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13 questions linked to/from How Special Relativity causes magnetism
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I'm very confused about why it is a consequence of special relativity.
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### Is a particular force different in different frames

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## Section: New Results

### Simplification of a constructive version of Tarski's system of geometry

Participant : Pierre Boutry.

Our long term project is to show the independence of all thirteen axioms in a variant of Tarski’s system of geometry. In the current situation, ten axioms have been checked to be independent using counter-models. Specific questions arise around the continuity axiom and decidability of equality between points. This is related to investigations concerning mutual interpretability with cartesian planes and an alternative system proposed by Michael Beeson.

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# Definition talk:Discontinuity of the First Kind

Equality of $f(x-)$ and $f(x+)$ (I assume this notation is understood by you) would not imply that the function is continuous at that point. See the function $f(x) = 1$ if $x = 0$, $f(x) = x$ else. This function is discontinuous at zero, though it certainly vanishes from either side. In general, this allows for removable discontinuities.
One may specify that $f(x-) \neq f(x+)$. This has the added benefit of differentiating the jump discontinuity from the removable discontinuity. One may wish to do this, as removable discontinuities are, more or less, trivial to deal with. Jump discontinuities that are not removable, may require a little more work.
On further thought, perhaps this definition should be changed to "discontinuity of the first kind", and jump discontinuity should have the added stipulation that $f(x-) \neq f(x+)$, so to give the graph that "jump" that one expects from the name. --Keith.U (talk) 13:31, 30 June 2016 (UTC)
It may even be worth separating out the definition for "discontinuity of the first kind" into a separate page. If you are following the exposition as given in a particular source (this approach is highly recommended -- it promotes a consistent approach) then it is best to use the terminology found in that source unless it clashes with other terminology found on this site, in which case a compromise will need to be sought. As the Rudin source has already been used by a few contributors who have posted up a few random disconnected results from it, it may be a profitable approach to work through that source methodically and ensure that everything posted up from there depends logically upon something else from there. Picking results and definitions from the middle of a work can leave such matters unresolved and thereby causing ambiguity and inaccuracy in $\mathsf{Pr} \infty \mathsf{fWiki}$. --prime mover (talk) 14:42, 30 June 2016 (UTC)


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Всего публикаций в данном разделе: 15825

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# Answer to Question #3871 in Statistics and Probability for Jon

Question #3871
Use pair data values and the summations given to
x:& 1 & & & & 3 & 5 & & 7 & & 9
y:& 10& & & 8& & 7& & & 4& & & 2

Σx= 25;& & & Σy= 31;& & & & & Σxy= 115:& & & & Σx² = 165

a. Find the equation for the least squares regression line.
b. Predict the value of y when x is 6 using your equation
1. The average x and y are
x = 25/5 = 5
y = 31/5 = 6.2
2. The regression coefficient is

$b = frac{frac{sum x_iy_i}{n}- ar{x}ar{y}}{frac{sum x_i^2}{n}-ar{x}^2} = frac{115/5 +5*6.2}{165/5 - 5^2} = -1$
thus the equation of the least squares would be

$(x-5) = -1(y - 6.2) \ y = -x + 11.2$
When x= 6,
y = -6 + 11.2 = 5.2

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• 研究报告 •

崂山次生林群落β多样性格局及其组分的驱动因素

1. 1中国林业科学研究院林业研究所， 国家林业和草原局林木培育重点实验室， 北京 100091； 2青岛农业大学园林与林学院， 青岛 266109）
• 出版日期:2020-10-10 发布日期:2021-04-09

Interpretation of environmental factors affecting beta diversity and its components of secondary forest in Lao Mountain.

JIANG Xiao-lei1,2, SUN Zhen-yuan1*, HAO Qing2, GUO Xiao2

1. (1Key Laboratory of Tree Breeding and Cultivation of State Forestry and Grassland Adminis tration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China; 2Qingdao Agricultural University, Qingdao 266109, China).
• Online:2020-10-10 Published:2021-04-09

Abstract: The aim of this study was to explore environmental factors affecting beta diversity and its components of secondary forest and to provide a theoretical basis for the protection and restoration of biodiversity in Lao Mountain. Based on the geographic coordinates and environmental factors of 69 sampling sites, using a Generalized Dissimilarity Modelling (GDM), we analyzed the relationship between beta diversity of different plant life types and the environmental factors. The relationship between different components of beta diversity and environmental factors was determined by Mantel test. The results of GDM showed that 10, 11 and 12 environmental variables affected beta diversity of trees, shrubs and herbs, with the explanation rate being 18.16%, 28.38%, and 43.61%, respectively. The effects of geo-distance, altitude and soil ammonium on the diversity of tree and shrub species were large, while the influence of geo-distance, pH and available potassium content on the diversity of herbage species was great. According to the results of beta diversity decomposition, the beta diversity of trees, shrubs and herbs of Lao Mountain’s forest community was dominated by turnover components. Geographical distance had a significant effect on the turnover of shrubs and herbs but did not affect that of trees. Altitude and soil nitrogen content significantly affected the turnover of trees and shrubs but did not affect herbs. Soil pH, organic matter content and available potassium had significant effects on the turnover of herbaceous species. Habitat filtration and dispersal restriction co-affected the beta diversity pattern, with higher impacts of habitat filtration. During the process of ecological restoration, attention should be paid to the effects of altitude, soil ammonium nitrogen content on woody species distribution and also to the effects of soil pH, organic matter content and available potassium content on herbaceous species distribution. Because the turnover components of beta diversity of trees, shrubs and herbs were much larger than that of nested components, species diversity conservation in the secondary forests of Lao Mountain should be carried out in an allround way.


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Brian and I drove to Philly on Sunday, December 3, 2017, to attend the first of two world premier performances and recording sessions for Symphony for a Broken Orchestra by composer David Lang, a luminary in the "new music" genre of classical music. I preceded this with studio interviews in Brooklyn with producer/engineers Arun Pandian and Devin Greenwood, the folks charged with recording and mixing stereo, surround, and 3D audio releases of this performance. Brian and I revisited Arun and Devin later for a photo shoot and follow up discussion about the mixing process. Symphony for a Broken Orchestra was commissioned by Temple Contemporary, Tyler School of Art at Temple University. Over 1000 physically broken classical and band instruments were collected from the attics and basements of various (many underserved) public high schools throughout Philadelphia, and a website was set up to raise awareness and solicit sponsors. Private and corporate sponsorship of individual instruments will pay for repair and reintegration of the instruments into the Philadelphia public school system's music programs. The Philadelphia Public School System's 2007 instrumental music budget was $1.3 million, but that same budget in 2015 had been reduced to$50,000. Fundraising efforts will expand with future commissions and performances in other major US cities with similar public school music budget cuts.

Before the Event

How did the two of you meet?

Arun Pandian: We met because a good friend of mine and Devin's is a songwriter named McGowan. [To Devin] How did you meet McGowan?

Devin Greenwood: Basically through John Legend's old Philly crew. He was writing songs with an Australian guy, who goes by the name of Old Man River, and he wanted both Arun and I to play on the project. We ended up forming a band, but this guy lives in Australia – halfway around the world! We were meeting and commiserating a lot, and eventually we became good friends.

AP: I'm from Texas. I originally went to school for finance. I landed the Miseducation tour with Lauryn Hill. I did that for a few years, and I got to play on both the record and the tour. Through that, I played with Santana, Mary J. Blige, and Nas. After that, I was in New York for a while. I did some off-Broadway performing with Savion Glover and Jared "Chocolate" Crawford, who I ended up producing in Philly for a record label, with Chris Schwartz of Ruffhouse Records, which The Fugees came out on. I was producing Chocolate on that label in Philly and then came back here to New York. After 9/11, I was at a crossroads. I asked the question, "Do I want to be a guitar player anymore, to tour with people and be on the road?" I decided that I didn't, that I'd rather control my own career. I started getting into sound, audio production, and engineering. I taught myself. I bought a piece of gear, learned how to use it, sold it, and got another piece of gear. I mentored with Henry Hirsch [Tape Op #56] at Waterfront Studios upstate, working with him as a producer. I also learned a lot by doing my own music. Since then, I've worked with artists like Norah Jones, Mumford & Sons, and Gabriel Garzón-Montano.

DG: Mine's a similar arc. I played classical piano as a kid, but ended up studying film in college, which is a real passion of mine. After film school though, I got a chance to tour with [vocalist] Donna Godchaux from The Grateful Dead. I did that for a couple of years when I was 22. Later on, I played with Lo Faber from God Street Wine, became Amos Lee's first music director, and did several years internationally with Melody Gardot, singing and playing [Hammond] B3. I was songwriting inbetween all that – Steve Forbert actually covered one of my songs – and I put out a record back in 2006 called Eye in the Cloud. Like so many singer-songwriters, I felt like I was on track to get signed, but at the same time I was starting to get into studio craft as well. Just outside of Philly there was a place where Shelly Yakus [Tape Op #31] used to work, called Scullville Studios. Shelly did all of the early Tom Petty, John Lennon, The Band, Dire Straits, U2, and so many famous songs that you know, like [Van Morrison's] "Moondance." He's a really great dude. I made my record there with Shelly and Rachel Alina. I learned about studio engineering and recording to tape, and I was seduced by this whole world. When I started producing local artists, and one of them got signed, people wanted me to make records for them. All of a sudden I was making money doing that. Around that same time, Norah Jones had me play on her third studio record [Not Too Late] and we were recording Amos Lee's first album as well at Sear Sound [#41] and the old Magic Shop [#66]. Before long I'd moved to New York and opened my own studio, The Honey Jar. Since then I've worked with people like Sufjan Stevens [#70], who was in the neighborhood, and Steve Reich [#15] – which is how I got into the new music world.

AP: I'm currently teaching audio engineering at Long Island University. A good friend of mine, from my Lauryn Hill days, is head of the recording technology program over there, and he approached me about teaching. It's right down the street from where we are now. My first inclination was to recoil back – I'm actually doing this for real. I don't want to go teach in a classroom. He said, "Oh, come on. You're really into this and you've got a studio. It'd be great for you to come talk to these kids." I did it, and I was surprised by how much I actually enjoyed it. After teaching at LIU, I started to think about doing a Master's in Music Technology or something like that. I ended up at NYU, studying in the Tonmeister program, where I got to learn about classical recording techniques, as well as 3D audio. Devin knew that I was doing this, so he told me about the broken instruments project, and he suggested we team up and record it. They weren't planning to record the concert. We partnered with Sennheiser. Speaking of, we'd like to thank David Missall and Axel Brisard for donating the microphones to record the event. We also have a GoPro Fusion camera that I purchased. I'm going to shoot 360 video of the performances as well, for the sound installation.

AP: There will be a vinyl 2-track stereo release of the 3D audio recording.

DG: We'll mix the 2-track record either here at the Honey Jar, or in Arun's studio, Signal Corps Recording. There will also be a sample bank, with avant-garde electronic music being made from those same sounds.

AP: And there will be the 3D sound installation, which I'll be mixing at NYU. They have the lab and the speaker setups that we need.

Let's talk a bit about how each of you transitioned from being touring musicians to producer/engineers.

AP: I'd been talking with Henry Hirsch; he knows an enormous amount about gear. He's been through it all, especially with Lenny Kravitz and the budgets they've had to buy and sell gear. Just talking to them about their vision for what type of gear to buy – the ideal type of gear that was contrary to what you see in most studios. Generally it boils down to one rule: the fewer electronics there are in the signal path, the better the audio. A lot of the older gear fits that mold. The more you put in the [signal chain], the more it stops sounding like what you were trying to capture in the first place. So, Henry gave me suggestions on what gear to buy. He said, "If you ever see a WSW, the Austrian branch of Siemens, you might want to grab it, because a lot of people don't know about it." That started a search. I had originally opened a studio where all I had were mic pres going straight into a Pro Tools system. I thought, "Oh, I don't need a console." Then I bought a summing mixer and there was a pretty drastic change to me in the depth of field. Then I figured, "Oh, great. Now I need to get a console." I remembered from my experience of going into studios as a producer before I knew what I was doing, the engineer would just spread out the music on the console. For the type of the music I loved, I needed to experience that sound.

DG: Once you experience it, it's hard to go back. It's very alluring. I learned about breaking out the tracks in a studio where we had this [Pro Tools] Mix Plus system. I was in there for the first time working on my own, and for whatever reason I said, "Let me break out the bass to its own channel." I'd been mixing and listening on two channels. I could hear it better and clearer. That was a unique, seminal experience, where it was so crystal clear what a specific piece of gear does.

AP: Henry suggested I find a console like this one. I saw it on eBay for \$3,000. The console was in parts. The guy hadn't put it together; he just shipped it to me. I was talking to techs, and they were quoting me these enormous prices to put it together. No one had the schematics, and nobody had any of the parts or the frame. I was on Gearslutz, and I found this guy named Damir Rogina from Croatia. He goes by the name Roginator – he's head of the MCI forum. He also knows everything there is to know about Siemens and WSW. He has all the schematics and all the extra parts. He even had an original frame. I flew him and his partner down, and they put the whole thing together, refurbished it, recapped it, and did everything in like two weeks.

What year is this console?

AP: Early '60s. Slowly, I realized how good it sounded and how well it's built. It's built beyond what anybody would normally need. It's basically [built for] Austrian government money for broadcast and military purposes.

It looks like a broadcast console.

AP: Yeah. It's a pretty heavy-duty build, and it's super simple. There's not a lot going on. It's just EQ, mic/line channels, five buses, three auxes, and that's about it. Once I had the console, I kept looking for other Siemens and WSW gear, and slowly I collected and refurbished some. They're built to a different rack standard from what we use in the US. I ended up finding Siemens racks, which I had customized by my friend, Steve Masucci [Tape Op #124]. There's another studio in Austria called Casino Baumgarten. They have a WSW tube console, which is where my tube pres came from. I slowly started to buy the gear and refine it down. It comes out of my desire for the studio to have a specific sound and direction. When people come here, they come here for that purpose.

DG: Arun turned me on to the idea of sticking to a brand and the way to use artistic limits to make a piece of art have more of a signature, by doing that with a studio. Instead of having all the cool gear, just curate it to one brand.

AP: It comes from the need to have your own sound. When we were artists, we were doing that. Why not do it on the engineering side of things? It doesn't make sense to cater to everybody's needs all the time. I think we have too many choices now with digital. Somebody has to make a decision somewhere. Might as well start with the gear.

How would you describe the sound that you've achieved?

AP: It goes along with what I was saying earlier – the simpler, the better. Not having too many choices. The console doesn't have the ability to re-route itself in 100 different ways. There's no recall; no fancy features. This keeps everything very simple. A lot of the gear back then wasn't made to necessarily have a vibe. It was made to translate in a euphoric way, to be very clean, and give you what you put into it as much as possible. That's kind of how I try to do things, to have some clarity and then have the depth.

DG: I was looking for the same thing, just a Class A discrete console. I actually had the same guy that Arun hired, Damir, come over and rebuild mine. I used to have the very first MCI 500 series – I had the prototype, and nobody had schematics. He rebuilt that, but it ended up being a bit of a nightmare. I wanted to downsize to something more manageable, so this is a Studer 189 MkII. It's a mobile recording console, originally used to record classical music from a truck. It's fully modular, so even while it's turned on and running, you can open it up and swap channels out.

That's insane!

DG: Here's the fader cassette [clicks out part of a channel strip while the console is on and playing background music in the control room]. You can pop the channel out, just like this.

And it doesn't affect whatever is running at the time?

DG: It doesn't affect what's going on. And it's double-sided, so there's actually [components] on each side of the board. They packed everything into a really small package in a way that only the Swiss know. The reverb input modules were all built for plate reverbs, so you've got send, as well as a low end rolloff for your send, and then you've got an EQ on your return. These change the actual distance on the plate. There's a proprietary connection on the back that lets you plug into the motor on the plate, so that you can change the reverb time right here.

Do you have that hooked up to your EMT plate?

DG: Not yet, but I will eventually. Obviously I'm not going to have four plates. The others will become stereo returns for something else. The other cool thing about this is that it interfaces with a Studer tape machine. I bought the tape machine – this 8-track deck. I can control the transport, arm the channels; and here are my eight returns.

What year is this console?

DG: 1974. Even the patchbay – everything is right here on the back. I'm a minimalist with the amount of gear that I want to use. I have that [Teletronix] LA-2A, which is a Rev 1 from the '60s, the very first edition of that model. There are four compressors in the console, as well as a plate reverb. That's my setup. My own musical style is cinematic and spacious, but also minimalist.

Let's talk about the Broken Orchestra project. I'm not that familiar with 3D sound myself.

AP: We've all experienced it, right? It's just spatial.

DG: In real life, it's 3D. If you're in the woods and hear the birds, it's 3D.

AP: On the recording/production side, we've mainly experienced sound in terms of left/right and front/back. Now we're adding the height channel to that.

By height channel, you mean up and down?

AP: Yeah. It's something that's always captured in the mics. An omni mic has sound that's captured in all directions, but it hasn't been the focus of recording production or the way speakers have been set up. The 5.1 is just the main layer around your ears, in the horizontal plane.

DG: In movie theaters, you've experienced surround sound, but someday it'll be in 3D.

Center channels lining the ceiling.

AP: Which they're already doing with Dolby Atmos.

The setup for the performance is in the round, but reversed. The audience is in the center, and the orchestra is the equivalent of stadium seating where the crowd would usually sit for theater in the round.

AP: Yes – the conductor is going to be dead center.

DG: [The conductor], Jayce Ogren, is a protege of the composer, David Lang. David will be there, but he's not conducting. Jayce is also a player, in that he's signaling how long the movements are going to last, how many hits there are going to be. He's bringing groups in and out the entire time.

Conductor Jayce Ogren

Funding will be used to restore whichever instruments can be restored, so that they can go back into the Philadelphia Public School system?

DG: My friend, Danielle [Birrittella, a singer known as DIA], coined a phrase yesterday – she calls it innovative philanthropy. It's about being creative at something that America has been pretty good at, which is bringing private money into the social system to do things that we care about. It's about bringing creativity into that process. I'd say that this kind of music in this scene relates to classical in the same way that Kandinsky or Jackson Pollock relates to Renaissance painting. It's [a postmodern] version of that music. It's a very different experience than hearing a Vivaldi piece. It's kind of psychedelic, actually.

AP: It's very 21st century, which is why 3D plays into this. It's sonic experimentation, using classical and jazz band instruments.

Some of the fundraising and promotional videos for the event show guys using mallets on broken cellos.

DG: When you take the strings off a broken cello, it becomes something else...

AP: A percussion instrument.

DG: Those guys are from Found Sound Nation, which is an artist collective I'm a part of. It does really great work all over the globe. FSN used to be the social arm of Bang On A Can, which was formed by David Lang, Julia Wolfe, and Michael Gordon to promote this kind of music.

Arun was telling me that there are going to be multimedia installations at performance spaces, or MOMA-type museums. This really is innovative philanthropy.

AP: That's why we got so excited about the project. It's a way to create art and help people at the same time.

DG: There will also be members of the Philadelphia Orchestra playing. The performances will be at the 23rd Street Armory in Philadelphia. The idea is you play the note where it would sound normally on the instrument, and if it's wrong, that becomes part of the piece. That's in the tradition of John Cage, who brought indeterminacy into music composition with rolling dice or using the I Ching to decide where notes go. That was a Modernist innovation, and it is very much a part of this music tradition which, again, is a very elitist tradition.

AP: There will be ten ensembles of musicians around the perimeter of the venue. Each ensemble is going to be a mixture of broken instruments. It won't just be violins or horns. From what I've been hearing in rehearsals, the piece jumps around between the different ensembles, doing all kinds of things. It's playing with the physical space.

How many people will be in the audience?

AP: 400 to 600. There will be 400 performers in the orchestra.

So the performers will be spread out more than the listeners, with physical space between the ensembles around the perimeter of the venue?

AP: Each ensemble will have its own group leader, and sometimes there will be call and response between the group leader and the ensemble. The group leader will be instructed to do something and then the ensemble will repeat it. Each group leader will play an instrument and will be directed by the master conductor.

Will the group leaders be children or professional musicians?

AP: Most likely professional musicians and teachers.

DG: The student players are in high school. The oldest adult players will be in their 80s and 90s.

Let's talk about the mic setup that you will be using to record the concert.

AP: Basically, [3D audio recording techniques] start with general stereo setups, like Blumlein, NOS, ORTF, or an X/Y spaced pair. People have taken those systems and turned them on their sides. What we're going to be using for this setup is an M/S setup that's been turned on its side, so it's now an M/Z setup.

DG: It's mid-side recording, turned perpendicularly.

AP: What you're getting when you're concerned with stereo playback is an inter-aural difference. You're concerned with the time differences between the left speaker and the right speaker, as well as the left ear and the right ear. When you're dealing with the up and down Z plane, it's mostly a spectral difference. It's the difference between us being closer to the floor, and the way sounds hit the floor and come back to us – and then certain sounds being above us. And not only the way they hit the pinna of our ears and bounce off our shoulders into our ears, but the spectral difference of the floor and the distance to the ceiling. That's what we're trying to capture with that Z-plane.

There is going to be a high ceiling at the armory. The rafters are going to create a reverb time that wouldn't happen in a studio or another smaller place.

AP: Exactly. We're just trying to capture that Z-plane and then translate it into the reproduction system.

DG: So, the omnis and cardioids are your left and right, which is all surround, and then you have your figure-8s facing up and down for the vertical plane.

AP: For us, it'll be cardioids on each of the ten ensembles, because we want to reject the audience [sound] from the back. Then it's going to be a figure-of-8 setup behind that, which will be facing up and down. We took the side and flipped it up.

Who determines what the front of the mid-side is?

AP: The mid. The mid is the front.

But if it's a 3D...

DG: There is no front. It depends on the orientation of the viewer once they're in the playback system for the installation.

Every person who sees the live performance will have a different listening experience, a totally different blend of the ten ensembles, with a slightly different experience of the reverb of the room.

AP: Absolutely. It's going to be our challenge then to break it down to stereo in some way that works for the vinyl release. The 3D installation will almost be easy, because we'll just set up speakers mimicking the mic pairs on each instrument. We're using a coincident 3D setup. People have also experimented with spaced setups. The setup I'm using was pioneered by Paul Geluso. There's another setup by David Bowles, called a Bowles array, and that's basically an NOS/ORTF setup turned on its side. Then Gregor Zielinsky of Sennheiser does a spaced-pair setup turned on its side. I'm giving you a breakdown essentially of where these ideas are coming from. When you're doing 3D audio recording, the thing you're challenged with is the direct sound you're getting from in front of you, as well as however you're getting the height channel. It's the correlation between those two channels, how they interfere with each other, and trying to reject the front signal and get as much reflection on the top and bottom as you can without smearing the two.

Does the playback format compensate for these variables and make some kind of "mix" decision? Did someone create a software crossover?

AP: No. Right now those three guys and their setups are the ways they've come up with to best deal with these problems. It's something that people are still researching and experimenting with. We are recording 3D in a huge, immersive setup. I don't know if there are a lot of those that have been done so far.

After the Event

Let's discuss the recordings that will be released of the concert. Will decisions be made to take a section from the first performance and a section from the second performance?

AP: I think we'll definitely consult with David Lang. He'll probably be there when we do the final mix. But it's not even for certain that we will comp between the two – that's just what we've been doing so far. One of the things that got him excited is the talk of doing an immersive installation experience.

There will be multiple mixes...

AP: There will definitely be a stereo mix. There will be some sort of surround mix, depending on how we want to do it. It's either going to be 9.1 or 7.1. And then there will be the installation, which is a whole separate thing.

The installation will be the 3D audio?

AP: Yeah.

Because of the limitations of most playback systems, even high end audiophile systems, I guess that most people will hear the final release as a surround mix?

AP: The surround mix will be a 3D audio surround mix. It'll be like an Auro-3D mix. It'll have a 5.1 main layer, and then you will have four height channels. We'll probably put it in some format like that. There are multiple formats that you can use that are equivalent to that. A lot of hi-fi people actually have the capability of splitting it out in such a way. The new Kraftwerk material was done in 9.1. I believe it was 3D, so it wasn't just a main layer. The format is usually on Blu-ray and there are two main proprietary systems for processing: Dolby Atmos and Auro 3D.

So, the decoding process will be where the choice of playback channels happens, per the speaker configuration of each listener's music system?

AP: Exactly. That's the point. If you only have 5.1, the mix will be decoded as 5.1.

A portion of the mix is happening at NYU's lab, where you're doing the Tonmeister program.

AP: They have a 20-speaker setup in the research lab, and a 10.2 set-up in Dolan, which is their main studio. It can get confusing, the way these speaker configurations are described. Like if you had a 9.1 system, it could be described as a 5.1 main layer with four up top, so it's 5.1.4. But some people call that 9.1

9.1 sounds cooler.

AP: But 9.1 could technically be a main layer with no 3D at all [all of the audio channels happening in the horizontal plane].

Now that the performance has taken place, what did you learn from the experience?

AP: On the recording side, it was shocking how smoothly things went, logistically. I'd say that the recording techniques we used were a good balance of not only the budget, but just logistically, setting everything up in the time that we had. It could have gotten really complicated with that many performers.

As an informed outsider, I thought it went about as smoothly as it could have possibly gone.

AP: It did. And after the fact, we haven't encountered anything that we regret or wish we'd done differently. The thing about doing it in the future is that if it's a new composer and setup, it's going to be totally different, so we'd approach it completely differently. This time, we curated it for the way that this specific event was set up. We were out of the way of the musicians and most of the audience. I think we were pretty inconspicuous.

DG: We had the overall technique planned out, but then we ended up putting spot mics here and there. I was walking around, looking for people with a combination of apparent competence, as well as a somewhat functional instrument, with a particular emphasis on low end. There wasn't a lot of low end in the recording, so we close mic'd some basses and bassoons. That came out really well and brought some bass into it. Those kinds of decisions were spontaneous.

AP: There were also dress rehearsals a couple of days before, so we had the benefit of modifying mic heights and placements.

Since it went so well, has there been any interest yet in doing it again someplace else?

AP: Yes! We're forming a non-profit organization to commission more compositions for broken school instruments in different cities. We hope to work with a whole range of New Music composers to shine a light on the current funding crisis for public school music departments, and to establish funds to keep working instruments in those schools.


text:
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Cristinel Stoica: on 10/5/12 at 20:04pm UTC, wrote Dear Hope and Jin, I think your essay is interesting, although I don't...

Christian Corda: on 10/5/12 at 13:16pm UTC, wrote Dears Hope and Jin, I enjoied in reading your interesting Essay. I am...

Sergey Fedosin: on 10/4/12 at 9:29am UTC, wrote If you do not understand why your rating dropped down. As I found ratings...

Jin He: on 9/22/12 at 21:24pm UTC, wrote Dear Pentcho Valev, Thanks for your kind words. In fact I learned about...

Pentcho Valev: on 9/22/12 at 16:54pm UTC, wrote Dear Jin He, Your cosmic breasts discovery is breathtaking. The FQXi...

Hoang Hai: on 9/19/12 at 15:50pm UTC, wrote Dear Very interesting to see your essay. Perhaps all of us are convinced...

Yuri Danoyan: on 8/25/12 at 20:37pm UTC, wrote hope he See my essay http://fqxi.org/community/forum/topic/1413 Specially...

Peter Jackson: on 8/23/12 at 16:24pm UTC, wrote Dear Hope and Jin Barred spirals were indeed unexplained in any secular or...

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FQXi FORUM
September 21, 2021

CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: Galaxy Anatomy: "Darwin Spirals" by Hope He and Jin He [refresh]

Author Hope He wrote on Jun. 22, 2012 @ 15:57 GMT
Essay Abstract

This whole universe is definitely not of `relativity'. The universe is hierarchical, absolutely. For example, the biosphere and all laboratorial experiments suffer the same effects of Earth's gravity. In the same time, gravity cannot be testified by scientists' laboratorial experiments, because the gravity between two protons is 10^40 times weaker than the electricity between the same two protons. The most independent system close to humans is the Solar system, which, however, is effectively an example of two-body system. All theories of gravity are derived from the two-body system. When applied to three or more bodies, the theories give chaotic results. This is similar to the situation that there exists no formula solution to any algebraic equation of order 5 or more. Scientists, however, use the two-body theories of Newton and Einstein to explain the whole universe. This is fundamentally wrong. Galaxies are relatively the independent many-body systems. Dr. He initiated the concept of rational structure and applied it to the study of galaxy structure in 2001. His result is neither a theory nor based on any two-body theory; it is based on the analysis of galaxy images.

Author Bio

Hope He is currently a high school student. She was born in China and came to the United States at the age of 5. She grew up in Alabama. Her main interests are math and science. She is the English editor of a new company in Wuhan, China. The company dedicated to the education and study on galaxies.

Alan Lowey wrote on Jun. 23, 2012 @ 10:53 GMT
Dear Hope and Jin,

I really enyoyed your essay. Well done on such a nice piece of work. It's the best one I've read so far. I liked the Darwin Spiral idea and also firmly believe in the necessity to understand these golden galaxy spirals. I *do* have an alternative solution for the conundrum, which is waiting to be approved and posted, fingers crossed. You should both be interested with luck. Cheers

report post as inappropriate
Author Hope He replied on Jun. 23, 2012 @ 15:04 GMT
Thank you for reading our essay! We will be looking forward to reading yours also. Good luck to you too!

Alan Lowey replied on Jun. 25, 2012 @ 09:40 GMT
Thanks!

report post as inappropriate

Alan Lowey replied on Jun. 29, 2012 @ 09:17 GMT
Hi again, I've been thinking about the barred spiral galaxy formation conundrum and have realised that my essay has a solution for you. It has been recently been approved and posted incidentally. I propose that there is non-spherical 'hidden exotic matter' (HEM) at the centre of the galaxies as well as at the centre of stars. The additional force on the plane of rotation of a galaxy or star would require a HEM shaped like a rugby ball ready to be kicked. This assumes that the exotic matter gravitation effect is dependent on it's cross sectional area. Hence a star on the rotational plane 'sees' more of the galactic centre HEM compared to a star high above or below the plane. If this galactic centre HEM is 'squashed' from the sides so that it becomes more lozenged shaped, then this also gives a greater x-sectional area in just a single bar, radiating out across the rotational plane. This model therefore predicts that the bar in a barred spiral is rotating at the same rate as the galactic centre HEM.

I'm keen to hear what you may have to think about these revolutionary ideas.

report post as inappropriate

J. C. N. Smith wrote on Jun. 23, 2012 @ 17:53 GMT
Hi,

Your essay is interesting and enjoyable to read. But please explain how your essay addresses the theme of this competition: which of our basic physical assumptions are wrong? That was not made clear to me. Thank you.

jcns

report post as inappropriate

Roger Schlafly wrote on Jun. 29, 2012 @ 19:38 GMT
Why do you say "This whole universe is definitely not of 'relativity'."? Is there some evidence or argument that is contrary to relativity?

report post as inappropriate
Alan Lowey replied on Jun. 30, 2012 @ 11:14 GMT
The essay is quite clear in my eyes. Dr Jin He works for a chinese company which models galaxy formation. He says that the spiral galaxies "are easy" but the barred spirals remain an enigma. I think that he intuitively believes that the 'standard model' has a fundamental problem, which is why Hope and Jin have entered the competition. A fact that is missing from the essay is that *at least* two thirds of spiral galaxies are barred spirals. This really means that modern science doesn't have the ability to model spiral galaxy formation, even with the existing dark matter halo hypothesis. I have given an alternative solution which has the ability to solve the barred spiral enigma as well as the ice age enigma.

report post as inappropriate

Author Hope He replied on Jun. 30, 2012 @ 14:56 GMT
Dear Roger Schlafly,

The big big evidence is the Machian Principle which was terminologied by Einstein.

Dear Alan Lowey and other friends,

We are very busy recently, and we can not visit this furums frequently. Even though we can find time, we may not have time to discuss the important and valuable issues with our friends.

Sorry for this.

Yours,

Hope He and Jin He

Alan Lowey replied on Jul. 3, 2012 @ 09:15 GMT
Nevermind. Your essay entry has been very useful to me, so thank you for that. I've even remembered that Mercury's precession can be accounted for by it's close proximity to the sun's exotic centre and irregularities. It's rise and fall about the ecliptic must take it into a region where it 'sees' a change in the sun's HEM cross sectional area imo.

Anyway, thanks again.

report post as inappropriate

Peter Jackson wrote on Aug. 23, 2012 @ 16:24 GMT
Dear Hope and Jin

Barred spirals were indeed unexplained in any secular or classical evolutionary theory, and a major bizarre anomaly, until quite recently, the enigma resolved courtesy of the discrete field model (DFM).

http://vixra.org/pdf/1102.0016v3.pdf

I should of course say that although web archived at vixra and the RAS this is still not yet published in a mainstream journal as reviewers reject anything that may be conceived as different from the current ruling paradigm. In fact the DFM derives the observed effects of SR direct from a quantum mechanism, but it is none the less unfamiliar.

To explain; after AGN (disc) accretion and (quasar) jetting the re-ionized matter lies in a line, or splayed butterfly shape (from spiral jets) ave some 2,000 light years long (1M at M87).

Due to intrinsic rotation in space the central 'virial' radius starts to rotate (Google Hawk) propagating matter at the circumference and 'trailing the arms. Inside this the remains of the jet arms form the bar, which rotates bodily, or in 'lock step', just like all spirals. This is on a NEW perpendicular axis to the old galaxy, explaining They then evolve into a full spiral, merge into a disc, accrete into a lenticular as the AGN grows, and the cycle restarts. I recently posted to Singh's essay blog I think why the cycle seems to be around 10-12Gyr. There are good NASA shots of Centuarus A undergoing the rapid accretion and jetting process. (Previously thought o be a collision!)

There are many other anomalies explained by this.

Some papers with previous interpretations but good evidence here, the first of intrinsic rotation;

Parra, F.I., Barnes, M., Catto, P.J., RPCtr. Th.Phys. Ox. CCFE. N.Fusion. 2011. http://arxiv.org/pdf/1102.4613.pdf

Zoltan H. Jimenez R. Bernardire.M. Constructing the cosmic evolution of quasars from the age distribution of local early-type galaxieshttp://arxiv.org/PS_cache/astroph/pdf/0610/0610723v1
.pdf

Wold,I. et al. Host galaxies of luminous quasars population synthesis of optical off-axis spectra.RAS Vol.408,Issue2,pp.713-730. http://adsabs.harvard.edu/abs/2010MNRAS.408..713W

Benton R, Metcalf, A. Constraints on Small-Scale Structures of Dark Matter from Flux Anomalies in Quasar Gravitational

Lenses http://arxiv/pdf/1007/1007.1599v1.pdf

DeBuhr, J., Quataert E., Ma, C.P., Galaxy-scale outflows driven by active galactic nuclei. MNRAS accepted. 30.10.2011

Martini ,P., Weinberg D.H.., Quasar Clustering and the Lifetime of Quasars 2001 ApJ 547 12 doi:10.1086/318331.

Porciani, C., Magliocchetti P., Norberg P., 2004 MNRAS 355 Issue 3, p1010-1030.

I do hope that is all enlightening. There is much more of course. Main paper now in review for 3rd time!

Best wishes

Peter

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Yuri Danoyan wrote on Aug. 25, 2012 @ 20:37 GMT
hope he

See my essay http://fqxi.org/community/forum/topic/1413

Specially post about the Universe as a logarithmic spiral.

report post as inappropriate

Hoang cao Hai wrote on Sep. 19, 2012 @ 15:50 GMT
Dear

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

report post as inappropriate

Pentcho Valev wrote on Sep. 22, 2012 @ 16:54 GMT
Dear Jin He,

Your cosmic breasts discovery is breathtaking. The FQXi community loves this kind of science. Your community rating is much greater than mine even now but I expect you to go to the top of the rating list.

Good luck, Pentcho Valev

report post as inappropriate

Jin He wrote on Sep. 22, 2012 @ 21:24 GMT
Dear Pentcho Valev,

In fact I learned about varying light speed etc on internet during 2007 when I found your posts and I sill remember your name.

report post as inappropriate

Sergey G Fedosin wrote on Oct. 4, 2012 @ 09:29 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
$R_1$
and
$N_1$
was the quantity of people which gave you ratings. Then you have
$S_1=R_1 N_1$
of points. After it anyone give you
$dS$
of points so you have
$S_2=S_1+ dS$
of points and
$N_2=N_1+1$
is the common quantity of the people which gave you ratings. At the same time you will have
$S_2=R_2 N_2$
of points. From here, if you want to be R2 > R1 there must be:
$S_2/ N_2>S_1/ N_1$
or
$(S_1+ dS) / (N_1+1) >S_1/ N_1$
or
$dS >S_1/ N_1 =R_1$
In other words if you want to increase rating of anyone you must give him more points
$dS$
then the participant`s rating
$R_1$
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

report post as inappropriate

Christian Corda wrote on Oct. 5, 2012 @ 13:16 GMT
Dears Hope and Jin,

I am going to give you a score 10.

Cheers,

Ch.

report post as inappropriate

Cristinel Stoica wrote on Oct. 5, 2012 @ 20:04 GMT
Dear Hope and Jin,

I think your essay is interesting, although I don't know much about galaxies. I hope you will find the recognition you deserve.

Best regards,

Cristi Stoica

report post as inappropriate


text:
# Euler straight line

(Redirected from Euler line)

The straight line passing through the point $H$ of intersection of the altitudes of a triangle, the point $S$ of intersection of its medians (the centroid), and the centre $O$ of the circle circumscribed to it. If the Euler line passes through a vertex of the triangle, then the triangle is either isosceles or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation

$$OH:SH=1:2$$

This line was first considered by L. Euler (1765).


text:
# How to combine bilateral blur passes?

I'm changing to bilateral blur for my SSAO implementation, just wondering how people usually combine the two passes - do you just add them together, or average, or something else?

I happen to be using Vulkan but don't think that should affect the answer too much.

• You apply one pass with your source image as input to get an intermediate image, then you apply the second pass using the intermediate image as input to get your output, no? Or am I misunderstanding the sense of "combining" that you mean? – DMGregory Dec 10 '20 at 12:26
• hmm I thought you blurred your source twice, once in each direction and then combined. I'll try what you're suggesting! – mike Dec 10 '20 at 13:28
• Ah, then I'll convert that comment to an answer. – DMGregory Dec 10 '20 at 13:31

You apply one pass with your source image as input to get an intermediate image.

Then you apply the second pass using the intermediate image as input to get your output.

   Input Image
↓
1st Pass
↓
Intermediate Image
↓
2nd Pass
↓
Output Image

• Works great as well, thanks! – mike Dec 11 '20 at 9:38


text:
# If $\sum_{n=0}^\infty a_n$ is convergent and $a_n \sim b_n$, is $\sum_{n=0}^\infty b_n$ also convergent?

If $\sum_{n=0}^\infty a_n$ is convergent and $a_n \sim b_n$ (e.g. $\lim_{n\to\infty} a_n/b_n = 1)$, is $\sum_{n=0}^\infty b_n$ also convergent?

• Do you assume $a_n > 0$? – Daniel Fischer Oct 25 '16 at 16:11
• @DanielFischer I'm not entirely sure of the consequences that assumption would make, so I'm inclined to say "maybe". – Frank Vel Oct 25 '16 at 16:14
• If you don't assume an eventually fixed sign, consider $b_n = \dfrac{(-1)^n}{\sqrt{n+1}} + \dfrac{1}{n+1}$. – Daniel Fischer Oct 25 '16 at 16:16

Assuming that all the terms are positive, the quotient $b_n/a_n$ converges to $1$ and hence is bounded, say by $k$. Then, $$\sum b_n\le\sum ka_n=k\sum a_n$$


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Given the right trapezoid calculate angle theta and the area of triangle hat(EAD), provided EA=4, AB=BC=CD=DA=2, AB_|_EC?

Jul 11, 2016

$\angle \theta = \arctan \left(\frac{\sqrt{3} - 1}{2}\right) \approx 0.350879411 r a d$

${S}_{E A D} = 2$

Explanation:

From right triangle $\Delta A B E$, knowing hypotenuse $A E = 4$ and cathetus $A B = 2$ we can find another cathetus:
$B E = \sqrt{{4}^{2} - {2}^{2}} = \sqrt{12} = 2 \sqrt{3}$.

In the right triangle $\Delta D C E$ cathetus $C D = 2$. Second cathetus $C E = C B + B E = 2 + 2 \sqrt{3}$

Now we can determine tangent of angle $\angle \theta$:
$\tan \left(\theta\right) = \frac{C D}{C E} = \frac{2}{2 + 2 \sqrt{3}} = \frac{1}{\sqrt{3} + 1} = \frac{\sqrt{3} - 1}{2}$
Angle $\angle \theta$ can be determined using an inverse function ${\tan}^{- 1} \left(\right)$ or, as it is often expressed, $\arctan \left(\right)$:
$\angle \theta = \arctan \left(\frac{\sqrt{3} - 1}{2}\right) \approx 0.350879411 r a d$

Area of triangle $\Delta E A D$ can be calculated using the length of its base $A D$ and altitude $D C$:

${S}_{E A D} = \frac{1}{2} \cdot A D \cdot D C = \frac{1}{2} \cdot 2 \cdot 2 = 2$


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# Derivative of Cosine

We shall prove formula for derivative of cosine function using by definition or first principle method.
Let us suppose that the function of the form $y = f\left( x \right) = \cos x$.

First we take the increment or small change in the function.

Putting the value of function $y = \cos x$ in the above equation, we get

Using formula from trigonometry, we have

Using this formula in equation (i), we get

Dividing both sides by $\Delta x$, we get

Taking limit of both sides as $\Delta x \to 0$, we have

Consider $\frac{{\Delta x}}{2} = u$, as $\Delta x \to 0$ then $u \to 0$, we get

Using the relation from limit $\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1$, we have

Example: Find the derivative of

We have the given function as

Differentiation with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\cos x = - \sin x$, we get


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## Abstract

(MECO) offloads intensive mobile computation to clouds located at the edges of cellular networks. Thereby, MECO is envisioned as a promising technique for prolonging the battery lives and enhancing the computation capacities of mobiles. In this paper, we consider resource allocation in a MECO system comprising multiple users that time share a single edge cloud and have different computation loads. The optimal resource allocation is formulated as a convex optimization problem for minimizing the weighted sum mobile energy consumption under constraint on computation latency and for both the cases of infinite and finite edge cloud computation capacities. The optimal policy is proved to have a threshold-based structure with respect to a derived offloading priority function, which yields priorities for users according to their channel gains and local computing energy consumption. As a result, users with priorities above and below a given threshold perform complete and minimum offloading, respectively. Computing the threshold requires iterative computation. To reduce the complexity, a sub-optimal resource-allocation algorithm is proposed and shown by simulation to have close-to-optimal performance.

[theorem]Acknowledgement [theorem]Axiom [theorem]Case [theorem]Claim [theorem]Conclusion [theorem]Condition [theorem]Conjecture [theorem]Criterion [theorem]Exercise [theorem]Notation [theorem]Problem [theorem]Solution [theorem]Summary

=24.5cm

## 1Introduction

The realization of Internet of Things (IoT) will connect tens of billions of resource-limited mobiles, e.g., mobile devices, sensors and wearable computing devices, to Internet via cellular networks. The finite battery lives and limited computation capacities of mobiles pose challenges for designing IoT. One promising solution is to leverage mobile-edge computing [1] and offload intensive mobile computation to nearby clouds at the edges of cellular networks, called edge clouds, with short latency, referred to as mobile-edge computation offloading (MECO). In this paper, we consider a MECO system with a single edge cloud serving multiple users and investigate the energy-efficient resource allocation.

Mobile computation offloading (MCO) (or mobile cloud computing) has been extensively studied in computer science, including system architectures [2], virtual machine migration [3] and server consolidation [4]. It is commonly assumed that the implementation of MCO relies on a network architecture with a central cloud (e.g., a data center). This architecture has the drawbacks of high overhead and long backhaul latency [5] and will soon encounter the performance bottleneck of finite backhaul capacity in view of exponential mobile traffic growth. These issues can be overcome by MECO based on a network architecture supporting distributed mobile-edge computing.

Resource allocation has been widely studied for various types of multiuser communication systems, e.g., TDMA (see e.g., [13]), orthogonal frequency-division multiple access (OFDMA) (see e.g., [14]) and code-division multiple access (CDMA) (see e.g., [15]). Note that all of them only focus on the radio resource allocation. In contrast, for newly proposed MECO systems, both the computation and radio resource allocation at edge clouds need to be jointly optimized for the maximum mobile energy savings, which makes the algorithmic design more complex.

This paper considers a multiuser MECO system based on TDMA. Consider both the cases of infinite and finite cloud computation capacities. The optimal resource-allocation policy is derived by solving a convex optimization problem that minimizes the weighted sum mobile energy consumption. Note that the consideration of MECO simplifies the problem formulation since the long backhaul latency and heavy overhead in central clouds can be neglected. To solve the problem, an offloading priority function is derived that yields priorities for users and depends on their channel gains and local computing energy consumption. Based on this, the optimal policy is proved to have an insightful threshold-based structure that determines complete or minimum offloading for users with priorities above or below a given threshold, respectively. Moreover, to reduce the complexity for computing the threshold, a simple sub-optimal resource-allocation algorithm is designed and shown to have close-to-optimal performance by simulation.

## 2System Model

The model of local computing is described as follows. Assume that the CPU frequency is fixed at each user and may vary over users. Consider an arbitrary time slot. Following the model in [12], let denote the number of CPU cycles required for computing -bit of input data at the -th mobile, and the energy consumption per cycle for local computing at this user. Then the product gives computing energy per bit. As shown in Fig. ?(b), mobile is required to compute -bit input data within the slot, out of which -bit is offloaded and -bit is computed locally. Then the total energy consumption for local computing at mobile , denoted as , is given by . Let denote the computation capacity of mobile that is measured by the number of CPU cycles per second. Under the computation latency constraint, . As a result, the offloaded data at mobile has the minimum size of with , where the function

Next, the energy consumption for offloading is modeled. Let denote the channel gain and the transmission power for mobile . Then the achievable rate, denoted by , is given as:

where is the variance of complex white Gaussian channel noise. The fraction of slot allocated to mobile for offloading is denoted as with , where corresponds to no offloading. For the case of offloading (), the transmission rate is fixed as since this is the most energy-efficient transmission policy under a deadline constraint. Define a function . It follows from that the energy consumption for offloading at mobile is

Note that if either or , is equal to zero.

Last, consider the edge cloud. It is assumed that the edge cloud has finite computation capacity, denoted as , measured as the maximum CPU cycles allowed for computing the sum offloaded data in each slot: . This constraint ensures low latency for cloud computing.

## 3Multiuser MECO: Problem Formulation

In this section, resource allocation for multiuser MECO is formulated as an optimization problem. The objective is to minimize the weighted sum mobile energy consumption: , where the positive weight factors account for fairness among mobiles. Under the constraints on time-sharing, cloud computation capacity and computation latency, the resource allocation problem is formulated as follows:

Several basic characteristics of Problem P1 are given in the following two lemmas.

See Appendix Section 7.1.

See Appendix Section 7.2.

Lemma ? shows that whether the cloud computation capacity constraint is satisfied determines the feasibility of this optimization problem, while the time-sharing constraint can always be satisfied and only affects the mobile energy consumption.

Assume that Problem P1 is feasible. The direct solution of Problem P1 using the dual-decomposition approach (the Lagrange method) requires iterative computation and yields no insight into the structure of the optimal policy. To address these issues, we adopt a two-stage solution approach that requires first solving Problem P2 below that relaxes Problem P1 by removing the constraint on the cloud computation capacity:

If the solution for Problem P2 violates the constraint on cloud computation capacity, Problem P1 is then incrementally solved building on the solution for Problem P2. This approach allows the optimal policy to be shown to have the said threshold-based structure and also facilitates the design of low-complexity close-to-optimal resource-allocation algorithm. It is interesting to note that Problem P2 corresponds to the case where the edge cloud has infinite computation capacity. The detailed procedures for solving Problems P1 and P2 are presented in the subsequent two sections.

## 4Multiuser MECO: Infinite Cloud Capacity

In this section, by solving Problem P2 using the Lagrange method, we derive a threshold-based policy for the optimal resource allocation. Moreover, the policy is simplified for several special cases.

To solve Problem P2, the Lagrange function is defined as

where is the Lagrange multiplier associated with the time-sharing constraint. For ease of notation, define a function . Let denote the solution for Problem P2 that always exists according to Lemma ?. Then applying KKT conditions leads to the following necessary and sufficient conditions:

Based on these conditions, the optimal policy for resource allocation is characterized in the following sub-sections.

Define an (mobile) offloading priority function, which is essential for the optimal resource allocation, as follows:

with the constant defined as

This function is derived by solving a useful equation as shown in the following lemma.

See Appendix Section 7.3.

The function generates an offloading priority value, , for mobile depending on corresponding variables quantifying fairness, local computing and channel. The amount of offloaded data by a mobile grows with an increasing offloading priority as shown in the next sub-section. It is useful to understand the effects of parameters on the offloading priority that are characterized as follows.

Lemma ? can be easily proved by deriving the first derivatives of with respect to each parameter. Moreover, it is consistent with the intuition that, to reduce energy consumption by offloading, the BS should schedule those mobiles having high computing energy consumption per bit (i.e., large and ) or good channels (i.e., large ).

### 4.2Optimal Resource-Allocation Policy

Based on conditions in - and Lemma ?, the main result of this section is derived, given in the following theorem.

See Appendix Section 7.4.

Theorem ? reveals that the optimal resource-allocation policy has a threshold-based structure when offloading saves energy. In other words, since the exact case of rarely occurs in practice, the optimal policy makes a binary offloading decision for each mobile. Specifically, if the corresponding offloading priority exceeds a given threshold, the mobile should offload all input data to the edge cloud; otherwise, the mobile should offload only the minimum amount of data under the computation latency constraint. This result is consistent with the intuition that the greedy method can lead to the optimal resource allocation.

### 4.3Special Cases

The optimal resource-allocation policies for several special cases considering equal weight factors are discussed as follows.

#### Uniform channels and local computing

Consider the simplest case where are identical for all . Then all mobiles have uniform offloading priorities. In this case, for optimal resource allocation, different mobiles can offload arbitrary data sizes so long as the sum offloaded data size satisfies the following constraint:

#### Uniform channels

Consider the case of . The offloading priority for each mobile, say mobile , is only affected by the corresponding local-computing parameters and . Without loss of generality, assume that . Then the optimal resource-allocation policy is given in the following corollary of Theorem ?.

The result shows that the optimal resource-allocation policy follows a greedy approach that selects mobiles in a descending order of energy consumption per bit for complete offloading until the time-sharing duration is fully utilized.

#### Uniform local computing

Consider the case of . Similar to the previous case, the optimal resource-allocation policy can be shown to follow the greedy approach that selects mobiles for complete offloading in the descending order of channel gain.

## 5Multiuser MECO: Finite Cloud Capacity

In this section, we consider the case of finite cloud computation capacity and analyze the optimal resource-allocation policy for solving Problem P1. The policy is shown to also have a threshold-based structure as the infinite-capacity counterpart derived in the preceding section. Both the optimal and sub-optimal algorithms are presented for policy computation.

### 5.1Optimal Resource-Allocation Policy

To solve the convex Problem P1, the corresponding Lagrange function can be written as

where is the Lagrange multiplier corresponding to the cloud computation capacity constraint. Using the above Lagrange function, it is straightforward to show that the corresponding KKT conditions can be modified from their infinite-capacity counterparts in - by replacing with , called the effective computation energy per cycle. The resultant effective offloading priority function, denoted as , can be modified accordingly from that in as

where . Based on above discussion, the main result of this section follows as shown below.

Computing the threshold for the optimal resource-allocation policy requires a two-dimension search over the Lagrange multipliers , using Algorithm ?. For an efficient search, it is useful to limit the range of and as follows.

See Appendix Section 7.5

Note that corresponds to the case of infinite cloud computation capacity and to the case where offloading yields no energy savings for any mobile.

### 5.2Sub-Optimal Resource-Allocation Policy

To reduce the computation complexity of Algorithm ? due to the two-dimension search, one simple sub-optimal policy is designed using Algorithm ?. The key idea is to decouple the computation and radio resource allocation. In Step , based on the approximated offloading priority in for the case of infinite cloud computation capacity, we allocate the computation resource to mobiles with high offloading priorities. Step optimizes the corresponding fractions of slot given offloaded data. This sub-optimal algorithm has low complexity requiring only a one-dimension search. Moreover, its performance is shown by simulation to be close-to-optimal in the sequel.

## 6Simulation Results

The simulation settings are as follows unless specified otherwise. The MECO system comprises mobiles with equal fairness weight factors, namely that for all such that the weighted sum mobile energy consumption represents the total mobile energy consumption. The time slot ms and channels are modeled as independent Rayleigh fading with average power loss set as . In addition, the variance of complex white Gaussian channel noise is W and the bandwidth Mhz. Consider mobile . The CPU computation capacity is uniformly selected from the set Ghz and the local computing energy per cycle follows a uniform distribution in the range J/cycle. For the computing task, both the data size and required number of CPU cycles per bit follow the uniform distribution with KB and cycles/bit. All random variables are independent for different mobiles, modeling heterogeneous mobile computing capabilities. Last, the cloud computation capacity is set as cycles per slot.

For performance comparison, a baseline equal resource-allocation policy is considered, which allocates equal offloading duration for mobiles satisfying and based on this, the offloaded data sizes are optimized.

Figure 1 shows the curves of total mobile energy consumption versus the time slot duration . Several observations can be made. First, the total mobile energy consumption reduces as the slot duration grows. Next, the sub-optimal policy computed using Algorithm ? is found to have close-to-optimal performance and yields total mobile energy consumption less than half of that for the equal resource-allocation policy. The energy reduction is more significant for a shorter slot duration since without the optimization on fractions of slot, the offloading energy of baseline policy grows exponentially with the decrease of allocated time fractions.

The curves of total mobile energy consumption versus the cloud computation capacity are displayed in Figure 2. It can be observed that the performance of the sub-optimal policy approaches to that of the optimal one when the cloud computation capacity increases and achieves substantial energy savings gains over the equal resource-allocation policy. Furthermore, the total mobile energy consumption is invariant after the cloud computation capacity exceeds some threshold (about ). This suggests that there exists some critical value for the cloud computation capacity, above which increasing the capacity yields no reduction on the total mobile energy consumption.

## 7conclusion

Consider a multiuser MECO system based on TDMA. This work shows that the optimal energy-efficient resource-allocation policy for clouds with infinite or finite computation capacities, is featured with a threshold-based structure. Specifically, the BS makes a binary offloading decision for each mobile, where users with priorities above or below a given threshold will perform complete or minimum offloading. Moreover, a simple sub-optimal algorithm is proposed to reduce the complexity for computing the threshold.

### 7.1Proof of Lemma

Since is a convex function, its perspective function, defined as , is still convex. Thus, the objective function, the summation of a set of convex functions, preserves the convexity. Combining it with the linear convex constraints leads to the desired result.

### 7.2Proof of Lemma

Whether Problem P1 is feasible depends on the following two key constraints: and . Assume is satisfied. Then it has

Thus, only when , Problem P1 is feasible.

### 7.3Proof of Lemma

First, we derive a general result that is the root of equation: with respect to as follows.

According to the definitions of and , it has

Thus, the solution for the general equation is

Note that to ensure in Problem P1, we need , which is equivalent to derived from . Then, by substituting and to and making arithmetic operations gives the desired result.

### 7.4Proof of Theorem

First, to prove this theorem, we need the following lemmas.

Since denotes the root of equation for , it has

Thus, based on the definition for Lambert function, we have . Then the desired result follows.

From , for , it has . Since the single-valued Lambert function is monotone increasing for , we can easily obtain the desired result.

Then, consider case 1) in Theorem ?. Note that for mobile , if and , it results in derived from . Thus, if these two conditions are satisfied for all , it leads to . For case 2), if there exists mobile such that or , it ensures . And the time-sharing constraint should be active since remaining time can be used for offloading so as to reduce the transmission energy. Moreover, consider each user . If , from and , should satisfy the following:

Using Lemma ? and Lemma ?, we have the following:

1. If , it has . Then, from , it gives From , and , it follows that .

2. If , it has .

3. If , it has . Combining it with leads to From , and , it follows that .

Furthermore, if , it has . Note that this case can be included in the scenario of with the definition of in .

Last, from , it follows that

where is derived using Lemma ?, completing the proof.

### 7.5Proof of Lemma

If there exists offloading mobile , it must satisfy and . Thus, considering all mobiles, it follows and The latter condition is equivalent to , completing the proof.

### References

1. M. Patel, B. Naughton, C. Chan, N. Sprecher, S. Abeta, A. Neal, et al., “Mobile-edge computing introductory technical white paper,” White Paper, Mobile-edge Computing (MEC) industry initiative, 2014.
2. H. T. Dinh, C. Lee, D. Niyato, and P. Wang, “A survey of mobile cloud computing: architecture, applications, and approaches,” J. Wireless Commun. and Mobile Computing, vol. 13, no. 18, pp. 1587–1611, 2013.
3. Z. Xiao, W. Song, and Q. Chen, “Dynamic resource allocation using virtual machines for cloud computing environment,” IEEE Trans. Parallel and Distributed Systems, vol. 24, pp. 1107–1117, Sep. 2013.
4. S. Srikantaiah, A. Kansal, and F. Zhao, “Energy aware consolidation for cloud computing,” in Proc. HotPower, vol. 10, pp. 1–5, 2008.
5. A. Ahmed and E. Ahmed, “A survey on mobile edge computing,” in Proc. IEEE Intl. Conf. Intel. Sys and Cont, 2016.
6. W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. O. Wu, “Energy-optimal mobile cloud computing under stochastic wireless channel,” IEEE Trans. Wireless Commun., vol. 12, no. 9, pp. 4569–4581, 2013.
7. C. You, K. Huang, and H. Chae, “Energy efficient mobile cloud computing powered by wireless energy transfer (extended version),” [Online]. Available: http://arxiv.org/abs/1507.04094.
8. Y. Mao, J. Zhang, and K. B. Letaief, “Dynamic computation offloading for mobile-edge computing with energy harvesting devices,” submitted to IEEE J. Select. Areas Commun., 2016.
9. X. Xiang, C. Lin, and X. Chen, “Energy-efficient link selection and transmission scheduling in mobile cloud computing,” IEEE Wireless Commmu. Letters, vol. 3, pp. 153–156, Jan. 2014.
10. S. Sardellitti, G. Scutari, and S. Barbarossa, “Joint optimization of radio and computational resources for multicell mobile-edge computing,” IEEE Trans. Signal and Info. Processing over Nectworks, Jun. 2015.
11. T. Zhao, S. Zhou, X. Guo, Y. Zhao, and Z. Niu, “A cooperative scheduling scheme of local cloud and internet cloud for delay-aware mobile cloud computing,” Proc. IEEE Globecom, pp. 1–6, 2015.
12. X. Chen, L. Jiao, W. Li, and X. Fu, “Efficient multi-user computation offloading for mobile-edge cloud computing,” IEEE Trans. Networking, vol. PP, pp. 1–1, Oct. 2015.
13. X. Wang and G. B. Giannakis, “Power-efficient resource allocation for time-division multiple access over fading channels,” IEEE Trans. Info. Theory, vol. 54, pp. 1225–1240, Mar. 2008.
14. C. Y. Wong, R. S. Cheng, K. B. Lataief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747–1758, Oct. 1999.
15. S.-J. Oh, D. Zhang, and K. M. Wasserman, “Optimal resource allocation in multiservice CDMA networks,” IEEE Trans. Wireless Commun., vol. 2, pp. 811–821, Jul. 2003.
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# Confectionery Crime Conundrum

Yesterday I received one of the strangest cases of my career as an I.S. PI (Insightful Sleuths Private Investigator). I was reclining in my chair, working on the latest crossword puzzle when I received a call out of the blue. "You've got to help us, Investigator!" cried the voice on the other end of the line. "We're fresh out of leads and the culprits have nearly escaped!"
Almost tearing my newspaper in excitement, I jumped out of my chair and reached out to my contact at the local police department. With their approval, I set off with a laptop and a spring in my step. When I reached the scene of the crime, I was briefed on the case by Deputy Dependable. The local candy shop had been robbed of its entire stock of confectionery (truffles, sugar cubes, macarons, cotton candy, and more). Although the careless criminals had left behind spades of evidence, there was just one problem: none of the police officers could understand what it meant! "There's something about this evidence that just doesn't square up," complained one of them. "This grid makes it look like they're plotting something."
"We must stop them!" Dep. Dep. exclaimed, "If we follow the right one of these paths, we might be able to get to them in time."
"Deputy, " I remarked, "this evidence doesn't square up, but it's absolutely enough to identify which way they went."

The criminals had left the following pieces of evidence behind:

1. A dead python, bent into an odd shape:

2. Incomplete math homework with too many unknowns:

$$c_{h, r} = 65 +\ S\ \%\ (n+1)$$

3. The following schematic:

4. A curiously folded note, reading:

Going outside is useless. Better look for clues inside!

5. Chain weapons strewn about outside, smeared with cotton candy:

6. Some sort of weird substitution cipher:

1996 48 1005 82
1996 142 1000 117
1996 286 502
1006, 997, 154, 1005
1998, 153, 498, 1996, 2011

7. A different kind of candy from the store on each possible route.

Which way did the sneaky sweet snatchers go, and why didn't the evidence square up?

Hint:

Two images specify puzzle types (both to be applied as sub-puzzles)

Hint #2:

To be or not to be, that is the question...

(Thanks to Stiv and M Oehm in chat and the partial answer, I just came in the end)

First-Trichain

(thanks to Stiv) We see that there are three chains in clue 5, hinting towards a Trichain puzzle. Note that the cotton candy are the same colour as 6 of the numbers in clue 3 - they are our Trichain clues. Solving it, we get this grid:

(Please refer to Stiv's partial answer to see how to solve it)

(thanks to Stiv) Clue 1 seems to indicate the presence of a Slitherlink, showing a snake 'slithering and linking back to its tail'. If we take the whole schematic in clue 3, it is impossible to solve. But what if we use our Trichain? Using the solution from our Trichain puzzle, we try to make a Slitherlink. If we use the unshaded clues, they give us an impossible puzzle, but if we use the shaded clues, they give us a valid Slitherlink:

(Please refer to Stiv's partial answer to see how to solve it)

Interpreting

With the solved Slitherlink, we can see it is a net of a cube with some gray lines inside.

Don't let that trick you, though!

Thanks to clue 4 and the 2nd hint, we realize that if we draw out spaces that are NOT used by the grey lines, we find this:

That's 5=2||, which can be read as $$S = 2^{11}$$, as confirmed by the OP

Last Steps

With that in mind, we can now progress onto the next step. $$S = 2^{11} = 2048$$, which we can plug into clue 2. Using S, we find the new formula: $$c_{h, r} = 65 + 2048 % (n + 1)$$. This had been somewhat puzzling since it was unclear what $$c_{h, r}$$ was intended to be. However, the penny dropped when Stiv recalled that ASCII characters begin at 65. $$c_{h, r}$$ must be a letter code!

Using this revelation to our advantage,

we apply the formula from clue 2 to clue 6, which gives us the following numbers:

116 104 101 121
116 111 111 107
116 104 101
99  117 98  101
114 121 117 116 101

Finally,

we can convert these numbers to ASCII, giving us the solution:

They went the cube route

Final thoughts

this evidence doesn't square up: Of course it doesn't! A cube is a 3D shape after all, while a square is a 2D shape. (OP edit: The cube net is what the pun about the evidence not squaring up was all about)

What is the "cube route"?: They took the route of the sugar cubes!
(Additional note from OP: another pun here about taking the cube root :P)

• Aha, well done to you and M Oehm for working through the last bits until the end :) Here's a +1 for your troubles...
– Stiv
Oct 2, 2021 at 12:46
• Congratulations! Those nasty nougat-nabbing ne'er-do-wells won't be getting away this time!
– Avi
Oct 2, 2021 at 14:00

Here's a partial answer explaining (in detail) what I believe to be the first few steps of the puzzle's solution. However, beyond that point I am as yet unsure how to proceed...

First off, consider the schematic in 3 and the image in 5 together. We can combine these...

...to solve a trichain puzzle. Note that there are three 'chain weapons' (suggesting 'tri-chain') and the colour of the cotton candy coating them matches the colour highlighting 6 particular cells of the grid - these are the trichain clues...

Step 1:

Black spaces can be shaded beside the diagonally adjacent 2 and 3. Their spaces can then be resolved.

Step 2:

Since R5C4 must be unshaded, this fixes the bottom-left corner by follow-on deductions...

Step 3:

R1C3 cannot be shaded, which fixes the 3-space at top-middle, and the rest follows.

It then makes sense to consider clues 1 and 2 together with what we have found.

Clue 1, depicting a snake eating its own tail to form a loop is highly suggestive of a slitherlink puzzle. The whole schematic, as depicted originally in clue 3, does not form the basis of a valid slitherlink puzzle as it stands (for many reasons - most notably the 0-3-3 formation in Row 4 which cannot be satisfied). Instead, we need some way to modify this schematic to enable a valid slitherlink game - how about using the output of our trichain puzzle?

If we consider using all the spaces that were unshaded (left diagram below), we end up with invalid slitherlink clue combinations (e.g. the triplet of 3's in the bottom-right). Meanwhile, if we use all of the clues that were shaded (right diagram below), we produce an alternative option that might be worth investigating...

(I initially attached too much significance to the fact that this grid only contains 0, 1 and 2 clues (no 3's), meaning it might contain a coded message in ternary. However, the two 101 triplets (equivalent to '10' in decimal) would clue two J's or K's (depending on whether you translate via A0Z25 or A1Z26) so this didn't look promising.)

(Also, in the very first version of my solution I made the mistake of assuming we had to add 1 to all of these numbers to make it solvable, misinterpreting the intentions of the mathematical note in clue 2 - that was an error, now fixed below!)

Solving this grid-deduction puzzle then...

Step 1:

Mark all the immediately known X's, where the loop cannot pass.

Now consider the two diagonally adjacent 1's in R2C6 and R3C7. The loop must pass to the Top or Left of the lower-right of these two 1's in order to connect with the other. We can therefore add X's to its Right and Bottom. This forces the line to pass to the Right of the nearby 2 clue, else we would end up with a loop end trapped in the upper-right corner.

In fact, (and I only spotted this afterwards) we couldn't put the loop to the Left of this 1-clue either, else it would have ended up needing two sides shading! We actually end up with several immediate knock-on deductions that complete resolve the right half of the grid:

Step 2:

Consider now the 1 in R3C4 - can we pass it to its Left? No - or the loop can never fully form:

How about to its Right? No - for the same reason:

Well, what about to it's Bottom then?! No! Again, the loop gets broken:

In which case, it must be passed to its Top! Simultaneously, to ensure we end up with a loop, the segment to the Top of the 1 clue in R5C3 must be shaded:

Step 3:

Finally, there is only one way to resolve the bottom part of the grid, the 1 clue in R4C2 must be passed via the Bottom (resolving the centre-left), and the top-left loop path can be deduced as a result:

What does this give us? Well, it looks a lot like:

the net of a cube! (This explains the pun in the question about the evidence 'squaring up'!)

Overlaying this on the original grid results in this:

The question then is where to go next? It seems that clue 4 is the most promising option...

since the talk of 'looking inside' suggests we should look inside the slitherlink loop / cube net for clues.

However, I'm not yet seeing how to interpret this information. Still thinking about that...

Clues so far unused: 2, 6 and 7.

• If we sum the numbers on each face (excluding those already used), it seems to clue three numbers on the left side of clue 2 (and partial information on the right side). Oct 1, 2021 at 15:05