If you recall, I found a great little flashlight a few months ago called the F1 by LED Lenser (http://www.gottahavegadget.com/ledlenser/). Let me introduce you to it's big brother the F1R by LED Lenser! This powerful little flashlight is even brighter than what I thought was one of the brightest flashlights I had ever seen. The F1R is 1000 Lumens (compared to 400 Lumens in the F1) which is ultra bright! However, that is not the best feature of this flashlight! The only drawback I found to the F1 was the battery life. The F1R comes with a rechargeable battery! As soon as I saw the email from LED Lenser that they had a new rechargeable, more powerful flashlight, I knew I had to have it! At full power, the Li-ion battery can last up to 3 hours! The housing is waterproof, and it weighs only 4.2 ounces. This powerful flashlight is easy to carry and puts amazing power right in your pocket. After use, simply take out the battery, throw it in the charger and you are ready to go for the next time you need it! LED Lenser is my first stop when I am looking for powerful light technology! Check them out on-line at www.ledlenser.com, you won't be disappointed!


text:
Q: Can't update iPhone apps After making an update to an iPad app I released some time ago, I've been getting reports that people are unable to actually update the app without deleting and re-installing. However, as far as I know, nothing in the update should be causing this. (All the update deals with is letting people email PDF documents, nothing major.) When people attempt to update, they're asked for their iTunes password, but after entering it, it merely goes back to the update screen and nothing happens. Additionally, it would seem that this only happens with my app, the people in question aren't having any issues with the other various apps on the App Store. Does anyone know what might be causing this and how I could fix it?
Thanks in advance!
(Also, if it matters, the app is a custom B2B app, the general public can't purchase it.)

A: I'm removing the text of my answer because it's so inaccurate it's embarrassing.  I mistook "B2B" for "Enterprise" and answered based off of that.  To make up for it, I'll look into the problem a bit more and if I find anything I will edit this answer accordingly.

Edit:
Okay, I can see why you put a bounty for this question on SO; there's not really any data on a problem like this anywhere.  Frankly, there's not much available information on B2B in general.  I'll post what I found anyway, in case it can be of any help to you.
I found the details reason behind Maggie's question, there.  Per Editing and Updating App Information:

Updates keep the same Apple ID and bundle ID, which means they are
  associated with your first version and free to your customers

Also, apparently, "You can't change the CFBundleIdentifier of a released app if you want to release updates for it, the App Store will automatically reject it when you upload." which is something I can vouch for, having experienced this with a normal app.  I do know that for a B2B app you do have to submit it to Apple for review, but I can't tell from the documentation I found if you need to actually submit it to the App Store, so it may not go through the various checks that normal apps go through, so this could be your problem.
Aside from that, according to the VPP guide, if your customers are installing the apps on the devices with Apple Configurator (broken right now, per app store reviews) the updates also have to be done with the Configurator.  You haven't said that Configurator was involved, but I did find this tidbit.

• Use Apple Configurator to install apps on new or supervised devices.
  Apple Configurator on a Mac makes it easy to mass configure and deploy
  devices that are centrally controlled. Redemption code spreadsheets
  acquired through the Volume Purchase Program can be imported by Apple
  Configurator, tracking the number of apps installed on each device. To
  update deployed apps using Apple Configurator, you must reconnect to
  the same Mac from which the apps were installed. Learn more at
  itunes.apple.com/us/app/apple-configurator

Anyway, good luck.  Wish I could be more help.

A: What you are describing (assuming that it is accurate) would certainly be a bug on Apple's side. If users are trying to update the app and the update is not being processed, then in one way or another that is a bug that Apple needs to address. Nothing that you do as a developer should be able to cause that situation to happen. I would suggest contacting Apple and possibly filing a bug report.

A: It seems that apple wants you to develop the Iphone apps in the latest build. Sometimes this cause issues between realeases (diferent versions of Itunes, OSX, IOS, etc) when you try to update your apps.
Try to publish the app in the latest version of xcode.
That happens a lot in iphone development testing.
Hope this help.

A: When updating an app, iOS looks for the bundleId and if there is another app with the same bundleId, it updates the  app with the highest version number. Maybe the version number is not set correctly or maybe people have issues because an other app (from the AppStore or an other B2B app) have the same bundleID but a higher version number.

A: I'm by far not an iPhone expert, but it seems something related might have been fixed in iOS 6.0.1.

Fixes a bug that prevents iPhone 5 from installing software updates
  wirelessly over the air

redpajama_set_name: RedPajamaStackExchange


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Tag: "bristol virginia utilities"
Two Publicly Owned Networks Move To Privatize
Posted August 7, 2017 by Lisa Gonzalez
If you're a regular reader at MuniNetworks.org, listen to our podcasts, or if you simply follow publicly owned network news, you know an increasing number of communities have decided to invest in local connectivity solutions in recent years. We've watched the number of "pins" on our community network map multiply steadily, but every now and then, a network drops off through privatization.
FastRoads Sold To N.H. Optical Systems
New Hampshire FastRoads received America Recovery and Reinvestment Act (ARRA), which combined with state funding, created the open access fiber optic network in the southwest section of the state. Over the next several years, the network expanded with private donations and local matching funds. Many of the premises that connected to the network had relied on dial-up before FastRoads came to town. But in part because state law makes bonding for network expansion difficult, Fast Roads will no longer be locally controlled.
The Monadnock Economic Development Corporation (MEDC), a nonprofit organization whose purpose is working to see like projects are completed that will improve economic development prospects in the region managed the project. MEDC contracted with another entity to maintain the network, which cost approximately $15,000 per month. Since they had achieved their core goal - the construction and launch of the network - MEDC had been looking for another entity to take over the network or to partner with them. They recently finalized a deal to sell the network to New Hampshire Optical Systems.
Back in 2013, Christopher spoke with Carole Monroe, who was the FastRoads Project CEO but has since moved on to ECFiber in Vermont. She described how the introduction of the network inspired incumbents to lower prices - a win for everyone, whether they connected to FastRoads or not. She also told us how community anchor institutions (CAIs) were getting better...
BVU OptiNet Update: Deal Coming Together
Posted April 24, 2017 by Lisa Gonzalez
A little over a year ago, we first shared the news about Bristol's decision to privatize its FTTH network, OptiNet. Virginia based Sunset Digital Communications offered to purchase the network for $50 million. The network has saved Bristol millions of dollars, stimulated economic development, and cut telecommunications costs for local residents and businesses. Nevertheless, after several corrupt officials drove the network into a dark period of scandal, all those advancements paled and Bristol was ready to sell the network.
After months of negotiations with BVU's partner in the Cumberland Plateau area service area, the details for the sale are coming together.
When There's A Partner
One of the last steps to completing the sale required approval from the Cumberland Plateau Company (CPC), which operates as a partner with BVU to bring connectivity to four additional counties in Virginia. As a partner with OptiNet in those areas, CPC owns approximately 50 percent of the assets.
When Sunset Digital offered $50 million for the BVU assets, CPC obtained the right of first refusal for the assets in the four counties where BVU and CPC work together as partners according to their contract.
Back in the fall of 2016, CPC was concerned about the legality and the details of the proposed transaction; they decided to wait for federal and state review before granting approval. Because the NTIA, the Economic Development Administration (EDA), and the Virginia Tobacco Commission provided grant funding to the CPC region for the deployment, the agencies needed to review and approve the proposal. The agencies approved the sale, but required that a large amount of BVU debt be paid. One of the claims that they required be paid was a claim for $8 million from CPC.
Approving The Offer
As part of the offer, Sunset promises to invest $6.5 million to connect more homes and businesses in the CPC region. They estimate CPC will gain about $21 million in revenue over 13 years while Sunset operates the network. CPC will retain ownership of its assets in the CPC service area and Sunset will transfer ownership of equipment in the CPC area to CPC.
After several rounds of...
Feasibility Study For Johnson City, TN
Johnson City Power Board (JCPB) in Tennessee began considering expanded uses for its fiber-optic infrastructure way back in 2009. After several stops and starts, the community is on track again, having just commissioned a Fiber and Wireless to the Premise (FTTP) Feasibility Study.
A Long Road
In 2009, when the municipal utility was installing fiber to substations they reviewed the idea of offering broadband to businesses and residents. Ultimately, they chose to focus on smart-grid development and save possible telecommunications offerings for some time in the future.
This isn't the first time the community of 63,000 has commissioned a feasibility study. In 2011, community leaders took the results from a study and decided a public-private partnership was the best route. The community is located between Bristol, Virginia, and Chattanooga, Tennessee - both communities with municipal fiber networks that have seen upticks in economic development. Competing for new businesses and retaining the ones they already have could not have been easy while sandwiched between the two communities with high-quality connectivity.
In 2012, Johnson City announced that it would be working with the BVU Authority in Bristol as a partner. Now that the BVU system will likely be sold to a private provider, Johnson City is back to square one, but with considerable experience in its pocket.
Asking For Input
As part if the study, JCPB has launched surveys on their website for residents and businesses; they're also making the surveys available through the mail. JCPB is asking the community to complete the surveys before the end of June.
From the JCPB survey page:
Over 1,000 communities nationwide have undergone similar evaluation processes and benefited from the information obtained from these types of surveys. Survey results have enabled key decision-makers within these municipalities to make more...
Discussing (Ranting) Consolidation - Community Broadband Bits Episode 209
Posted July 5, 2016 by Christopher Mitchell
In celebration of Independence Day, we are focused this week on consolidation and dependence. At the Institute for Local Self-Reliance, we are very focused on independence and believe that the consolidation in the telecommunications industry threatens the independence of communities. We doubt that Comcast or AT&T executives could locate most of the communities they serve on a blank map - and that impacts their investment decisions that threaten the future of communities.
So Lisa Gonzalez and I talk about consolidation in the wake of Google buying Webpass and UC2B's partner iTV-3 selling out to Countrywide Broadband. And we talk about why Westminster's model of public-private partnership is preferable to that of UC2B.
We also discuss where consolidation may not be harmful and how the FCC's order approving the Charter takeover of Time Warner Cable will actually result in much more consolidation rather than new competition.
Read the transcript from this show here.
This show is 18 minutes long and can be played below on this page or via iTunes or via the tool of your choice using this feed.
You can download this mp3 file directly from here. Listen to other episodes here or view all episodes in our index.
Thanks to Fifes and Drums of the Old Barracks for the music, licensed using Creative Commons. The song is "Cork Hornpipe."
Sale of OptiNet: BVU Caught Between Virginia's Rock And A Hard Place
Posted February 12, 2016 by Lisa Gonzalez
For more than a decade, the people of Bristol, Virginia have enjoyed what most of us can only dream about - fast affordable, reliable, connectivity. In recent days, we learned that Bristol Virginia Utilities Authority (BVU) has entered into a deal to sell its OptiNet triple-play fiber network to a private provider. The deal is contingent on approval by several entities.
As we dig deeper into the situation, we understand that troubles in southwestern Virginia and Bristol have led to this decision. Nevertheless, we urge the Bristol community to weigh the long-term consequences before they sacrifice OptiNet. Once you give up control, you won't get it back.
"...A Few Bad Apples..."
If the people of Bristol surrender this valuable public asset to the private market, they run the risk of undoing 15 years of great work. None of this is a commentary on the private provider, Sunset Digital Communications, which may be a wonderful company. The problem is that Sunset will be making the decisions in the future, not the community.
OptiNet has helped the community retain and create jobs, attracting and retaining more than 1,220 well-paying positions from Northrup Grumman, CGI, DirecTV, and Alpha Natural Resources. Businesses have cut Internet access and telecommunications costs. Officials estimate around $50 million in new private investment and $36 million in new annual payroll have come to the community since the development of OptiNet. The network allowed public schools to drastically reduce telecommunications expenses and introduce gigabit capacity long before such speeds were the goal among educators.
Schools and local government saved approximately $1 million from 2003 - 2008. Subscribers have saved considerably as well, which explains OptiNet's high take rate of over 70 percent. Incumbent telephone provider Sprint (now CenturyLink) charged phone rates 25 percent higher than OptiNet in 2003. The benefits are too numerous to mention in one short story.
However, BVU is emerging from a dark period marked by corrupt management. This sad reality actually makes its considerable achievements all the more remarkable. Last...
Chris Joins Sarah Morris on Oregon Radio to Discuss Cost of Connectivity
Posted November 20, 2014 by Lisa Gonzalez
This is the third year the Open Technology Institute (OTI) at New America Foundation studied the cost and quality of connectivity in the U.S. Once again, the results indicate we trail behind peer countries. On November 11th, Chris joined Sarah Morris, one of the report authors, to discuss the report's findings, municipal networks, and how Title II reclassification may change the landscape. They joined Dave Miller for the Think Out Loud program on Oregon Public Broadcasting.
In addition to detailed data analysis on where the best speeds and prices are, The Cost of Connectivity 2014 provides reviews of several other papers from sources such as Akamai, the FCC, and the American Enterprise Institute.
Some notable findings from the report:
The average cost of plans in nearly every speed tier studied for the report was higher in the U.S. than in Europe.
Cities considered speed leaders have consistently increased speed offerings on an annual basis. In places where the speed was not increased, as in Lafayette, rates decreased. Almost half of the speed leaders cities offer gigabit speeds.
OTI made special note of the success of municipal networks in places where traditional providers are not willing to invest:
Although there are many examples of successful locally-owned networks, we focus on Chattanooga, TN; Bristol, VA; and Lafayette, LA, which now offer some of the fastest and most affordable high-speed residential products available in the country despite the fact that they have some of the lowest population densities among the cities we survey.
All three cities offer gigabit speeds that place them on par with Hong Kong, Seoul, Tokyo, and Zürich in our speed leaders rankings, although the prices vary from city to city, and are ahead of the major incumbent ISPs in the U.S. In fact, the only other provider that offers gigabit speeds in the cities we surveyed is Google Fiber, which sells 1 Gbps service in Kansas City, KS, and Kansas City, MO, for the same price as EPB in Chattanooga, TN. By contrast, Verizon's top tier is a 500 Mbps symmetrical connection that is available to some residents of New York, NY; Washington, DC...
Chattanooga Profiled in Al Jazeera America
Posted June 25, 2014 by Lisa Gonzalez
"There are companies that do what we do, but we can do it in hours, and they can take weeks," said Posey. "Anywhere else, it would take a lot more time and a lot more money ... Chattanooga is essential to our business model."
Al Jazeera America's Peter Moskowitz recently spoke with Clay Posey, one of the entrepreneurs flocking to Chattanooga for the network. Posey works in one of the startup incubators there, Co.Lab, developing his idea for pre-operative models that allow surgeons to prepare before operating on patients.
While Chattanooga may not be the norm and may not be an easy venture for every municipality, it lifts the bar. From the article:
"Whenever a corporation like Comcast wants to do something like raise prices, we can point at Chattanooga and say, 'Why can't we have something like that?'" said Christopher Mitchell, head of the community broadband networks initiative at the nonprofit Institute for Local Self-Reliance. "It establishes a baseline or at least an aspirational standard."
The article describes lobbying efforts by large corporate providers designed to stop the municipal networks model. Another Chattanooga entrepreneur told Moskowitz:
"Having public or quasi-public Internet service providers is a good solution to consolidation because they most likely won't be sold," said Daniel Ryan, a local Web developer who helped run the digital operation of Barack Obama's 2012 presidential campaign. "Do I think if every city did this, Comcast would go out of business? No. But it means there will always be competition."
Moskowitz included a brief historical summary of the network, its contribution to the electric utility, and the challenges created by state barriers. He included our Community Broadband Networks map.
For more detail on Chattanooga's fiber network, download our case study Broadband at the Speed of Light: How Three Communities Built Next-Generation Networks. The case study also covers the communities of Bristol, Virginia and Lafayette, Louisiana. We...
Bristol Virginia Utilities to Offer Home Management Services
BVU in Bristol is now offering Quantum Home, a security and home management system that uses the community's publicly owned fiber network. The system allows home owners to also manage lights, temperature, and appliances from anywhere using a computer, tablet, or smartphone. For a quick video demo, check out the BVU website.
Installation costs range between $200 to $2,000 for installation and monthly charges are $39.95 - $49.95. Comcast offers a similar service, Xfinity Home, and requires installation fees to be paid in full when the system is installed. BVU plans to allow customers to amortize the installation fees over 12 months if they wish.
BVU launched OptiNet in 2001 and offers reliable triple-play at affordable prices in Bristol and surrounding areas. We talked with Jim Baller about the history of publicly owned networks in Episode #57 and Episode #63 of the Community Broadband Bits podcast. BVU's OpiNet played a prominent role as one of the first publicly owned completely fiber triple-play networks.
Tricities.com reporter David McGee recently attended a BVU Board of Directors meeting where management described the new service.
"This is an exciting new service that is actually in the market and we've already been selling it," [Authority interim CEO Mike] Bundy told the board... "It will be not just home security but home automation. It's cutting-edge technology."
Jim Baller on the History of Municipal Networks, Part II on Community Broadband Bits Podcast
Posted September 10, 2013 by Christopher Mitchell
Jim Baller is back again for the second show in our series on the history of municipal broadband networks. He is the President of the Baller Herbst Law Group in Washington, DC, and a long time advocate for both community owned networks specifically and better access to the Internet for all more generally.
We kicked this history series off on Episode 57 where we talked about some of the early community cable networks and how federal law came to allow states to preempt local authority.
In this episode, we talk about the early FTTH networks in Chelan, Washingon; Bristol, Virginia; Kutztown, Pennsylvania; and Dalton, Georgia. We also talk about how the states began restricting local investments, particularly in Pennsylvania under pressure from Verizon.
We will continue the series in subsequent episodes. We want your feedback and suggestions for the show - please e-mail us or leave a comment below. Also, feel free to suggest other guests, topics, or questions you want us to address.
Read the transcript of this episode here.
This show is 18 minutes long and can be played below on this page or subscribe via iTunes or via the tool of your choice using this feed.
Listen to previous episodes here. You can can download this Mp3 file directly from here.
Find more episodes in our podcast index.
Thanks to Break the Bans for the music, licensed using Creative Commons.
Jim Baller Discusses Municipal Broadband History - Community Broadband Bits Episode #57
Posted July 30, 2013 by Christopher Mitchell
Jim Baller has been helping local governments to build community owned networks for as long as they have been building them. He is the President of and Senior Principal of the Baller Herbst Law Group in Washington, DC. Jim joins us for Episode #57 of the Community Broadband Bits podcast to discuss some of the history of community owned networks.
Jim has a wealth of experience and helped in many of the most notable legal battles, including Bristol Virginia Utilities and Lafayette.
We start by noting some of the motivations of municipal electric utilities and how they were originally formed starting in the late 19th century. But we spend the bulk of our time in this show focusing on legal fights in the 90's and early 2000's over whether states could preempt local authority to build networks.
In our next interview with Jim, we'll pick up where we left off. If you have any specific thoughts or questions we should cover when we come back to this historical topic, leave them in the comments below or email us.
You can learn more about Jim Baller on his website at Baller.com.
Read the transcript from this episode here.
We want your feedback and suggestions for the show - please e-mail us or leave a comment below. Also, feel free to suggest other guests, topics, or questions you want us to address.
This show is 30 minutes long and can be played below on this page or subscribe via iTunes or via the tool of your choice using this feed. Search for us in iTunes and leave a positive comment!
In the case of muni systems, which are not-for-profit enterprises, one measure of "success" is defined as the level of their "take rate" – that is, the percentage of potential subscribers who are offered the service that actually do subscribe. Nationwide, the take rates for retail municipal systems after one to four years of operation averages 54 percent. This is much higher than larger incumbent service provider take rates, and is also well above the typical FTTH business plan usually requiring a 30-40 percent take rate to "break even" with payback periods.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section*{Abstract}
\begin{quote}
{\footnotesize In this paper, we study biharmonic Riemannian submersions. We first derive bitension field of a general Riemannian submersion, we then use it to obtain biharmonic equations for Riemannian submersions with $1$-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. These are used to construct examples of proper biharmonic Riemannian submersions with $1$-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions. }

\end{quote}

\maketitle

\section{Introduction}

 A map $\phi:(M, g)\longrightarrow (N, h)$ between
Riemannian manifolds is a {\em biharmonic map} if
$\phi|\Omega$ is a critical point of the bienergy functional
\begin{equation}\nonumber
E_{2}\left(\phi,\Omega \right)= \frac{1}{2} {\int}_{\Omega}
\left|\tau(\phi) \right|^{2}{\rm d}v_g
\end{equation}
for every compact subset $\Omega$ of $M$, where
$\tau(\phi)={\rm Trace}_{g}\nabla {\rm d} \phi$ is the
tension field of $\phi$. Using the first variational formula
(see \cite{Ji}) one obtains the biharmonic map equation
\begin{equation}\label{BI1}
\tau^{2}(\phi):=-\triangle^{\phi}(\tau(\phi)) - {\rm
Trace}_{g} R^{N}({\rm d}\phi, \tau(\phi)){\rm d}\phi
=0,
\end{equation}
where
\begin{equation}\notag
\triangle^{\phi}=-{\rm Trace}_{g}(\nabla^{\phi})^{2}= -{\rm
Trace}_{g}(\nabla^{\phi}\nabla^{\phi}-\nabla^{\phi}_{\nabla^{M}})
\end{equation}
is the Laplacian on sections of the pull-back bundle $\phi^{-1}
TN$ and $R^{N}$ is the curvature operator of $(N, h)$ defined by
$$R^{N}(X,Y)Z=
[\nabla^{N}_{X},\nabla^{N}_{Y}]Z-\nabla^{N}_{[X,Y]}Z.$$

Biharmonic Riemannian submersions were first studied by Oniciuc in \cite{On} where he proved that if a Riemannian submersion $\phi: (M^{m}, g)\longrightarrow (N^n, h)$ has basic tension field, i.e.,  $\tau(\phi)(p)=\tau(\phi)(q)$ provided $\phi(p)=\phi(q)$, (or the tension field is constant along the fibers), then
\begin{align*}
\tau_2(\phi)={\rm Tr}_h(\nabla^N)^2\tau(\phi)+\nabla^N_{\tau(\phi)}\tau(\phi)+{\rm Ric}^N(\tau(\phi)).
\end{align*}
 In particular, it was proved in  \cite{On} that if the tension field $\tau(\phi)$ of a Riemannian submersion $\phi: (M^{m}, g)\longrightarrow (N^n, h)$ is a unitary Killing vector field on $N$, then it is a biharmonic map.\\

The biharmonic map equations for a more general class of maps, i.e., conformal submersions (or even more generally, horizontally weakly conformal maps), were obtained independently in \cite{BFO} and \cite{LO}. As an application of these equations, a family of biharmonic Riemannian submersions were constructed in \cite{LO}. Later in \cite{WO}, biharmonic Riemannian submersions from $3$-dimensional spaces with $1$-dimensional fibers were studied using the so-called integrability data. It was proved in \cite{WO} that  a Riemannian submersion from a $3$-dimensional space form onto a surface is biharmonic if and only if it is harmonic, i.e., the fibers have to be totally geodesic.\\

In a recent paper \cite{GO}, the authors found many examples of biharmonic Riemannian submersions in their study of generalized harmonic morphisms which are maps bewteen Riemannian manifolds that pull back local harmonic functions to local  biharmonic functions.
\begin{example}(see \cite{GO} for details)
The map $\phi:\mbox{${\mathbb R}$}^4\setminus\{(0, 0, 0, x): x\in \mbox{${\mathbb R}$}\}=M^4\longrightarrow \mbox{${\mathbb R}$}^2$ with $\phi(x_1,\cdots, x_4)=(\sqrt{x_1^2+x_2^2+x_3^2\,}, x_4)$ is a proper biharmonic Riemannian submersion with the mean curvature vector field of fibers being  basic vector field. In fact, using cylindrical coordinates $(r, \theta, \varphi, x_4)$ the Riemannian submersion can be expressed as $\phi (r, \theta, \varphi, x_4)=(r, x_4)$, and the tension field of the Riemannian submersion is $\tau(\phi)={\rm d}\phi(\frac{2}{r}\frac{\partial}{\partial r})$ and the mean curvature vector field of the fibers is $\mu=-\frac{1}{r}\frac{\partial}{\partial r}$.
\end{example}
\begin{example}(see \cite{GO} for details)
 Let $\mathbb{R}^2_{+}=\{ (x,y)\in \mbox{${\mathbb R}$}^2: y>0\}$ denote the upper-half plane and $C$ be a positive constant. Then the Riemannian submersion defined by the projection of the warped product
 \begin{align}\notag
\pi  : ( \mathbb{R}^2_{+} \times \mathbb{R} , dx^2 +
dy^2+Cy^4 dz^2) &\to (\mathbb{R}^2_{+} ,dx^2 + dy^2) \\\notag
\pi(x,y,z) =(x,y),
\end{align}
is a proper biharmonic Riemannian submersion with the tension field $\tau(\pi)=\frac{2}{y}{\rm d}\pi(\frac{\partial}{\partial y})$ and the mean curvature vector field of fibers $\mu=-\frac{2}{y}\frac{\partial}{\partial y}$, which is basic.
\end{example}

Note that there are many Riemannian submersions ( e.g., those given by the projections of a twisted product onto its first factor studied in Corollary \ref{Twisted}) whose mean curvature vector fields are not basic.

In this paper, we study biharmonicity of a general  Riemannian submersion. We first derive the bitension field of a general Riemannian submersion, we then use it to obtain biharmonic equations for Riemannian submersions with $1$-dimensional fibers generalizing a result in \cite{WO} and Riemannian submersions with basic mean curvature vector fields of fibers. These are used to construct examples of proper biharmonic Riemannian submersions with $1$-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions.

\section{Biharmonic equations for Riemannian submersions}
Biharmonic equations for Riemannian submersions can be obtained as a special case of biharmonic equations for conformal submersion obtained in \cite{BFO} and \cite{LO}. However, the equation obtained this way seems to be hard to apply. For the convenience of applications, we first derive the following form of the  bitension field of a Riemannian submersion and then use it to obtain biharmonic equation for Riemannian submersions with $1$-dimensional fibers or with  basic mean curvature vector field of fibers.\\

\begin{theorem}\label{MT}
Let $\phi: (M^m,g)\longrightarrow (N^n,h)$ be a Riemannian submersion between Riemannian manifolds, then the bitension field of $\phi$ is given by
\begin{eqnarray}\notag
\tau_{2}(\phi)&=&-(m-n){\rm d}\phi \Big(  \sum_{i=1}^n\big\{\nabla^{M}_{e_{i}}(\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}- \nabla^{M}_{(\nabla^{M}_{e_i}e_i)^{\mathcal{H}}}\mu+[\mu, (\nabla^{M}_{e_i}e_i)^{\mathcal{V}}]\big\}\\\notag
&&+\sum_{s=n+1}^m\{ [[\mu,e_s],e_s] +[\mu,(\nabla^{M}_{e_s}e_s)^{\mathcal{V}}]\}-(m-n)\nabla^{M}_{\mu}\mu \Big)\\\label{EQ1}
&&-(m-n){\rm Ricci}^N(\rm d\phi(\mu)),
\end{eqnarray}
where $\{e_{i},e_{s}\}_{i=1,\dots,n, s=1,\dots,m-n}$ is a local orthonormal frame on the total space $M$ with $e_{i}$ horizontal and $e_{s}$ vertical,  $\mu = \tfrac{1}{m-n} \sum_{s=1}^{m-n} (\nabla_{e_{s}} e_{s})^{\mathcal{H}}$
is  the mean curvature vector field of the fibers of the Riemannian submersion.
\end{theorem}
\begin{proof}

It is well known (see e.g., \cite{BW1}, Lemma 4.5.1) that for a Riemannian submersion $\phi: (M^m,g)\longrightarrow (N^n,h)$ and any horizontal vector fields  $X,Y$ and vertical vector fields $V, W$, we have
\begin{equation}\label{cases1}
\begin{cases}
(\nabla{\rm d }\phi)(X,Y)=0,\\
(\nabla{\rm d }\phi)(V,W)=-\rm d\phi(\nabla^{M}_{V}W),\\
(\nabla{\rm d }\phi)(X,V)=-\rm d\phi(\nabla^{M}_{X}V),\\
\end{cases}
\end{equation}
for all $X, Y \in \Gamma(\mathcal{H})$ and $V, W \in \Gamma(\mathcal{V})$.\\

One can easily check that  the tension field of the Riemannian submersion is given by
\begin{equation}
\tau(\phi) = -(m-n)d\phi(\mu).
\end{equation}

By the definition of the second fundamental form of the map $\phi$ and (\ref{cases1}) we have
\begin{eqnarray}\notag
\nabla^{\phi}_{e_{i}} d\phi(\mu) &=& d\phi(\nabla^{M}_{e_{i}}\mu)=d\phi\big((\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}\big),\\ \label{GD1}
\nabla^{\phi}_{e_{i}} \nabla^{\phi}_{e_{i}} d\phi(\mu) &=&\rm d\phi(\nabla^{M}_{e_{i}}(\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}),\\\label{GD2}
-\nabla^{\phi}_{\nabla^{M}_{e_i}e_i} d\phi(\mu) &=&-\rm d\phi(\nabla^{M}_{(\nabla^{M}_{e_i}e_i)^{\mathcal{H}}}\mu)+d\phi([\mu, (\nabla^{M}_{e_i}e_i)^{\mathcal{V}}]).
\end{eqnarray}
For vertical vector field $e_s$  on $M$, since $\mu$ is horizontal, we have
\begin{align}\label{GD3}
 \nabla^{\phi}_{e_{s}} d\phi(\mu)=-d\phi([\mu,e_s]).
\end{align}
Using (\ref{GD3}) and (\ref{cases1}), we have
\begin{align}\label{GD4}
\nabla^{\phi}_{e_s}\nabla^{\phi}_{e_{s}} d\phi(\mu)&=-\nabla^{\phi}_{e_s}d\phi([\mu,e_s])=d\phi([[\mu,e_s],e_s]).
\end{align}
Using (\ref{cases1}) again, we obtain
\begin{align}\label{GD5}
-\nabla^{\phi}_{\nabla^{M}_{e_s}e_s} d\phi(\mu)&=-\rm d\phi(\nabla^{M}_{(\nabla^{M}_{e_s}e_s)^{\mathcal{H}}}\mu)+d\phi([\mu,(\nabla^{M}_{e_s}e_s)^{\mathcal{V}}]).
\end{align}
Finally, the curvature term in the bitension field formula for a Riemannian submersion becomes
\begin{align}\notag
-\sum_{i=1}^m {\rm R}^{N}(\rm d\phi(e_i), \tau(\phi))\rm d\phi(e_i)&=(m-n)\sum_{i=1}^n R^{N}(\rm d\phi(e_i), \rm d\phi(\mu))\rm d\phi(e_i)\\\label{Ric}
&=-(m-n){\rm Ricci}^N(\rm d\phi(\mu)).
\end{align}
Substituting (\ref{GD1}), (\ref{GD2}), (\ref{GD4}), (\ref{GD5}), and (\ref{Ric}) into the bitension field formula we have
\begin{eqnarray}\notag
\tau_{2}(\phi)&=&-(m-n){\rm d}\phi \Big(  \sum_{i=1}^n\big\{ \nabla^{M}_{e_{i}}(\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}
- \nabla^{M}_{(\nabla^{M}_{e_i}e_i)^{\mathcal{H}}}\mu+[\mu, (\nabla^{M}_{e_i}e_i)^{\mathcal{V}}]\big\}\\\notag
&&+\sum_{s=n+1}^m\{ [[\mu,e_s],e_s] +[\mu,(\nabla^{M}_{e_s}e_s)^{\mathcal{V}}]\}-(m-n)\nabla^{M}_{\mu}\mu \Big)\\\label{EQ1}
&&-(m-n){\rm Ricci}^N(\rm d\phi(\mu)),
\end{eqnarray}
from which the theorem follows.
\end{proof}

\begin{corollary}\label{Co1}
Let $\phi: (M^m,g)\longrightarrow (N^n,h)$ be a Riemannian submersion with basic mean curvature vector field of the fibers. Then it is biharmonic if and only if
\begin{eqnarray}\notag
&&{\rm d}\phi \Big(  \sum_{i=1}^n\big\{\nabla^{M}_{e_{i}}(\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}- \nabla^{M}_{(\nabla^{M}_{e_i}e_i)^{\mathcal{H}}}\mu\big\}
-(m-n)\nabla^{M}_{\mu}\mu \Big)+{\rm Ricci}^N(\rm d\phi(\mu))=0,
\end{eqnarray}
where $\{e_{i},e_{s}\}_{i=1,\dots,n, s=1,\dots,m-n}$ is a local orthonormal frame on the total space $M$ with $e_{i}$ horizontal and $e_{s}$ vertical,  $\mu = \tfrac{1}{m-n} \sum_{s=1}^{m-n} (\nabla_{e_{s}} e_{s})^{\mathcal{H}}$
is  the mean curvature of the fibers of the Riemannian submersion.
\end{corollary}
\begin{proof}
It is well known (see e.g., \cite{ON}) that for a basic vector field $X$ and a vertical vector field $V$ of a Riemannian submersion, $[X, V]$ is vertical and hence $\rm d \phi([X, V])=0$. From this and Theorem \ref{MT} we obtain the corollary.
\end{proof}

Notice that for any basic vector fields $X, Y$ of a Riemannian submersion $\phi:(M^m,g)\longrightarrow (N^n,h)$, i.e., horizontal vector fields that are $\phi$-related to vector fields $\bar{X}, \bar{Y}$ which means ${\rm d}\phi(X)=\bar{X}$ and ${\rm d}\phi(Y)=\bar{Y}$. In this case, it is well known (see e.g., \cite{ON}) that $\nabla^M_XY$ is basic and $\phi$-related to $\nabla^N_{\bar X}{\bar Y}$ so
\begin{equation}\label{Ba}
{\rm d}\phi(\nabla^M_XY)=\nabla^N_{\rm d\phi(X)}\rm d\phi(Y).
\end{equation}

Since we can always choose a local orthonormal frame $\{e_i, e_s\}$ adapted to a Riemannian submersion so that $\{e_i: i=1, 2,\dots , n\}$ are basic vector fields, we can use Corollary \ref{Co1} and (\ref{Ba}) to have the following
\begin{corollary}\label{Co2}
Let $\phi: (M^m,g)\longrightarrow (N^n,h)$ be a Riemannian submersion with basic mean curvature vector field of the fibers. Then it is biharmonic if and only if
\begin{eqnarray}\label{Bas}
{\rm Tr}_h\,(\nabla^{N})^2\tau(\phi)
+\nabla^{N}_{\tau(\phi)}\tau(\phi) +{\rm Ricci}^N(\tau(\phi))=0.
\end{eqnarray}
\end{corollary}

\begin{remark}
One can check that the notion of ``the tension field of a Riemannian submersion being basic" defined in \cite{On} is equivalent to the mean curvature vector field of the fibers of the Riemannian submersion being basic. Thus, our Corollary \ref{Co2} recovers Theorem 4.1 in \cite{On}.
\end{remark}

\begin{theorem} \label{MT2}
The Riemannian submersion
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +e^{2\lambda} g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
defined by the projection of a warped product onto its first factor is biharmonic if and only if
\begin{equation}\label{W-P}
{\rm grad}\, \Delta\lambda+2{\rm Ricci}^{M}({\rm grad}\,\lambda)+\frac{n}{2}{\rm grad}\,|{\rm grad}\lambda|^{2}=0,
\end{equation}
where ${\rm grad}$ and $\Delta$ are the gradient and the Laplace operators on $(M^m, g_M)$.
\end{theorem}

\begin{proof}

First of all, one can easily see that if $\lambda$ is constant, then the Riemannian submersion $\phi$ is harmonic since it has minimal fibers. In this case, we also have (\ref{W-P}). Now suppose that $\lambda$ is not constant. Let  $P=(M\times N, g_M+e^{2\lambda}g_N)$ be the warped product of $(M^m, g_M)$ and $(N^n, g_N)$ with the warping function $f=e^{\lambda}$. We denote by $\nabla^{P}$, $\nabla^{M}$ and $\nabla^{N}$  the connections of the warped product $(P, g=g_M+e^{2\lambda}g_N)$, the base  $(M^m, g_M)$ and the fiber $(N^n, g_N)$ spaces respectively. Then we have
\begin{equation}\label{Prod}
\begin{cases}
 \nabla^{P}_XY=\nabla^{M}_XY,\;\;\; \nabla^{P}_XV=\nabla^{P}_VX=X(\lambda)V\;\; \\
 \nabla^{P}_VW=-g(V,W){\rm grad}\,\lambda+\nabla^{N}_VW\\
\end{cases}
\end{equation}
where for all $X, Y\in \mathfrak{X}(M)$ and $V, W \in \mathfrak{X}(N)$.\\

Let $\{e_{1},...,e_{m},e_{m+1},...,e_{m+n}\}$ be a local orthonormal frame on the warped product  $P=(M\times N,  g_M +e^{2\lambda} g_N)$ such that $\{e_{1},...,e_{m}\}$ are horizontal
and $\{e_{m+1},...,e_{m+n}\}$ are vertical. Clearly this orthonormal frame is adapted to the Riemannian submersion $\phi$. Using (\ref{Prod}) we obtain the tension field of the Riemannian submersion
\begin{eqnarray}\label{GD40}
\tau(\phi) =-n{\rm d}\phi (\mu)=-\sum_{s=m+1}^{m+n}{\rm d} \phi(\nabla^{P}_{e_{s}}e_{s})=n\, {\rm d} \phi({\rm grad}\lambda)=n\, {\rm grad}\lambda,
\end{eqnarray}
where the last equality was obtained by using the fact that $\phi$ is a projection and hence ${\rm d} \phi(X)=X$ for any horizontal vector field $X$.\\

Clearly, in this case, the mean curvature vector field $\mu=-{\rm grad} \lambda$ of the fibers of the Riemannian submersion is basic. So, by Corollary \ref{Co2} and (\ref{GD40}), the Riemannian submersion $\phi$ is biharmonic if and only if

\begin{eqnarray}\label{GD41}
{\rm Tr}_{g_M}\,(\nabla^{M})^2 {\rm grad}\lambda
+n \nabla^{M}_{{\rm grad}\lambda}{\rm grad}\lambda +{\rm Ricci}^M({\rm grad}\lambda)=0.
\end{eqnarray}

We can check (see e.g., \cite{BFO}) that
\begin{equation}\label{GD42}
{\rm Tr}_{g_M} (\nabla^M)^{2}({\rm grad}\,\lambda)={\rm grad}\, \Delta\lambda+{\rm Ricci}^{M}({\rm grad}\lambda).
\end{equation}

On the other hand, it is  the well known  (see e.g., \cite{Pe}) that
\begin{equation}\label{GD43}
\nabla^{M}_{{\rm grad}\lambda}{\rm grad}\lambda=\frac{1}{2}{\rm grad} (|{\rm grad}\lambda|^{2}).
\end{equation}

Substituting (\ref{GD42}) and (\ref{GD43}) into  (\ref{GD41}) we obtain Equation (\ref{W-P}), which completes the proof of the theorem.
\end{proof}
\begin{remark}
 It is interesting to note that in their study of biharmonic maps from warped product manifolds, Balmus-Montaldo-Oniciuc \cite{BMO} introduced the notion of {\em biharmonic warping function} defined to be a warping function $f$ so that the identity map ${\rm Id}:M\times_{f}N\equiv (M\times N, g_M+f^2g_N)\longrightarrow (M\times N, g_M+g_N)$, ${\rm Id}(x,y)=(x,y)$ is a proper biharmonic map for some manifold $N$ of dimension greater than zero. They also proved that the projection
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +f^2 g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
 is  proper biharmonic if and only if
\begin{equation}
{\rm Tr}_g\nabla ^2 {\rm grad}\, \ln \,f+{\rm Ricci}^{M}({\rm grad}\, \ln \,f)+\frac{n}{2}{\rm grad}\,|{\rm grad}\, \ln \,f|^2=0.
\end{equation}
(ii)  Note also that the biharmonic equation for the projection of a warped product in the form of (\ref{Bas}) was obtained in \cite{Ur1} as in this case the mean curvature vector field of the Riemannian submersion is basic. So our Theorem \ref{MT2} gives an improvement of the equation obtained in \cite{Ur1}.\\
\end{remark}

\begin{corollary}
If $ (M^m, g_M)$ is an Einstein Manifold with ${\rm Ricci}^{M}=ag_M$, then the Riemannian submersion
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +e^{2\lambda} g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
defined by the projection of a warped product onto its first factor is biharmonic if and only if
\begin{equation}\label{Einstein}
 \Delta\lambda+2a\lambda+\frac{n}{2}|{\rm grad}\lambda|^{2}=C,
\end{equation}
where $C$ is a constant and $\Delta$ is the Laplace operator on $(M^m, g_M)$.
\end{corollary}
\begin{proof}
If $ (M^m, g_M)$ is an Einstein Manifold with ${\rm Ricci}^{M}=a\,g_M$, then one can easily check that ${\rm Ricci}^{M}({\rm grad}\lambda)=a\, {\rm grad}\lambda$. Substituting this into (\ref{W-P}) we conclude that the Riemannian submersion
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +e^{2\lambda} g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
defined by the projection of a warped product onto its first factor is biharmonic if and only if
\begin{equation}\notag
 {\rm grad}\Big(\Delta\lambda+2a\lambda+\frac{n}{2}|{\rm grad}\lambda|^{2}\Big)=0,
\end{equation}
from which we obtain the corollary.
\end{proof}

\begin{corollary}\label{MT3}
 The Riemannian submersion
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +e^{2\lambda} g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
defined by the projection of a warped product onto a compact manifold with negative Ricci curvature is  biharmonic if and only if
it is harmonic.
\end{corollary}
\begin{proof}
By Theorem \ref{MT2}, the Riemannian submersion
\begin{align}\notag
\phi: ( M^{m} \times N^n , g_M +e^{2\lambda} g_N) \longrightarrow (M^m , g_M),\;\;
\phi(x,y) =x
\end{align}
defined by the projection of a warped product onto its first factor is biharmonic if and only if
\begin{equation}\label{W1}
{\rm grad}\, \Delta\lambda+2{\rm Ricci}^{M}({\rm grad}\,\lambda)+\frac{n}{2}{\rm grad}\,|{\rm grad}\lambda|^{2}=0.
\end{equation}
On the other hand, it is well known that for any function $\lambda$ on $M$, we have the following Bochner formula
\begin{equation}\notag
\frac{1}{2} \Delta|{\rm grad}\, \lambda|^2=|\nabla{\rm d}\, \lambda|^2+ \langle {\rm grad}\, \lambda, {\rm grad}\, \Delta \lambda\rangle +{\rm Ricci}^{M}({\rm grad}\,\lambda, {\rm grad}\,\lambda).
\end{equation}
Substituting the term ${\rm grad}\, \Delta\lambda$ from (\ref{W1}) into the Bochner formula we have
\begin{equation}\label{W2}
\frac{1}{2} \Delta|{\rm grad}\, \lambda|^2=|\nabla{\rm d}\, \lambda|^2 - \frac{n}{2} \langle {\rm grad}\, \lambda, {\rm grad}\,|{\rm grad}\lambda|^{2}\rangle -{\rm Ricci}^{M}({\rm grad}\,\lambda, {\rm grad}\,\lambda).
\end{equation}
If $\lambda$ is constant, then the Riemannian submersion is harmonic and the corollary follows. Now suppose $\lambda$ is not a constant, then, since $M$ is compact the function $|\nabla{\rm d}\, \lambda|^2$ attains its maximum at some point $p\in M$. Evaluating both sides of Equation (\ref{W2}) at the point $p$ we see that the right-hand side is positive  by the fact that $({\rm grad}\,|{\rm grad}\lambda|^{2})(p)=0$ and the assumption of negative Ricci curvature whilst the left-hand side is non-positive since the Laplacian being the trace of the Hessian of $|{\rm grad}\, \lambda|^2$, which is semi-negative definite at $p$. The contradiction shows that $\lambda$ has to be a constant on $M$. This completes the proof of the corollary.

\end{proof}

\begin{example}
For any constants $a_i>0, \,b_i$,  $i=1,2,...,n,$  let $\kappa_i(x_i)=-\frac{2}{x_i+b_i}$,  $\kappa_i(x_i)=a_i\tan\big(\frac{a_i}{2}(x_i+b_i)\big),$ or
$\kappa_i(x_i)=\frac{a_i(1+e^{a_i(x_i+b_i)})}{1-e^{a_i(x_i+b_i)}},$ and let $\lambda(x_1,x_2,...,x_n)=- \sum^{n}_{i=1} \int\kappa_i(x_i)dx_i,$ then the Riemannian submersion
\begin{align}\notag
\pi  :\big(  \mathbb{R}^{n} \times \mathbb{R} , g_0 +e^{2\lambda(x)} dt^2\big) \longrightarrow (\mathbb{R}^n ,g_0),\;\;\pi(x,t) =x
\end{align}
given by the projection of a warped product onto its first factor is a proper biharmonic map. In particular, the projections
 \begin{align}\notag
\pi  :\big( (0,\infty)^n \times \mathbb{R} , g_0 +(x_1 x_2\cdots x_n)^4 dt^2\big) \longrightarrow ((0,\infty)^n ,g_0),\;\;\pi(x,t) =x,
\end{align}
and
 \begin{align}\notag
\pi  :\big( (0,\infty)^n \times \mathbb{R} , g_0 +(\sinh x_1 \sinh x_2\cdots \sinh x_n)^2 dt^2\big) \longrightarrow ((0,\infty)^n ,g_0),\;\;\pi(x,t) =x,
\end{align}
are both biharmonic Riemannian submersions.

In fact, since the base manifold is an open submanifold of a Euclidean space which is an Einstein space with $a=0$, we use this and Equation (\ref{Einstein}) with $n=1$ to conclude that the Riemannian submersion $\pi$ is biharmonic if and only if
\begin{equation}\label{EinP1}
 \Delta\lambda+\frac{1}{2}|{\rm grad}\lambda|^{2}=C,
\end{equation}
where the Laplacian and the gradient operators are defined by the Euclidean metric. If we look for the special solutions of the form $\lambda=-\sum^{n}_{i=1} \int\kappa_i(x_i)dx_i$, then a straightforward computation shows that Equation (\ref{EinP1}) is equivalent to
\begin{align}\label{P20}
 -\sum_{i=1}^n\big( \kappa^{\prime}_i-\frac{\kappa^{2}_i}{2}\big)=C.
\end{align}
Since $\kappa_i=\kappa_i(x_i)$, one can easily check that (\ref{P20}) is equivalent to

\begin{align}\label{P2}
 \kappa^{\prime}_i-\frac{\kappa^{2}_i}{2}=C_i,  \ i=1,2,...,n,
\end{align}
which can be solved by the following cases.\\

\textbf{Case I.} For $C_i=0$,  we solve (\ref{P2}) to have solutions
$$
\kappa_i=-\frac{2}{x_i+b_i},\ \ \ i=1,2,...,n,
$$
for some constant $b_i.$

\textbf{Case II.} For  $C_i=\frac{a_i^2}{2}>0$ with constant $a_i>0$, then it is easily chcked that (\ref{P2}) can be solved to have
$$
\kappa_i=a_i\tan\Big(\frac{a_i}{2}(x_i+b_i)\Big),\ \ \ i=1,2,...,n,
$$
for some constant $b_i.$

\textbf{Case III.} For $C_i=-\frac{a_i^2}{2}<0$, we solve (\ref{P2}) to obtain

$$
\kappa_i=\frac{a_i(1+e^{a_i(x_i+b_i)})}{(1-e^{a_i(x_i+b_i)})},\ \ \ i=1,2,...,n,
$$
for some constant $b_i.$\\

From these, we obtain the example.
\end{example}

\section{Biharmonic Riemannian submersions with $1$-dimensional fibers}

 In this section, we will follow the idea from \cite{WO} to describe biharmonicity of a Riemannian submersion with $1$-dimensional fibers
from a generic $(n+1)$-manifold by using the integrability data of a
special orthonormal frame adapted to a Riemannian submersion.  We also use the biharmonic equation derived to study biharmonicity of a Riemannian submersion defined by the projection of a twisted product onto it first factor.\\

Let $\phi:( M^{n+1} , g)\longrightarrow (N^n, h)$ be a Riemannian
submersion. A local orthonormal frame is said to be {\bf adapted to
the Riemannian submersion} $\phi$ if the vector fields in the frame
that are tangent to the horizontal distribution are basic (i.e.,
they are $\phi$-related to a local orthonormal frame in the base
space). Such a frame always exists (cf. e.g., \cite{BW1}). Let
$\{e_1,\ldots,e_n, e_{n+1}\}$ be an orthonormal frame adapted
to $\phi$ with $e_{n+1}$ being vertical. Then, it is well known (see
\cite{ON}) that $[e_i,e_{n+1}]$ $(i=1, 2, \ldots, n)$ are vertical
and $[e_i,e_j]$ $(i, j=1, 2, \ldots, n)$ are $\phi$-related to
$[\varepsilon_i, \varepsilon_j]$, where $\{\varepsilon_i, \ldots,
\varepsilon_n\}$ is an orthonormal frame in the base manifold. If we
assume that
\begin{equation}\label{RS22}
[\varepsilon_i,\varepsilon_j]=F_{ij}^k\varepsilon_k,
\end{equation}
for $F^k_{ij}\in C^{\infty}(N)$ and use the notations
$f^k_{ij}=F^k_{ij}\circ \phi,\;\forall\; i, j, k=1, 2, \ldots, n$. Then, we have
\begin{equation}\label{LB}
\begin{cases}
[e_i,e_{n+1}]=\kappa_ie_{n+1},\\
[e_i,e_j]=f^k_{ij} e_k-2\sigma_{ij}e_{n+1},\;\;\; i, j=1, 2, \ldots,
n,
\end{cases}
\end{equation}
where $\kappa_i,\,{\rm and}\; \sigma_{ij} \in C^{\infty}(M)$. We
will call $ \{f^k_{ij},  \kappa_i,\,\sigma_{ij}\}$ {\bf the
integrability data} of the adapted frame of the Riemannian
submersion $\phi$. It follows from (\ref{LB}) that
\begin{eqnarray}
f_{ij}^k=-f_{ji}^k\; \;{\rm and}\;\sigma_{ij}=-\sigma_{ji},\;\;\forall\; i,j, k=1,2, \cdots, n.
\end{eqnarray}

\begin{theorem}\label{1D}
 Let $\phi:(M^{n+1},g)\longrightarrow (N^n,h)$ be a Riemannian
submersion with the adapted frame $\{e_1,\ldots, e_{n+1}\}$ and the
integrability data $ \{f^k_{ij}, \kappa_i, \sigma_{ij}\}$. Then,
the Riemannian submersion $\phi$ is biharmonic if and only if
\begin{eqnarray}\label{1DE}
&&\Delta\kappa_k+\sum^{n}_{i,j=1}\Big(2e_i(\kappa_j)P^{k}_{ij}+\kappa_j\big[ e_i (P^{k}_{ij})+P^{l}_{ij}P_{il}^k -\kappa_iP_{ij}^k - P_{ii}^lP_{lj}^k\big] \Big)\\\notag
&&+{\rm Ricci}^N(\rm d\phi(\mu), {\rm d}\phi(e_k))=0, \;\;k=1,2,\ldots, n,
\end{eqnarray}
where $P_{ij}^k=\frac{1}{2}(-f_{ik}^j-f_{jk}^i +f_{ij}^k)$ for all $ i,j, k=1,2, \cdots, n$ and $\Delta$ denotes the Laplacian of $(M^{n+1}, g)$.
\end{theorem}
\begin{proof}
Let $P_{ij}^k=\frac{1}{2}(-f_{ik}^j-f_{jk}^i +f_{ij}^k)$ for all $ i,j, k=1,2, \cdots, n$. Then, one can easily check that
\begin{eqnarray}
&&P_{ij}^k =P_{ji}^k+f_{ij}^k, \;\; P_{ii}^k=-f_{ik}^i.
\end{eqnarray}
A straightforward computation using (\ref{LB})
and Koszul formula gives
\begin{eqnarray}\label{NAi}
\nabla_{e_{i}} e_{j} &=& P_{ij}^ke_k-\sigma_{ij}e_{n+1}\;\;\;{\rm  for\; any\; } i, j=1,2, \cdots, n,\\\label{NAs}
\nabla_{e_{n+1}} e_{n+1} &=& \sum_{i=1}^n\kappa_ie_i.
\end{eqnarray}

The mean curvature vector field of the fibers of the Riemannian  submersion $\phi$ is given by
\begin{eqnarray}\label{MC}
\mu=(\nabla_{e_{n+1}}e_{n+1})^{\mathcal{H}}=\sum_{i=1}^n\kappa_ie_i.
\end{eqnarray}
A straightforward computation using (\ref{LB}), (\ref{NAi}), (\ref{NAs}) and (\ref{MC}) yields
\begin{eqnarray}\notag
 && \sum_{i=1}^n\big\{\nabla^{M}_{e_{i}}(\nabla^{M}_{e_{i}}\mu)^{\mathcal{H}}\big\}^{\mathcal{H}}=\big\{\sum^{n}_{i,j=1}\nabla^{M}_{e_i}\Big(e_i(\kappa_j)+\sum^{n}_{l=1}\kappa_lP^{j}_{il}\Big)e_j\big\}^{\mathcal{H}}\\\notag
&&=\sum^{n}_{i,k=1}e_i\Big(e_i(\kappa_k)+\sum^{n}_{j=1}\kappa_jP^{k}_{ij}\Big)e_k+\sum^{n}_{i,j=1}\Big(e_i(\kappa_j)+\sum^{n}_{l=1}\kappa_lP^{j}_{il}\Big)P_{ij}^ke_k\\\label{GD20}
&&=\sum^{n}_{k=1}\Big(\sum^{n}_{i=1}e_ie_i(\kappa_k)+2\sum^{n}_{i,j=1}e_i(\kappa_j)P^{k}_{ij}+\sum^{n}_{i,j=1}\kappa_j(e_i P^{k}_{ij})+\sum^{n}_{i,j=1}\kappa_jP^{l}_{ij}P_{il}^k\Big)e_k\\
&&  \big\{- \nabla^{M}_{(\nabla^{M}_{e_i}e_i)^{\mathcal{H}}}\mu\big\}^{\mathcal{H}}=-\sum_{i,k,j=1}^n\big( P_{ii}^j(e_j\kappa_k)+\kappa_j P_{ii}^lP_{lj}^k  \big)e_k\\
&&  [\mu, (\nabla^{M}_{e_i}e_i)^{\mathcal{V}}]=0,\; {\rm since}\; (\nabla^{M}_{e_{i}}e_{i})^{\mathcal{V}}=0,\\
&&  [[\mu,e_{n+1}],e_{n+1}] =\sum_{k=1}^n e_{n+1}(e_{n+1}\,\kappa_k)e_k\\
&&  [\mu,(\nabla^{M}_{e_{n+1}}e_{n+1})^{\mathcal{V}}]=0,\; {\rm since}\; (\nabla^{M}_{e_{n+1}}e_{n+1})^{\mathcal{V}}=0,\\
&& \big\{-\nabla^{M}_{\mu}\mu \big\}^{\mathcal{H}}=-\sum_{i,k=1}^n \kappa_i\Big(e_i(\kappa_k)+\sum_{j=1}^n \kappa_jP_{ij}^k )\Big)e_k\\
&& {\rm Ricci}^N(\rm d\phi(\mu))=\sum_{k=1}^n{\rm Ricci}^N(\rm d\phi(\mu), {\rm d}\phi(e_k)){\rm d}\phi(e_k).
\end{eqnarray}

Using these and Theorem \ref{MT} we conclude that the Riemannian submersion $\phi$ is biharmonic if and only if

\begin{eqnarray}\notag
&&\sum^{n}_{k=1}\Big(\sum^{n}_{i=1}e_ie_i(\kappa_k)+2\sum^{n}_{i,j=1}e_i(\kappa_j)P^{k}_{ij}+\sum^{n}_{i,j=1}\kappa_j(e_i P^{k}_{ij})+\sum^{n}_{i,j=1}\kappa_jP^{l}_{ij}P_{il}^k\\\notag
&&  -\sum_{i,j=1}^n\big( P_{ii}^j(e_j\kappa_k)+\kappa_j P_{ii}^lP_{lj}^k \big) + e_{n+1}(e_{n+1}\,\kappa_k)-\sum_{i=1}^n \kappa_ie_i(\kappa_k)-\sum_{i,j=1}^n \kappa_i\kappa_jP_{ij}^k \\\notag
&&+{\rm Ricci}^N\big(\rm d\phi(\mu), {\rm d}\phi(e_k)\big)\Big){\rm d}\phi(e_k)=0.
\end{eqnarray}
This is equivalent to
\begin{eqnarray}\notag
&&\sum^{n+1}_{i=1}e_ie_i(\kappa_k) -\sum_{i=1}^n P_{ii}^j(e_j\kappa_k) -\sum_{i=1}^n \kappa_ie_i(\kappa_k)\\\label{GD30}
&&+\sum^{n}_{i,j=1}\Big(2e_i(\kappa_j)P^{k}_{ij}+\kappa_j(e_i P^{k}_{ij})+\kappa_jP^{l}_{ij}P_{il}^k -\kappa_jP_{ij}^k - \kappa_j P_{ii}^lP_{lj}^k\Big)\\\notag
&&+{\rm Ricci}^N(\rm d\phi(\mu), {\rm d}\phi(e_k))=0, \;\;k=1,2,\ldots, n
\end{eqnarray}
since $\{{\rm d}\phi(e_k)|\;k=1,2,\ldots, n\}$ is an orthonormal frame on $\phi^{-1}TN$.
Finally, a straightforward computation of the Laplacian $\Delta\kappa_k$ shows that Equation (\ref{GD30}) is equivalent to the following
\begin{eqnarray}\notag
&&\Delta\kappa_k+\sum^{n}_{i,j=1}\Big(2e_i(\kappa_j)P^{k}_{ij}+\kappa_j\big[ e_i (P^{k}_{ij})+P^{l}_{ij}P_{il}^k -\kappa_iP_{ij}^k - P_{ii}^lP_{lj}^k\big] \Big)\\\notag
&&+{\rm Ricci}^N(\rm d\phi(\mu), {\rm d}\phi(e_k))=0, \;\;k=1,2,\ldots, n,
\end{eqnarray}
from which we obtain the theorem.
\end{proof}

\begin{remark}
One can check that when $n=2$,  the integrability data is given by
\begin{eqnarray}\notag
f_{12}^1&=& f_1,\; f_{12}^2=f_2,\\\notag
P^{2}_{11}&=&-f_1, \; P^{2}_{21}=-f_2, \; P^{1}_{12}=f_1, \; P^{1}_{22}=f_2,\; {\rm all\;other}\; P_{ij}^k=0.
\end{eqnarray}
Substituting this and ${\rm Ricci}^N(\rm d\phi(\mu), {\rm d}\phi(e_k))=\kappa_k{\rm K}^N$ into Equation (\ref{1DE}), we obtain that a Riemannian
submersion $\phi:(M^{3},g)\longrightarrow (N^2,h)$  is biharmonic if and only if
\begin{eqnarray}
&&\Delta\kappa_1+(e_1\kappa_2)f_1+(e_2\kappa_2)f_2+e_1(\kappa_2  f_1)+e_2(\kappa_2 f_2)\\\notag
&& -\kappa_1\kappa_2f_1-(\kappa_2)^2f_2 -\kappa_1(-{\rm K}^N+f_1^2+ f_2^2)=0,\\
&&\Delta\kappa_1-(e_1\kappa_1)f_1-(e_2\kappa_1)f_2-e_1(\kappa_1  f_1)-e_2(\kappa_1 f_2)\\\notag
&& +\kappa_1\kappa_2f_2+(\kappa_1)^2f_1 -\kappa_2(-{\rm K}^N+f_1^2+ f_2^2)=0.
\end{eqnarray}
This is exactly Proposition 2.1 in \cite{WO} of which our Theorem \ref{1D} gives a generalization.
\end{remark}

\begin{corollary}\label{Twisted}
 Let $g_0$ be the standard Euclidean metric on $\mbox{${\mathbb R}$}^n$. Then, the Riemannian submersion defined by the projection of the twisted product
\begin{align}\notag
\phi  : ( \mathbb{R}^{n} \times \mathbb{R} , g_0 +e^{2\lambda(x,t)} dt^2) &\to (\mathbb{R}^n ,g_0),\;\; \phi(x,t) =x
\end{align}
 is biharmonic if and only if its tension field viewed as a map
 \begin{equation}
 \tau(\phi):( \mbox{${\mathbb R}$}^n \times \mathbb{R} , g_0+e^{2\lambda(x,t)}  dt^2) \longrightarrow\mbox{${\mathbb R}$}^n, \;\tau(\phi)(x,t)=(\kappa_1(x,t),\kappa_2(x,t),\ldots, \kappa_n(x,t))
 \end{equation}
  is harmonic, where $\tau(\phi)=\sum^n_{i=1}\kappa_i(x,t){\rm d}\phi(e_i)$ is the tension field of the Riemannian submersion with respect to an adapted frame.
 \end{corollary}
\begin{proof}
In this case, we can check that the orthonormal frame
$\{e_i=\frac{\partial}{\partial x_i}|\, i=1,\ldots, n,\,e_{n+1}= e^{-\lambda}\frac{\partial}{\partial t}\} $  is adapted to
the Riemannian submersion $\phi$ with $ d\phi(e_i)=\varepsilon_i, i=1,
2,...,n$ and $e_{n+1}$ being vertical, where
$\varepsilon_i=\frac{\partial}{\partial x_i}, i=1,2,...,n$ form an orthonormal frame on the
base space $(\mathbb{R}^n ,g_0)$.  An easy  computation gives
 \begin{align}\notag
 [e_i,e_j]=0,\ \  i,j=1,2,...,n,\;\; [e_i,e_{n+1}]=\kappa_ie_{n+1},
\end{align}
where $\kappa_i=-\lambda_{x_i}$. It follows that the integrability data in this case are given by
\begin{align*}
f_{ij}^k=\sigma_{ij}=0,\;\; \forall\; i,j,k=1,2,\ldots, n.
\end{align*}
It follows that
\begin{equation}\label{111}
\begin{array}{lll}
P_{ij}^{k}=\sigma_{ij}=0 ,\;\; \forall\; i,j,k=1,2,\ldots, n.
\end{array}
\end{equation}

Using these and Theorem \ref{1D} we conclude that the Riemannian submersion defined by the twisted product onto its first factor is biharmonic  if and only if
\begin{equation}\label{Delta}
\Delta \kappa_i=0,\forall\; i=1,2,\ldots, n.
\end{equation}
This means exactly that the tension field of the Riemannian submersion viewed as a map
 \begin{equation}
 \tau(\phi):( \mbox{${\mathbb R}$}^n \times \mathbb{R} , g_0+e^{2\lambda(x,t)}  dt^2) \longrightarrow\mbox{${\mathbb R}$}^n, \;\tau(\phi)(x,t)=(\kappa_1,\kappa_2,\ldots, \kappa_n)
 \end{equation}
  is harmonic.
 \end{proof}

\begin{remark}
(i) We would like to point out that the mean curvature vector field of the Riemannian submersion studied in Corollary \ref{Twisted}  is $\mu=-\sum_{i=1}^n\frac{\partial \lambda(x, t)}{\partial x_i}e_i$, which is not  a basic vector field.\\
(ii) One can also check that
\begin{align}\label{correction}
\Delta_{M}\kappa_i&=\sum^{n}_{j=1}\frac{\partial^{2} \kappa_i}{\partial {x_j}^{2}}+e^{-2\lambda}\frac{\partial^2\kappa_i}{\partial {t^2}}-\sum^{n}_{j=1}\kappa_j\frac{\partial\kappa_i}{\partial x_j}-e^{-2\lambda}\frac{\partial \lambda}{\partial t}\frac{\partial \kappa_i}{\partial t}.
\end{align}
\end{remark}

\begin{ack}
Both authors would like to thank C. Oniciuc for some useful comments and questions that help to clarify some statements of  the original manuscript. The second author also wants to thank Changyou Wang for some constructive discussions and sharing his insight on Equation (\ref{Einstein}) in the paper and Fred Wilhelm for some useful communication on constructions of Riemannian submersions.
\end{ack}

redpajama_set_name: RedPajamaArXiv


The chapter title is a Chinese idiom that you can search up in Google. It refers to a shrewish wife bellowing, and the husband who is henpecked.
Sometimes, rumors spread very quickly. Such as the new rumor of the love and hate between Dai Rong Princess and Jia You Emperor. The main idea was that Dai Rong Princess coveted the prosperity of Da Long, and not wanting to return to the poor country of Jing, tried all she could to become one of the Emperor's feiqin. Even more extreme was that Dai Rong Princess recommended herself in front of the Emperor and Empress. But who knew that she was rejected multiple times by the Emperor. Due to her actions, the Emperor became dissatisfied with the country of Jing and even rejected the request from the Crown Prince of Jing for an audience.
There were others that said it was previously Dai Rong Princess who targeted the Empress and the honorable officials because she wanted to enter the palace as fei, so she specially spread rumors to blight the Empress' reputation.
In the subconsciousness of the people in the Jing City of Da Long, they looked down at the kind of small country that was Jing. Previously, when they gossiped about how exceptional Dai Rong Princess was, no one ever said that Dai Rong Princess was better than the Empress, they were only speculating that the Emperor wanted something fresh and fancied Dai Rong Princess.
When what had happened in the Imperial Palace reached the ears of the people of Da Long, there was an explosion. The princess of Jing was shameless, the Emperor didn't even want the princess when she recommended herself, she couldn't compare to a finger of the Empress. Maybe she had nothing inside, and wanted to grab onto their Da Long. Furthermore, others started to say that this princess of Jing had ulterior motives and wanted to enter the Imperial Palace of Da Long to give birth to a son and take Da Long's throne, to conquer Da Long's lands.
Dai Rong shook her head as she rejected: "No, the Qin Family said that the Emperor treats the Empress well only because of the Tian Family, so it's impossible!
Dai Rong froze. A beat later, she looked in disbelief at Tu Er: "Why do you know so much about Da Long, you ……" Did he already scheme for the position of Crown Prince beforehand, so that's why he knew so much about Da Long?
Dai Rong staggered. For what seemed to be the first time, she looked at her useless and average gege, but after looking at the other's slightly smiling face, she couldn't say anything.
By this time, the members of the Qin Family were so frightened, their faces were the color of dirt. Qin furen didn't dare to stop her daughter's mouth anymore, and could only tremble as she knelt at the side.
When Qu Qing Ju heard something this hilarious, she instantly laughed out loud. She didn't have any more interest in talking with Qin Bai Lu. How strange was her brain that she would push all the misfortunes onto her. Qu Qing Ju did not believe that she herself had such great abilities. Or was it also her who'd ordered He Yuan to divorce Qin Bai Lu?
How xiandi decreed the marriage, it was xiandi who made the decision. Could she have forced xiandi to make a decision?
By now, the members of the Qin Family wanted to strangle Qin Bai Lu. If it wasn't for the furen of the head of the family, who was strong in her conduct, they wouldn't have tolerated a divorced woman living in their ancestral home. If they knew beforehand that she would say such insane things, do insane things, even if they had to start a conflict, they would have sent her away rather than let her drag down the entire clan.
He Heng's face was frigid as ice. No man was willing to hear about how his beloved woman was being thought of by another man. Furthermore, this man had nearly taken over his position.
Qin Bai Lu had pushed all of their conflicts onto He Yuan. Then, Qu Qing Ju would purposefully distort it into Qin Bai Lu being jealous of her receiving the Emperor's affections. The two matters seemed to be the same matter, but the feeling it gave others was quite different.
She wanted He Heng to know. The person she wanted to capture was him. The person she paid attention to was him. Only like this would there be no danger left hidden. It would even increase the trust and affection He Heng had towards her.
"You!" Qin Bai Lu hadn't thought Qu Qing Ju would give such a speech. She opened her mouth to start cursing but Qin furen by her side unexpectedly threw a heavy slap on her face, throwing her onto the ground.
Other than the crying of Qin furen, there were no other sounds in the hall. Qin Bai Lu covered her face, not looking at her own mother, but at the Emperor and Empress sitting in front of her, her eyes full of hate. Grabbing out the sharp silver hairpin out of her hair, she suddenly stood and rushed towards Qu Qing Ju.
"Niangniang!" On the side, Mu Jin was so frightened that she threw herself from behind Qu Qing Ju to halfway across Qu Qing Ju. There was a sound of porcelain breaking, and then she heard a smothered grunt from nearby. Then the entire hall started to make a racket.
Raising her head in puzzlement, she saw Qin Bai Lu lying on the side in a sorry state, the Emperor in front of the Empress, one of the corners of his robe creased.
The entire hall instantly became silent, all those that were shocked, that had been pushing Qin Bai Lu down, been asking for mercy. At one point, it was as though their necks were being strangled so that they were unable to make a sound.
Mu Jin stared with wide eyes and mouth at the Empress flipping out. As she watched the Emperor unconsciously step back as the Empress ranted at him, her shoulders couldn't help but shake as she swallowed.
At the side, Qian Chang Xin, Ming He and the others also stood with dumbstruck eyes. Who had seen the Empress be so valiant?! Who had even seen the Empress scold the Emperor? As the personal attendants of the Emperor, they should stop the finger that the Empress was poking onto the Emperor's chest, should reproach the Empress for her disrespectful conduct, but …… for some reason, they didn't dare speak, and didn't dare to even breathe loudly.
After the Qin Family was criticized, they finally managed to react. All of them banged their heads on the ground, begging for the Emperor to spare their lives. For their family to have someone who'd tried to kill the Empress, that was a serious crime worthy of killing off their nine generations.
Qu Qing Ju pushed away his hand, directly sitting back on the chair. She picked up a teacup and lowered her head to drink, not looking at him again.
Qian Chang Xin and Ming He silently retracted their left foot which they had mustered up the courage to extend out. Since even the Emperor who had been scolded was full of smiles, as servants, what could they say?
But the Empress when she was angry was really too frightening. Not even the roars of the lioness from He Dong could compare.
"Who's worried about you?!" Qu Qing Ju pushed him. Seeing that it was futile, she didn't waste any more energy, allowing him to embrace her.
"Alright, I know I was wrong. Don't be angry. Let's go see Tun Tun, he's certainly missing us." Even though he had been scolded, He Heng felt happy. If it wasn't out of concern for him, based on Qing Ju's normal behavior, she wouldn't have acted so inappropriately.
It was somewhat useless point, but this feeling was still really …… refreshing.
Everyone gave their agreement. Mu Jin looked with trepidation at the hall behind her, and couldn't resist wiping the fine sweat on her forehead.
Empress was really very ……amazing!
Thanks for the update. The title is so apt. The lioness from Hedong roars. QQJ is a lioness. HH though scolded is happy.
hmph, yeah, that dai rong isn't worthy of bdcoming a ruler, her brother's better!
oh, the qin family…qin bai lu…really? she's totally insane…when she tried to kill qing ju, i was so shocked—! but then, i was even more shocked by qing ju's outburst, scolding he heng?!that was utterly hilarious, ahahaha! definitely a faved chapter, lol!
don't tell me that he heng developed a certain disease like Klaus from certain novel I know? 😰 don't tell me he likes being scolded?
Not sure what novel you are talking about… … but He Heng likes that Qu Qing Ju is concerned for him.
I can't stop laughing when I read this chapter. Aww.
I cried while Yanou was scolding Zifeng while crying.
This is the first time that QQJ really lost her poise…err….temper….and because of her fear that HH got hurt… really, action speaks louder than words…. HH offering his body to save QQJ, that's LOVE… QQJ spewing steam with trembling hands to boot, that is LOVE… and QBL?….she finally caused her death and the downfall of her entire clan… such evil woman…. thanks for the update…. this was a really exciting chapter….
QQJ well done! You even kicked someone and everyone was afraid of you! Mu Jin, nice reaction! She will admire the empress even more! How glad I am that QQJ has such loyal people around her!
Omg that's is too funny LOL 😂😂😂 also that's wat she gets!! QBL's stupidity is laughable. He Heng is just too funny so loveable!😍😍😍 Thank you for the chapter g really don't want it to end at all!!!
My brain was a bit slow in getting what happened, but when I realized what she did and why she shouted, I started tearing up. GOOD CHAPTER!!
Oh wow! She finally snapped!
hah, sure DR can enter the palace if her country can rival Da Long. but, too bad, a princess from a country solely and heavily relying on Da Long's favors. taking her as a feipin only because she is beautiful equals can't really turn down requests from the Jing Country, why bother with such a troublesome matter.
Thanks. Dai Rong is just beyond stupid. Jing will definitely suffer because of her.
QBL is insane and delusional. I love that QQJ was so worried about HH that she forgot propriety and scolded HH. It's so cute how he's happy at being scolded because it shows how much she cares about him.
Trying to figure out who in the Qin family will be executed, everyone in the ancestral home? 9 generations?
Also I hope HY won't get in trouble due to wat QBL said… That would be really bad since he already gave up the throne and is just living this life away not causing any trouble like he used to!
Had to read it 3 times. Teared up on this one, such good reaction! You know they completely have each other now, this chapter just solidified it. Not sure if QQJ will give any mercy to the women this time, anger has been shown! See what happens.
Let's flip out! Throw the table!!
Don't underestimate the power of a modern woman yo!
Thank you…..they really love and care deeply about each other. DR Princess was misled by the Qin family. Its truly loveable that the empress lose her composure and scolded the emperor disregarding protocol and all…..
And dai rong is soo vain and marcisstic. Thinking her beauty is enough. Really wanna slap this girl.
Lol … oh well HH really should feel happy instead of angry … QQJ while feeling worried and angry still remember to admire that he's got long legs! 😅 But yeah can't blame the guy for seeing her so flustered and losing her composure for the first time and out of worry for him. How sweet is that?
Yea….QQJ finally expressed how she feel for HH through her action. Now we only need QQJ to say that she love HH that is a start.
QQJ's angry act assured HH and us how much she cared and maybe love HH even though she won't admit it herself. Before it only looks like HH cared so much for QQJ but not the other around, but this chapter confirms us on her feelings on her hubby the Emperor.
As for QBL, she deserve it. Too delusional to the point of insanity. DR princess, please learn or your beloved country will suffer. HH will never let go of anyone who dared to even slight his beloved.
The number of chapters is getting closer to the end.. That makes me a bit sad….
amazing chapter ! this is the first time QQJ reacted so strongly for HH ! that let us see that she really cares about him .
And that princess … she gets stupider and stupid er !
No, it's not the first time. She panicked when he was "attacked" by "assassins" whenI they stayed at the hot springs. She ran out without even brushing her hair.
sigh qin family is doomed and QQJ's reaction after the near assassination was refreshing!
Love this chapter. This really shows that QQJ finally truly reciprocates HH's feelings. She wouldn't have reacted this way otherwise especially with the control she has. So happy for HH. He has been looking for proof of QQJ's affections and he finally got it.
Thank you for the update. She really is awesome. See the lioness roar.
No idea what QBL's mum was waiting for, as soon as QBL opened her mouth to utter that she should have been married to HH, if I were her mother I would have slapped her until she loses the function of her mouth. If she wants to die, no problem but she can't drag the family with her. However her elder are partly at fault, so they get what they sow.
Lol!! Da empress flipped, I was so scared for HH. I thought he was stabbed by the silver pin.
This series should be named "the hen pecked husband is a virtuous emperor" our lioness is awesome!!!
This is my favorite chapter from this novel XD, must be the fourth time I've read through it, thank you for translating this beautiful piece of literature!
QBL is just like QYS (or was it QSY the younger sister given to Rui) they both think that if it wasn't for QQJ's existence they would be living a good life now. Such delusional women. If it wasn't for QQJ I don't think HH would even be Emperor. At first he didn't peg me to be that ambitious of the throne.


text:
Securing Our Digital Democracy
Lessons on cybersecurity and our democratic systems from Alex Stamos at Code for America Summit
Since leaving the technology industry, former Facebook and Yahoo Chief Security Officer Alex Stamos has become a leading voice and researcher in new and emerging threats to democracy in a fully digital age. His candid talk on the mainstage covered what's keeping him up at night when it comes to securing our digital democratic systems and suggestions on what we can do to protect them.
"We in tech said, 'We are going to change the world.' We deploy these incredibly powerful technologies and gave a huge amount of individual choice to people without thinking through all the consequences."
Alex spoke about the advantages and disadvantages that the United States faces when it comes to conflicts in cyberspace. While we may have the best offensive capabilities on the planet, the openness of our society and the dependency of our economy on technology puts us at risk. Our free and open elections are the bedrock of our democracy, but also make us particularly vulnerable to cyber attacks. And our hugely powerful tech companies are facing the issue of public legitimacy. These companies have hard equities that need to be balanced: between privacy and safety, and between authenticity and the ability to give someone the power of anonymous speech.
"We need to figure out how to measure whether or not our products are making the world a better place, and not just do people like them."
While Alex's expertise is in subjects that err towards "doom and gloom," he found a more hopeful atmosphere at Summit: one that comes from being in a room full of people dedicated to making government work better, to making our democracy fairer and more representative, and to putting their talents to good use. Because to protect our digital democracy, government, the tech sector, and citizens will need to work together.
To hear the presentation in full, watch the video or read the transcript below.
Hey everybody. Appreciate Jen, Dan, and Corey inviting me to be the Cassandra to follow her uplifting speech there. But if you're good at a tone, then why play against type, right? It's probably just better to stick with what I'm good at, which is doom and gloom. Being a little facetious. There's a lot of wonderful things going on. I'm really glad to be here because it's great to be among a group of people who are dedicated to making our government work better, to making our democracy more representative and fairer for our citizens, and to be with people who are motivated by the right goals. This is not a group of people who are here to make money or optimize ads, which are the groups I'm often around. This is a group of people who want to put their talents to good use.
So I thought I could talk a little bit, though, about some of the risks we are facing from a cybersecurity perspective when it comes to our society overall, and specifically our elections, because there's a lot of things that we're going to need your help on. A lot of things that we've got to work on. The first thing I want to say is that the unfortunate fact is that conflict in cyberspace is naturally something that reduces the advantages the United States has built over our adversaries over the last hundred years. Now, that being said, the United States has undoubtedly the best offensive capabilities on the planet. When you look at the offensive capabilities of groups like NSA and Cyber Command, especially combined with our allies in the Five Eyes, the United States is undoubtedly the 800 pound gorilla when it comes to offensive capability, but the use of those capabilities in the US is actually relatively restrained versus elsewhere in the world.
There's a funny irony here that, people that hack on behalf of the United States are socialists, and people that hack in Russia and China are capitalists. And that if you are hacking on behalf of the national interest of the United States, you're a civilian employee of NSA, you're a military employee of NSA or Cyber Command, or you work for a defense contractor like a General Dynamics or a Raytheon, but you're being paid by the American taxpayer. Where a number of countries have created economic incentives for groups to form to build their own capabilities to fund themselves, who then do hacking on behalf of their national interests as a side project, as a exchange for getting cover and protection from the government. So that puts us at a natural disadvantage in that those capabilities that we have paid for as American taxpayers are rightfully reserved for a much more narrow set of goals than perhaps are seen as national goals and other countries.
Another issue is the nature of our society and our economy makes us vulnerable. We have the world's most technology dependent economy, and the result of that is that makes it much less robust against this kind of attack. So an example of people who were robust against attack was Ukraine. There was a massive attack against the Ukrainian power grid in 2015, almost certainly by elements of the Russian military, and 230,000 people lost power. But they only lost power for six hours and it's because the Ukrainian power grid, while modern in some ways, is still at its base, a massively over engineered, Soviet-era, analog system, and so they could unplug all of their computers and go throw these huge physical switches and turn the power back on. We don't have that here. If the power went out here in Oakland, PG&E doesn't have a huge Soviet switch from 1957 that they can go flip. We have built our power grid in the name of efficiency and control in a way that is completely dependent upon computer control.
We're also uniquely vulnerable to information warfare because of the openness of our society. I want to just do a quick review of what happened in 2016 from my perspective and why it was specifically targeting United States. So there's four main areas of Russian information warfare in the 2016 election.
The first was the social media hacking, the online troll campaign, which took advantage of the fact that we have social networks that don't license people. You don't have to show an ID to get a SIM and to create an account on a social media network like some other countries do. We allow people to have open and free discussion on our social media sites and we also have a very vibrant discussion in which groups of people can push political ideas that were once considered out of bounds of acceptable political discourse, have now become centered in our discourse, and it's those specific movements, for example, in 2016, the Black Lives Matter movement, that were hijacked by the Russians and tried to be subverted into a tool against our own democracy. So the openness we give people to find and to push these issues that are really important to them, also makes us vulnerable.
There was the hack and leak campaign by a different part of Russia, in this case, the GRU, the main intelligence directorate, and that was a direct technical attack. There was a core cybersecurity issue in the security of DNC systems, and the fact that a number of people who used to be super important, who are still super important, but they used to be in the government and protected by professional IT folks, are now just grandfathers with their own Gmail accounts, that those people are legitimate targets for military actors. That is the cyber component, but the real amplification of the hack and leak campaign, the ability for the Russians to change the entire conversation around the election, was really a vulnerability that is part of our free press, and that we don't have a National Secrets Act (although it looks like the DOJ is going to try to backdoor one in through the Espionage Act attack against Julian Assange, which is another issue) but for the most part, we don't enforce national security laws against legitimate journalists.
Every single day we read important stories from leaked information, and every once in awhile, we read really important story that comes from information that was highly classified and we don't put journalists in jail for that and we don't make them give up their sources. But the fact that we have this open and free information economy and that we protect the press in United States also creates this fundamental vulnerability, that if you have hacked information, you can strategically leak little parts of that in a way that changes the overall conversation, with the knowledge that the American media ecosystem, that some part of it will cover the story and that means the entire ecosystem will eventually have to cover that.
We had direct attacks against our election infrastructure, which relies on the fact that we have a distributed federal system. There's something like 10,000 authorities who run election systems in the United States and that is very difficult for us to protect.
Then the fourth category is what is called gray propaganda, which is the fact that we don't license the press and we don't disallow foreign press from operating the United States. So groups like RT and Sputnik are allowed to be on cable stations at Best Westerns across the country, because we don't control whether or not the media, even if we know it's controlled by US adversaries, whether or not they're allowed to have the freedom of speech that we give to our own media. So this overall issue is a set of fundamental issues in American democracy, that those fundamental weaknesses are the weaknesses that we cherish, that we call strengths. I think we should see them as strengths, but it also means that there is a fundamental vulnerability in our system that we're going to have to be really careful about.
The other big issue, obviously that we're going through is the US tech industry is going through a real crisis of public legitimacy. We in tech said, we are going to change the world. We didn't really believe it when we said that. That's usually a phrase that's used to get venture capital or get 22 year olds to sign up at a recruiting fair, but it turns out maybe we should have listened a little better to when we said we'd change the world. We deploy these incredibly powerful technologies and gave a huge amount of individual choice to people without thinking through all the consequences. There are some legitimately hard problems here, and there's some really hard equities that need to be balanced. Equities between privacy and safety, equities between authenticity and the ability to give people anonymous or pseudo-anonymous speech. But the fact is that the companies who are balancing this are doing so generally in a manner that is not open to debate or transparent and they're acting in a pseudo-governmental manner, without the trappings of legitimacy that we often see from democratic governments.
The other issue here is that, until recently, the major tech companies have resisted responsible regulation. Clearly these multinational companies, it is difficult for them to say that they want to be regulated by any one country, but the fact that they have successfully resisted regulation in the United States, has built a huge back pressure elsewhere in the world for regulation that is perhaps not that well perceived. So one of the things that I've learned over the last couple of years is that people in the media, people in government, react really well to the truth and people saying, instead of everything is great and wonderful, which is the standard line you'll hear from any major tech company is to say, the world's actually a difficult place. We operate at a scale at which we know that there will be hundreds of thousands or millions of people who are going to use our product and who are going to try to use it for some kind of harm.
Harm that varies from abusive individuals all the way up until trying to ferment religious violence or celebrate mass casualty attacks, that all of these harms are real things. We're going to deal with it and we want to have a partnership in doing so.
Then the third major weakness we have right now is the United States does not really have a national cyber security strategy or a rational institutional framework. We have hyper competent offensive units in the United States, like I said, and the NSA and Cyber Command are very, very good at what they do. But all of their incentives are lined up on the offensive side. So you don't get the secret photo of you shaking Obama's hand in the Oval Office because you patched 100,000 servers. You get that photo because you blew up Iranian centrifuges. So the entire incentive structure for our offensive units is on the offensive side.
That is reflected in how the government handles vulnerabilities it finds, what it does after those vulnerabilities leak out to the public, and the fact that these units spend a lot of their intellectual time and capital on finding new bugs, more than they do on defending the US. We have defensive units in DHS. They're highly focused on critical infrastructure, which is great. It's also difficult to say that they have the breadth and depth of experience that we have in the offensive side in places like NSA. So effectively, we have the FBI as our defensive cyber security agency in the United States. There's a lot of great people at the FBI who are very, very smart and talented. But as law enforcement, they are trained to watch something bad happen, take very detailed notes, and to indict those people two years later.
That is a model that doesn't work in the cybersecurity world. It is not a model that allows you to lean into these threats and to have a whole of government response. So unfortunately the people who were supposed to pull this all together in the White House got fired. There's some hyper really competent people. So as a result, we don't really have a national strategy of how we're going to deal with these issues. So some things I think we need to think about as a country and an industry and some things we could work on together.
First, tech industry needs to get our shit together, to be honest. We need to think adversarially in everything we do. So whenever we're building a product, we have to educate ourselves in the ways that those products or similar products were misused in the past and prevent those kinds of known bad things.
We also have to think a little more creatively about how they could possibly use in the future. There's an interesting parallel here between what we're going through in the last couple of years and what happened in the run up to 9/11. If you read the 9/11 report, there's a section all about institutionalizing imagination, that there is a lack of imagination in the defensive units in the government and of thinking about the different things they would do if they were a adversary that had the same lack of resources that the US adversary did at the time, al Qaeda. We need to do the same thing in tech and that we've got to be much more imaginative about putting ourselves in the place of the people who want to cause harm with our products and to think and use our imaginations about how that might happen.
We need to figure out how to measure whether or not our products are making the world a better place and not just do people like them. Effectively, most of the things that tech companies measure to design their products come down to customer satisfaction. Do you like this product? So there's another product that has really good short term customer satisfaction. It's called heroin. If you give somebody a shot of heroin and you asked them 30 seconds later, are you enjoying my product? The answer is inevitably yes, if they can talk at all.
The customer satisfaction does not capture the actual impact of that product on that person or the society overall. That is one of our fundamental issues in Silicon Valley, is we don't have the metrics to measure whether or not we're actually making people better informed, whether they're happier after they use our product, not just addicted to using it in the short term. That short term metric focus is one of the things that got us into these troubles. We need to have predictability and transparency in these decisions we make. We're making really important decisions in tech. Decisions like what is acceptable boundaries of parody and satire in political speech in the United States? That is an incredibly difficult issue, and those decisions need to be made in a very transparent and rational manner, so that people get a little bit of predictability, that it's not just a crapshoot of what decisions can be made in the secret board room on any of these issues.
When we do make these decisions, we need to really base them upon the individual rights and safety, and we need to have a good foundation of when companies decide to restrict the freedom of individuals, they're doing so for a definable impact. Not because it helps the company with PR in the short term, but because it actually makes the community healthier or it helps keep people safe in the longer term. We in tech need to figure out some kind of creative way to finance journalism, especially local journalism, in the long run. There's a lot of issues have led to the decline of local journalism, but the fact is that we're at a place where a huge amount of money has been diverted into the tech industry, and that is having real impacts on the ability of people to stay educated. I'm really afraid that we're going to move to a world where you pay for good journalism and crap journalism is free and that is a really bad world for our democracy.
It's something that is not in the best interest of these companies. It's great for billionaires to go buy outlets and to support them for awhile. I'm glad that some billionaires are doing that, but that is not a sustainable solution to figure out a way to share in some of this revenue. We need tech to work with Congress on some reasonable regulatory framework. So I don't have time to go into a lot of the details here, but my colleagues at Stanford and I are publishing 108 pages. So in academia, that's considered a short, white paper. I'm getting used to how much people can write in the weekend in academia. So we have the short beach read, 108 pages, on the election coming out on June 6th, and we are going to be looking at all kinds of areas of attack against the election.
It is effectively a followup on the Mueller report, with recommendations of how do we deal with the facts of what happened that Mueller laid out? So we'll have a lot more detail in there, but some of the things that I'd like to preview now is one thing we need is we need reasonable federal US privacy regulation.
We need a federal privacy regulator who is competent and has the ability to interpret laws to make them easy to use. One of the real challenges for people trying to operate in Europe right now is that nobody really knows what European privacy law says, because it has to be interpreted by 28 different regulators and those regulators are loath to give you a get out of jail free card before there's all the litigation that they want to go through over the next 10 years. That's a really bad thing for startups. If you're a 20 person company, you shouldn't have to have a full time privacy lawyer on staff, trying to decide how the privacy law applies to every single decision. We need to have a privacy law that gives people predictability, so that they can make the right decisions that are privacy preserving, that are good for users, but that they also have the confidence that when they make those decisions that they will not be punished if the goalposts change later.
We need cyber security legislation that helps small and medium businesses. We talk all the time about these attacks against the big companies. But the truth is the big companies have to take care of themselves. There's a whole host of organizations in United States who are completely screwed if they go up against one of these adversaries. That is a serious issue that we don't talk about enough. Around elections, we need to pass something like the Honest Ads Act, with a lot more teeth around non-electionary ads. So this is something we go into great detail in the report, but one of the challenges we have right now is the interference by folks like the Russians, is not actually classified as political advertising under current US law because it doesn't say vote for or vote against. It is about issues. So we need to control that issue advertising, restricted its hyper targeting of it, have real transparency requirements. There've been changes here, but those changes have all been voluntary, and they've only been done by a handful of companies. So we need to bring some kind of standard to the entire advertising ecosystem.
As companies ask for regulation, we need them to do so compliant with a consistent human rights framework. That's one of the real fears I have about what's going on right now, is that pivoting to push all of these problems and decisions on the governments means that a number of governments that are not respectful of human rights are also going to be making some of these decisions. So when we say there needs to be more regulation, we need to put an asterisk on that, saying more regulation compliant with basic human rights standards. Because we don't want the things that we do in the US and other developed democracies to be used as an excuse to turn the internet into a tool of oppression globally.
In the US, we need to have a defensive cyber security agency that just does defense and no law enforcement. Other democracies have this. In Germany to have this thing called the BSI, in France, they have a group called Anssi, but the truth is that, in other countries there are coordinating bodies that have the ability to plan ahead and to coordinate an all of government response on all of these issues.
We need to work towards international norms that are better. The truth is, is we're in this weird world where nation states can do things to each other that, if done in the physical world, would be considered an act of war, but in the online world, we just consider it a Tuesday. That norm is really, really dangerous. It's really dangerous for us as a country, but it's dangerous for citizens globally, because the number of attacks that are done by countries against their own citizens is going up continuously, and allowing these norms to exist where you can hack journalists, hack activists, hack students, and that's just considered a normal part of managing the democracy. That is a completely inappropriate norm to exist.
So there's some things that we've got to do in academia and we're working on this. We need to change how we educate people. So one of the things we're doing at Stanford is we're teaching cybersecurity classes for non CS majors, because we need to have people from all kinds of backgrounds and disciplines in the computer security world to help inform these decisions. These are not just technical problems, they are sociology problems, psychology problems, political science problems. We're also building tools to study these issues. We're building this thing called the Internet Observatory, and the goal there is to build the technology tools and processes necessary to support good public policy and social science research. We're educating the next generation. We have a Masters in cybersecurity policy at Stanford now. So a number of people here might be interested in making a career pivot. This is a great opportunity for both current undergrads and mid career professionals to study both the technical and the policy sides of cybersecurity.
We have the tools and we have the talent, to quote my favorite '80's movie. We have to execute properly and we need to execute as a society overall. One of the things that really makes me afraid right now is that it seems that everybody is lined up in their corners and we can't have that. We need civil society, we need citizens, corporations, and government all working on this together, because in the long run, making the United States secure against cyber attack is not just good for us, but it will set good standards for how people around the world are treated, and for the protection of democracies against autocrats. We really, are war against creeping autocracy. We've got to keep that in mind when we make these decisions. So hopefully we can find ways to work together on this because I'm really afraid that if we don't put all of our best people behind it, we are going to fall behind.
That being said, groups like this make me confident that we're on the right path, so I want to thank you guys. I want to thank Jen and the rest of the team for having me here. I hope you have a great Summit. Thank you very much.
Summit 2023 Call for Proposals
Announcing Code for America Summit 2023
Leading the Field: Kelly Benton


text:
Set of configurations used to alter the usual behavior of sending the request. These can be defined in a collection at Item or ItemGroup level which will be inherited if applicable.

These are used to remove the dependency of agent specific configurations to make collection behave similarly irrespective of the agent used to execute the collection.

The behavior can be protocol specific or requester options. Currently supported protocol profile behaviors are:

- `strictSSL: Boolean`

Enable or disable certificate verification.

- `followRedirects: Boolean`

Follow HTTP 3xx responses as redirects.

- `maxRedirects: Number`

Set maximum number of redirects to follow.

- `disableBodyPruning: Boolean`

Control request body pruning for following methods: ```GET, COPY, HEAD, PURGE, UNLOCK```

- `disableUrlEncoding: Boolean`

Disable percent encoding of auth, path, query and fragment URL segments.

- `disabledSystemHeaders: Object`

Disable system headers which get added implicitly.

```javascript
{
    'cache-control': true,
    'postman-token': true,
    'content-type': true,
    'content-length': true,
    'accept-encoding': true,
    'connection': true,
    'host': true
}
```

- `insecureHTTPParser: Boolean`

Use an insecure HTTP parser that accepts invalid HTTP headers.

- `followOriginalHttpMethod: Boolean`

Redirect with the original HTTP method, by default redirects with HTTP method GET.

- `followAuthorizationHeader: Boolean`

Retain `authorization` header when a redirect happens to a different hostname.

- `removeRefererHeaderOnRedirect: Boolean`

Removes the `referer` header when a redirect happens.

- `tlsPreferServerCiphers: Boolean`

Use the server's cipher suite order instead of the client's during negotiation

- `tlsDisabledProtocols: Array`

the SSL and TLS protocol versions to disabled during negotiation

- `tlsCipherSelection: Array`

Order of cipher suites that the SSL server profile uses to establish a secure connection

**A collection with protocol profile behaviors:**

```javascript
{
  "info": {
    "name": "protocolProfileBehavior",
    "schema": "https://schema.getpostman.com/json/collection/v2.1.0/collection.json"
  },
  "item": [{
    "name": "GET request with body using disableBodyPruning",
    "request": {
      "method": "GET",
      "body": {
        "mode": "raw",
        "raw": "foo=bar"
      },
      "url": "localhost:3000"
    },
    "protocolProfileBehavior": {
      "disableBodyPruning": true
    }
  }],
  "protocolProfileBehavior": {
    "followRedirects": false,
    "strictSSL": true
  }
}
```

redpajama_set_name: RedPajamaGithub