## Are there fluid tripod heads made for panning with a lightweight, compact camera?

A video camera tripod is different than a still photograph tripod, because it needs some resistance when panning and tilting.

It is normally more sturdy because it is expected to stay in place while you move the head. A light one will move when you pan. It does not really matter if the camera is light or heavy.

Of course if the camera is heavier you need one more reliable, in materials and construction.

If you just need a tripod for still photography, just buy one you like. But feel it first, do not buy something “cheap” just because.

## compactness – Does a compact Lie group have finitely many conjugacy classes of maximal Abelian Lie subgroups?

Let $$G$$ be a compact Lie group. An Abelian Lie subgroup $$A leq G$$ is a maximal Abelian Lie subgroup if, for any Abelian Lie subgroup $$A’$$ such that $$A leq A’ leq G$$, then $$A’ = A$$.

Of course any maximal torus of $$G$$ (there is only one, up to conjugacy classes) is a maximal Abelian Lie subgroup, but there are other ones too, for example the Klein 4-group in $$mathrm{SO}(3)$$.

What I’m wondering is if the number of conjugacy classes of maximal Abelian Lie subgroups of any compact Lie group $$G$$ is always finite?

## oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.

## settings – Missing HSPA/LTE bands on Xperia Z3 Compact D5803?

With the recent official Lineage build for Z3C, I imported a cheap 5803 from Hong Kong to play around with. Installed Lineage, and noticed I was only getting an Edge connection on T-Mobile. Re-installed a “stock” ROM via Flashtool/Xperifirm, and the problem persisted. I’ve had two 5803’s in the past, with the same carrier, and had no problems with the data connection. Going into the “Configuration” service menu, I looked at the available bands, and it seems like lots are missing (basing my expectations on FrequencyCheck). Am I confused, or is there something wrong/different with this phone?

## dashboard – How to show progress bar and percentage value in a compact space?

By trying to keep the number in the bar, users are potentially getting information less quickly, which goes against what a dashboard seeks to achieve: Insight of status at a glance.

You can get more contrast by pulling the number up, and making it larger. Then, display progress as a contrasting line, reserving the color only for the values that are progressing.

I’m not sure what your other constraints are, but if the purple needs more contrast, you can darken the text and the bar, but they can work visually as one unit.

It’s easier to read a prominent ‘7%’, than to calculate the fill position in the progress bar. Your current design is the other way around: the bar is prominent, but I strain to read the text.

Rather than two purple hues, gray represents the absence of completion.

## measure theory – Polish compact groups locally acting on standard Lebesgue spaces

Yes $$G$$ is a countable discrete group, then we can consider the change of Bernoulli $$2 ^ G$$. $$G$$ acts on $$2 ^ G$$ via shift, and leaving $$mu$$ be the product of $$(1/2, 1/2)$$-measure in each coordinate, then $$(2 ^ G, mu)$$ is a Borel probability measure, essentially free, preserving the action of $$G$$ on a standard Lebesgue space.

My question is whether there is an analogue for locally compact Polish groups. More specifically, if $$G$$ is a locally compact Polish group, $$G$$ admit a Borel probability measure, essentially free, preserving the action on a standard Lebesgue space?

## Compact space integration of signed radon measurements in the Sobolev \$ W ^ {- 1, q} \$ space of Evans paper; Does it work in a space dimension?

Context: I am working on a PDE problem where I have an approximate sequence of measured value functions and I have to incorporate it compactly in a negative Sobolev space $$W ^ {- m, q}$$ over the bounded interval $$mathbb {R}$$. I am especially interested in spaces where $$q = 2$$. I only found such integration in the only theorem of the article:

Evans – Weak convergence methods for nonlinear partial differential equations, 1990.

Theorem 6 (Compactness of measurements, page 7): Suppose the sequence $${ mu_k } _ {k = 1} ^ { infty}$$ is delimited by $$mathcal {M} (U)$$, $$U subset mathbb {R} ^ n$$. so $${ mu_k } _ {k = 1} ^ { infty}$$ is precompressed in $$W ^ {- 1, q} (U)$$ for each $$1 leq q <1 ^ *$$.

Here $$mathcal {M} (U)$$ represents the space of the Radon measurements signed on $$U$$ of finite mass, $$U subset mathbb {R} ^ n$$ is an open, bounded and smooth subset of $$mathbb {R} ^ n, n geq 2$$ and $$1 ^ * = frac {n} {n-1}$$ represents a Sobolev conjugate.

The identical theorem (Lemma 2.55, page 38) is given in the book: Malek, Necas, Rokyta, Ruzicka – Weak and measured solutions to evolutionary PDE, 1996, with the difference that instead of $$1 leq q <1 ^ *$$, there is written $$1 leq q < frac {n} {n-1}$$ (Here it is not explicitly written that $$n geq 2$$).

My question: Theorem 6 works in one dimension ($$n = 1$$)? Simply put, do we have a compact footprint of space $$mathcal {M} (U)$$ in space $$W ^ {- 1, q} (U)$$, or $$U subset mathbb {R}$$?

• I guess if we have a compact integration in $$W ^ {- 1, q} (U)$$, we also have it in the $$W ^ {- m, q} (U), m geq 1$$?
• Are there other measurement spaces (for example, finite positive measurement space $$mathcal {M} _ +$$, probability measure space with finite first moment $$Pr_1$$, etc.) which are compactly integrated into certain negative spaces of Sobolev $$W ^ {- m, q} (U)$$?

I think if we use the definition of the Sobolev conjugate: $$frac {1} {p ^ *} = frac {1} {p} – frac {1} {n}$$, we get for $$p = 1, n = 1$$ the $$frac {1} {1 ^ *} = frac {1} {1} – frac {1} {1} Rightarrow 1 ^ * = infty$$. So we would have Theorem 6 (maybe) to work for each $$1 leq q < infty$$ (then for $$q = 2$$ as well)? If we use $$p ^ * = frac {np} {n-p}$$ we would have for $$n = 1,$$ $$p ^ * = frac {p} {1-p}$$ and here we couldn't take $$p = 1$$ and get $$p ^ *$$.

I don't usually take care of measured and negative Sobolev spaces, so I don't know much about them. Help would be great and I really need it. And any additional references in addition to the two mentioned above would be nice. Thanks in advance.

## functional analysis – Results on: (Path / Initial condition) – A dependent variant of the exponential map generates compact diffeomorphisms

Let $$M$$ to be a connected and simply connected Riemannian variety, not compact, and suppose that $${V_p } _ {p in M}$$ is a family of vector fields on $$M$$ indexed by $$M$$. Suppose further that the card
$$p to V_p (x),$$
is smooth and defines the differential equation
$$f_t & # 39; = V_p (f_t), , f_0 = p. qquad (*)$$
So in $$(*)$$ there is an explicit dependency $$p$$ both in the CI and in the "dynamics" of ODE itself. So let's re-label this explicit dependency $$p$$ of a solution to $$(*)$$ by $$f_ {t, p}$$.

Let $$X$$ be the collection of all smooth vector fields with compact support on $$M$$ such as

• A solution $$f_ {t, p}$$ exists in time $$1$$ for each $$p$$,
• Card taking $$(V_p) _ {p in M}$$ at the time $$1$$ solution of $$(*)$$ is a diffeomorphism,
• $$p to V_p (x)$$ is not constant for at least a value of $$x in M ​​$$.

Let's denote this last card by $$tilde {Exp}: X to Diff_ {c, 0} (M)$$. We know, see this article for references, that if there is no dependence on $$p$$ then $$Exp$$ generates $$Diff_ {c, 0} (M)$$. Are such results known in this case, that is to say where the vector fields depend not trivially on integrated circuits?

## fa.functional analysis – Definition of Lyapunov exponents for compact operators

There is the following well-known result from Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exhibitors:

Let $$H$$ be a $$mathbb R$$-Hilbert space, $$A_n in mathfrak L (H)$$ be compact and $$B_n: = A_n cdots A_1$$ for $$n in mathbb N$$. Let $$| B_n |: = sqrt {B_n ^ ast B_n}$$ and $$sigma_k (B_n)$$ denote the $$k$$e greatest singular value of $$B_n$$ for $$k, n in mathbb N$$. Yes $$limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 tag1$$ and $$frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for all} k in mathbb N tag2,$$ then

1. $$| B_n | ^ { frac1n} xrightarrow {n to infty} B$$ for some non-negative and self-adhesive compacts $$B in mathfrak L (H)$$.
2. $$frac { ln sigma_k (B_n)} n xrightarrow {n to infty} Lambda_k: = left. begin {cases} gamma ^ {(k)} – gamma ^ {(k -1)} & text {, if} gamma ^ {(i)}> – infty \ – infty & text {, otherwise} end {cases} right } tag2$$ for everyone $$k in mathbb N$$.

question 1: I have seen this result in many textbooks, but I have wondered why it is stated this way. First of all, isn't it $$(2)$$ clearly equivalent to $$frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3$$ for some people $$lambda_i$$ for everyone $$k in mathbb N$$ which in turn is equivalent to $$sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4$$ for some people $$mu_i ge0$$ for everyone $$k in mathbb N$$? $$(4)$$ seems to be much more intuitive than $$(3)$$, since no $$lambda_i$$, But $$mu_i = e ^ { lambda_i}$$ are precisely Lyapunov's exponents of the limit operator $$B$$. Am I missing something? The definition of $$Lambda_i$$ (which is equal to $$lambda_i$$) seems odd to me.

question 2: What is the interpretation of $$B$$? Usually I look at a discrete dynamic system $$x_n = B_nx_0$$. What is $$B$$ (or $$Bx$$) tells us about the asymptotic behavior / evolution of orbits?

## What are the minimum lighting conditions necessary for good autofocus in compact cameras (sub DSLR)?

By good autofocus, I mean stable focusing in 3 seconds or less.

By compact cameras, I mean for example the Canon SX.

And I'm interested in a measurable criterion (preferably in Lux).

Edit:

I'm not looking for a specific answer for a specific model. Just for an order of magnitude – like 100 Lux is usually enough.