## 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.

## ca.classical analysis and odes – Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn’t seem to be an answer. I apalogize in advance for cross posting.

Let $$Erightarrow X$$ be a holomorphic vector bundle over a compact Riemann surface with a holomorphic connection $$nabla:Erightarrow Eotimes K$$, where $$K$$ is the canonical bundle of $$X$$. Since the holomorphic connection is necessarily flat, its sheaf of local holomorphic sections $$mathcal{E}$$ defines a (holonomic) $$D$$-module. Every holonomic D-module is locally cyclic, i.e. for any point $$z_0$$ there exists a neighborhood $$U$$ s.th. $$mathcal{E}(U)$$ has a cyclic generator as a $$D$$-module (see e.g. Proposition 3.1.5. in Björk: Analytic $$D$$-Modules and Applications). Suppose we are given a coordinate $$z$$ on $$U$$ and identify $$D(U)cong D_1$$, with $$D_1=mathbb{C}leftlbrace z rightrbrace leftlangle partial_z rightrangle$$ (differential operators with coefficients in convergent power-series). So locally it holds $$mathcal{E}(U)cong D_1/ I$$, where $$I$$ is the ideal of differential operators annihilating the cyclic generator. This ideal is in general generated by two elements $$P,Q$$, with $$P$$ an operator of smallest possible degree in $$I$$ and furthermore $$I/D_1P$$ is of torsion type, i.e. for any $$Din I$$ it holds $$z^nDin D_1P$$ for some $$n$$ (Proposition 5.1.4 and Remark 5.1.5 in Björk: Analytic $$D$$-Modules and Applications). This implies for the dual $$D$$-module $$hom_{D_1}(D_1/I,mathbb{C}leftlbrace zrightrbrace)=leftlbrace fin mathbb{C}leftlbrace zrightrbrace , middle|, Pf=Qf=0rightrbrace=leftlbrace fin mathbb{C}leftlbrace zrightrbrace , middle|, Pf=0rightrbrace$$.

So far so good. Now on $$U$$ the holomorphic connection reads $$nabla|_U=partial+A$$ with $$A$$ some matrix of holomorphic functions.The dual bundle naturally comes with a holomorphic connection, too, which in local coordinates takes the form $$partial-A^T$$. The whole discussion above shows that locally flat sections ($$(partial-A^T)Y=0$$) are in one to one correspondence with solutions of $$Pf=0$$.

On the other hand there is Deligne’s lemma of a cyclic vector. One way to formulate it, is to say that locally on a coordinate neighborhood $$U$$, for a vector bundle with holomorphic connection there exists $$Gin mathrm{GL}(n,mathcal{O}(U))$$, s.th.
$$begin{equation} partial_z G-G A^T=tilde{A}G end{equation}$$ with $$tilde{A}$$ in companion form. Here $$partial-A^T$$ is the local form of the holomorphic connection. But in general the non unit entries $$a_i$$ in $$tilde{A}$$ are only meromorphic and $$G$$ might not be invertible as a holomorphic matrix.

It is clear that a system of linear differential equations $$partial_z Y=A^T Y$$ with $$A^T$$ in companion form corresponds to a single $$n$$-th order scalar differential equation $$Qf=0$$. So from Deligne’s cyclic vector lemma I get an $$n$$-th order scalar differential equation, but the corresponding differential operator might not be in $$D_1$$, but in $$mathbb{C}leftlbrace zrightrbrace (z^{-1})leftlangle partial_zrightrangle$$.

Q: Is there any relation between the differential operator I get from the discussion in the first paragraph applied to the dual bundle and the differential equation I get from Deligne’s cyclic vector lemma?

I guess they are the same, maybe after imposing further constraints on the open $$U$$. It might very well be that the relation is obvious and just shows my lack of understanding.

## 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.

## Measurement theory: locally compact Polish groups operating in standard Lebesgue spaces

Yes $$G$$ is a discrete accounting group, so one can consider Bernoulli's turn $$2 ^ G$$. $$G$$ acts on $$2 ^ G$$ in turn, and leaving $$mu$$ be the product of the $$(1/2, 1/2)$$-measured at each coordinate, then $$(2 ^ G, mu)$$ is a Borel probability measure, essentially free, that preserves the action of $$G$$ in a standard Lebesgue space.

My question is whether there is any analogue of this for locally compact Polish groups. More precisely, if $$G$$ is a locally compact Polish group, does $$G$$ Admitting an essentially free Borel probability measure that preserves action in a standard Lebesgue space?

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

Background: I work on a PDE problem where I have a rough sequence of functions with measure values ​​and I need to compactly embed it in some negative space of Sobolev $$W ^ – m, q}$$ in the limited interval in $$mathbb {R}$$. I am mainly interested in spaces where $$q = 2$$. I found just one of those inlays in the article's only theorem:

Evans – Weak Convergence Methods for Nonlinear Partial Differential Equations, 1990.

Theorem 6 (Compactness for measurements, page 7): suppose the sequence $${ mu_k } _ {k = 1} ^ { infty}$$ is limited in $$mathcal {M} (U)$$, $$U subset mathbb {R} ^ n$$. So $${ mu_k } _ {k = 1} ^ { infty}$$ it is pre-compact in $$W ^ – 1, q} (U)$$ for each $$1 leq q <1 ^ *$$.

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

The identical theorem (Lemma 2.55, page 38) is given in the book: Malek, Necas, Rokyta, Ruzicka – Weak and Measured Value Solutions for Evolutionary PDE, 1996, with a difference that instead of $$1 leq q <1 ^ *$$, there it is written $$1 leq q < frac {n} {n-1}$$ (here it is not explicitly written that $$n geq 2$$)

My question: How Theorem 6 works in one dimension ($$n = 1$$)? I mean we have a compact space inlay $$mathcal {M} (U)$$ in the space $$W ^ – 1, q} (U)$$, where $$U subset mathbb {R}$$?

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

I think if we use Sobolev's conjugate definition: $$frac {1} {p ^ *} = frac {1} {p} – frac {1} {n}$$, we arrived by $$p = 1, n = 1$$ the $$frac {1} {1 ^ *} = frac {1} {1} – frac {1} {1} Rightarrow 1 ^ * = infty$$. So we would have that theorem 6 (maybe) works for every $$1 leq q < infty$$ (and then for $$q = 2$$ also)? If we use $$p ^ * = frac {np} {n-p}$$ we would have for $$n = 1,$$ $$p ^ * = frac {p} {1-p}$$ and here we couldn't drink $$p = 1$$ and get $$p ^ *$$.

I don't usually deal with Sobolev spaces with measure values ​​and negatives, so I don't know much about them. Help with this would be great and I definitely need it. And any additional reference in addition to the two mentioned above would be good. Thanks in advance.

## actual analysis: let \$ (X, d) \$ be a compact metric space. Then \$ (X, d) \$ is complete and limited.

Definition: a metric space $$(X, d)$$ it is said to be compact if each sequence supports a converging subsequence.

I have problems with the following result:

Leave $$(X, d)$$ Being a compact metric space. So $$(X, d)$$ it is both complete and limited.

So far I have been able to demonstrate that $$(X, d)$$ is completed as follows.

Yes $$X$$ is compact, so each sequence $$x_ {n} in X$$ supports a convergent subsequence.

In particular, if $$x_ {n}$$ is Cauchy, then admits a convergence subsequence, which also makes it convergent.

What worries me is the second part. How do we show that it is bounded?

Could someone help me with this?

## GN general topology: separation of compact sets into locally compact spaces

Not all locally compact Haussdorff spaces are known to be normal, see for example

here

But it seems like the following is true, I just want to make sure I don't make any mistakes:

Motto Leave $$X$$ be a locally compact Haussdorff space and leave $$K, W subset X$$ be compact with $$K cap W = emptyset$$. So there are open sets $$K subset U, W subset V$$ such that $$U cap V = emptyset$$.

Proof:

We have $$K subset (X backslash W)$$. As $$K$$ it's compact $$X backslash W$$ is open and $$X$$ is a locally compact Haussdorff space, according to Rudin's Theorem 2.7, "Real and complex analysis" we can find an open set $$U$$ (with compact closure) so that $$K subset U subset bar {U} subset X backslash W$$.

Leave $$V: = X backslash bar {U}$$. So $$V$$ It is open and meets the above conditions.

qed

Is this correct? I'm a little uncomfortable, especially since the test only uses the fact that $$W$$ is closed, compactness is only necessary to $$K$$.

## Fa functional analysis – Definition of Lyapunov exponents for compact operators

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

Leave $$H$$ be a $$mathbb R$$-Hilbert space, $$A_n in mathfrak L (H)$$ be compact and $$B_n: = A_n cdots A_1$$ for $$n in mathbb N$$. Leave $$| B_n |: = sqrt {B_n ^ ast B_n}$$ and $$sigma_k (B_n)$$ denote the $$k$$th largest 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 $$frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for everyone} k in mathbb N tag2,$$ so

1. $$| B_n | ^ { frac1n} xrightarrow {n to infty} B$$ for some non-negative and self-attached compact $$B in mathfrak L (H)$$.
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 all $$k in mathbb N$$.

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

Question 2: What is the interpretation of $$B$$? I'm usually seeing a discrete dynamic system $$x_n = B_nx_0$$. That makes $$B$$ (or $$Bx$$) tell us about the asymptotic behavior / evolution of the orbits?