclusions between $L^p$ spaces.

Theorem. Let $mu(X)<infty$. Then $$1le p le qleinftyimplies L^q(X,mathcal{A},mu)subseteq L^p(X,mathcal{A},mu) $$

Def. Let $Omegasubseteqmathbb{R}^n$ be an open subset, $fcolonOmegato (-infty,+infty)$ q.o defined. We said that the function $f$ is locally integrable in $Omega$ if $fin L^1(G,mathcal{L}(mathbb{R}^n)cap G,lambda)$ for all $Ginmathcal{L}(mathbb{R^n})$ such that $overline{G}subseteqOmega.$

In the definition $lambda$ is the Lebesgue measure on $mathbb{R}^n$.

We denote the set of locally integrable function with $L^1_{text{loc}}$

I must prove that $$L^p(Omega)subseteq L^1_{text{loc}}(Omega)quadtext{for all};pin(1,+infty)$$

using the previous theorem.

Naturally $$L^1(Omega)subseteq L^1_{text{loc}}(Omega)$$

Now I don’t know how to proceed. Could anyone give me a suggestion? Thanks!

How can I measure the exact range of focus of a given fixed focus webcam?

The concept of depth of field is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that depth-of-field will vary based upon a change to any of the following factors: focal length, aperture, magnification/display size, viewing distance, etc.

There’s only one distance that is in sharpest focus. Everything in front of or behind that distance is blurry. The further we move away from the focus distance, the blurrier things get. The questions become: “How blurry is it? Is that within our acceptable limit? How far from the focus distance do things become unacceptably blurry?”

What we call depth of field (DoF) is the range of distances in front of and behind the point of focus that are acceptably blurry so that to our eyes things still look like they are in focus.

The amount of depth of field depends on two things: total magnification and aperture. Total magnification includes the following factors: focal length, subject/focus distance, enlargement ratio (which is determined by both sensor size and display size), and viewing distance. The visual acuity of the viewer also contributes to what is acceptably sharp enough to appear in focus instead of blurry.

The distribution of the depth of field in front of and behind the focus distance depends on several factors, primarily focal length and focus distance.

The ratio of any given lens changes as the focus distance is changed. Most lenses approach 1:1 at the minimum focus distance. As the focus distance is increased the rear depth of field increases faster than the front depth of field. There is one focus distance at which the ratio will be 1:2, or one-third in front and two-thirds behind the point of focus.

At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.

At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.

For why this is the case, please see:

Why did manufacturers stop including DOF scales on lenses?
Is there a ‘rule of thumb’ that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920×1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?
As well as this answer to Simple quick DoF estimate method for prime lens

ux field – How to measure where your site/app ranks in Anderson’s UX pyramid?

It would be quite difficult to get the same level of granularity as the Anderson’s UX pyramid, but the way that the different levels of user experience is ranked gives us a clue as to how we can possibly go about it as a starting point.

I suggest that the ‘chasm’ that is difficult to cross, which is at the level of CONVENIENT allows you to at least work out which side your organisation’s products and services lie. And if you look immediately below CONVENIENT there is USABLE, which is a relatively well-defined quality that can be measured semi-quantitatively in many different ways (look for questions relating to measuring usability or usability testing).

There are more formal processes to dissect the various levels of experiences, again not at the granularity that the pyramid describes, but a starting point would be the Kano’s model of customer satisfaction that defines ‘Delighters’ which can help you see if there are elements of your products/services that lie in the DESIRABLE level.

How can are measure the exact range of focus of a given fixed focus webcam?

By moving objects around the camera I can see that they get very blurry at 10cm and less sharp at 3-4m away from the camera but how can I measure the exact range of focus? If I check the sharpness of document at different distances – when it’s too close it’s blurry but when it’s far it’s not sharp probably due to the limited resolution.

(Actually this webcam is manual focus so basically I want to calibrate it).

I only know the horizontal view angle (60 degrees) and can guess the sensor size (1/2.3-1/4.5″).

I am also curious to know whether it’s possible to measure focal length or

geometric measure theory – Minimal cones and homology spheres

Let $Sigma subset mathbf{S}^{n}$ be a codimension one, embedded minimal surface in the round $n$-dimensional sphere. Let moreover $mathbf{C} = mathbf{C}(Sigma)$ be the minimal cone in $mathbf{R}^{n+1}$ whose link is $Sigma$. This is called a homology sphere if it has the same homology as $mathbf{S}^{n-1}$, that is $H_0(Sigma,mathbf{Z}) = mathbf{Z} = H_{n-1}(Sigma,mathbf{Z})$ and $H_i(Sigma,mathbf{Z}) = 0$ for all other $i$.

  • The topology of this link—in the sense of whether or not is a homology sphere—is relevant in some constructions. For example White (1) constructs complete minimal surfaces asymptotic to a given area-minimising $mathbf{C}$ if its link $Sigma$ is not a homology sphere (along with some other hypotheses). White points out that all area-minimising cones (known at the time) had links that were not homology spheres.
  • On the other hand, in a series of papers Hsiang constructed examples of non-equatorial minimal hyperspheres in $mathbf{S}^n$. Although I may be misinterpreting Hsiang’s results, as far as I understand none of the corresponding cones are known to be area-minimising, but Hsiang and Sterling (2) proved that some of them are stable, for a range of dimensions that is large—roughly $n geq 20$ if I am not mistaken—and conjectured unbounded.

Question 1. Does there exist an example of a homology sphere $Sigma^{n-1}$ minimally embedded in $mathbf{S}^n$ so that $mathbf{C} = mathbf{C}(Sigma)$ is area-minimising? For instance is any of the examples of Hsiang–Sterling known (or suspected) to be minimising?

Question 2. In the other direction, is there an example of a homology sphere $Sigma$ so that $mathbf{C}(Sigma)$ is stable but not area-minimising? Morally speaking, should this be the rule or the exception?

(1) White. New applications of mapping degrees to minimal surface theory. J. Differential Geom. 29 (1989). no. 1, 143-162.

(2) Hsiang and Sterling. On the construction of nonequatorial minimal hyperspheres in $S^n(1)$ with stable cones in $mathbf{R}^{n+1}$. Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 24, Phys. Sci., 8035-8036.