#ifndef __REDIR_TCP_H__
#define __REDIR_TCP_H__

#include "cstring.h"
#include "redirect.h"
#include "utility.h"

class CRedirectTCP;
class CRedirectTCP_XmitThread;
class CRedirectTCP_RecvThread;

class CRedirectTCP_RecvThread : public CRedirectBase
{
public:
	CRedirectTCP_RecvThread();
	virtual void StartRedirect();

	CRedirectTCP			*m_pRedirTCP;

	CSocket					 m_sClientSocket;
	CSocket					 m_sServerSocket;
};

class CRedirectTCP_XmitThread : public CRedirectBase
{
public:
	CRedirectTCP_XmitThread();
	virtual void StartRedirect();

	CRedirectTCP_RecvThread	*m_pRedirTCP_Recv;
};

class CRedirectTCP : public CRedirectBase
{
public:
	CRedirectTCP();
	virtual void StartRedirect();

	CSocket					 m_sListenSocket;
};

#endif // __REDIR_TCP_H__


Because of the wide TV-like screen, the classroom overlooking Lewisohn lawn might feel a bit like a living room, if it weren't for the16 students and two professors in front of it. Here in 308 Lewisohn, technology plus traditional lecture equals a new kind of learning experience for students, and a creative but surprisingly natural teaching experience for professors.

308 Lewisohn is one of the newest smart classrooms on the Columbia campus and the only one created by the Columbia Center for New Media Teaching and Learning (CCNMTL). It is an experimental classroom designed to help faculty use technology to enrich the learning process by engaging students in analytic and creative situations. Here the technology of fortune 500 boardrooms is at work for the benefit of education and everything, from wireless access to the easy-to-move tables, is an element added to bring endless possibilities to teaching and learning.

"The purpose of the experimental digital classroom is have a laboratory for testing new classroom technologies that is also part of the living enterprise of the University," said Frank Moretti, executive director of CCNMTL. "We have been able to learn directly from our faculty collaborators the efficacy of different technologies as they are being used in classroom practice."

Peter Awn, religion professor and dean of the School of General Studies (GS), and Marc Meyer, adjunct professor of religion, both experienced in teaching with multimedia, were among the faculty to sign up when the CCNMTL smart room opened last semester. In their team-taught, seminar-style GS class, Ancient and Classical Traditions, they used the smartboard to access multimedia resources, online documents and interactive tools that were built for outside class use as study environments or resources. The technology did not dominate the classroom or dwarf the professors, it was just another tool.

"What we are not doing is asking students to give up their books, but rather to use these resources to enhance their studies. And they work so well together to accomplish that goal," said Awn. "By using technology in this course we can address one of the great drawbacks of studying texts from diverse time periods and cultures out of historical context. Web resources become study environments in which to locate these texts and ideas historically."

Easy and unobtrusive, the smart board enhanced the teaching and learning experience in the course by displaying existing multi-media resources that apply to the texts studied so that the entire class could view them and comment together. The teaching and learning process equally engaged the professors/guides and the students/knowledge explorers, who were able to creatively use the resources before them.

Marc Meyer and John Frankfurt use technology to team-teach a GS colloquium

"To make a student-centered class, you can't come up with an agenda and then resist deviating from it," said Meyer. "Peter and I are of the opinion that even though they don't have a wealth of experiential knowledge, that doesn't mean that students shouldn't be involved in developing the curriculum. To use technology in the classroom in this manner is to discover the process of designing curriculum in the moment."

Awn and Meyer enhanced the natural rhythm of the class using the technology. While Awn speaks about the influence of St. Augustine on the Judeo Christian tradition, Meyer might be tracing invisible lines on the smartboard screen to scroll through the online multimedia study environment On Christian Doctrine that he helped CCNMTL to build. There he could quickly access texts that applied to the day's discussion topic. Meyer might find a video clip from a BBC production that related to the discussion on St. Augustine, click on it and ask the students to comment on how it applies to what they have read. Or the smartboard might be entirely forgotten, as the natural discourse between several students carried the discussion in a completely new direction.

But at any given point, Meyer could scroll through an online resource, find a passage from an ancient text that applies to the discussion and ground it again in text by inviting the students to read and comment on the text together.

"The purposeful use of technology should be as seamless a process as possible," said Meyer. "It is as if you have a chef in the kitchen who doesn't think about the technology assisting him or her, but just uses it intuitively. It takes a time commitment and practice to get there, but the benefits to the student are tangible because you can see in the middle of a discussion how it can enhance understanding of the subject."

In Awn and Meyer's classroom, the smart board was used in a variety of ways. The group could also post or access different resources on the same subject together on the screen and easily compare them in class. Or video, audio, image and text could be pulled up as a point of departure for conversation, which helped to engage student interest through the sight and sound.

In addition to content-rich CCNMTL-developed resources, such as On Christian Doctrine, Awn and Meyer also used the smart board to employ CCNMTL-developed tools to enhance learning. Third Space, an online work space where video clips can be annotated and shared over the Internet, enabled students to view video of Joseph Campbell's acclaimed series The Power of Myth, and post on the class bulletin board responses, comments, and even their favorite clip demonstrating a theme discussed in class. Both professors and students also used the CU Analyzer, a tool that enables the user to easily compile a multimedia presentation or product by pulling items off the Internet and organizing them for presentation. Professors used the tool to organize and prepare discussions with multi-media resources and students used it to prepare their final class project.

"It is almost like my class now has a desktop," said one faculty member in an evaluation of the space last semester. "I go to class and I bring my briefcase, now my briefcase is in the classroom and everyone can look at it."


2004 Tour Down Under

The 2004 Tour Down Under, known as Jacob's Creek Tour Down Under for sponsorship reasons, was the sixth edition of the Tour Down Under stage race. It took place from 20 January to 25 January  in and around Adelaide, South Australia and was the first major stage race of the season. The race was won by Patrick Jonker, who rode for Team UniSA.

Stages

Stage 1
20 January 2004 – Adelaide - Adelaide,

Stage and General Classification after Stage 1

Stage 2
21 January 2004 – Norwood to Kapunda,

Final standings

General classification

Points Classification

King of the Mountains classification

Young Riders' classification

References

Category:Tour Down Under
Tour Down Under
Tour Down Under
Category:2004 in Oceanian sport


Q:

How can I convert the value that more than 999 number in PHP Excel as number

I'm stuck with this for a few days about PHPExcel.
Why when it comes to 1000 the value convert to text automatically?
But It's normal if in the cell containing the number 0-999.
I have my code and the result below.
Thanks in advance.

A:

Please try this
$objPHPExcel->getActiveSheet()->getStyle('H'.$row)->getNumberFormat()->setFormatCode('#,##0.00');


Atlanta’s ace, Julio Teheran, has a career 3.34 ERA, a good mark even for this pitching-friendly era. This year’s numbers, at least in some ways, are the best of his career. He’s a 25-year-old with less than $30 million in guaranteed money remaining on his contract, which runs through 2020 if you count the low-cost team option for that season. If the Braves make him available before the deadline, he’d instantly become one of the most valuable assets on the market.

But is Teheran an ace, like the Braves are likely to price him? Or is he more of a good pitcher on a nice run? Since assuming regular major-league duties in 2013, he’s 44th in pitching Wins Above Replacement despite being 11th in innings pitched, as he’s accrued value by staying healthy and racking up innings, but not dominating in the traditional walk, strikeout, and home run categories.

But this year, once again, Teheran is dealing, and he’s doing so without strikeout, walk, or home-run rates that would appear on the first page of each leaderboard. For teams hoping to acquire the Braves ace, they’ve got to be wondering who he will be on their team; the guy who looks great by ERA or the guy who looks like an innings-eater by FIP?

It’s instructive to look at his ranks in a few categories. Let’s start with the main factors included in the more popular ERA estimators. In the interest of smoothing over some of the peaks and valleys of his career, let’s establish 2013 as the beginning point for this investigation.

Julio Teheran, Ranked In a Few Key Categories Stat Value Rank K% 21.5% 42 BB% 6.7% 61 K-BB% 14.8% 42 HR/9 1.07 96 GB% 37.8% 125 Since 2013, qualified starters, n=140.

Teheran has made it this far with slightly above-average strikeout and walk rates, but considering that his ERA over that time frame ranks 30th, the peripherals don’t match the outcomes. So let’s venture further afield. There are plenty of stats that inform projection systems which don’t show up in all of our ERA estimators, so let’s throw these into the mix and see if something pops.

Julio Teheran, Ranked in Ancillary Categories Stat Value Rank FB Velo 90.9 90 PU% 4.3% 22 BABIP 0.271 9 Soft% 17.2% 81 Ideal Angle Exit Velo 92.4 93 Since 2013, qualified starters, n=140.

PU% = IFFB*FB, or pop-ups divided by balls in play. Average = 3.4%

Ideal launch angles (10-30 degrees) are best for homers and line drives.

Now here are a couple things that stand out. Like pop ups. And batting average on balls in play. He’s been good at those things, and they can drive a good ERA.

The spread on pop ups isn’t very large, though. First-best (Chris Young) is separated from 140th-best (Jaime Garcia) by 6.5 percentage points. Still, the pop-up figure helps explain Teheran’s small BABIP number. If we gave Teheran a league-average pop-up rate since 2013, we’d have to put 68 more hits on his ledger, and his BABIP would soar to .301, or basically league average.

Even though the spread isn’t huge, pop-up rate (pop ups divided by all balls in play) is decently sticky year to year. It has a 0.63 year-to-year correlation, which is much better than the 0.49 number for home runs. In other words, you would make a safer bet to bet on Teheran to continue getting more pop ups than if you bet on him to continue giving up more homers than average. Re-rank Teheran for strikeouts plus pop ups minus walks, and there he is at 32nd in our sample, in line with his ERA. Just outside the top 30 again, though.

On the other hand, given that Teheran has below-average velocity for a righty and lives high in the zone for those pop ups, it’s probably also true that he will continue giving up homers. Derek Carty found that it takes 2.7 years for home-run rate to stabilize, and we’ve got those years for Teheran, too. You can’t ignore the home runs.

And let’s not forget that pop ups are now included in pitching WAR as an automatic out, equal to a strikeout. So a FIP-based WAR isn’t ignoring the pop-ups; he’s getting credit for those.

But when you look at FIP alone, you see the huge gap, with a 54 point gap between his ERA and FIP, thanks in part to all those popups. But how much of that gap should we expect to continue going forward?

Let’s look at pitchers similar to Teheran based on this ability to induce pop-ups. The top 10% in pop-ups since we started tracking the stat in 2007 have had ERAs that were 1% better than league average, while their FIPs have been 5% worse. That’s not Teheran-like — his career gap is 15 percentage points — but let’s look at the guys that have had 600-inning FIP and ERA gaps with great pop-ups, and what they did during and after that stretch.

There are 22 pitchers that have achieved three years of an FIP-ERA gap over 0.4 with a pop-up rate over 4%. Ten of them appeared on the list once and never returned. But we can look at the 19 not named Julio Teheran, Noah Lowry, or Wei-Yin Chen and see how they faired the next three years after they first showed up on the list.

Pop-Up Heavy FIP-Beaters, Before and After Time Frame Ave IP Weighted PU Weighted ERA Weighted FIP Weighted Diff During Stretch 598 5.0% 3.68 4.24 -0.56 After 446 4.8% 4.22 4.44 -0.22 Minimum 600 IP during FIP-beating stretch.

Minimum -0.4 ERA-FIP.

Minium 4% pop-up rate.

n in both buckets = 19

Would you look at that. The pitchers got worse, but, as a group, they maintained a noticeable gap between their ERA and FIP. And of course they were going to get worse — they got older! Some of that regression was simply age-related decline, while Teheran is still young. But it is worth noting that they didn’t sustain the same size gap that they posted during their initial run.

The pitchers that fell apart and had a worse ERA than FIP were either near the end of their careers (Al Leiter, Russ Ortiz, and Hideo Nomo) or started to lose their ability to get pop-ups at the rate that would include them in the sample (Ramon Ortiz, Randy Wolf, and Jeremy Hellickson).

Jake Peavy is probably the best comp for Teheran on the list, once you consider age, quality, and arsenal. He had a 3.69 FIP, 3.13 ERA, and a 4.2% pop-up rate built mostly on good command and a great slider. Matt Cain is another good one (3.43 ERA, 3.86 FIP, 4.6% pop-ups). Cain and Peavy both both had stellar ERAs in their next three seasons, which provides a lot of hope for Teheran boosters. There are examples on both sides of the argument.

It is worth mentioning that those pitchers remained in their own home parks for those next three seasons, though. And that the last Brave righty with similar stats didn’t survive a park change, or hasn’t so far. Shelby Miller (3.27 ERA, 3.87 FIP, 4.0% pop-ups) has had mechanical problems, and his arsenal is fairly different, but he makes our ‘before’ list, and the after doesn’t look great.

As long as Teheran maintains his ability to get the pop-up, he should be fine, even if maybe not quite as good as his ERA would suggest. And if you look at pitchers most like Julio Teheran — pitchers with a three-year demonstrated ability to get the pop-up — they’re at least a good bet for the next three years. Even more so when you consider the Braves righty is only 25 years old. Here’s the whole table of comparable pitchers.

If we assume that Teheran’s youth will allow him to regress less than most of these guys, maybe his true-talent ERA gap is something like 0.3 runs per nine innings, and with a ~3.90 FIP projection, that would put him as a 3.60 ERA guy going forward, in the NL anyway. It’s a little closer to 4.00 in the AL, once you adjust for league quality and the DH.

That’s not a dominating frontline starter, but given Teheran’s youth and controllable years, he remains a very valuable piece. He’s not likely to keep beating his FIP by this degree, but he is likely to keep beating it by a decent margin, and that makes him a quality mid-rotation arm. A team shouldn’t pay ace prices to land Teheran based on his ERA, but he’s a good pitcher on a great contract, and so the line for his services will be quite long.


Youri Tielemans was in the headlines last night after scoring a stunning strike for Belgium’s U21s against Spain.

From a good 30 yards out, Tielemans let fly with a wicked strike that kissed the roof of the net and gave Spain’s goalkeeper absolutely no chance of stopping it.

As far as technique goes, it was as good a strike as you’re likely to see, which is all the more impressive when you consider Tielemans is just 17.

There’s no doubt that plenty of scouts from European teams will be keeping a closer eye on Tielemans after his goal last night but back in 2013, he made it clear which club he had an affection towards as well as Anderlecht.

“It’s my goal to be in the senior squad this season,” he said.

“Are Anderlecht really my dream club? Sure, though I also love Manchester United.”

Back in April, Tielemans also pulled off a superb through ball to show off his passing range:

Tielemans is considered a fine prospect and became the youngest Belgian player to play in the UEFA Champions League when he made his debut at the age of 16 years, 4 months and 25 days, which is a testament to how highly rated he is.

Timothy Fosu-Mensah was the first youngster Louis van Gaal brought into the club but with goals like that, Tielemans will certainly keep himself in the limelight.

Image: dhnet


John Barrowman injured in Jack and the Beanstalk pantomime horse fall Published duration 5 January 2013

media caption John Barrowman talks about his love of panto during rehearsals for Jack and the Beanstalk in December

Actor John Barrowman has been injured during a performance of Jack and the Beanstalk at a Glasgow theatre.

The Doctor Who and Torchwood star was taken to hospital from the SECC Clyde Auditorium on Friday as a precaution. His injuries were not serious.

Barrowman had fallen off a live horse during the matinee show, which also stars the Krankies.

The Glasgow-born actor returned to the stage for a matinee performance on Saturday afternoon.

Speaking ahead of Saturday's show, Barrowman told BBC Scotland he was "pleased to be back" but was still not feeling 100%.

He added: "I just felt the horse jitter underneath me, then the back end lifted up and then the next thing I knew I was in the air. I landed hard in the middle of the stage.

"I got back on with the show, but the shooting pain just went through my legs and my back. I said to the audience "ladies and gentlemen, I have to stop the show, I need to be checked out".

"Then I went round the back and basically collapsed and I was put in a stretcher and strapped up. It was quite scary."

Barrowman's understudy - his nephew Greg Barrowman - stepped in to finish the show.

Janette Tough, "Wee Jimmy" Krankie, was almost killed in an on-stage accident in the city in 2004.

She suffered a fractured skull after falling from a mechanical beanstalk at the Pavilion Theatre.

Jack and the Beanstalk is due to end its run on Sunday.


Q:

Direct2D - Nothing is being drawn to the screen, something wrong in the way I'm handling arrays in C++

Hello everyone and thank you for looking. This is a follow up to the original question posted here.
I have a base class that I define thusly:
class DrawableShape
{
public:
    virtual HRESULT DrawShape(ID2D1HwndRenderTarget* m_pRenderTarget)
    {
        return S_OK;
    }
};

I have two classes that extend this class, both are similar, so I'm listing one:
class MyD2DEllipse : public DrawableShape
{
private:
    D2D1_ELLIPSE data;
public:
    MyD2DEllipse();
    HRESULT DrawShape(ID2D1HwndRenderTarget* m_pRenderTarget);
};

The DrawShape function is implemented like this:
HRESULT MyD2DEllipse::DrawShape(ID2D1HwndRenderTarget* m_pRenderTarget)
{
    HRESULT hr = E_FAIL;
    ID2D1SolidColorBrush* m_pBrush;
    hr = m_pRenderTarget->CreateSolidColorBrush(
                D2D1::ColorF(D2D1::ColorF::OrangeRed),
                &m_pBrush
                );
    m_pRenderTarget->DrawEllipse(&data, m_pBrush, 10.f);
    return hr;
}

I want to draw a random number of ellipses and rectangles to the screen, so I first find out those random numbers, create an array of DrawableShape with that size (since I can't allocate objects dynamically in C++), replace the parent objects with the child objects, and then call the draw function on the array, randomly again. Here's what my code looks like:
    HRESULT Demo::OnRender()
    {
        HRESULT hr = S_OK;
        hr = CreateDeviceResources();

        if (SUCCEEDED(hr))
        {
            m_pRenderTarget->BeginDraw();
            m_pRenderTarget->SetTransform(D2D1::Matrix3x2F::Identity());
            m_pRenderTarget->Clear(D2D1::ColorF(D2D1::ColorF::White));

            // Decide on # of primitives
        randEllipse = 1 + (rand() % 5);
        randRectangle = 1 + (rand() % 5);
        totalPrimitives = randEllipse + randRectangle;

        DrawableShape *shapes;
        shapes = new MyShape[totalPrimitives];

        for (int i=0; i<randEllipse; i++)
        {
            MyEllipse ellipse1;
            shapes[i] = ellipse1;
        }
        for (int i=randEllipse; i<(randEllipse + randRectangle); i++)
        {
            MyRectangle rect1;
            shapes[i] = rect1;
        }

        for (int i=0; i<totalPrimitives; i++)
        {
            hr = shapes[i].DrawMyShape(m_pRenderTarget);
        }
        hr = m_pRenderTarget->EndDraw();
    }
}

That should've worked, but it doesn't. Also, after writing this out, I realize that I'm better off creating the array in some sort of init function, and calling the draw on the array in the OnRender function. Please help!!

EDIT: Okay I've got the shapes working with pointers, the problem is the construction of the array. So I have something like this:
    MyD2DRectangle rect1;
    MyD2DEllipse ell1;

    DrawableShape *shape1 = &rect1;
    DrawableShape *shape2 = &ell1;
    shape1->DrawShape(m_pRenderTarget);
    shape2->DrawShape(m_pRenderTarget);

That seems to work by itself. How can I create the array of DrawableShape without slicing?

A:

shapes is an array of MyShape instances.  When you say shapes[i] = ellipse1; or shapes[i] = rect1;, you are losing the subclass data as part of this assignment, which is known as slicing in C++.
As such, each call to shapes[i].DrawMyShape(m_pRenderTarget); is just returning S_OK as defined in MyShape.
In order to properly use polymorphism in C++, you need to use pointers or references to MyShape instances (usually allocated using new).  If you are not allowed to do this (homework?), then you need to find a way to do this without polymorphism.


Mirror’s Edge Catalyst’s City of Glass will Expand as You Progress Through the Story

Mirror’s Edge Catalyst’s senior producer Sara Jansson, in an interview with Xbox One News, revealed that Catalyst will not be be completely open from the beginning. The City Of Glass will rather be expanding as we progress through the story and unlock new gadgets.

Jansson explains:

“It’s an expanding city that will open up more and more as you move through the story, a critique of the first game was that it was a little bit too difficult for a lot of people or a bit too frustrating, and we believe that just throwing players into a huge city to do what they want would have been a little intimidating, especially as our gameplay is pretty different from many other games.”

The City of Glass will expand as players progress through the story

In Mirror’s Edge Catalyst, you will experience the rise of Faith, a daring Runner who uncovers a hidden truth within the walls of the city of Glass. Through her journey, Faith will explore every corner of the city, from the highest, most pristine skyscrapers, to the dark and gritty tunnels that lie beneath. Players will need to master the environment, using their momentum and timing their combat to fight the oppression of the ruling elite, uncovering the dark secrets behind the city’s beautiful façade. Mirror’s Edge Catalyst gives players the freedom to explore the entire city, discovering a variety of new activities, like ‘Dash’, races from point A to B, environmental puzzles and story missions to complete at their own pace.

Mirror’s Edge Catalyst removes the ability to use guns

The first Mirror’s Edge, while it did not rely on gunplay, the player was given the option to use guns if the opportunity present itself. Mirror’s Edge Catalyst, takes away that option.

“We’ve completely removed that aspect of the game. You can’t even pick them up.”

EA and developer DICE have created a new combat system that emphasizes melee combat, and Jansson continues:

“Faith will fight, but she doesn’t kill,”

“Now the fighting is more of an extension of the movement. It builds on the flow. It fits a lot better with the game mechanics. When Faith is in flow, when she stitches together move after move without failing and keeps her momentum going, she’s actually invulnerable to bullets. That can keep her out of harm’s way. It’s only when she stops that she can get hurt.”

Mirror’s Edge Catalyst will be available on February 23, 2016, at retail and for digital download on Xbox One, PlayStation 4, and Origin for PC.


<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE catmetadata SYSTEM "http://www.gentoo.org/dtd/metadata.dtd">
<catmetadata>
	<longdescription lang="en">
		The net-fs category contains network filesystem packages.
	</longdescription>
	<longdescription lang="de">
		Die Kategorie net-fs enthült Netzwerkdateisysteme.
	</longdescription>
	<longdescription lang="es">
		La categoría net-fs contiene paquetes de sistemas de ficheros en red.
	</longdescription>
	<longdescription lang="ja">
		net-fsカテゴリーにはネットワーク・ファイルシステムのパッケージが含まれています。
	</longdescription>
	<longdescription lang="nl">
		De net-fs categorie bevat netwerk-bestandssystemen.
	</longdescription>
	<longdescription lang="vi">
		Nhóm net-fs chứa các gói hệ tập tin mạng.
	</longdescription>
	<longdescription lang="it">
		La categoria net-fs contiene pacchetti per i filesystem di rete.
	</longdescription>
	<longdescription lang="pt">
		A categoria net-fs contém pacotes para sistemas de arquivos de
		rede.
	</longdescription>
	<longdescription lang="pl">
		Kategoria net-fs zawiera pakiety związane z NFS, sieciowym systemem
		plików.
	</longdescription>
</catmetadata>


Oxidation of Abeta and plaque biogenesis in Alzheimer's disease and Down syndrome.
The processes involved with beta-amyloid (Abeta) degradation and clearance in human brain are not well understood. We hypothesized that the distribution of oxidatively modified Abeta, as determined by an affinity-purified antibody in the entorhinal and frontal cortices of Alzheimer's disease (AD), Down syndrome (DS), nondemented elderly control cases, and canine brain, would provide insight into the mechanisms of Abeta accumulation. Based upon plaque counts, oxidized Abeta was present within 46-48% of diffuse and primitive plaques and 98% of cored plaques. Dense punctate deposits of oxidized Abeta were distributed throughout the neuropil in AD and DS brains but were also present within controls with mild neuropathology and isolated cognitive impairments. Confocal studies indicate that punctate oxidized Abeta deposits were within activated microglia. Oxidatively modified Abeta may reflect the efforts of microglial cells to take up and degrade Abeta. Oxidative modification of Abeta may be an early event in Abeta pathogenesis and may be important for plaque biogenesis.


By the tender age of 13, I considered myself to be quite a metal buff: I had already determined my favorite band, Avenged Sevenfold, live at the Vans Warped Tour. Unfortunately, my parents were reluctant to send their teen headfirst into a mosh pit. But after putting my head into the latest live event virtual reality apps using both the Oculus Go and Samsung Gear VR headsets, I’m already convinced that this particular form of teen angst will soon be a thing of the past.

Two companies have already made it their mission to take customers on tour with their favorite artist or get courtside with their favorite home team without having to set foot outside (or shell out thousands of dollars). For a subscription or access fee and the cost of a VR headset, MelodyVR and NextVR will teleport you from crowd surfing at a 2Chainz show to being ten feet away from Ariana Grande.

Steven Hancock, COO of MelodyVR, tells Inverse they could introduce a product that ensures the metalhead youths of the future won’t miss out on the show of a lifetime.

MelodyVR: Bringing You on Tour

The first app Inverse tested was MelodyVR, an effort launched by former music industry types and technologists who have spent the last few years selling artists and record labels on the potential of VR while building their content library (they also needed to figure out how to unobtrusively shoot 3D footage at live shows). The result is an impressive an iTunes-like platform for purchasing VR live concerts (or individual songs a la carte). The service launched in earnest in May.

“Whether it’s a sold-out show, the tickets are too expensive, a minor wants to see it, or an artist simply doesn’t go on tour in a certain part of the world, we want MelodyVR to bridge that gap,” MelodyVR’s Hancock says.

As soon I pulled the Oculus Go headset over my eyes, I was struck by the attention to detail. The surreal, immersive white menu screen evoked the Memory Corridor from the Assassin’s Creed franchise or the “Construct” in the Matrix.

The obvious difference was that instead of recreating my memories or summoning firearms, I could choose to jump into, say, a recording of an Imagine Dragons concert. MelodyVR has secured deals with three major record labels, giving users access to a plethora of pre-recorded performances and even private acoustic shows. But the best part about it isn’t the wide selection — the app currently offers shows from 40 different artists ranging from Cyprus Hill to the London Symphony Orchestra — it’s the views.

Indeed, our favorite part of MelodyVR was its 360-degree view that enables multiple viewing angles. This means you can toggle between hovering above the audience, standing on stage, and sitting front row. These are all perspectives that even the most seasoned concert-goer won’t have access to at a single performance.

But can watching a pre-recorded concert in VR really replace the energy of actually attending a concert? Maybe not entirely – especially not the sweating – but MelodyVR is sure as hell trying. Throw in a pair of eardrum-busting headphones and a few plastic-cup filled beers and you’ll feel a lot closer to a live music experience than you’d ever thought possible while chilling in your living room.

This year they also plan to launch a live-streaming service, that would virtually transport users to concerts as they’re happening live. This way fans that couldn’t make it to the venue don’t have to wait for it to be uploaded onto the app. Down the line, they’re going to begin letting users tune into live or past shows with their friends in an attempt to recreate the feel of a road trip to a concert.

NextVR: Taking You Courtside

But while MelodyVR is focusing on getting raving fans in front of their heartthrobs, NextVR is capitalizing on getting diehard sports fans in front of their idols.

MelodyVR can flex its deals with music labels, but NextVR has dominated the sports broadcasting field. The California-based company has partnerships with the NBA, NFL, the International Champions Cup, and yes, even the WWE to give subscribers a first-person view of some of the world’s most exclusive sporting events.

Certain tickets to the NBA Finals cost more than what I would make in three years, but the NextVR app let me watch the final minutes of 2018’s Game 4 from court side and right above the rim. As with MelodyVR, NextVR is also experimenting with ways to leverage the technology to give fans access they couldn’t hope for otherwise.

After the game was done and dusted, for example, NextVR enabled me to follow the teams into their locker rooms. I got to watch LeBron James give a sullen interview and Stephen Curry pop a bottle of champagne, both of whom seemed like they were within a foot away from me. Normally you’d need to wait for a post-game interview to hear your favorite athlete’s thoughts, but watching it in VR lets you see their emotions as they unravel behind the scenes.

This works for all of the sports NextVR has to offer, but there is still one missing detail that keeps this product from being as immersive as it could be: the lack of the 360-degree view. MelodyVR is already making headway toward giving its experience a sense of the crowd’s energy, but no matter what angle you’re looking at the court, field, or pitch you’ll only be able to see a 180-degree camera angle.

That said, NextVR is trailblazing in live broadcasted games, while MelodyVR is still working to get this feature up and running. Last summer, NextVR offered a selection of live soccer matches to tune into during the International Champions Cup — a series of friendly club soccer games.

As VR technology becomes cheaper to access and companies continue to improve what they’re able to capture and broadcast to users’ headset, we could see a future where you can catch your favorite band on tour from your couch. This way, if VR ever becomes as ubiquitous as smartphones, every performance or sporting event would be just a click away.


Intrusion sensors are widely used in security systems to monitor the boundaries of a well defined area in order to detect the presence, location and motion of people and vehicles. Exemplary sites that may benefit from the use of such security systems are national borders, boundaries of military installations, nuclear power plants, prisons, and businesses. A number of existing intrusion sensors based on seismic, ultrasonic, infrared, and magnetic technologies have been employed, but are unfavorable for numerous reasons. These existing systems are expensive, difficult to conceal, have high false alarm rates, and are capable of providing coverage for only a limited portion of a perimeter.
By way of illustration, in the particular application of intrusion sensing to the monitoring of national borders to detect and apprehend illegal drug smugglers or illegal aliens, the border terrain under surveillance may be vast and rugged. The sensor system employed in this application must not only detect an intrusion but must also be able to determine the location of the intrusion along the monitored border. The sensor system must also be easily concealable to prevent tampering. Furthermore, such a system must have and maintain a record of low false alarms. Conventional intrusion sensing systems are not able to meet such stringent requirements or cannot provide satisfactory performance without incurring prohibitive costs.
Accordingly, it is desirable to provide an intrusion sensing system which provides intrusion detection as well as the location of intrustion. It is further desirable that such sensing system be deployable in a reasonable manner over a vast area, such as for the purpose of monitoring a national border.


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The 24-year-old actor, who found his breakout role – or roles, since he was playing twins Cameron and Tyler Winklevoss – in “The Social Network,” tells PopcornBiz he’s on the brink of closing the deal to portray the early Federal Bureau of Investigation’s number two man.

“We are in, I'd say, final negotiation phases for doing Clint Eastwood's new project 'J. Edgar,'" says Hammer, “about J. Edgar Hoover who founded the FBI, and it's very exciting. I believe Leonardo Di Caprio will be playing J. Edgar Hoover, and I'm in talks to play Clyde Tolson, who is his best friend and confidant and, some say lover, through his life.”

Hammer says he’s always been fascinated by the old school FBI, “this sort of private, militarized detective force that was almost run by one man. How did it get that powerful? I guess he held onto the power by having blackmail on everybody.”

He says he’s thrilled at the prospect of working with the iconic Eastwood. “I couldn't think of another director that I'd be happier going to after working with David Fincher, to be perfectly honest. I think he's incredible, and he has a career that I could not venerate or want to emulate more.”

As for DiCaprio: “To be honest, he’s one of the best actors of our generation, so to get to work with him - I hope I don't look like an idiot! Ever since 'What's Eating Gilbert Grape' I've, like, studied that guy.”

Since catching attention in “The Social Network,” Hammer reveals that his own social networking has been dramatically reduced.

“After the movie came out I just started to get bombarded with Friend requests from people and messages – a lot of people I didn't know. So I just canceled my Facebook after I finished. It died a noble death. Facebook and I broke up.”


/**
 * Copyright (c) 2000-present Liferay, Inc. All rights reserved.
 *
 * This library is free software; you can redistribute it and/or modify it under
 * the terms of the GNU Lesser General Public License as published by the Free
 * Software Foundation; either version 2.1 of the License, or (at your option)
 * any later version.
 *
 * This library is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
 * details.
 */

package com.liferay.ide.server.ui.cmd;

import com.liferay.ide.server.remote.IRemoteServerWorkingCopy;

import org.eclipse.osgi.util.NLS;

/**
 * @author Gregory Amerson
 */
public class SetLiferayPortalContextPathCommand extends RemoteServerCommand {

	public SetLiferayPortalContextPathCommand(IRemoteServerWorkingCopy server, String liferayPortalContextPath) {
		super(server, Msgs.setLiferayPortalContextPath);

		this.liferayPortalContextPath = liferayPortalContextPath;
	}

	public void execute() {
		oldLiferayPortalContextPath = server.getLiferayPortalContextPath();

		server.setLiferayPortalContextPath(liferayPortalContextPath);
	}

	public void undo() {
		server.setLiferayPortalContextPath(oldLiferayPortalContextPath);
	}

	protected String liferayPortalContextPath;
	protected String oldLiferayPortalContextPath;

	private static class Msgs extends NLS {

		public static String setLiferayPortalContextPath;

		static {
			initializeMessages(SetLiferayPortalContextPathCommand.class.getName(), Msgs.class);
		}

	}

}


Q:

How to load Zend 2 classes on autoloader

Previously, i am using zend 1.x. I am able to autoload zend classes using the below code.
// autoload class from Zend Lib
require_once ABSPATH.'/classes/Zend/Loader/Autoloader.php';
$loader = Zend_Loader_Autoloader::getInstance();
 try{
// database connection
$dbo = Zend_Db::factory('pdo_mysql', array(
        'host'     => DB_HOST,
        'username' => DB_USER,
        'password' => DB_PW,
        'dbname'   => DB_PREFIX.DB_NAME
    ));
$dbo->getConnection();
// save database adapter for easy usage in other classes
Zend_Db_Table::setDefaultAdapter($dbo);
Zend_Registry::set('db', $dbo);

}catch(Zend_Db_Adapter_Exception $e){
print $e;
 //header("Location: http://www.google.com/error/");
}

I am upgrading to zend 2 as the classes might be better. May i know how do i autoload them?

A:

If you are just using ZF2 as a standalone library without employing the full MVC framework, then autoloading is fairly straightforward:

Make sure that the Zend directory is on your php include_path.
Push an autoloader using spl_autoload_register

The following is essentially what Zend\Loader\StandardAutoloader::loadClass() does when functioning as a fallback autloader:
spl_autoload_register(function($class) {
    $f = stream_resolve_include_path(str_replace('\\', '/', $class) . '.php');
    if ($f !== false) {
        return include $f;
    }
    return false;
});

This would use the PSR-1 autloading mechanism for all classes, not just the Zend classes.
Alternatively, you could just do the following:
require_once 'Zend/Loader/StandardAutoloader.php';
$autoloader = new Zend\Loader\StandardAutoloader(array(
    'fallback_autoloader' => true,
));
$autoloader->register();

As above, this will apply PSR-1 autoloading to all classes. If you want this mechanism for the Zend classes only, then pass 'fallback_autoloader' => false.


This invention relates to an automatic frequency control circuit for synchronizing the horizontal deflection current in a television receiver with the horizontal synchronous signal contained in a video signal, such as a broadcast color television signal or a video signal from a video tape recorder.
Conventional automatic frequency control (AFC) circuit includes a voltage-controlled oscillator (VCO) whose frequency is controlled in accordance with a phase error of a flyback pulse (FBP) generated from the horizontal deflection current, with respect to the horizontal sync signal (HSYNC) signal. When the AFC circuit is formed of digital circuits, the phase error is detected with a resolution of the operating clock period. That is, the phase error smaller than the operating clock period could not be detected or corrected.
One way of increasing the accuracy of the phase error detection is to increase the frequency of the operating clock. However, to attain an accuracy adequate for horizontal synchronization in a television receiver, the frequency of the oscillating clock needs to be extremely high, and producing such a high frequency is practically impossible.
Another problem relates to compromise between the response and stability. To increase the response (promptness) in eliminating the phase error, the time constant of an integrator included in the AFC circuit may be reduced. This however lowers the stability, and the AFC circuit is susceptible to jitter or noises, and the stability of the displayed image is lowered.


---
abstract: 'We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle ${{\mathfrak E}}$ on a compact Kähler manifold, with structure group a connected linear algebraic reductive group $G$, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.'
date: Revised 29 June 2014
title: |
    Approximate Hermitian-Yang-Mills structures\
    on semistable principal Higgs bundles
---

SISSA preprint 03/2014/mate

[^1]

[Ugo Bruzzo]{}\
Scuola Internazionale Superiore di Studi Avanzati,\
Via Bonomea 265, 34136 Trieste, Italia;\
Istituto Nazionale di Fisica Nucleare, Sezione di Trieste\
E-mail: [bruzzo@sissa.it]{}\
[Beatriz Graña Otero]{}\
Departamento de Matemáticas, Pontificia Universidad Javeriana,\
Cra. 7$^{\hbox{\tiny \rm ma}}$ N$^{\mbox{\tiny\rm o}}$ 40-62, Bogotá, Colombia\
E-mail: [bgrana@javeriana.edu.co]{}

Introduction
============

The notion of slope stability introduced by Mumford and Takemoto [@MF; @Take] allows one to construct good moduli spaces of vector bundles (or, more generally, of coherent sheaves) on projective varieties. These moduli spaces usually are separated schemes of finite type, and, if suitable conditions are satisfied, they are fine, e.g., they carry universal objects on them. The definition of slope stability makes sense also on a compact Kähler manifold, and one can construct corresponding moduli spaces also in this case. However, for bundles on a compact Kähler manifold, there is also a differential-geometric analogue of slope stability, namely, the existence of a bundle metric which satisfies a certain condition, called the Hermitian-Yang-Mills condition: the mean curvature of the Chern connection singled out by the metric is a (constant) multiple of the identity endomorphism [@koba; @Lub83; @Don85; @Don87]. Actually, it turns out that, for an irreducible bundle, stability is equivalent to the existence of an Hermitian-Yang-Mills metric: this is the celebrated Hitchin-Kobayashi correspondence. A similar situation prevails in the presence of a Higgs field, for some other kinds of decorated bundles [@S88; @S92; @Ban; @Sch], in the case of principal bundles [@RamaSubra] and principal Higgs bundles [@AnBis; @BisSchu].

The weaker property of semistability is equivalent to the existence of an Hermitian-Yang-Mills metric in an approximate sense. Let us clarify the relation between (semi)stability for principal or vector bundles on compact Kähler manifolds, with or without Higgs fields, and the existence on these bundles of (approximate) Hermitian-Yang-Mills structures. After realizing that the sheaf of sections of a holomorphic Hermitian (ordinary) vector bundle satisfying the Hermitian-Yang-Mills condition is polystable (i.e., it is a direct sum of stable sheaves having the same slope) [@koba; @Lub83], a converse result was proved. First Donaldson in the projective case [@Don85; @Don87], and then Uhlenbeck and Yau in the compact Kähler case [@UY], showed that a stable bundle admits a (unique up to homotheties) Hermitian metric which satisfies the Hermitian-Yang-Mills condition. One can also show that if an Hermitian bundle satisfies the Hermitian-Yang-Mills condition in an approximate sense, then it is semistable, while the converse was proved in [@koba] for $X$ projective and in [@Jacob12] in the compact Kähler case.

Later Simpson [@S88; @S92] proved a Hitchin-Kobayashi correspondence for Higgs vector bundles. Given a Higgs bundle equipped with an Hermitian metric, one defines a natural connection (that we call the *Hitchin-Simpson connection*); when the latter satisfies the Hermitian-Yang-Mills condition, then the Higgs bundle is polystable, and *vice versa*. Our main aim is to provide a suitable definition of approximate Hermitian-Yang-Mills structure for principal Higgs bundles and show that whenever a Higgs principal bundle satisfies an approximate Hermitian-Yang-Mills condition, it is semistable, and [*vice versa.*]{}

In this paper, after reviewing some basic definitions and properties, we give appropriate definitions of the notions of semistability and approximate Hermitian-Yang-Mills structure for principal Higgs bundles. In particular, one says that a principal Higgs $G$-bundle admits an approximate Hermitian-Yang-Mills structure if for every value of a positive real parameter $\xi$, there is a reduction $\sigma_{\xi}$ of the structure group to a maximal compact subgroup such that the mean curvature of the Hitchin-Simpson connection given by the reduction approximates a central element in the Lie algebra of $G$ up to $\xi$. Then one shows that a principal Higgs $G$-bundle admits an approximate Hermitian-Yang-Mills structure if and only if it is semistable. A sketch of the proof of this result is as follows. If the principal Higgs bundle ${{\mathfrak E}}=(E, \phi)$ admits an approximate Hermitian-Yang-Mills structure, one shows that the same holds for the adjoint Higgs bundle $\Ad({{\mathfrak E}})$; the latter then is semistable [@BG2], so that ${{\mathfrak E}}$ is semistable as well. The proof of the converse result relies on an analysis of the flow of the Donaldson functional defined on the space of Hermitian metrics on the Higgs bundle $\Ad({{\mathfrak E}})$, showing that the flow preserves the condition that an Hermitian metric on $\Ad({{\mathfrak E}})$ comes from a reduction of the structure group of ${{\mathfrak E}}$ to a maximal compact subgroup $K$ (this implements in the case of principal Higgs bundles the ideas used in [@BisJacStem] for principal bundles, but we provide here a more detailed description of this technique).

Some related work has been done in [@BBJS], where the existence — in a suitable sense — of a limit Hermitian-Yang-Mills connection on unstable Higgs principal $G$-bundles is established.

Preliminaries {#Sec:2}
=============

We give the basic definitions concerning Higgs bundles. Let $X$ be an $n$-dimensional compact  Kähler manifold, with Kähler form $\omega$. Unless otherwise stated, $X$ will be assumed to be connected.

Given a coherent sheaf $F$ on $X$, we denote by $\deg(F)$ its degree $$\deg(F) = \int_X c_1(F)\cdot \omega^{n-1}$$ and if $r=\rk(F)>0$ we introduce its *slope* $$\mu(F) = \frac{\deg(F)}{r}.$$

A Higgs sheaf ${{\mathfrak E}}$ on $X$ is a coherent sheaf $E$ on $X$ endowed with a morphism $\phi \colon E \to E \otimes \Omega_X^1$ of ${\mathcal O}_X$-modules such that the morphism $\phi\wedge\phi : E \to E
\otimes \Omega^2_X$ vanishes, where $\Omega_X^i$ is the sheaf of holomorphic $i$-forms on $X$. A Higgs subsheaf $F$ of a Higgs sheaf ${{\mathfrak E}}=(E,\phi)$ is a subsheaf of $E$ such that $\phi(F)\subset F\otimes\Omega_X^1$. A Higgs bundle is a Higgs sheaf ${{\mathfrak E}}$ such that $E$ is a locally-free ${\mathcal O}_X$-module.

There exists a stability condition for Higgs sheaves, analogous to that for ordinary sheaves, which makes reference only to $\phi$-invariant subsheaves.

A Higgs sheaf ${{\mathfrak E}}=(E,\phi) $ on $X$ is semistable (resp. stable) if $E$ is torsion-free, and $\mu(F)\le \mu(E)$ (resp. $\mu(F)< \mu(E)$) for every proper nontrivial Higgs subsheaf $F$ of ${{\mathfrak E}}$. Finally, ${{\mathfrak E}}$ is polystable if it is a direct sum of stable Higgs sheaves with the same slope.

Connections and fibre metrics {#Sec:0}
-----------------------------

Let ${{\mathfrak E}}= (E, \phi)$ be a Higgs bundle over a compact Kähler manifold $X$ equipped with an Hermitian fibre metric $h$. Then there is on $E$ a unique connection $D_{(E,h)}$ which is compatible with both the metric $h$ and the holomorphic structure of $E$ [@Ati]. This is often called the *Chern connection* of the Hermitian bundle $(E,h)$.

Now let $\bar\phi$ be the adjoint of the morphism $\phi$ with respect to the metric $h$, i.e., the morphism $\bar\phi: E
\rightarrow \Omega^{0,1}_X\otimes E $ such that $$h(s, \phi(t)) =
h(\bar\phi(s), t)$$ for all sections $s,t$ of $E$. It is easy to check that the operator $$\label{e:simpsonconnection}
{\mathcal{D}}_{({{\mathfrak E}},h)} = D_{(E,h)} + \phi + \bar\phi$$ defines a connection on the bundle $E$. One should notice that this connection is neither compatible with the holomorphic structure of $E$, nor with the Hermitian metric $h$.

\[simpsonconnection\] The connection is called the *Hitchin-Simpson connection* of the Hermitian Higgs bundle $({{\mathfrak E}},h)$. Its curvature will be denoted by ${\mathcal R}_{({{\mathfrak E}},h)} = {\mathcal{D}}_{({{\mathfrak E}},h)}
\circ {\mathcal{D}}_{({{\mathfrak E}},h)}$.

If ${{\mathfrak E}}$ is a Higgs line bundle the notion of Hermitian flatness coincides with the usual one.

We shall denote by $\mathcal{K}_{({{\mathfrak E}},h)}\in\End(E)$ the mean curvature of the Simpson connection. This is defined as usual: if we consider wedging by the Kähler 2-form $\omega$ as a morphism $\mathcal A^p\to
\mathcal A^{p+2}$ (where $\mathcal A^p$ is the sheaf of ${\mathbb C}$-valued smooth $p$-forms on $X$), and denote by $\Lambda: \mathcal A^p\to
\mathcal A^{p-2}$ its adjoint, then $\mathcal{K}_{({{\mathfrak E}},h)}= i
\Lambda {\mathcal R}_{({{\mathfrak E}},h)}$. We shall regard the mean curvature as a bilinear form on sections of $E$ by letting $$\mathcal{K}_{({{\mathfrak E}},h)}(s,t)= h(\mathcal{K}_{({{\mathfrak E}},h)}(s),t)\,.$$

Let us recall, for comparison and further use, the form that the Hitchin-Kobayashi correspondence acquires for Higgs bundles [@S92 Thm. 1].

A Higgs vector bundle ${{\mathfrak E}}=(E, \phi)$ over a compact K[ä]{}hler manifold is polystable if and only if it admits an Hermitian metric $h$ such that the mean curvature of the Hitchin-Simpson connection of $({{\mathfrak E}},h)$ satisfies the Hermitian-Yang-Mills condition $$\mathcal{K}_{({{\mathfrak E}},h)}= c \cdot \Id _E$$ for some constant real number $c$.

The constant $c$ is related to the slope of $E$: $$\label{c}
c=\frac{2n\pi}{n! \vol(X)} \mu(E)$$ where $n=\dim(X)$ and $$\vol(X) = \frac{1}{n!}\int_X \omega^n\,.$$

Approximate Hermitian-Yang-Mills structures and semistability
-------------------------------------------------------------

As we already noted, and in analogy with the case of ordinary bundles, the semistability of a Higgs bundle may be related to the existence of an approximate Hermitian-Yang-Mills structure, which is introduced by replacing the Chern connection by the Hitchin-Simpson connection.

We say that a Higgs bundle ${{\mathfrak E}}=(E, \phi)$ has an *approximate Hermitian-Yang-Mills structure* if for every positive real number $\xi$ there is an Hermitian metric $h_{\xi}$ on $E$ such that $$\vert \mathcal{K}_{({{\mathfrak E}},h)}-c \cdot \Id _E\vert  < \xi \,.$$

Here $\vert\,\vert$ is the norm defined on a self-adjoint endomorphism of $E$ as $$\vert \psi \vert^2  = \operatorname{max}_{X} \operatorname{tr} \psi^2\,.$$ The constant $c$ is again given by equation .

The following result was proved in [@BG2].

\[ss\] \[approHYM\] A Higgs bundle ${{\mathfrak E}}=(E, \phi)$ on a compact Kähler manifold admitting an approximate Hermitian-Yang-Mills structure is semistable.

The converse implication was proved in [@Cardona-I] when $\dim(X)=1$, and in [@LiXi12] for arbitrary dimension. More exactly,

[@LiXi12] \[LiXi\] If ${{\mathfrak E}}=(E, \phi)$ is a semistable Higgs bundle on a compact Kähler manifold, then it admits an approximate Hermitian-Yang-Mills structure.

Evolution equation and Donaldson functional
-------------------------------------------

A way of proving Theorem \[LiXi\] is to use a version of the so-called Donaldson functional for Higgs bundles. This was introduced by Simpson in [@S88] and was further studied in [@Cardona-I; @Cardona-II; @LiXi12]. Let us denote by $\mathcal H^+(E)$ the space of Hermitian metrics on the vector bundle $E$. This is a Hilbert manifold modelled on the space $\mathcal H(E)$ of Hermitian endomorphism of $E$ suitably completed to an $L^2$ Hilbert space. After fixing a reference metric $k$ in $\mathcal H^+(E)$ one defines a functional on $\mathcal H^+(E)$ by letting $${\mathcal L}(h,k)=\int_X \left[ \left(i \int_0^1 tr(h_t{^{-1}}\partial h_t \cdot \mathcal R_t)dt \right)- \frac{c}{n} \log(\det(k^{-1}h)\,\omega \right]\wedge
\frac{ \omega^{n-1}}{(n-1)!}$$ where $c$ is the constant in Equation , and $h_t$ is any smooth curve in $\mathcal H^+(E)$ joining $h$ to $k$; the value of the functional turns out to be independent from the choice of this curve. Note that here $ \mathcal R_t$ is the curvature of the Hitchin-Simpson connection associated with the metric $h_t$ and the Higgs field $\phi$. The flow of this functional (the “Donaldson heat flow”) $$h_t^{-1}\partial_t h_t =-\nabla {\mathcal L}=  -(\mathcal K_t -c \, \operatorname{Id}_E)$$ defines a parabolic partial differential equation which for any initial datum $h_0$ in $\mathcal H^+(E)$ has a unique smooth solution $h_t$ defined for all $t\ge 0$.

Collecting results from [@S88; @LiXi12] one has:

\[Dona\] Let ${{\mathfrak E}}$ be a Higgs vector bundle over a compact Kähler manifold $X$ with Kähler form $\omega$, and let $h_t$ be the solution of the Donaldson heat flow with reference metric $k$ and initial condition $h_0$. Then:

1.  the function ${\mathcal L}(h_t,k)$ is a monotone decreasing function of $t$;

2.  if ${\mathcal L}(h_t,k)$ is bounded from below, then $\lim_{t\to+\infty} \vert \mathcal K_t - cI_t\vert = 0$;

3.  if ${{\mathfrak E}}$ is semistable, ${\mathcal L}(h_t,k)$ is bounded from below.

This Theorem implies Theorem \[LiXi\]. Moreover, if ${{\mathfrak E}}$ is stable, this can be used to prove that ${{\mathfrak E}}$ admits an Hermitian-Yang-Mills metric.

Semistable principal bundles {#Sec:2}
----------------------------

We recall some basics about principal bundles, notably a definition of (semi)stability for principal bundles (basic references about this topic are [@Rama; @BalSes]).

Let $X$ be a compact Kähler manifold and $G$ a reductive algebraic group over ${\mathbb C}$. A principal $G$-bundle $E$ on $X$ is a complex manifold with a free action of $G$ such that $E/G \simeq X$ and the projection $\pi:E \to X$ is locally isotrivial (locally trivial up to an étale cover).

If $\rho\colon G \to \operatorname{Aut}(Y)$ is a representation of $G$ as automorphisms of a variety $Y$, we may construct the associated bundle $E(\rho)=E\times_\rho Y$, the quotient of $E\times Y$ under the action of $G$ given by $(u,y)\mapsto (ug,\rho(g^{-1})y)$ for $g\in G$. If $Y={{\mathfrak g}}$ is the Lie algebra of $G$, and $\rho$ is the adjoint action of $G$ on ${{\mathfrak g}}$, one gets the adjoint bundle of $E$, denoted by $\Ad E $. Another important example is obtained when $\rho$ is given by a group homomorphism $\lambda\colon G\to G'$; in this case the associated bundle $E'=E\times_\lambda G'$ is a principal $G'$-bundle. We say that the structure group $G$ of $E$ has been extended to $G'$.

If $E$ is a principal $G$-bundle on $X$, and $F$ a principal $G'$-bundle on $X$, a morphism $E\to F$ is a pair $(f,f')$, where $f'\colon G\to G'$ is a group homomorphism, and $f\colon E\to F$ is a morphism of bundles on $X$ which is $f'$-equivariant, i.e., $f(ug)=f(u)f'(g)$. Note that this induces a vector bundle morphism $\tilde f\colon \Ad E \to\Ad(F)$ given by $\tilde
f(u,\alpha)=(f(u),f'_\ast(\alpha))$, where $f'_\ast\colon{{\mathfrak g}}\to{{\mathfrak g}}'$ is the morphism induced on the Lie algebras. As an example, consider a principal $G$-bundle $E$, a group homomorphism $\lambda\colon G\to G'$, and the extended bundle $E'$. There is a natural morphism $f\colon E\ \to E'$ defined as $f=\operatorname{id}\times\lambda$ if we identify $E$ with $E\times_GG$.

If $K$ is a closed subgroup of $G$, a *reduction* of the structure group $G$ of $E$ to $K$ is a principal $K$-bundle $F$ over $X$ together with an injective $K$-equivariant bundle morphism $F \to E$. Let $E(G/K)$ denote the bundle over $X$ with standard fibre $G/K$ associated to $E$ via the natural action of $G$ on the homogeneous space $G/K$. There is an isomorphism $E(G/K)\simeq E/K$ of bundles over $X$. Moreover, the reductions of the structure group of $E$ to $K$ are in a one-to-one correspondence with sections $\sigma\colon X \to E(G/K)\simeq E/K$.

We first recall the definition of semistable principal bundle when the base variety $X$ is a curve. Let $T_{E/K,X}$ be the vertical tangent bundle to the bundle $\pi_K\colon E/K \to X$.

Let $E$ be a principal $G$-bundle on a compact Riemann surface $X$. We say that $E$ is stable (semistable) if for every proper parabolic subgroup $P\subset G$, and every reduction $\sigma\colon X \to E/P$, the pullback $\sigma^\ast (T_{E/P,X})$ has positive (nonnegative) degree.

When $X$ is a higher dimensional variety, the definition must be somewhat refined; the introduction of an open dense subset whose complement has codimension at least two should be compared with the definition of (semi)stable vector bundle, which involves non-locally free subsheaves (which are subbundles exactly on open subsets of that kind).

Let $X$ be a compact Kähler manifold. A principal $G$-bundle $E$ on $X$ is stable (semistable) if and only if for any proper parabolic subgroup $P\subset G$, any open dense subset $U\subset X$ such that $\operatorname{codim}(X-U)\ge 2$, and any reduction $\sigma\colon U \to (E/P)_{\vert U}$ of $G$ to $P$ on $U$, one has $\deg \sigma^\ast (T_{E/P,X}) > 0$ ($\deg \sigma^\ast (T_{E/P,X}) \ge 0$). \[higher\]

One should note that a line bundle defined on an open dense subset of $X$, whose complement has codimension 2 at least, extends uniquely to the whole of $X$, so that we may consistently consider its degree. This is discussed in detail in [@RamaSubra], see also [@koba], Chapter V.

Principal Higgs bundles {#phb}
-----------------------

We switch now to principal Higgs bundles. Let $X$ be a compact Kähler manifold, and $G$ a reductive complex algebraic group.

A principal Higgs $G$-bundle ${{\mathfrak E}}$ is a pair $(E,\phi)$, where $E$ is a principal $G$-bundle, and $\phi$ is a global holomorphic section of $\Ad E \otimes\Omega^1_X$ such that $[\phi,\phi]=0$ in $\Ad E\otimes\Omega_X^2$.

When $G$ is the general linear group, under the identification $\Ad E \simeq
\End(V)$, where $V$ is the vector bundle corresponding to $E$, this agrees with the usual definition of Higgs vector bundle.

A morphism between two principal Higgs bundles ${{\mathfrak E}}=(E,\phi)$ and ${{\mathfrak E}}'=(E',\phi')$ is a principal bundle morphism $f\colon E\to E'$ such that $(f_\ast\times \operatorname{id})(\phi)=\phi'$, where $f_\ast\colon \Ad E \to\Ad(E')$ is the induced morphism between the adjoint bundles.

Let $L$ be a closed subgroup of $G$, and $\sigma\colon X \to
E(G/L)\simeq E/L$ a reduction of the structure group of $E$ to $L$. One has a principal $L$-bundle $F_\sigma$ on $X$ and a principal bundle morphism $i_\sigma\colon F_\sigma\to E$ inducing an injective morphism of bundles $\Ad(F_\sigma) \to \Ad E $. Let $\Pi_\sigma\colon  \Ad E \otimes
\Omega^1_X\to(\Ad E /\Ad(F_\sigma))\otimes \Omega^1_X $ be the induced projection.

A section $\sigma\colon X\to E/L $ is a [*Higgs reduction*]{} of $(E,\phi)$ if $\phi\in\ker \Pi_\sigma$.

When this happens, the reduced bundle $F_\sigma$ is equipped with a Higgs field $\phi_\sigma$ compatible with $\phi$ (i.e., $(F_\sigma,\phi_\sigma)\to (E,\phi)$ is a morphism of principal Higgs bundles).

Let us again consider the case when $G$ is the general linear group $Gl(n,{\mathbb C})$, and let us assume that $P$ is a (parabolic) subgroup such that $G/P$ is the Grassmann variety $\operatorname{Gr}_k({\mathbb C}^n)$ of $k$-dimensional quotients of ${\mathbb C}^n$. If $V$ is the vector bundle corresponding to $E$, a reduction $\sigma$ of $G$ to $P$ corresponds to a rank $n-k$ subbundle $W$ of $V$, and the fact that $\sigma$ is a Higgs reduction means that $W$ is $\phi$-invariant, i.e., $\phi(W)\subset W\otimes \Omega^1_X$.

We introduce a notion of semistability for principal Higgs bundles (which is equivalent to the one given in Definition 4.6 in [@AnBis]). We give it here when $X$ is a curve; this is generalized to higher dimensional varieties as one does for principal bundles, see Definition \[higher\].

\[Hstabcurv\] Let $X$ be a compact Riemann surface. A principal Higgs $G$-bundle ${{\mathfrak E}}=(E,\phi)$ is stable (resp. semistable) if for every proper parabolic subgroup $P\subset G$ and every Higgs reduction $\sigma\colon X\to E/P$ one has $\deg \sigma^\ast (T_{E/P,X})> 0$ (resp. $\deg \sigma^\ast (T_{E/P,X})\ge 0$).

Hermitian-Yang-Mills structures for principal Higgs bundles {#Sec:3}
===========================================================

We review some facts about connections on principal bundles, covering also the case when a Higgs field is present. Let $X$ be an $n$-dimensional Kähler manifold, $G$ a complex reductive linear algebraic group and $\pi: {{\mathfrak E}}\to X$ a principal Higgs $G$-bundle. Let $K$ be a maximal compact subgroup of $G$. Note that the Lie algebra ${{\mathfrak g}}$ of $G$ admits an involution $\iota$, called the *Cartan involution*, whose +1 eigenspace is isomorphic to the Lie algebra $\mathfrak k$ of $K$. If ${{\mathfrak E}}=(E,\phi)$ is a principal Higgs $G$-bundle, we may extend $\iota$ to an involution on the sections of the bundle $\Ad E \otimes\mathcal A^1$ (where $\mathcal A^1$ is the bundle of complex-valued smooth differential 1-forms) by letting $$\iota(s\otimes\eta) =- \iota(s)\otimes\bar\eta\,.$$

Given a reduction $\sigma$ of the structure group of $E$ to $K$, there is a unique connection $D_{\sigma}$ on $E$ which is compatible with the complex structure of $E$ and with the reduction [@KobaNomi2]. By analogy with the vector bundle case, we call it the *Chern* connection associated with the reduction $\sigma$. The Higgs field may be used to introduce another connection $$D_{\sigma,\phi} = D_\sigma + \phi + \iota(\phi)$$ which we call the *Hitchin-Simpson connection* of the triple $({{\mathfrak E}},\sigma)=(E,\phi,\sigma)$.

To state our results we need the notion of *polystable* principal Higgs bundle. We may associate with every character $\chi$ of $K$ a line bundle $L_\chi=E\times_\chi {\mathbb C}$ on $E/K$, where we regard $E$ as a principal $K$-bundle on $E/K$. We call $\mu_\sigma$, the slope of a reduction $\sigma$, the group homomorphism $\mu_\sigma\colon \mathcal X(K) \to {\mathbb Q}$ (where $ \mathcal X(K) $ is the group of characters of $K$) which to any character $\chi$ associates the degree of the line bundle $\sigma^\ast(L_\chi^\ast)$ [@Rama].

\[admi\] A reduction $\sigma$ of the structure group of $G$ of a principal Higgs $G$-bundle ${{\mathfrak E}}$ to a parabolic subgroup $P\subset G$ is said to be *admissible* if $\mu_\sigma(\chi)=0$ for every character of $\chi$ of $P$ which vanishes on the centre of $G$.

A principal Higgs $G$-bundle ${{\mathfrak E}}$ is said to be *polystable* if there is a parabolic subgroup $P$ of $G$ and a Higgs reduction $\sigma$ of the structure group of $E$ to a Levi subgroup $L$ of $P$ such that

1.  the reduced principal Higgs $L$-bundle ${{\mathfrak E}}_\sigma$ is stable;

2.  the principal Higgs $P$-bundle obtained by extending the structure group of ${{\mathfrak E}}_\sigma$ to $P$ is an admissible reduction of the structure group of ${{\mathfrak E}}$ to $P$ (cf. Definition \[admi\]).

Also in this case one has a *Hitchin-Kobayashi correspondence* [@AnBis]. We say that a reduction $\sigma$ of the structure group $G$ of a principal Higgs $G$-bundle ${{\mathfrak E}}$ to a maximal compact subgroup $K$ is *Hermitian-Yang-Mills* if there is an element $\tau$ in the centre $\mathfrak z$ of the Lie algebra ${{\mathfrak g}}$ of $G$ such that $$\mathcal K_{\sigma,\phi} = \tau$$ where $ \mathcal K_{\sigma,\phi} $ is the *mean curvature* of the Hitchin-Simpson connection.[^2]

[@BisSchu] \[HK\] A principal Higgs $G$-bundle ${{\mathfrak E}}$ is polystable if and only if it admits an Hermitian-Yang-Mills reduction to a maximal compact subgroup $K\subset G$.

This notion of polystability extends the one holding for Higgs vector bundles, i.e., a Higgs vector bundle is polystable if it is a direct sum of stable Higgs vector bundles having the same slope.

The proof of the following result is implicitly contained in [@AnBis].

A principal Higgs bundle is polystable if and only if its adjoint bundle is polystable. \[polyrem\]

A principal Higgs $G$-bundle ${{\mathfrak E}}$ over $X$ is said to admit an *approximate Hermitian-Yang-Mills structure* if there exist an element $\tau \in {{\mathfrak z}}$, where ${{\mathfrak z}}$ is the center of ${{\mathfrak g}}$, and for every $\xi > 0$, a reduction $\sigma_{\xi}$ of $G$ to $K$, such that $$|\mathcal K_{\sigma_{\xi},\phi}-\tau|< \xi.$$

Here the norm $\vert\,\cdot\,\vert$ on the sections of $\Ad E$ is defined as $$\vert\psi\vert= \operatorname{max}_{x\in X} \sqrt{\kappa(\psi,\iota(\psi))}$$ where $\kappa$ is an ad-invariant nondegenerate bilinear form on $\mathfrak g$ which extends the Killing-Cartan form on the semisimple part of $\mathfrak g$.

Existence of approximate Hermitian-Yang-Mills structures {#Sec:3}
========================================================

In this Section we prove the following result.

\[MainThm\] A principal Higgs $G$-bundle ${{\mathfrak E}}$ over $X$ is semistable if and only if it admits an approximate Hermitian-Yangs-Mills structure.

We start with some preliminary constructions. To deal with the approximate Hitchin-Kobayashi correspondence for ${{\mathfrak E}}$, it is convenient to associate a metric to every reduction $\sigma$ of the structure group $G$ of ${{\mathfrak E}}$ to a maximal compact subgroup $K$. To this end we consider the adjoint representation $\Ad\colon G \to Gl(\mathfrak g)$ and choose a maximal compact subgroup $\tilde K$ of $Gl(\mathfrak g)$ containing $\Ad(K)$. Note that the principal Higgs bundle obtained by extending the structure group of ${{\mathfrak E}}$ to $Gl(\mathfrak g)$ via the morphism $\Ad$ is isomorphic to the Higgs bundle of linear frames $L(\Ad {{\mathfrak E}})$ of the adjoint Higgs bundle $(\Ad E, \tilde{\phi})$, where $\tilde{\phi}$ is the Higgs map on $\Ad E $ via the morphism $(\Ad \times id)_\ast$. The reduction $\sigma$ induces an hermitian metric $h_\sigma$ in $\Ad E$. The adjoint morphism $\Ad$ also induces a morphism $\Ad\colon \Ad E_\sigma\to \End(\Ad E)$, and therefore also a morphism $\Ad \colon\Omega^2( \Ad E_\sigma)\to \Omega^2(\End(\Ad E))$. It is easy to check that this morphism identifies the curvature of the Chern connection $D_\sigma$ with the curvature of the Chern connection of the metric $h_\sigma$, which are elements in $\Omega^2( \Ad E_\sigma)$ and $\Omega^2(  \End(\Ad E))$, respectively. So the curvature of the Hitchin-Simpson connection $D_{\sigma, \phi}$ on ${{\mathfrak E}}$ goes through the $\Ad$ morphism to the curvature of the Hitchin-Simpson connection ${\mathcal{D}}_{(\sigma, h_\sigma)}$ associated to the Hermitian metric $h_\sigma$ on $\Ad E$.

Assume that a metric $h$ on $\Ad E$ has a connected holonomy group. Then $h$ comes from a reduction of $G$ to $K$ according to the previous construction if and only if the curvature of its Chern connection lies in the image of $\Ad$. \[p1\]

Let $H(h)$ be the holonomy group of the Chern connection of $h$. After fixing a point $u$ in the principal bundle $L(\Ad E)$, the latter has a reduced bundle $P(h,u)$ with structure group $H(h)$. The curvature of the Chern connection acting on the tangent vectors to $L(\Ad E)$ generates the Lie algebra of the holonomy group $H(h)$. Under our assumptions, this implies that the Lie algebra of $H(h)$ is contained in the Lie algebra of $K$ (more precisely, it is contained in the image of the Lie algebra of $K$ under the adjoint map). Since the holonomy group $H(h)$ is connected, it is contained in $K$.

This implies that the holonomy bundle $P(h,u)$ can be regarded as a reduction $E'$ of the structure group of $E$ to $H(h)$. Moreover, $E'$ induces a reduced subbundle $E''\subset E$ with structure group $K$.[^3] By construction the corresponding reduction of $E$ to $K$ induces in $\Ad E$ the metric $h$.

Note that if ${{\mathfrak E}}=(E,\phi)$ is a Higgs bundle with a metric $h$, the curvature of the Hitchin-Simpson connection $\mathcal D_{h,\phi}$ lies in the image of $\Ad$ if and only if the curvature of the Chern connection $D_h$ does.

[[@Nom56]]{} There exist reductions of $E$ to $K$ whose holonomy group is $K$. \[goodhol\]

Note that since $K$ is a maximal compact subgroup of a connected reductive group over ${\mathbb C}$, it is connected.

Let $h_t$ the solution of the Donaldson flow with initial datum $h_0$. If the holonomy of $h_0$ is connected, the same is true for the holonomy group of $h_t$ for every $t \ge 0$. \[connected\]

Realize $H(h_0)$ as a subgroup of $Gl({{\mathfrak g}})$ by choosing a point $u\in L(\Ad E)$. Define a map $f\colon H(h_0) \times {\mathbb R}_{\ge 0} \to Gl({{\mathfrak g}})$ as follows. For $(g,t)\in H(h_0) \times {\mathbb R}_{\ge 0}$ represent $g$ by a loop $\gamma\colon [0,1]\to X $ based at the base point of the fibre containing $u$; consider the curve $\tilde\gamma$ obtained in $L(\Ad E)$ by horizontally lifting $\gamma$ starting from $u$ using the Chern connection of the metric $h_t$. Let $g_t\in Gl({{\mathfrak g}})$ be such that $\tilde\gamma(t) = u\,g_t$. Then $f(g,t)=g_t$. This is independent of the choice of $\gamma$ and is a continuous map, hence its image for fixed $t$ is connected. But the latter is exactly the holonomy group $H(h_t)$.

Let $\mathcal H^+(\Ad E)$ be the space of Hermitian metrics on $\Ad E$. Let $\mathcal H^+_c(\Ad E)$ be the subspace of $\mathcal H^+(\Ad E)$ formed by the metrics whose holonomy group is connected. In view of Lemma \[connected\], one can restrict to this space without loss of generality. We also note that $\mathcal H^+_c(\Ad E)$ is an open submanifold of $\mathcal H^+(\Ad E)$. Finally, let $\mathcal H^+_K(\Ad E)$ be the subspace of $\mathcal H^+_c(\Ad E)$ formed by the metrics that come from reduction of ${{\mathfrak E}}$ to $K$.

\[sub\] $\mathcal H^+_K(\Ad E)$ is a closed submanifold of $\mathcal H^+_c(\Ad E)$.

We give only a sketch of the proof. By using an Ad-invariant bilinear form on the Lie algebra $\mathfrak g$ (the Killing-Cartan form on the semisimple part of $\mathfrak g$ and any nondegenerate bilinear form on the radical of $\mathfrak g$), the Kähler metric on $X$, and integration on $X$, one can endow the vector space $\Omega^2(\End(\Ad E))$ with a scalar product;[^4] then we can orthogonally split this space into the image of the adjoint map Ad and its complement $\mathrm{Ad}^\perp$. We denote by $r:\Omega^2(\End(\Ad E)) \to \mathrm{Ad}^\perp$ the projection and introduce a map $$\varpi \colon \mathcal H^+_c(\Ad E) \to \mathrm{Ad}^\perp$$ which to a metric $h$ on $E$, having connected holonomy group, associates the projection $r(\mathcal R_{h,\phi})$ onto $ \mathrm{Ad}^\perp$ of the curvature of the Hitchin-Simpson connection $\mathcal D_{h,\phi}$. By Proposition \[p1\], $\mathcal H^+_K(\Ad E)$ coincides with the locus $\varpi^{-1}(0)$. By studying this map one proves the claim.

\[remains\] There exist metrics $h_0$ on $\Ad E$, coming from a reduction of ${{\mathfrak E}}$ to $K$, such that, denoting by $h_t$ the solution of the Donaldson flow with initial datum $h_0$, for every $t\ge 0$ the metric $h_t$ comes from a reduction of ${{\mathfrak E}}$ to $K$.\[preserve\]

We take an $h_0$ induced by a reduction $\sigma$ of ${{\mathfrak E}}$ to $K$ whose holonomy group is $K$ (cf. Prop. \[goodhol\]). If the gradient vector field of the Donaldson functional is tangent to the submanifold $\mathcal H^+_K(\Ad E)$, the Donaldson flow remains inside the latter. So we need to prove that claim. Under our assumptions, the gradient flow of the Donaldson functional at a point $h \in \mathcal H^+(\Ad E)$ is proportional to the mean curvature $\mathcal K_{h,\phi}$ of the Hitchin-Simpson connection $\mathcal D_{h,\phi}$, which takes values in the image of Ad. On the other hand, the normal vector field to the submanifold $\mathcal H^+_K(\Ad E)$ is orthogonal to the image of Ad for what we saw in Lemma \[sub\]. This implies that the gradient flow is contained in the submanifold $\mathcal H^+_K(\Ad E)$.

We can now prove Theorem \[MainThm\]. Following the argument in [@BisJacStem] we can reduce to the case of semisimple structure group. If $G$ is semisimple, the center ${{\mathfrak z}}$ of the Lie algebra ${{\mathfrak g}}$ is trivial and the formula in the thesis becomes $$|\mathcal K_{\sigma_\xi,\phi}|< \xi\,.$$

If the principal Higgs $G$-bundles ${{\mathfrak E}}$ admits an approximate Hermitian-Yang-Mills structure, then the adjoint bundle $\Ad({{\mathfrak E}})$ also does, as we noted above. By Theorem \[approHYM\] on [@BG2] we conclude that $\Ad({{\mathfrak E}})$ is semistable. But this holds true if and only if ${{\mathfrak E}}$ is semistable.

For the converse implication, assume that ${{\mathfrak E}}$ is semistable, so that $\Ad({{\mathfrak E}})$ is semistable as well. We consider the Donaldson flow for the Higgs bundle $\Ad {{\mathfrak E}}$ with initial metric $h_0$ that comes from a reduction of ${{\mathfrak E}}$ to $K$ and has connected holonomy group. By Theorem \[remains\], the ensuing flow remains inside the submanifold $\mathcal H^+_K(\Ad E)$. In view of Theorem \[Dona\], this induces an approximate Hermitian-Yang-Mills structure on $\Ad {{\mathfrak E}}$ which can be regarded as approximate Hermitian-Yang-Mills structure on ${{\mathfrak E}}$.

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[^1]: This research was partly supported by GNSAGA-Istituto Nazionale per l’Alta Matematica and PRIN “Geometria delle varietà algebriche.”

[^2]: Note that if $\tau$ is an element in the centre $\mathfrak z$ of $\mathfrak g$, the constant equivariant morphism $f_\tau\colon E \to \mathfrak g$, $f_\tau(u) =\tau$, defines a section of $\Ad E$.

[^3]: This is a general feature: if $E\to X$ is a principal Higgs $G$-bundle, $G'\subset G''\subset G$ are nested subgroups, and ${{\mathfrak E}}'$ is a reduction of the structure group of $E$ to $G'$, the latter corresponds to a section $X \to E/G'$; composing with the morphism $E/G'\to E/G''$, this gives a reduction $E''$ of $E$ to $G''$. Note that $E'' \simeq E' \times_{G'}G''$.

[^4]: Actually this space should be suitably completed using Sobolev norms, but we shall omit these details here.


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Neurologic complications of bacterial endocarditis have been observed for centuries but its management has remained challenging at all times. The cerebrovascular complications of this disorder are the most feared and difficult to address. The management of mycotic aneurysms, recent ischemic/hemorrhagic strokes with and without brain abscesses, and mechanical valve patients continues as an ongoing challenge. Literature continues to appear, providing new alternatives in treatment and hope for improved therapy.


Q:

Cleaning up SQL query with nested query and inner join [Rephrase]

Been trying to reintroduce myself to SQL through some practice questions I've developed for myself, but struggling to find a better way of approaching the following problem:
playlists
id  title
1   Title1
2   Title2

playlist_clips
id  playlist_id clip_id
1   Title1      3
2   Title2      1

playlist_tags
playlist_id tag_id
1           1
1           2
2           2

Clips and Tags are two entirely separate tables, and I am using the playlist_tags and playlist_clips to connect them to the playlists table, to represent the two-way one-to-many relationships.
I wanted to select all the playlists that have a given title, and have ALL of the tags provided in the query (in this example [1, 2]), not just "at least one of them"; additionally, I want to query information from the playlist_clips table, including the number of clips, total playlist duration, etc.
This is what I've come up with:
select p_clips.* from
    (
        p.id, p.title, CAST(COALESCE(sum(pc.duration_seconds), 0) as UNSIGNED) as total_duration_in_seconds, count(pc.id) as number_of_clips
        from playlists p
        left join playlist_clips pc on p.id = pc.playlist_id
        where p.title like "Test1"
        group by id
    ) as p_clips

    inner join

    (
        select *
        from playlists p
        left join playlist_tags pt on p.id = pt.playlist_id
        where pt.tag_id in (1, 2)
        group by id
        having count(*) = 2
    ) as p_tags

    on p_clips.id = p_tags.id

Whilst, from my testing I've found this to work, it doesn't look particularly elegant, and I also assume it's not terribly efficient performance-wise. (I've removed irrelevant parameters from the code for this example, such as select parameters.)
What would be a cleaner way of approaching this, or at the least, a more optimized approach?
Expected Result:
id  title
260 Title1

EDIT: I apologize for my initial confusing post, I've tried to clean up my tables and the information they contain.

A:

Assuming you want summary information from both the clips and the playlist:
select p.*
from playlists p join
     (select pt.playlist_id
      from playlist_tags pt
      where pt.tag_id in (1, 2)
      group by id
      having count(*) = 2
     ) pt
     on p.id = pt.playlist_id join
     (select pc.playlist_id, group_concat(pc.clip_id) as clips
      from playlist_clips pc
      group by pc.playlist_id
     ) pc
     on p.id = pc.playlist_id
where p.title like 'Test1';

Note:  This puts all the clip information into one column.