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In the early days of mobile, debugging was quite a challenge. Sure, you could get ahold of a device and perform a quick visual assessment, but what would you do after discovering a bug?
With a distinct lack of debugging tools, developers turned to a variety of hacks. In general, these hacks were an attempt to recreate a given issue in a desktop browser and then debug with Chrome Developer Tools or a similar desktop toolkit. For instance, a developer might shrink the size of the desktop browser's window to test a responsive website or alter the user agent to spoof a particular mobile device.
To put it bluntly, these hacks don't work. If you're recreating issues on the desktop, then you can't be certain that any of your fixes will work. This means you'll be constantly bouncing back and forth between the mobile device and the hacks in your desktop browser.
Fast forward to today, when we have a robust suite of debugging tools that provide meaningful debugging information directly from a physical device. Best of all, you can use the same desktop debugging tools that you know and love, all on an actual mobile device.
In this article, we'll explore a variety of emulators and simulators that you can use for quick and easy testing. Then, we'll look at remote debugging tools, which enable you to connect a desktop computer to a mobile device and leverage a rich debugging interface.
Testing on real physical devices always pays off. But that doesn't mean you shouldn't also test on emulators and simulators. These virtual environments not only expand your testing coverage to more devices, but also are a quick and easy way to test small changes on the fly.
To test iOS devices, such as the iPhone and iPad, you have a number of options, most notably Apple's official iOS Simulator. Included as part of Xcode, this simulator enables you to test across different software and hardware combinations, but only from a Mac.
Although iOS Simulator is difficult to find, using it is fortunately easy. Simply open up Safari in the simulator and test your website. You can switch between different iPhone and iPad devices, change the iOS version, rotate the viewport and more.
Android also provides an emulator. Luckily, this one is cross-platform. Unfortunately, setting it up is a bit of a pain.
First, download the bundle that includes Android Development Tools (ADT) for Eclipse and the Android software development kit (SDK). Next, follow Google's instructions to install the SDK packages, making sure to install the default selections as well as the "Intel x86 Emulator Accelerator (HAXM installer)". You'll also need to track down HAXM — search your Mac forIntelHaxm.dmg or your PC for IntelHaxm.exe, and run the file to install it.
Installing the Android SDK packages: HAXM improves the performance of the emulator.
The "Device Definitions" tab provides preset AVDs. Use one of them or create your own.
Set whatever you like for the CPU, and set "No skin" and "Use host GPU." Now you can run the virtual device and use Android's browser to test your website.
Finally, you'll probably want to learn some keyboard commands to better interact with the emulator.
Note: Manymo is an alternative, in-browser Android emulator. You can even embed it in a web page, which is pretty darn cool.
Emulators and simulators are useful, but they're not 100% accurate. Always test on as many real devices as possible.
That doesn't mean you need to buy a hundred phones and tablets. You can take advantage of remote testing resources, which provide a web-based interface to interact with real physical devices. You'll be able to interact with a phone remotely and view any changes in the screencast that is sent back to your machine.
If you want to test a Samsung device, such as the Galaxy S5, you can do so for free using Samsung's Remote Test Lab, which enables you to test on a wide selection of Samsung devices.
Additionally, you can use the resources in Keynote Mobile Testing. They're not cheap, but the number of devices offered is pretty astonishing, and you can test a handful of devices for free.
Note: If you're looking to get your hands on real devices, Open Device Lab can point you to a lab in your area, where you can test on a range of devices for free.
Remote debugging addresses a variety of the challenges presented by mobile debugging. Namely, how do you get meaningful debugging information from a small and relatively underpowered device?
Remote debugging tools provide an interface to connect to a mobile device from a desktop computer. Doing this, you can debug for a mobile device using the development tools on a more powerful, easier-to-use desktop machine.
With the release of iOS 6.0, Apple introduced a tool that enables you to use desktop Safari's Web Inspector to debug iOS devices.
Now, in the "Develop" menu you should see your iOS device, along with any open pages in mobile Safari.
Once your iOS device is connected, you'll see it in the "Develop" menu.
Select one of these pages, and you'll have a wide range of developer tools at your fingertips. For example, try out the DOM Inspector, which enables you to tap DOM elements on your mobile device and see debugging information on the desktop.
Web Inspector in desktop Safari is inspecting this iPhone.
You don't need a physical iOS device to use remote debugging. You can also debug instances of iOS Simulator.
Note: Much like iOS Simulator, you can only do remote debugging for iOS on Mac OS X.
Similar to iOS, Android has a remote debugging solution. The tools in it enable you to debug an Android device from a desktop machine using Chrome's Developer Tools. Best of all, Android's remote debugging is cross-platform.
First, go to "Settings" → "About Phone" on your Android 4.4+ phone (or "Settings" → "About Tablet"). Next, tap the "Build Number" seven (7) times. (No, I'm not joking. You'll see a message about being a developer at the end.) Now, go back to the main settings and into "Developer Options." Here, enable "USB debugging," and you're all set.
Go into your desktop Chrome browser, and type about:inspect in the address bar. Enable "Discover USB devices," and you'll see your device in the menu.
Once you enable "Discover USB devices," you'll see a list of devices connected remotely to Chrome, along with a list of debuggable web pages or apps for each device.
Here, the DOM Inspector in the desktop browser is remotely inspecting a page on the Android device.
Note: You can also remotely debug with the Android emulator.
You now know how to remotely debug a variety of devices. But if you want to debug iOS on Windows or on Linux or debug other devices, such as Windows Phone or BlackBerry, then try Weinre, which works on any device.
Note: Make sure to insert your own IP in the command above. You can find your IP on a Mac using the command ipconfig getifaddr en0 and on Windows using ipconfig.
Next, go to the development server that is outputted by Weinre in the console (in my case, it'slocalhost:8080). Here, look at the "Target Script" section, and grab the <script> tag. You'll need to include that on whichever pages you want to debug.
The Weinre development server gives you the client-side script to embed, along with a link to the debugging interface.
Finally, click on the link at the top of this page for the user interface for debugging clients (in my case, it's http://localhost:8080/client/#anonymous). Now, once you open the page in your device, you should see it in the list of targets.
Note: If you're having trouble connecting a device to your localhost, consider setting up a public tunnel with ngrok.
Weinre's debugging interface provides a link to each debuggable target.
Here, Weinre is debugging iOS with the DOM Inspector.
Note: Ghostlab is another remote testing option that supports multiple platforms.
In this article, we've learned how to set up a robust testing suite using a combination of physical devices, emulators, simulators and remote testing tools. With these tools, you are now able to test a mobile website or app across a wide variety of devices and platforms.
We've also explored remote debugging tools, which provide useful information directly from a mobile device. Hopefully, you now realize the benefits of remote debugging for mobile. Without it, we're really just taking stabs in the dark.


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text:
United States Secretary of Labor Alexander Acosta stepping down, Trump says
U.S. Secretary of Labor Alex Acosta speaks during a press conference July 10, 2019 at the Labor Department in Washington, DC. (Photo by Alex Wong/Getty Images)
WASHINGTON - (AP) -- Labor Secretary Alexander Acosta said Friday he is resigning following renewed scrutiny of his handling of a 2008 secret plea deal with wealthy financier Jeffrey Esptein , who is accused of sexually abusing dozens of underage girls.
President Donald Trump, with Acosta at his side, made the announcement as he left the White House for a trip to Wisconsin and Ohio. The president said Acosta had been a "great" labor secretary.
"I hate to see this happen," Trump said. He said he did not ask Acosta to leave the Cabinet.
Acosta said his resignation would be effective in seven days. Acosta said he didn't think it was right for his handling of Epstein's case to distract from his work as secretary of labor.
"My point here today is we have an amazing economy and the focus needs to be on the economy job creation," Acosta said.
Acosta was the U.S. attorney in Miami when he oversaw a 2008 non-prosecution agreement Jeffrey Epstein. Epstein avoided federal charges, plead guilty to state charges and served 13 months in jail. Similar charges recently filed against Epstein by federal prosecutors in New York had put Acosta's role in the 2008 deal under renewed scrutiny.
Top Democratic lawmakers and presidential candidates had demanded that Acosta resign over his handling of the agreement, which a federal judge has said violated federal law because Acosta did not notify Epstein's victims of the arrangement. The Justice Department has been investigating.
Trump had initially defended Acosta but said he'd look "very closely" his handling of the 2008 agreement.
The deal came under scrutiny earlier this year following reporting by the Miami Herald.
Epstein, 66, reached the deal to secretly end a federal sex abuse investigation involving at least 40 teenage girls that could have landed him behind bars for life. He instead pleaded guilty to state charges, spent 13 months in jail, paid settlements to victims and is a registered sex offender.
Acosta had attempted to clear his name, and held a news conference -- encouraged by Trump -- to defend his actions. In a 50-plus-minute lawyerly rebuttal, Acosta argued his office had secured the best deal it could at the time and was working in the victims' best interests.
"We did what we did because we wanted to see Epstein go to jail," he said, refusing to apologize for his actions. "We believe that we proceeded appropriately."
Pressed on whether he had any regrets, Acosta repeatedly suggested that circumstances had changed since then.
"We now have 12 years of knowledge and hindsight and we live in a very different world," he said. "Today's world treats victims very, very differently," he said.
After federal attorneys in New York announced the new charges against Epstein this week, Acosta tweeted that he was "pleased" by their decision.
"The crimes committed by Epstein are horrific," Acosta tweeted. "With the evidence available more than a decade ago, federal prosecutors insisted that Epstein go to jail, register as a sex offender and put the world on notice that he was a sexual predator."
"Now that new evidence and additional testimony is available, the NY prosecution offers an important opportunity to more fully bring him to justice," he said.
Acosta, the nation's 27th labor secretary, took on the role officially in early 2017, leading a sprawling agency that enforces more than 180 federal laws covering about 10 million employers and 125 million workers. He was confirmed in the Senate 60-38.
But Acosta had frustrated some conservatives who had been pushing for his ouster long before the Epstein uproar. Among their frustrations were Acosta's decisions to proceed with several employment discrimination lawsuits and to allow certain Obama holdovers to remain on the job.
Acosta is a former federal prosecutor and civil rights chief. Before joining the administration he was dean of the Florida International University law school.
© 2019 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.
'This is a misguided idea': UFO whistleblower Bob Lazar warns people not to storm Area 51
By Kelly Taylor Hayes
Bob Lazar, a physicist who claims to have worked with alleged alien technology and exposed Area 51 to the world in 1989, issued a warning to anyone attempting to actually storm the classified U.S. Air Force base.
More than a million people have signed up to attend a joke Facebook event planning a raid on Area 51 in southern Nevada to "see them aliens."
The now-viral event, titled, "Storm Area 51, They Can't Stop All of Us," states that the group of alien hunters will meet at 3 a.m. on Sept. 20 near the top-secret base to coordinate a plan of attack to reveal the truth behind the curtain.
By Ryan Gaydos | Fox News
Retired boxer Pernell "Sweet Pea" Whitaker, who held world titles at four weight classes, has been killed after being hit by a car, Virginia Beach police said Monday.
An emergency call came in Sunday night about a vehicle-pedestrian accident and when officers arrived at the scene they located an adult male who was hit by a vehicle, police said in a news release.
Whitaker, 55, of Norfolk, Va., was identified as the victim. Police said he died at the scene.
Trump moves to end asylum protections for most Central American migrants at US-Mexico border
(AP) -- The Trump administration is moving to end asylum protections for most Central American migrants.
A rule published Monday bars migrants from seeking asylum in the United States if they've traveled through another country first.
Tens of thousands of migrant families from Central America travel through Mexico to the U.S. each month, many claiming asylum. The Trump administration claims families are taking advantage of legal loopholes it says allow migrants a free pass to the country while they wait out phony asylum requests.

redpajama_set_name: RedPajamaCommonCrawl


This is not the newest "black hat" SEO trick. And in no way am I implying Google Adwords (paid search) is the same or even related to organic content. Even though now you can track results using metrics, Adwords and organic still are different animals.
While there are other ways to get SEO top ranking in Google, this tactic is by far the least understood.
Advertising is advertising. No matter what channel your use, you are paying for impressions and hopefully a few conversions too.
I happen to think, as many of my peers do, that digital advertising using a service such as Adwords yields a far better bang for your buck compared to print or radio advertising—but paid search is still advertising, not some third marketing realm.
With Adwords you pay for keyword ranking. The more you pay, the better you rank. When you stop paying, you stop ranking; your ads disappear and whatever traction you've gained goes away—immediately.
Organic content is just that: You publish website content, blog content, post on social media, YouTube, etc. When Google or some other search engine finds your content, it gets ranked.
Simple? Not really. It would be simple if we knew what Google's ranking algorithm looked like, but we don't, so we rely on experience, marketing research, and a little guesswork.
While it shouldn't happen, we've seen organic ranking increase by using both Adwords and organic in a digital marketing strategy.
The screen shot below was from a client who, at the time we took over their marketing, had done very little organic content marketing but had been running Adwords campaigns. The irony is that the screen shot is not from an Adwords or PPC report, it's from Hubspot's organic Keyword Tool.
So how did this company get #1 organic ranking on so many keywords? In this case, the organic results come from executing Adwords campaigns while inadvertently using similar keywords on their website. Is this crazy? To some degree, yes.
How can this work? Simple. Adwords builds website traffic—very much like social media does. Any way you can drive traffic to your website will build ranking. The challenge is: how to do this efficiently and predictably.
Pillar content is comprised of one main "killer" blog article linking back to several associated lesser blog articles.
Start with a main theme that would be of interest to your key prospects (personas).
Research and choose a keyword associated with your main theme.
Add associated (child) blog topics and choose keywords for them.
A topic cluster is simple: It's a series of posts or pages interlinked together. Thus, when Google crawls one of those posts or pages, the search engine robot can find your other, similar content easily.
A topic cluster should also represent quality content. Google is very interested in delivering quality content because quality is a reflection on the search engine's capability of finding what you're looking for.
If someone searches the Internet and fails to find what they're looking for, Google looks bad. Google is in the advertising business and their sales ability comes from successful searches.
I don't want to go into detail on how to run an Adwords campaign. But I will interject and say this: You'll save money and get better results by hiring an agency to manage your Adwords campaigns. There are reasons for this.
How potato chips are cooked.
Where to buy potato chips.
The main secret to using both Adwords and organic to boost your ranking is to: Loop in one pillar content link to a site page that contains an SEO keyword you want to rank for.
Write a main blog article on "how potato chips are made." Add other articles to create a topic cluster, linking them together.
Add a static page to your website on "potato chips manufacturing" using an SEO-rich keyword on that page.
Add (link) that static page to your pillar content strategy.
Use the exact keyword from your static page in an Adwords campaign.
Thus, you are buying keyword ranking for the same keyword you are using in your blog and website content.
Sounds crazy and maybe elaborate, but try it for a while and watch your organic keyword ranking grow. Be sure to watch your keyword metrics over time.
Beware that if you only focus on Adwords ranking and not on creating keyword rich organic content, your success will be dependent on the dollars you spend on advertising. In short, you'll gain results, but with worse ROI than if you created SEO rich organic content as well.
Sorry to cite a cliche, but it's the same old, "you can do something or you can do it well."
What to know more? Have a conversation with us!


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NTSB: Engineer sped 14 times in week before derailment
William Rockefeller exceeded speed limit on four of six runs a week before Bronx crash.
NTSB: Engineer sped 14 times in week before derailment William Rockefeller exceeded speed limit on four of six runs a week before Bronx crash. Check out this story on lohud.com: http://lohud.us/1vDSVp3
Khurram Saeed, ksaeed@lohud.com Published 6:48 p.m. ET Oct. 3, 2014 | Updated 10:36 p.m. ET Oct. 3, 2014
In this photo taken on Sunday, Dec. 1, 2013, Metro North Railroad engineer William Rockefeller is wheeled on a stretcher away from the area where the commuter train he was operating derailed in the Bronx borough of New York.(Photo: AP Photo/Robert Stolarik)
The Metro-North Railroad engineer who sped off the tracks last December causing a deadly derailment in the Bronx had exceeded the speed limit several times on the Hudson Line in the week before the crash, a federal analysis released Friday shows.
Using event recorder data, investigators found that William Rockefeller went over the speed limit by at least 5 mph 14 times during four of six runs in the last week of November. Precise information about how fast he was going on those occasions was not included in the 46 documents released Friday by the National Transportation Safety Board.
The NTSB is investigating the Dec. 1 derailment that killed four people, including Jim Lovell, 58, of Cold Spring and James Ferrari, 59, of Montrose, and injured dozens. There were an estimated 115 people on board.
On that day, Rockefeller's train was traveling at 82 mph when it flew off a curve that had a 30 mph speed limit. Rockefeller told investigators he had fallen into a trance-like state and he was later found to have suffered from severe sleep apnea that went undiagnosed before the derailment.
The NTSB expects to issue its final report on the cause of the crash next month.
In January, representatives from the NTSB, Metro-North and the Federal Railroad Administration analyzed event recorder data from various locomotives on the Hudson Line prior to the December derailment. A random sample of trains operated by engineers other than Rockefeller found speeding at different locations between Grand Central Terminal and Poughkeepsie.
Of the 18 runs reviewed from Nov. 16 to Nov. 30, there were 20 speeding incidents over nine runs. Those trains went from 5 mph to 12 mph over the posted speed limit, including in some spots where trains were allowed to travel as fast as 80 mph.
Metro-North spokeswoman Marjorie Anders said "much has changed" in the past 10 months.
"Since then, Metro-North has implemented an aggressive speed compliance program, including more random downloads, more radar checks and more on-board supervision," Anders said. "It has been very effective in reducing overspeed events."
In August, she said the railroad conducted 79 speed checks by radar and 86 event recorder downloads. Of these 165 tests, there was one speeding incident which was less than 10 mph over the limit. There are also signal-enforced restrictions now in place that automatically slow trains at curves and bridges.
Metro-North President Joseph Giulietti has spent much of the year stressing safety. He has lowered on-time performance goals, which were once a hallmark and source of pride for the Metropolitan Transportation Authority commuter railroad.
Another document released by the NTSB on Friday listed the crews' cellphone calls and texts and appeared to confirm that the engineer and conductors were not using phones at the time of the crash. The FRA said earlier there was no evidence the crash had anything to do with phone use.
Other documents reported on the condition of the derailed Metro-North cars. There were blood-stained seats, floors and doors. Several rail cars lost almost all their windows on one side and were littered with dirt, rocks and even tree branches when they came to rest.
The new documents include a quote from a New York City police detective who entered one of the damaged cars right after the accident.
"There was people's personal items and there was people laying around," he told the NTSB. "There was not one window left on the train."
In a document relating to an accident in March, the Metro-North engineer on a train that killed an electrician on the tracks in Manhattan told investigators that the electrician never looked up despite the fact that he was blowing the train's horn.
William Maher said he was coming out of a tunnel when he noticed three workers too close to the track. He said two got out of the way, but one, later identified as James Romansoff, 58, of Yonkers, leaned or fell into the train's path.
Maher told investigators that track workers used to stop what they were doing when a train approached, acknowledge the engineer, and go back to work once the train passed. He said most workers now continue what they are doing without ever looking up.
"To me," he said, "that's a little discomforting."
The Associated Press contributed information to this article.
Twitter: @ksaeed1
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text:
\section{Introduction}\label{sec-1}

Hida's white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals
of Brownian motion \cite{hida,huang,kuo,obata}. In 1988, Ito \cite{ito} introduced his analysis of generalized Poisson functionals, which can be viewed as
a theory of infinite dimensional calculus on generalized functionals of Poisson martingale.
It is known that both Brownian motion and Poisson martingale are continuous-time normal martingales.
There are theories of white noise analysis for some other continuous-time processes (see, e.g., \cite{albe, barhoumi, di, hu,lee}).

Discrete-time normal martingales \cite{privault} also play an important role in many theoretical and applied fields.
For example, the classical random walk (a special discrete-time normal martingale) is used to establish functional central limit theorems
in probability theory \cite{dud,rud}.
It would then be interesting to develop a theory of infinite dimensional calculus on generalized functionals
of discrete-time normal martingales.

Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale satisfying some mild conditions.
In a recent paper \cite{wang-chen}, we constructed generalized functionals of $M$, and introduced a transform, called the Fock transform,
to characterize those functionals.

In this paper, we apply the Fock transform \cite{wang-chen} to investigate generalized functional sequences of $M$.
First, by using the Fock transform, we obtain a necessary and sufficient condition for a generalized functional sequence of $M$ to be strong convergent.
Then we introduce a type of generalized martingales associated with $M$, called $M$-generalized martingales, which are a special class of generalized functional sequences of $M$
and include as a special case the classical martingales with respect to the filtration generated by $M$.
We establish a strong-convergent criterion in terms of the Fock transform for $M$-generalized martingales. Some other convergence criteria are also obtained.
Finally we show some applications of our main results.

Our one interesting finding is that for an $M$-generalized martingale, its strong convergence is just equivalent to its strong boundedness.

Throughout this paper, $\mathbb{N}$ designates the set of all nonnegative integers and $\Gamma$ the finite power set of $\mathbb{N}$, namely
\begin{equation}\label{eq-1-1}
\Gamma = \{\,\sigma \mid \text{$\sigma \subset \mathbb{N}$ and $\#(\sigma) < \infty$} \,\},
\end{equation}
where $\#(\sigma)$ means the cardinality of $\sigma$ as a set.
In addition, we always assume that $(\Omega, \mathcal{F}, P)$ is a given probability space with $\mathbb{E}$ denoting the expectation with respect to $P$.
We denote by $\mathcal{L}^{2}(\Omega, \mathcal{F}, P)$ the usual Hilbert space of
square integrable complex-valued functions on $(\Omega, \mathcal{F}, P)$
and use $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ to mean its inner product and norm, respectively.
By convention, $\langle\cdot,\cdot\rangle$ is conjugate-linear in its first argument and linear in its second argument.

\section{Generalized functionals}\label{sec-2}

Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale on $(\Omega, \mathcal{F}, P)$
that has the chaotic representation property and
$Z=(Z_n)_{n\in \mathbb{N}}$ the discrete-time normal noise associated
with $M$ (see Appendix). We define
\begin{equation}\label{eq-2-1}
   Z_{\emptyset}=1;\quad Z_{\sigma} = \prod_{i\in \sigma}Z_i,\quad \text{$\sigma \in \Gamma$, $\sigma \neq \emptyset$}.
\end{equation}
And, for brevity, we use $\mathcal{L}^2(M)$ to mean the space of square integrable functionals of $M$, namely
\begin{equation}\label{eq-2-2}
  \mathcal{L}^2(M) = \mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P),
\end{equation}
which shares the same inner product and norm with $\mathcal{L}^2(\Omega, \mathcal{F}, P)$, namely $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$.
We note that $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ forms a countable orthonormal basis for $\mathcal{L}^2(M)$ (see Appendix).

\begin{lemma}\label{lem-2-1}\cite{wang-z}
Let $\sigma\mapsto\lambda_{\sigma}$ be the $\mathbb{N}$-valued function on $\Gamma$ given by
\begin{equation}\label{eq-2-3}
\lambda_{\sigma}=
\left\{
  \begin{array}{ll}
    \prod_{k\in\sigma}(k+1), & \hbox{$\sigma\neq \emptyset$, $\sigma\in\Gamma$;}\\
    1, & \hbox{$\sigma=\emptyset$, $\sigma\in\Gamma$.}
  \end{array}
\right.
\end{equation}
Then, for $p>1$, the positive term series $\sum_{\sigma\in\Gamma}\lambda^{-p}_{\sigma}$ converges and moreover
\begin{equation}\label{eq-2-4}
\sum_{\sigma\in\Gamma}\lambda^{-p}_{\sigma}\leq \exp\bigg[\sum_{k=1}^{\infty}k^{-p}\bigg]<\infty.
\end{equation}
\end{lemma}

Using the $\mathbb{N}$-valued function defined by (\ref{eq-2-3}), we can construct a chain of Hilbert spaces consisting of functionals of $M$ as follows.
For $p\geq 0$, we define a norm $\|\cdot\|_p$ on $\mathcal{L}^2(M)$ through
\begin{equation}\label{eq-2-5}
  \|\xi\|_{p}^2=\sum_{\sigma\in \Gamma}\lambda_{\sigma}^{2p}|\langle Z_{\sigma}, \xi\rangle|^{2},\quad \xi \in \mathcal{L}^2(M)
\end{equation}
and put
\begin{equation}\label{eq-2-6}
  \mathcal{S}_p(M) = \big\{\, \xi \in \mathcal{L}^2(M) \mid \|\xi\|_{p}< \infty\,\big\}.
\end{equation}
It is not hard to check that $\|\cdot\|_{p}$ is a Hilbert norm and $\mathcal{S}_p(M)$ becomes a Hilbert space
with $\|\cdot\|_{p}$. Moreover, the inner product corresponding to $\|\cdot\|_{p}$ is given by
\begin{equation}\label{eq-2-7}
  \langle \xi,\eta\rangle_p
  = \sum_{\sigma\in \Gamma}\lambda_{\sigma}^{2p}\overline{\langle Z_{\sigma},\xi\rangle} \langle Z_{\sigma}, \eta\rangle,\quad
  \xi,\, \eta \in \mathcal{S}_p(M).
\end{equation}
Here $\overline{\langle Z_{\sigma},\xi\rangle}$ means the complex conjugate of $\langle Z_{\sigma},\xi\rangle$.

\begin{lemma}\label{lem-2-2}\cite{wang-chen}
For each $p\geq 0$, one has $\{Z_{\sigma}\mid \sigma\in\Gamma\} \subset \mathcal{S}_p(M)$ and moreover the system
$\{\lambda^{-p}_{\sigma}Z_{\sigma}\mid \sigma\in\Gamma\}$ forms an orthonormal basis for $\mathcal{S}_p(M)$.
\end{lemma}

It is easy to see that $\lambda_{\sigma}\geq 1$ for all $\sigma\in \Gamma$. This implies that $\|\cdot\|_p \leq \|\cdot\|_q$
and $\mathcal{S}_q(M)\subset \mathcal{S}_p(M)$
whenever $0\leq p \leq q$. Thus we actually get a chain of Hilbert spaces of functionals of $M$:
\begin{equation}\label{eq-2-8}
 \cdots \subset \mathcal{S}_{p+1}(M) \subset \mathcal{S}_p(M)\subset  \cdots \subset \mathcal{S}_1(M) \subset \mathcal{S}_0(M)=\mathcal{L}^2(M).
\end{equation}
We now put
\begin{equation}\label{eq-2-9}
  \mathcal{S}(M)=\bigcap ^{\infty}_{p=0}\mathcal{S}_{p}(M)
\end{equation}
and endow it with the topology generated by the norm sequence $\{\|\cdot \|_{p}\}_{p\geq 0}$.
Note that, for each $ p\geq 0$, $\mathcal{S}_p(M)$ is just the completion of $\mathcal{S}(M)$ with respect to $\|\cdot\|_{p}$.
Thus $\mathcal{S}(M)$ is a countably-Hilbert space \cite{becnel, gelfand-vi}.
The next lemma, however, shows that $\mathcal{S}(M)$ even has a much better property.

\begin{lemma}\label{lem-2-3}\cite{wang-chen}
The space $\mathcal{S}(M)$ is a nuclear space, namely for any $p\geq 0$,
there exists $q> p$ such that the inclusion mapping $i_{pq}\colon \mathcal{S}_{q}(M) \rightarrow \mathcal{S}_p(M)$
defined by $i_{pq}(\xi)=\xi$ is a Hilbert-Schmidt operator.
\end{lemma}

For $p\geq 0$, we denote by $\mathcal{S}_p^*(M)$ the dual of $\mathcal{S}_p(M)$ and $\|\cdot\|_{-p}$
the norm of $\mathcal{S}_p^*(M)$. Then $\mathcal{S}_p^*(M)\subset \mathcal{S}_q^*(M)$ and
$\|\cdot\|_{-p} \geq \|\cdot\|_{-q}$ whenever $0\leq p \leq q$.
The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see, e.g., \cite{becnel} or \cite{gelfand-vi}).

\begin{lemma}\label{lem-2-4}\cite{wang-chen}
Let $\mathcal{S}^*(M)$ the dual of $\mathcal{S}(M)$ and endow it with the strong topology. Then
\begin{equation}\label{eq-2-10}
  \mathcal{S}^*(M)=\bigcup_{p=0}^{\infty}\mathcal{S}_p^*(M)
\end{equation}
and moreover the inductive limit topology on $\mathcal{S}^*(M)$ given by space sequence $\{\mathcal{S}_p^*(M)\}_{p\geq 0}$ coincides with the strong topology.
\end{lemma}

We mention that, by identifying $\mathcal{L}^2(M)$ with its dual, one comes to a Gel'fand triple
\begin{equation}\label{eq-2-11}
\mathcal{S}(M)\subset \mathcal{L}^2(M)\subset \mathcal{S}^*(M),
\end{equation}
which we refer to as the Gel'fand triple associated with $M$.

\begin{lemma}\label{lem-2-5}\cite{wang-chen}
The system $\{Z_{\sigma} \mid \sigma \in \Gamma\}$ is contained in $\mathcal{S}(M)$ and moreover it serves as a basis in $\mathcal{S}(M)$ in the sense that
\begin{equation}\label{eq-2-12}
  \xi = \sum_{\sigma \in \Gamma} \langle Z_{\sigma}, \xi\rangle Z_{\sigma}, \quad \xi \in \mathcal{S}(M),
\end{equation}
where $\langle\cdot,\cdot\rangle$ is the inner product of $\mathcal{L}^2(M)$ and the series converges in the topology of $\mathcal{S}(M)$.
\end{lemma}

\begin{definition}\label{def-2-1}\cite{wang-chen}
Elements of $\mathcal{S}^*(M)$ are called generalized functionals of $M$, while elements of $\mathcal{S}(M)$ are called testing functionals of $M$.
\end{definition}

Denote by $\langle\!\langle \cdot,\cdot\rangle\!\rangle$
the canonical bilinear form on $\mathcal{S}^*(M)\times \mathcal{S}(M)$, namely
\begin{equation}\label{eq-2-13}
  \langle\!\langle \Phi,\xi\rangle\!\rangle = \Phi(\xi),\quad \Phi\in \mathcal{S}^*(M),\, \xi\in \mathcal{S}(M),
\end{equation}
where $\Phi(\xi)$ means $\Phi$ acting on $\xi$ as usual.
Note that $\langle\cdot,\cdot\rangle$ denotes the inner product of $\mathcal{L}^2(M)$, which is different from
$\langle\!\langle \cdot,\cdot\rangle\!\rangle$.

\begin{definition}\label{def-2-2}\cite{wang-chen}
For $\Phi \in \mathcal{S}^*(M)$, its Fock transform is the function $\widehat{\Phi}$ on $\Gamma$ given by
\begin{equation}\label{eq-2-14}
  \widehat{\Phi}(\sigma) = \langle\!\langle \Phi, Z_{\sigma}\rangle\!\rangle,\quad \sigma \in \Gamma,
\end{equation}
where $\langle\!\langle \cdot,\cdot\rangle\!\rangle$ is the canonical bilinear form.
\end{definition}

It is easy to to verify that,
for $\Phi$, $\Psi \in \mathcal{S}^*(M)$, $\Phi=\Psi$ if and only if $\widehat{\Phi}=\widehat{\Psi}$.
Thus a generalized functional of $M$ is completely determined by its Fock transform.
The following theorem characterizes generalized functionals of $M$ through their Fock transforms.

\begin{lemma}\label{lem-2-6}\cite{wang-chen}
Let $F$ be a function on $\Gamma$. Then $F$ is the Fock transform of an element $\Phi$ of $\mathcal{S}^*(M)$ if and only if it satisfies
\begin{equation}\label{eq-2-15}
  |F(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma
\end{equation}
for some constants $C\geq 0$ and $p\geq 0$.
In that case, for $q> p+\frac{1}{2}$, one has
\begin{equation}\label{eq-2-16}
  \|\Phi\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}}
\end{equation}
and in particular $\Phi \in \mathcal{S}_q^*(M)$.
\end{lemma}

\section{Convergence theorems for generalized functional sequences}\label{sec-3}

Let $M=(M_n)_{n\in \mathbb{N}}$ be the same discrete-time normal martingale as described in Section~\ref{sec-2}.
In the present section, we apply the Fock transform (see Definition~\ref{def-2-2}) to establish convergence theorems for generalized functionals of $M$.

In order to prove our main results in a convenient way, we first give a preliminary proposition,
which is an immediate consequence of the general theory of countably normed spaces, especially nuclear spaces \cite{becnel, gelfand-shi, gelfand-vi},
since $\mathcal{S}(M)$ is a nuclear space (see Lemma~\ref{lem-2-3}).

\begin{proposition}\label{prop-3-1}
Let $\Phi$, $\Phi_n \in \mathcal{S}^*(M)$, $n\geq 1$, be generalized functionals of $M$. Then the following conditions are equivalent:

(i)\ \ The sequence $(\Phi_n)$ converges weakly to $\Phi$ in $\mathcal{S}^*(M)$;

(ii)\ \ The sequence $(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^*(M)$;

(iii)\ \ There exists a constant $p\geq 0$ such that $\Phi$, $\Phi_n \in \mathcal{S}_p^*(M)$, $n\geq 1$,
and the sequence $(\Phi_n)$ converges to $\Phi$ in the norm of $\mathcal{S}_p^*(M)$.
\end{proposition}

Here we mention that ``$(\Phi_n)$ converges strongly (resp. weakly) to $\Phi$'' means that $(\Phi_n)$ converges to $\Phi$ in the strong (resp. weak) topology of $\mathcal{S}^*(M)$.
For details about various topologies on the dual of a countably normed space, we refer to \cite{becnel, gelfand-shi}.

The next theorem is one of our main results, which offers a criterion in terms of the Fock transform for checking whether or not a sequence in $\mathcal{S}^*(M)$ is strongly convergent.

\begin{theorem}\label{thr-3-1}
Let $\Phi$, $\Phi_n \in \mathcal{S}^*(M)$, $n\geq 1$, be generalized functionals of $M$.  Then the sequence $(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^*(M)$ if and only if it satisfies:
\begin{enumerate}
  \item[(1)] $\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma)$ for all $\sigma\in \Gamma$;
  \item[(2)] There are constants $C\geq 0$ and $p\geq 0$ such that
\begin{equation}\label{}
   \sup_{n\geq 1}|\widehat{\Phi_n}(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
The ``only if '' part. Let $(\Phi_n)$ converge strongly to $\Phi$ in $\mathcal{S}^*(M)$. Then, we obviously have
\begin{equation*}
\widehat{\Phi_n}(\sigma)=\langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle
\rightarrow
\langle\!\langle \Phi,Z_{\sigma}\rangle\!\rangle=\widehat{\Phi}(\sigma),\quad \sigma\in \Gamma,
\end{equation*}
because $\{Z_{\sigma}\mid \sigma \in \Gamma\}\subset \mathcal{S}(M)$ and $(\Phi_n)$ also converges weakly to $\Phi$.
On the other hand, by Proposition~\ref{prop-3-1}, we know that there exists $p\geq 0$ such that
$\Phi$, $\Phi_n \in \mathcal{S}_p^*(M)$, $n\geq 1$,
and $(\Phi_n)$ converges to $\Phi$ in the norm of $\mathcal{S}_p^*(M)$,
which implies that $C\equiv \sup_{n\geq 1}\|\Phi_n\|_{-p} < \infty$.
Therefore
\begin{equation*}
  \sup_{n\geq 1}|\widehat{\Phi_n}(\sigma)|
   = \sup_{n\geq 1}|\langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle|
   \leq \sup_{n\geq 1}\|\Phi_n\|_{-p}\|Z_{\sigma}\|_p
   = C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation*}

The ``if'' part. Let $(\Phi_n)$ satisfy conditions (1) and (2). Then,
by taking $q > p+ \frac{1}{2}$ and using Lemma~\ref{lem-2-6}, we get
\begin{equation}\label{eq-3-2}
  \sup_{n\geq 1}\|\Phi_n\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}},
\end{equation}
in particular $\Phi_n \in \mathcal{S}_q^*(M)$, $n\geq 1$. On the other hand, $\{Z_{\sigma} \mid \sigma \in \Gamma\}$ is total in $\mathcal{S}_q(M)$,
which, together with (\ref{eq-3-2}) as well as the property
\begin{equation*}
  \langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle
   =\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma)
   = \langle\!\langle \Phi,Z_{\sigma}\rangle\!\rangle,\quad \sigma\in \Gamma,
\end{equation*}
implies that
$\Phi \in \mathcal{S}_q^*(M)$ and
\begin{equation*}
\langle\!\langle \Phi_n,\xi\rangle\!\rangle \rightarrow \langle\!\langle \Phi,\xi\rangle\!\rangle,\quad \forall\, \xi\in \mathcal{S}_q(M).
\end{equation*}
Thus $(\Phi_n)$ converges weakly to $\Phi$ in $\mathcal{S}^*(M)$, which together with Proposition~\ref{prop-3-1} implies that
$(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^*(M)$.
\end{proof}

In a similar way we can prove the following theorem, which is slightly different form Theorem~\ref{thr-3-1}, but more convenient to use.

\begin{theorem}\label{thr-3-2}
Let $(\Phi_n) \subset \mathcal{S}^*(M)$ be a sequence of generalized functionals of $M$.  Suppose
$\big(\widehat{\Phi_n}(\sigma)\big)$ converges for all $\sigma\in \Gamma$, and moreover there are constants $C\geq 0$ and $p\geq 0$ such that
\begin{equation}\label{}
   \sup_{n\geq 1}|\widehat{\Phi_n}(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation}
Then there exists a generalized functional $\Phi\in \mathcal{S}^*(M)$ such that $(\Phi_n)$ converges strongly to $\Phi$.
\end{theorem}

\section{$M$-generalized martingales and their convergence theorems}

In this section, we first introduce a type of generalized martingales associated with $M$,
which we call $M$-generalized martingales, and then we use the Fock transform to a give necessary and sufficient condition for such a generalized martingale to be strongly convergent. Some other convergence results are also obtained.

For a nonnegative integer $n\geq 0$, we denote by $\Gamma\!_{n]}$ the power set of $\{0,1,\cdots, n\}$, namely
\begin{equation}\label{eq-4-1}
  \Gamma\!_{n]}=\{\,\sigma \mid \sigma\subset \{0,1, \cdots, n\}\,\}.
\end{equation}
Clearly $\Gamma\!_{n]} \subset \Gamma$. We use $\mathbf{I}_{n]}$ to mean the indicator of $\Gamma\!_{n]}$, which is a function on $\Gamma$ given by
\begin{equation}\label{eq-4-2}
  \mathbf{I}_{n]}(\sigma)=
  \left\{
    \begin{array}{ll}
      1, & \hbox{$\sigma \in \Gamma\!_{n]}$;} \\
      0, & \hbox{$\sigma \notin \Gamma\!_{n]}$.}
    \end{array}
  \right.
\end{equation}

\begin{definition}\label{def-4-1}
A sequence $(\Phi_n)_{n\geq 0} \subset \mathcal{S}^*(M)$ is called an $M$-generalized martingale if it satisfies that
\begin{equation}\label{eq-4-3}
  \widehat{\Phi_n}(\sigma) = \mathbf{I}_{n]}(\sigma)\widehat{\Phi_{n+1}}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,
\end{equation}
where $\mathbf{I}_{n]}$ mean the indicator of $\Gamma\!_{n]}$ as defined by (\ref{eq-4-2}).
\end{definition}

Let $\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$ be the filtration on $(\Omega, \mathcal{F}, P)$  generated by $Z=(Z_n)_{n\geq 0}$, namely
\begin{equation}\label{}
  \mathcal{F}_n = \sigma\{Z_k \mid 0\leq k \leq n\},\quad n\geq 0.
\end{equation}
The following theorem justifies Definition~\ref{def-4-1}.

\begin{theorem}\label{thr-4-1}
Suppose $(\xi_n)_{n\geq 1}\subset \mathcal{L}^2(M)$ is a martingale with respect to filtration $\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$.
Then $(\xi_n)_{n\geq 1}$ is an $M$-generalized martingale.
\end{theorem}

\begin{proof}
By the assumptions, $(\xi_n)_{n\geq 1}$ satisfies that the following conditions
\begin{equation}\label{eq-4-5}
  \xi_n = \mathbb{E}[\xi_{n+1}\,|\, \mathcal{F}_n],\quad n\geq 0,
\end{equation}
where $\mathbb{E}[\,\cdot \mid\! \mathcal{F}_n]$ means the conditional expectation given $\sigma$-algebra $\mathcal{F}_n$.
Note that
\begin{equation*}
\mathbb{E}[Z_{\tau}\,|\, \mathcal{F}_n] = \mathbf{I}_{n]}(\tau)Z_{\tau},\quad  \tau\in \Gamma,
\end{equation*}
which, together with (\ref{eq-4-5}) and the expansion
$\xi_{n+1} = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle Z_{\tau}$,
gives
\begin{equation*}
  \xi_n = \mathbb{E}[\xi_{n+1}\,|\, \mathcal{F}_n]
        = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle \mathbb{E}[Z_{\tau}\,|\, \mathcal{F}_n]
        = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle \mathbf{I}_{n]}(\tau)Z_{\tau}.
\end{equation*}
Taking Fock transforms yields
\begin{equation*}
  \widehat{\xi_n}(\sigma)
        = \sum_{\tau \in \Gamma}\langle \xi_{n+1}, Z_{\tau}\rangle \mathbf{I}_{n]}(\tau)\widehat{Z_{\tau}}(\sigma)
        = \langle  \xi_{n+1}, Z_{\sigma}\rangle \mathbf{I}_{n]}(\sigma)
        = \mathbf{I}_{n]}(\sigma)\widehat{\xi_{n+1}}(\sigma),
\end{equation*}
where $\sigma \in \Gamma$. Thus $(\xi_n)_{n\geq 1}$ is an $M$-generalized martingale.
\end{proof}

The next theorem gives a necessary and sufficient condition in terms of the Fock transform for an $M$-generalized martingale to be strongly convergent.

\begin{theorem}\label{thr-4-2}
Let $(\Phi_n)_{n\geq 1} \subset \mathcal{S}^*(M)$ be an $M$-generalized martingale. Then the following two conditions are equivalent:
\begin{enumerate}
  \item[(1)] $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^*(M)$;
  \item[(2)] There are constants $C\geq 0$ and $p\geq 0$ such that
\begin{equation}\label{}
   \sup_{n\geq 1}|\widehat{\Phi_n}(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
By Theorem~\ref{thr-3-1}, we need only to prove ``$(2)\Rightarrow (1)$''.
Let $\sigma \in \Gamma$ be taken. Then by the definition of $M$-generalized martingales (see Definition~\ref{def-4-1}) we have
\begin{equation*}
  \widehat{\Phi_m}(\sigma) = \mathbf{I}_{m]}(\sigma)\widehat{\Phi_{m+k}}(\sigma),\quad m,\, k\geq 0.
\end{equation*}
Now take $n_0\geq 0$ such that $\sigma\in \Gamma_{n_0]}$. Then $\mathbf{I}_{{n_0}]}(\sigma)=1$ and moreover
\begin{equation*}
  \widehat{\Phi_{n_0}}(\sigma)
   = \mathbf{I}_{{n_0}]}(\sigma)\widehat{\Phi_{n}}(\sigma)
   = \widehat{\Phi_{n}}(\sigma),\quad n>n_0,
\end{equation*}
which implies $\big(\widehat{\Phi_n}(\sigma)\big)$ converges. Thus, by Theorem~\ref{thr-3-2}, $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^*(M)$.
\end{proof}

\begin{theorem}\label{thr-4-3}
Let $D$  be a subset of $\mathcal{S}^*(M)$. Then the following two conditions are equivalent:
\begin{enumerate}
  \item[(1)] There is a constant $p\geq 0$ such that $D$ is contained and bounded in $\mathcal{S}_p^*(M)$;
  \item[(2)] There are constants $C\geq 0$ and $p\geq 0$ such that
\begin{equation}\label{}
   \sup_{\Phi \in D}|\widehat{\Phi}(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
Obviously, condition (1) implies condition (2). We now verify the inverse implication relation. In fact, under condition (2), by using Lemma~\ref{lem-2-6} we have
\begin{equation*}
 \sup_{\Phi \in D} \|\Phi\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}},
\end{equation*}
where $q>p+\frac{1}{2}$, which clearly implies condition (1).
\end{proof}

The next theorem shows that for an $M$-generalized martingale, its strong (weak) convergence is just equivalent to its strong (weak) boundedness.

\begin{theorem}\label{thr-4-4}
Let $(\Phi_n)_{n\geq 1} \subset \mathcal{S}^*(M)$ be an $M$-generalized martingale. Then the following conditions are equivalent:
\begin{enumerate}
  \item[(1)] $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^*(M)$;
  \item[(2)] $(\Phi_n)_{n\geq 1}$ is weakly bounded in $\mathcal{S}^*(M)$;
  \item[(3)] $(\Phi_n)_{n\geq 1}$ is strongly bounded in $\mathcal{S}^*(M)$;
  \item[(4)] $(\Phi_n)_{n\geq 1}$ is bounded in $\mathcal{S}_p^*(M)$ for some $p\geq 0$.
\end{enumerate}
\end{theorem}

\begin{proof}
Clearly, conditions (2), (3) and (4) are equivalent each other because $\mathcal{S}(M)$ is a nuclear space (see Lemma~\ref{lem-2-3}).
Using Theorems~\ref{thr-4-2} and \ref{thr-4-3}, we immediately know that conditions (1) and (4) are also equivalent.
\end{proof}

\section{Applications}

In the last section we show some applications of our main results.

Recall that the system $\{Z_{\sigma}\mid \sigma\in \Gamma\}$ is an orthonormal basis of $\mathcal{L}^2(M)$. Now if we write
\begin{equation}\label{eq-5-1}
  \Psi_n^{(0)} = \sum_{\tau \in \Gamma\!_{n]}}Z_{\tau},\quad n\geq 0,
\end{equation}
then $\big(\Psi_n^{(0)}\big)_{n\geq 0} \subset \mathcal{L}^2(M)$, and moreover $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is a martingale with respect to filtration
$\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$. However,  $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is not convergent in $\mathcal{L}^2(M)$ since
\begin{equation}\label{eq-5-2}
  \|\Psi_n^{(0)}\|= \sqrt{\#(\Gamma\!_{n]})} = 2^{\frac{n+1}{2}} \rightarrow \infty\quad (\text{as $n\rightarrow \infty$}),
\end{equation}
where $\#(\Gamma\!_{n]})$ means the cardinality of $\Gamma\!_{n]}$ as a set and $\|\cdot\|$ the norm in $\mathcal{L}^2(M)$.

\begin{proposition}\label{prop-5-1}
The sequence $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ defined above is an $M$-generalized martingale, and moreover it is strongly convergent in $\mathcal{S}^*(M)$.
\end{proposition}

\begin{proof}
According to Theorem~\ref{thr-4-1}, $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is certainly an $M$-generalized martingale.
On the other hand, in viewing the relation between the canonical bilinear form on $\mathcal{S}^*(M)\times \mathcal{S}(M)$ and the inner product in $\mathcal{L}^2(M)$, we have
\begin{equation}\label{eq-5-3}
  \widehat{\Psi_n^{(0)}}(\sigma)
   = \langle\!\langle \Psi_n^{(0)}, Z_{\sigma} \rangle\!\rangle
   = \langle \Psi_n^{(0)}, Z_{\sigma} \rangle
   = \mathbf{I}_{n]}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,
\end{equation}
which implies that
\begin{equation*}
  \sup_{n\geq 0}\big|\widehat{\Psi_n^{(0)}}(\sigma)\big| \leq C \lambda_{\sigma}^p,\quad \sigma \in \Gamma
\end{equation*}
with $C=1$ and $p=0$. It then follows from Theorem~\ref{thr-4-2} that $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is strongly convergent in $\mathcal{S}^*(M)$.
\end{proof}

Recall that \cite{wang-chen}, for two generalized functionals $\Phi_1$, $\Phi_2 \in \mathcal{S}^*(M)$, their convolution $\Phi_1\ast\Phi_2$ is defined by
\begin{equation}\label{}
 \widehat{ \Phi_1\ast\Phi_2}(\sigma) = \widehat{ \Phi_1}(\sigma)\widehat{\Phi_2}(\sigma),\quad \sigma \in \Gamma.
\end{equation}

The next theorem provides a method to construct an $M$-generalized martingale through the $M$-generalized martingale$\big(\Psi_n^{(0)}\big)_{n\geq 0}$ defined in (\ref{eq-5-1}).

\begin{theorem}\label{thr-5-1}
Let $\Phi \in \mathcal{S}^*(M)$ be a generalized functional and define
\begin{equation}\label{}
  \Phi_n =  \Psi_n^{(0)} \ast\Phi,\quad n\geq 0.
\end{equation}
Then $(\Phi_n)_{n\geq 0}$ is an $M$-generalized martingale, and moreover it converges strongly to $\Phi$ in $\mathcal{S}^*(M)$.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem-2-6}, there exist some constants $C\geq 0$ and $p\geq 0$ such that
\begin{equation}\label{eq-5-6}
  |\widehat{\Phi}(\sigma)| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation}
On the other hand, by using (\ref{eq-5-3}), we get
\begin{equation}\label{eq-5-7}
  \widehat{\Phi_n}(\sigma)
  =\widehat{\Psi_n^{(0)}}(\sigma)\widehat{\Phi}(\sigma)
  =\mathbf{I}_{n]}(\sigma)\widehat{\Phi}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,
\end{equation}
which, together with the fact $\mathbf{I}_{n]}(\sigma)\mathbf{I}_{n+1]}(\sigma)=\mathbf{I}_{n]}(\sigma)$, gives
\begin{equation*}
  \widehat{\Phi_n}(\sigma)= \mathbf{I}_{n]}(\sigma)\widehat{\Phi_{n+1}}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0.
\end{equation*}
Thus $(\Phi_n)_{n\geq 0}$ is an $M$-generalized martingale. Additionally, it easily follows from (\ref{eq-5-6}) and (\ref{eq-5-7}) that
$\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma)$ for each $\sigma \in \Gamma$ and
\begin{equation*}
  \sup_{n\geq 0}|\widehat{\Phi_n}(\sigma)|
    =\sup_{n\geq 0}\big[\mathbf{I}_{n]}(\sigma)\big]|\widehat{\Phi}(\sigma)|
  \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.
\end{equation*}
Therefore, by Theorem~\ref{thr-3-2}, we finally find $(\Phi_n)_{n\geq 0}$ converges strongly to $\Phi$.
\end{proof}

\section*{Appendix}

In this appendix, we provide some basic notions and facts about discrete-time normal martingales. For details we refer to \cite{privault,wang-lc}.

Let $(\Omega, \mathcal{F}, P)$ be a given probability space with $\mathbb{E}$ denoting the expectation with respect to $P$.
We denote by $\mathcal{L}^{2}(\Omega, \mathcal{F}, P)$ the usual Hilbert space of
square integrable complex-valued functions on $(\Omega, \mathcal{F}, P)$
and use $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ to mean its inner product and norm, respectively.
\vskip2mm

\noindent {\bf Definition A.1.}\label{def-A-1}
A stochastic process $M=(M_n)_{n\in \mathbb{N}}$ on
$(\Omega, \mathcal{F}, P)$ is called a discrete-time normal martingale if it is square integrable and satisfies:
\begin{enumerate}[(i)]
  \item $\mathbb{E}[M_0 | \mathcal{F}_{-1}] = 0$ and $\mathbb{E}[M_n | \mathcal{F}_{n-1}] = M_{n-1}$
 for $n\geq 1$;
  \item $\mathbb{E}[M_0^2 | \mathcal{F}_{-1}] = 1$ and $\mathbb{E}[M_n^2 | \mathcal{F}_{n-1}] = M_{n-1}^2 +1$
for $n\geq 1$,
\end{enumerate}
where $\mathcal{F}_{-1}=\{\emptyset, \Omega\}$,
$\mathcal{F}_n = \sigma(M_k; 0\leq k \leq n)$ for $n \in \mathbb{N}$ and $\mathbb{E}[\cdot | \mathcal{F}_{k}]$ means the conditional expectation.
\vskip2mm

Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale on $(\Omega, \mathcal{F}, P)$.
Then one can construct from $M$ a process $Z=(Z_n)_{n\in \mathbb{N}}$ as

\begin{equation*}
Z_0=M_0,\quad Z_n = M_n-M_{n-1},\quad  n\geq 1. \eqno(\mathrm{A}.1)
\end{equation*}
It can be verified that $Z$ admits the following properties:
\begin{equation*}
    \mathbb{E}[Z_n | \mathcal{F}_{n-1}] =0\quad \text{and}\quad  \mathbb{E}[Z_n^2 | \mathcal{F}_{n-1}] =1,\quad  n\in \mathbb{N}. \eqno(\mathrm{A}.2)
\end{equation*}
Thus, it can be viewed as a discrete-time noise.
\vskip2mm

\noindent {\bf Definition A.2.} \label{def-A-2}
Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale. Then the process $Z$ defined by (A.2) is called the discrete-time normal noise associated
with $M$.
\vskip2mm

The next lemma shows
that, from the discrete-time normal noise $Z$, one can get an orthonormal system in $\mathcal{L}^2(\Omega, \mathcal{F}, P)$,
which is indexed by $\sigma \in \Gamma$.
\vskip2mm

\noindent {\bf Lemma A.1.}
Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale and $Z=(Z_n)_{n\in \mathbb{N}}$ the discrete-time normal noise associated with $M$.
Define $Z_{\emptyset}=1$, where $\emptyset$ denotes the empty set, and
\begin{equation*}
    Z_{\sigma} = \prod_{i\in \sigma}Z_i,\quad \text{$\sigma \in \Gamma$, $\sigma \neq \emptyset$}. \eqno(\mathrm{A}.3)
\end{equation*}
Then $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ forms a countable orthonormal system in
$\mathcal{L}^2(\Omega, \mathcal{F}, P)$.
\vskip2mm

Let $\mathcal{F}_{\infty}=\sigma(M_n; n\in \mathbb{N})$, the $\sigma$-field over $\Omega$ generated by $M$.
In the literature, $\mathcal{F}_{\infty}$-measurable functions on $\Omega$ are also known as functionals of $M$.
Thus elements of $\mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P)$ can be called square integrable functionals of $M$.
For brevity, we usually denote by $\mathcal{L}^2(M)$ the space of square integrable functionals of $M$, namely
\begin{equation*}
\mathcal{L}^2(M)=\mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P). \eqno(\mathrm{A}.4)
\end{equation*}
\vskip2mm

\noindent {\bf Definition A.3.} \label{def-A-3}
The discrete-time normal martingale $M$ is said to have the chaotic representation property if
the system $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ defined by (A.3) is total in $\mathcal{L}^2(M)$.
\vskip2mm

So, if the discrete-time normal martingale $M$ has the chaotic representation property, then the system $\{Z_{\sigma}\mid \sigma \in \Gamma\}$
is actually an orthonormal basis for $\mathcal{L}^2(M)$,
which is a closed subspace of $\mathcal{L}^2(\Omega, \mathcal{F}, P)$ as is known.
\vskip2mm

\noindent{\bf Remark A.1.}
\'{E}mery \cite{emery} called a $\mathbb{Z}$-indexed process $X=(X_n)_{n \in \mathbb{Z}}$ satisfying (A.2)
a novation and introduced the notion of the chaotic representation property for such a process.

\section*{Acknowledgement}

This work is supported by National Natural Science Foundation of China (Grant No. 11461061).

redpajama_set_name: RedPajamaArXiv


text:
/* -*- Mode:C++; c-file-style:"gnu"; indent-tabs-mode:nil -*- */
/**
 * Copyright (C) 2013 Regents of the University of California.
 * @author: Yingdi Yu <yingdi@cs.ucla.edu>
 * @author: Jeff Thompson <jefft0@remap.ucla.edu>
 * See COPYING for copyright and distribution information.
 */

#include <ndn-cpp-dev/security/identity-certificate.hpp>

using namespace std;

namespace ndn {

bool
IdentityCertificate::isCorrectName(const Name& name)
{
  int i = name.size() - 1;

  string idString("ID-CERT");
  for (; i >= 0; i--) {
    if(name.get(i).toEscapedString() == idString)
      break;
  }

  if (i < 0)
    return false;

  int keyIdx = 0;
  string keyString("KEY");
  for (; keyIdx < name.size(); keyIdx++) {
    if(name.get(keyIdx).toEscapedString() == keyString)
      break;
  }

  if (keyIdx >= name.size())
    return false;

  return true;
}

void
IdentityCertificate::setPublicKeyName()
{
  if (!isCorrectName(getName()))
    throw Error("Wrong Identity Certificate Name!");

  publicKeyName_ = certificateNameToPublicKeyName(getName());
}

bool
IdentityCertificate::isIdentityCertificate(const Certificate& certificate)
{
  return dynamic_cast<const IdentityCertificate*>(&certificate);
}

Name
IdentityCertificate::certificateNameToPublicKeyName(const Name& certificateName)
{
  int i = certificateName.size() - 1;
  string idString("ID-CERT");
  bool foundIdString = false;
  for (; i >= 0; i--) {
    if (certificateName.get(i).toEscapedString() == idString)
      {
        foundIdString = true;
        break;
      }
  }

  if(!foundIdString)
    throw Error("Incorrect identity certificate name " + certificateName.toUri());

  Name tmpName = certificateName.getSubName(0, i);
  string keyString("KEY");
  bool foundKeyString = false;
  for (i = 0; i < tmpName.size(); i++) {
    if (tmpName.get(i).toEscapedString() == keyString)
      {
        foundKeyString = true;
        break;
      }
  }

  if(!foundKeyString)
    throw Error("Incorrect identity certificate name " + certificateName.toUri());

  return tmpName.getSubName(0, i).append(tmpName.getSubName(i + 1, tmpName.size() - i - 1));
}

}

redpajama_set_name: RedPajamaGithub


The information day covered the priorities and main aspects of the 2019 CEF Transport call, evaluation and award process, and also gave tips on how to prepare submit a successful proposal. Interactive Q&A sessions took place during the information day.
If you followed, even partially, the virtual info day, then please take a minute to fill our feedback survey - it will help us improve future events and better answer your needs!
The event was live streamed and the recording is available at the link below.


Often referred to as "a breath of fresh air", WORDZ is a gifted lyricist, poet and song writer, delivering authentic HipHop & soul.
Under the mentorship of an underground North Philly rap group named, Beautiful Black Children aka BBC, WORDZ began his professional recording career in the mid 90s at the age of 15. His first album, 'Mental Replays', was released in 2003, where he gained popularity by selling his music on street corners in downtown Philly.
With the release of 10 albums total currently, WORDZ is featured in numerous showcases and concerts throughout the US and continues to provide listeners with music that is artistically pure and organic. He has shared the stage with Legends such as Black Thought, Sonia Sanchez, Guru and The Floacist and has performed on 106th and Park.
Today, the talented artist is dedicated to putting the art back into HipHop, with dreams of touring internationally. WORDZ continues to Iive his life spreading a message of peace, love and HipHop 'by any means necessary'.


Hyperflex's new Voodoo HFZ brings major advancements in cold water wetsuit technology. Highlighting Voodoo is the introduction of AtomicFoam™ Super Stretch Neoprene, which we internally-lined with our new AQ6™ thermal fabric. Additionally, the Chest/Back/Hood is equipped with our new ReactorCore+™ insulation panels. Voodoo HFZ also benefits from our new Cocoon™ Entry Point & Closure System design. All seams are glued and blind stitched and sealed with our FusionWeld™ liquid tape. Simply put, Voodoo increases warmth without increasing thickness allowing you to surf lighter, longer, and warmer.
infused with a higher count of molecular air cells achieving a warmer insulation rating for the same thickness and weight.
a soft, thin fibrous fabric which internally lines 90% of the suit which increases warmth, comfort, and has quick dry properties while also allowing for maximum stretch.
A light weight honeycombed hollow membrane with a closed-cell outer mesh skin with a heat boosting & moisture-wicking micro fleece interior lining that acts like a miniature factory which dramatically increases heat applied to blood before it is pumped through the body.
features a high-grade stainless steel slider for corrosion resistance and an improved zipper tooth design making it virtually waterproof.
applied as a liquid which penetrates seams and fibers forming a molecular bond which prevents water exchange and increases seam strength and durability.
we secured the hood cinch cord to the hood to prevent it from flapping around in your face. No more poking your eye out while you shred.
designed with a singular wide-hatch opening which allows you to slide in and out of your suit with ease while sealing water out with an anti-flush head gusset panel.
all critical seam intersections are strengthened with heat-applied reinforcement tabs.


Pat Petrie collecting some hardware. Pat and his father Sam built some of the most competitive modified stock cars to race at Lakeside Speedway. I believe Pat won two Championships in the modified division and was a champion in the Compact Division. The Compact Division was a prelude to the Late Model division. Now days, Pat lives in Las Vegas, Nevada. He and his sons still compete in auto racing.
Some of my childhood hero's!
Can't take credit for the photos,just some I've collected over the years,truly grateful to those responsible for capturing these Hill Climb greats!


I know a significant percentage of people who play The Sims 3 don't like English in their game. We all have our weird rules for what we will and will not download. I, personally, won't download anything that has photographic images or art that is too real. Unless I really like it, of course.
Anyway, the whole "Keep Calm and Carry On" thing is pretty huge, and, thus, there are limitless parodies floating around for it. I have included most of my favorites, and, while I was at it, made three more art sets using images from the Keep Calm Gallery.
these are awesome. thanks so much.


Does risk management add value?
We have been advising large businesses and government entities on enterprise risk management (ERM) for many years. Managers often ask us to justify why they should invest in ERM and how to demonstrate its value in measurable terms.
This guidance note distils some of the empirical evidence on the benefits an organisation should expect from an effective ERM framework and process.
There is clear evidence that companies with risk management processes generate better financial and business outcomes than those without. There is also clear evidence that companies with higher levels of risk management maturity perform better than those with lower levels of maturity.


Expect to pay more for your coffee, but worth it! They make their own syrups and pride themselves in their craft. Also offer delicious breakfast options - I especially love their sausage (local farm) and biscuits made fresh each morning.


text:
Q: List of all objects with the current number of used custom fields You can view custom object field limits by going on:
App > Customize > Object > Limits
Is there a way of getting a list of all objects with the current number of used custom fields?
EDIT: I know there are app exchange tools but I cannot use them - I need a soql quer/ report or a SF native tool (no installations or third parties)

A: I'll convert this comment into an answer.
You can obtain this data rather clumsily with the Tooling API in the Developer Console. Enter the following in the Query Editor, and check the "Use Tooling API" box:
SELECT count(DeveloperName), TableEnumOrId FROM CustomField GROUP BY TableEnumOrId ORDER BY count(DeveloperName) DESC

That'll get you the objects with the highest count of custom fields at the top. Note that it supplies a mix of standard object names and custom object Ids, each one with the count of custom field attached to it. To find the name of a given Custom Object's Id, you can issue another query:
SELECT DeveloperName FROM CustomObject WHERE Id = 'THE_ID'

Or, of course, you can build yourself a translation table with
SELECT Id, DeveloperName FROM CustomObject

A: If you can run apex:
Integer count = 0;
Map<String, Integer> custFieldsPerObject = new Map<String, Integer>();
map<String, Schema.SObjectType> GlobalMap = Schema.getGlobalDescribe();
    for (Schema.SObjectType Obj : GlobalMap.values()) {
        Schema.DescribeSObjectResult ObjDesc = Obj.getDescribe();
        Integer customFieldCount = 0;
        for (SObjectField aField : objDesc.fields.getMap().values()) {
            if (aField.getDescribe().isCustom()) {
                customFieldCount++;
            }
        }
        custFieldsPerObject.put(ObjDesc.getName(), customFieldCount);

        if(ObjDesc.iscustom()){
           count ++;
           system.debug('Object Name: ' + ObjDesc.getName());
        }
    }
System.debug('Custom Object '+ count);
System.debug('Custom fields per Object '+ JSON.serializePretty(custFieldsPerObject));
Integer maxFields = 0;
String maxFieldsObject = '';
for (String objectName : custFieldsPerObject.keySet()) {
    Integer numberOfCustFields = custFieldsPerObject.get(objectName);
    if (numberOfCustFields > maxFields) {
        maxFields = numberOfCustFields;
        maxFieldsObject = objectName;
    }
}
System.debug('Max Custom fields: ' + maxFields + ' on ' + maxFieldsObject);

redpajama_set_name: RedPajamaStackExchange


text:
Playbox UK announced 'Bang Bang Racing' game for Android device with Tegra
British developers Playbox has announced that it was expanding into the Android platform, with a twist: the Bang Band Racing game will NVIDIA's Tegra chip need to execute. The game was originally made for the Xbox 360 and PlayStation 3 and Playbox says it is the game optimized for Tegra. For example, it uses NVIDIA's PhysX technology, a physics engine, which is known for its PC hardware physics acceleration.
The game is not available for download, but the 1280 × 800 screen shots look very nice. We can not wait to see how fast the framerate is. There is no official release date other than "soon", but you must keep an eye on Tegra Zone, a free app from NVIDIA, because it probably means "any day now."
Developers have complained about Android fragmentation as of late. One could say that the PC is too fragmented, smartphones and tablets have special share of problems. Example, different chips support different types of texture compression, which means that you have many sets, more storage, which is not ideal for downloads.
By only supporting Nvida Tegra's Playbox can get off to an easier start and at the same time use the special effects that may or may not work on competing hardware. Of course, NVIDIA's marketing support contribute significantly to the Bang Bang Racing launch. It is likely that Playbox will be a version that will run on non-Tegra hardware at some point release, but in the meantime, Tegra 2 is the only option.
Categories: Android, Gaming, News, Software
Tags: Andoid, android platform, bang bang, Bang Bang Racing, game developer, NVIDIA, PhysX, Playbox, racing game, Tegra, UK
Xiaomi Mi3 with Tegra 4 or Snapdragon 800 launched; Specs and Price
Vodafone UK HTC Sensation Smartphone released; Free on Monthly Plans Starting at £35
Vodafone BlackBerry Bold 9900 White version on Sale in UK
ViewSonic ViewPad 10Pro Windows and Android Dual-Boot Oak Trail Tablet Specs


text:
Q: Value of v-text-field returns null when text area has a value I'm trying to get the value from vuetify's v-text-field. The problem I am having is that I thought by typing something in, the field will automatically assign whatever is typed to vue-text-field's value. However this doesn't seem to be the case. Below is the code directly from vuetify with some additional code attempting to extract the value:
<template>
  <v-card
    class="overflow-hidden"
    color="purple lighten-1"
    dark
  >
    <v-toolbar
      flat
      color="purple"
    >
      <v-icon>mdi-account</v-icon>
      <v-toolbar-title class="font-weight-light">
        {{setterTitle}}
      </v-toolbar-title>
      <v-spacer></v-spacer>
      <v-btn
        color="purple darken-3"
        fab
        small
        @click="isEditing = !isEditing"
      >
        <v-icon v-if="isEditing">
          mdi-close
        </v-icon>
        <v-icon v-else>
          mdi-pencil
        </v-icon>
      </v-btn>
    </v-toolbar>
    <v-card-text>
      <v-text-field ref="rowCount"
        :disabled="!isEditing"
        :value="2"
        autofocus
        color="white"
        label="Number of Rows"
      ></v-text-field>
      <v-autocomplete
        :disabled="!isEditing"
        :items="states"
        :filter="customFilter"
        color="white"
        item-text="name"
        label="Number of Columns"
      ></v-autocomplete>
    </v-card-text>
    <v-divider></v-divider>
    <v-card-actions>
      <v-spacer></v-spacer>
      <v-btn
        :disabled="!isEditing"
        color="success"
        @click="save"
      >
        Save
      </v-btn>
    </v-card-actions>
    <v-snackbar
      v-model="hasSaved"
      :timeout="2000"
      absolute
      bottom
      left
    >
      Your profile has been updated
    </v-snackbar>
  </v-card>
</template>

<script>
  export default {
    props: {
      setterTitle: String
    },
    data () {
      return {
        hasSaved: false,
        isEditing: null,
        model: null,
        states: [
          { name: 'Single', abbr: 'S', id: 1 },
          { name: 'Double', abbr: 'D', id: 2 },
          { name: 'Triple', abbr: 'T', id: 3 },
          { name: 'Custom', abbr: 'C', id: 0 },
          // { name: 'New York', abbr: 'NY', id: 5 },
        ],
      }
    },

    methods: {
      customFilter (item, queryText, itemText) {
        const textOne = item.name.toLowerCase()
        const textTwo = item.abbr.toLowerCase()
        const searchText = queryText.toLowerCase()
        console.log(textOne, textTwo,itemText)
        return textOne.indexOf(searchText) > -1 ||
          textTwo.indexOf(searchText) > -1
      },
      save () {
        this.isEditing = !this.isEditing
        this.hasSaved = true
        console.log(this.$refs.rowCount.value)
      },
    },
  }
</script>

As you can see, in the v-text-field tag I added a ref. I then tried to use that reference to extract the value from the save() method. Provided that I also have :value="2" assigned, this.$refs.rowCount.value always return 2. However if I don't have the value property set as 2 then this.$refs.rowCount.value returns null even though I've typed something in the v-text-field. I'm guessing I am misunderstanding something here. Maybe when I type to the field, it doesn't automatically assign what is typed to the value?  Thank you for your help.
I've attached the api for v-text-field https://vuetifyjs.com/en/api/v-text-field/#component-pages

A: v-text-field is designed as common custom input control in Vue - it has a value prop and a input event which means it can by used with v-model - see the docs how that works
As a user, you do this:
<template>
  <v-text-field v-model="text" />
</template>
<script>
export default {
  data: function() {
    return {
      text: '' // initial value
    }
  }
}
</script>

To your question - value is a prop. All props in Vue are one way only - parent can send data to the component (and change that data in the future) but child component cannot change value of the prop it receives. That's why when you type something into the textbox the v-text-field must use an input event to tell the parent component that something has changed and that parent should update "it's own source" of value prop...
v-model is Vue's "syntactic sugar" how to work with these two more easily...


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