## The reason behind IPv6 adoption rate dramatical drop in China according to Google measurements?

Google has an IPv6 measurement page that reports that their numbers report on the percentage of users that access Google over IPv6.

According to the report by Jan 2020 0.3% of users in China used IPv6 to access Google

However, looking at this metric in dynamic we see the substantial drop starting from June 2019.

I failed to find any solid news that may cause such behavior. I have two hypotheses in mind.

1. Also as it is a percentage metric, they can adjust their calculation on the total internet penetration rate in China.
2. Previously open discussions between netizens took place on Google Plus groups. In April 2019,
Chinese-language blogs, forums, and groups. For obvious reasons,
discussions must be hosted outside China, and posters must register under pseudonyms. So probably that caused the shift from Google services but I hardly believe that it may cause such plummet.

## 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.

## set theory – Stationary accuracy of ultra-powers by low-order measurements

Assume $$U$$ is a normal ultrafilter on $$kappa$$ of Mitchell order zero, and let $$j_U: V to M$$ be the associated incorporation. Is there a non stationary $$X subseteq kappa ^ +$$ such as $$X in M ​​$$ and $$M models X$$ is stationary?

Note that if $$W$$ is a normal measurement derived from an embedding $$i: V to N$$ or $$mathcal P ( kappa ^ +) subseteq N$$, then $$W$$ gives a correct stationary ultrapower, hence the restriction to a low Mitchell order.

## fa. functional analysis – Decomposition of the radon measurements space while respecting the fractional harmonic capacity?

It is well known that there is a generalization of Lebesque's decomposition theorem as follows.

Any non-negative Radon measure can be broken down only into absolutely continuous and singular terms with respect to the harmonic capacity. The absolute continuous term itself can be decomposed (not only) into a function in L ^ 1 and a function in H ^ {- 1} (dual of H_0 ^ 1).

I will be thanked if someone could help me. Is there some kind of decomposition for fractional harmonic capacity?

## gps – how to calculate long and lat location of raw measurements on android phones?

I am doing research on GNSS I used the GNSS recorder to collect measurements but I couldn't understand how these measurements turned into long and lat location? The image below of the measurement attributes that I have collected with the GNSS recorder.

## reference request – Law of large numbers for random Dirac measurements

assume $${X_1, … X_n }: Omega to mathbb {R} ^ p$$ to be i.i.d. random vectors with common probability law / measure $$p$$, that is to say. $$Prob (X_i ^ {- 1} (E)) = p (E) forall E subset mathbb {R} ^ p$$ Measurable Borel.

Consider Dirac's random measurements $$delta_ {X_i}$$, and their mean, which is a random probability measure over $$mathbb {R} ^ p$$, Defined by $$frac {1} {n} sum_ {i = 1} ^ {n} delta_ {X_i}$$. I would like to know if $$frac {1} {n} sum_ {i = 1} ^ {n} delta_ {X_i}$$ weakly converges to deterministic measurement $$p$$,

that is to say for each function continuous and limited $$f: mathbb {R} ^ p to mathbb {R}$$, must you have:

$$frac {1} {n} sum_ {i = 1} ^ {n} {f (X_i)} to int _ { mathbb {R} ^ p} f (x) dp (x)$$
as probability convergence of a sequence of random variables $$frac {1} {n} sum_ {i = 1} ^ {n} {f (X_i)}$$?

ON A related note, I would also like to know whether the following is true or not:

If a sequence of random measures converges to a deterministic probability measure, is it equivalent to having the same convergence almost surely? This question is motivated by the fact that when a sequence of random variables converges to a probability constant, convergence is a.s.

P.S. I understand that this question could be elementary for many of you, so some references would be greatly appreciated!

## Convex Analysis – Extreme points of a set of measurements with a given barycenter

Let $$X$$ to be a compact convex metric subspace of a locally convex Hausdorff topological vector space, $$x_0 in X$$, and $$P$$ to be the space of all Borel probability measures on $$X$$ with barycenter $$x_0$$.

Question: What are the extreme points of $$P$$?

In the particular case of $$p in P$$ whose support has a finite dimension range, it is easy to deduce that $$p$$ is extreme if and only if his support is affinely independent. This, in particular, fully characterizes the extreme points if $$X$$ is of finite dimension.

I wonder if there is an equally simple / geometric characterization for the general case.

## GPS – Application Using Geometric Measurements

I am doing an application that would be for a client when I decided to put two 30cm. I could generate a calculation between these companies approximate distances from fixed coordinate coordinates with the name of the relevant moderates via the customer's GPS if it gets closer the point appeared in the application someone knows it the quickly generate in Java

## Equivalent distance measurements?

That's maybe a ridiculous question and an apology to all the mathematicians here who think so, but that's …

√ (a ^ 2 + b ^ 2 + c ^ 2) = | a | + | b | + | c |

? ..

## Non-commutative geometry – Equivalence of two approaches of transversal measurements for a foliation

Assume that $$(V, F)$$ is a flaky variety. There are three approaches equivalent to the notion of cross measurement as described in this book (see pages 65-69). I would like to understand the last line of section 5$$alpha$$ where it is stated that ,, it is easy to check, as in the case of flows, that $$Lambda$$ meets the definition 2 & # 39 ;. The context is as follows: we start with a closed current $$C$$ (of degree $$= dim F$$) positive in the sense of the sheet and with the help of this current defines a measure $$mu_U$$ (locally on $$U$$, domain of the foliation sheet) on the plate set by the fromula
$$langle C, omega rangle = int Big ( int _ { pi} omega Big) of mu_U ( pi).$$
Once we have this measurement, we can define $$Lambda (B): = int Map (B cap pi) of mu_ {U} ( pi)$$ for any cross Borel $$B$$ (ie Borel subassembly $$B subset V$$ as for each sheet $$L$$ of foliation $$B cap L$$ is at most countable.

Why $$Lambda$$ satisfied $$Lambda (B) = Lambda ( psi (B))$$ for any injection of Borel $$psi$$ which keeps the leaves.

I guess that should somehow stem from the state of $$C$$ closed, but I do not know how to perform the calculations (for example, a problem I encountered is that $$Lambda$$ is defined locally and I do not see how to move from $$U$$ to another table of foliation $$U$$ (which can happen for the general $$psi$$).