A fire extinguisher is a device used to put out a fire, often in an emergency situation. They consist of a pressurised container of chemicals that when discharged can put out a fire. It is important to familiarise yourself with the use of fire extinguishers in your vicinity, as improper or untimely use may be counterproductive.
The first version of the modern fire extinguisher was invented in the United Kingdom by Captain George Manby in 1816, consisting of a copper vessel of 3 gallons (13.6 litres) of pearl ash (potassium carbonate)solution under compressed air pressure.
The late 19th century saw the invention of the Soda-Acid extinguisher where a cylinder contained 1 or 2 gallons of water that had sodium bicarbonate mixed in it. Suspended in the cylinder was a phial containing concentrated sulphuric acid. Activating the extinguisher by striking a plunger to break the phial, or inverting the extinguisher to release a lead bung from the phial (depending on type), released the acid which mixed with the bicarbonate solution to generate carbon dioxide gas that expelled the water under pressure through a nozzle or short length of hose.
Around 1912 Pyrene pioneered the carbon tetrachloride or CTC extinguisher, where the liquid was expelled from a brass or chrome container by handpump, usually of 1 imperial quart (1.1 L) or 1 imperial pint (0.6 L) capacity but also made in up to 2 imperial gallon (9 L) size, onto a fire. The CTC vapourised and extinguished the flames by chemical reaction. This extinguisher was suitable for liquid and electrical fires and was popular in motor vehicles for the next 60 years. The vapour and combustion by-products were highly toxic and deaths did occur from using these extinguishers in confined spaces.
A fire extinguisher from Germany.
Class A: Fires involving common combustibles such as wood, grass, paper, plastic's and rubber.
Class B: Fires involving flammable liquids such as kerosene, diesel fuel, oil and petrol.
Class C: Fires involving flammable gasses like LPG, Natural gas, and acetylene.
Class D: Fires involving flammable metals such as magnesium.
Class E: This is not a class of fire per se but is used to refer to electrical fires.
Class F: Fires involving preheated fats and oils such as fires in a deep fat fryer.
In the United States, the National Fire Protection Association has created five classes of fire extinguishers, A, B, C, D, and K.
Class A extinguishers are used for putting out fires of general combustibles such as wood or paper. These usually contain a mixture of water and compressed gas. The symbol is a triangle around the letter A.
Class B extinguishers are intended for use on liquids or grease and are distinguished by a square around the letter B.
Class C extinguishers are used on electrical fires. Their mixture is non-conducting to ensure the user is not electrocuted.
Class D extinguishers are used on fires of flammable metals and certain other exotic chemicals. They are designated with a star around the letter D. These are rarely encountered and much more expensive than class A, B, or C extinguishers. Class D extinguishers are specialised for a particular hazard, and require additional operator training.
Class K extinguishers are used in Kitchens and environments with the high probability of grease fires. These are rarer because the category has only existed since 1998, so many kitchens are still equipped with Class B extinguishers.
Water is the most common chemical for class A fires and is quite effective as one would imagine. Most water based extinguishers also contain traces of other chemicals to prevent the extinguisher rusting. Some also contain wetting agents which help the water penetrate deep into the burning material and cling better to steep surfaces. Water works to extinguish a fire by simply cooling it below the ignition point, although large amounts of water can also exclude oxygen. However, water will merely exacerbate other fire classes. For instance, water sprayed over burning liquid petroleum merely spreads the flames around.
Foams are commonly used on class B fires, and are also effective on class A fires. These are mainly water based, with a foaming agent so that the foam can float on top of the burning liquid to exclude oxygen. Ordinary foams are designed to work on nonpolar flammable liquids such as petrol (gasoline), but may break down too quickly in polar liquids such as alcohol or glycol. Facilities which handle large amounts of flammable polar liquids use a specialised "alcohol foam" instead. Alcohol foams must be gently "poured" across the burning liquid. If the fire cannot be approached closely enough to do this, they should be sprayed onto an adjacent solid surface so that they run gently onto the burning liquid. Ordinary foams work better if "poured" but it is not critical.
A "protein foam" was used for fire suppression in aviation crashes until the 1960s development of "light water", also known as "Aqueous Film-Forming Foam" (or AFFF). Carbon dioxide (later sodium bicarbonate) extinguishers were used to knock down the flames and foam used to prevent re-ignition of the fuel fumes. "Foaming the runway" can reduce friction and sparks in a crash landing, and protein foam continued to be used for that purpose, although FAA regulations prohibited reliance upon its use for suppression.
BC powder is either sodium bicarbonate or potassium bicarbonate and calcium carbonate, finely powdered and propelled by carbon dioxide or nitrogen. Similarly to halon (see below) - but less effectively - these powders actually interrupt the chemical processes of the fire. They also provide some cooling and exclusion of oxygen, although this effect is relatively weak. These powders thus provide rapid knockdown of flame fronts, but may not keep the fire suppressed. Consequently, they are often used in conjunction with foam for attacking large class B fires. BC extinguishers are often kept in small vehicles since they provide good knockdown of a rapidly flaring class B fire, from a small package.
ABC powder is monoammonium phosphate and/or ammonium sulfate. As well as suppressing the flame in the air, it also melts at a low temperature to form a layer of slag which excludes oxygen from the fuel. For this reason it can also be effective against class A fires. ABC powder is usually the best agent for fires involving multiple classes. However it is not effective against three dimensional class A fires, or those with a complex or porous structure. Foams or water are better in those cases.
Both types of powders can also be used on class C fires, but provide a significant cleanup and corrosion problem that is likely to make the electrical equipment unsalvageable.
Most class F (class K in the US) extinguishers contain a solution of potassium acetate, sometimes with some potassium citrate or potassium bicarbonate. The extinguishers spray the agent out as a fine mist. The mist acts to cool the flame front, while the potassium salts saponify the surface of the burning cooking oil, producing a layer of foam over the surface. This solution thus provides a similar blanketing effect to a foam extinguisher, but with a greater cooling effect. The saponification only works on animal fats and vegetable oils, so class F extinguishers cannot be used for class B fires. The misting also helps to prevent splashing the blazing oil.
Halons are very versatile extinguishers. They will extinguish any type of fire except class D and are highly effective even at quite low concentrations. They are the only fire extinguishing agents that are quite suitable for discharge in aircraft (carbon dioxide is too toxic in confined spaces, and other materials pose a corrosion hazard to the aircraft). They work by breaking the chemical reaction of the fire. Halons are chlorofluorocarbons and are being phased out for more environmentally-friendly alternatives. Halon fire extinguishers may cost upwards of 800 USD due to production and import restrictions.
In the UK and Europe Halons were made illegal at the end of 2003, except for certain specific aircraft and law enforcement uses.
Recently Dupont has begun marketing several nearly saturated fluorocarbons under the trademarks FE-13, FE-25, FE-36, FE-227, and FE-241. These materials are claimed to have all the advantageous properties of halons, but lower toxicity, and zero ozone depletion potential. They require about 50% greater concentration for equivalent fire quenching.
METL-X, finely powdered sodium chloride (salt) propelled by carbon dioxide or argon. After a crust at least 50 mm (2 inches) thick has formed over the burning metal, it is picked up with a long handled shovel and placed in a bucket of salt or very dry sand, and additional agent poured on top. Suitable for sodium, potassium, magnesium, titanium, aluminium, and most other reactive bulk metal fires, but not lithium or finely powdered metals.
Finely powdered graphite, applied with a long handled scoop, is preferred for fires in fine powders of reactive metals, where the blast of pressure from an extinguisher may stir up the powder and cause a dust explosion. Graphite both smothers the fire and conducts away heat.
Finely powdered copper propelled by compressed argon is the currently preferred method for lithium fires. It smothers the fire, dilutes the fuel, and conducts away heat. It is capable of clinging to dripping molten lithium on vertical surfaces. Graphite can also be used on lithium fires but only on a level surface.
Other materials sometimes used include powdered sodium carbonate, powdered dolomite and argon gas.
Very dry sand may be used to smother a metal fire if nothing better is available. It should be applied with a long-handled shovel to avoid the operator receiving flash burns, then the molten mass transferred to a bucket of dry sand. Note that even the smallest trace of moisture may result in a steam explosion, spattering burning molten metal around. For this reason salt is sometimes preferred, since it is more obvious if it becomes damp.
To operate safely and effectively, extinguishers should be subject to regular maintenance by a competent person and most countries in the world require this maintenance as part of fire safety legislation.
Lack of maintenance can lead to an extinguisher not discharging when required, or worse still, rupturing when pressurised. Deaths have occurred, even in recent times, from corroded extinguishers exploding.
Overhaul:-- CO2 extinguishers, due to their high operating pressure, are subject to pressure vessel safety legislation and must be hydraulic pressure tested & date stamped every 10 years.
In the US, maintenance requirements can be more stringent with 6 monthly servicing and 5 yearly hydraulic pressure testing for all types of extinguisher.
This page was last modified 00:09, 16 Jun 2005.
This page has been accessed 6699 times.


Minx nails. Don't the photographer at my graduation today fully appreciated, but I simply didn't care. Think I'm in love. Oh dear, this could get out of hand. Well at least I will have gorge nails to look at everyday!
Ruth Writes is a 30-something Liverpool based blog covering mid-size fashion, pescatarian food and beauty.


Stopped for lunch with my family of four. Rast was sefrved cold with a lot of bother. Overpriced soft drinks and a 'disconnected' rude landlady. Poor choice of wines (no merlot), ciders and beers. Avoid if you can.


By Melusina Bisson. Golf Club. Published at Thursday, March 22nd, 2018 11:01:09 AM.
By Viollette Provencher. Golf Club. Published at Thursday, March 22nd, 2018 11:01:05 AM.
By Vignette Beaule. Golf Club. Published at Thursday, March 22nd, 2018 11:01:08 AM.


Yes, we can break everything in 3.x but ... do we want to start from scratch?
logging libraries because each framework is using a different one.
is huge advantage and I don't want to loose it.
> not when looking at recent activity of the past years.
> to work as wanted.
> If we want to use something which is longer on the market, then slf4j.
>> currently either Log4j, Logback, java.util.logging or to SLF4j.
>>>> LoggerFactory - the rest is delegated to given logging library.
>>>>>>> in, I'd have to fill out the forms to become a committer on Struts.
>>>>>>> would I find that information?
>>>>>> custom solution. Please can we explore this topic a bit?


text:
Rassie Erasmus Faces a Monumental Task
Kyle Dutton
JOHANNESBURG, SOUTH AFRICA - MARCH 01: Rassie Erasmus is announced as the new Springbok coach during the Announcement of New Springbok Management at SuperSport Studios on March 01, 2018 in Johannesburg, South Africa. (Photo by Sydney Seshibedi/Gallo Images)
Any person taking over at the helm of the Springboks, as Director of Rugby and Head Coach, finds themselves in a very tough position. Fans will want results from the beginning, but this won't be sustainable if Springbok rugby stays in the dark ages in terms of structures. Somehow these two goals need to go hand in hand, which isn't easy to accomplish, and Rassie Erasmus will need some staunch support from the team he has surrounded himself with.
Rassie Erasmus: a Monumental Task
There is no doubting the raw talent that South African rugby has at its disposal. Genetically blessed with physical specimens that easily match or supersede those of other nations, plus school coaches who nurture rugby players in allowing them to express themselves on the field. No matter which schoolboy team is sent to international competitions, like the Sanix Tournament in Japan, they usually end up coming up trumps.
Following on from schoolboy level, there are local competitions like Varsity Cup and Gold Cup, which should allow for this talent to further shine through, and in most cases it does. The beauty of these competitions is that these players are treated as amateurs, either holding jobs outside of the game or studying towards degrees in all sorts of fields. More emphasis should be placed on these amateurs and semi-pros!
The Professional Structure
The problem area lies in the professional space. There are over 950 contracted rugby players in the country – which is probably double the amount needed. Compare that with the 200-odd that current Six Nations Champions, Ireland, have on their national books; and you'll be able to put this in perspective.
This causes a whole host of potential problems. Firstly, there is a plethora of youngsters trying to make it as professionals in unions that don't allow for progression. They lie stagnant, waiting for an opportunity that may never come. They then struggle to transition out of the sport when they decide to 'retire'.
Loftus Morrison of Leopards in action with Grant Janke and Reg Muller of Valke during the Absa Currie Cup Division 1 Semi Final match between Leopards and Hino Valke at Profert Olen Park on October 02, 2015 in Potchefstroom, South Africa. (Photo by Gallo Images/Getty Images)
The majority of owners, managers and coaches of these unions have been there since the 1980s and they haven't changed the way they manage or coach rugby players. Agents also want to keep the players in the system as they only earn money if the player keeps playing, which is sad.
Big lumps of human end up getting taught to run at the man instead of at the space. Skill levels that were honed at schoolboy level are left to fade away into the sunset and the rugby public then wonder why the lucky break for these players doesn't come.
Changing the structure
This is where Rassie and his team should focus. To fundamentally change the way rugby is structured at a professional, semi-professional and amateur level that allows for talent to be nurtured the whole way through. At the same time, more focus needs to be put on the life of a player outside of the sport so that we are not faced with so many stories of failure post retirement.
As a suggestion; they should reduce the number of unions to the six biggest catchment areas. Each union then has a cap on the number of players they can contract, to begin with it could sit at 100 across all age groups. This gives us 600 contracted rugby players.
The rest then filter down to their local clubs or universities. All contracted players coming back from injury must follow a process of playing for their local team (club or university) before being brought back into the professional fold. Imagine what this would do for the semi-professional and amateur fan base and viewership!
Changing the structure, both politically and managerially in any institution in South Africa is very difficult, but this in the only way to ensure that any future Springbok team consistently ranks in the top three in the world.
Previous articleIsrael Folau story growing day-by-day
Next articleBrad Shields considered to play for England during June Internationals
Kyle is based in Johannesburg and is a passionate sports fan. A former finance professional, he decided to change tack slightly, combining his love of sport with his previous career. As a Director at The Sport Exchange, he looks to provide a holistic service for managing athletes, specifically focusing on the transition into life after sport.

redpajama_set_name: RedPajamaCommonCrawl


Is one dead pixel a returnable offense?
Discussion in 'Buying Tips and Advice' started by ThatWasAwkward, Feb 4, 2006.
I just got my 17" Core Duo iMac yesterday. It's pretty great overall, but there is one tiny dead pixel off on the right side of the screen. It's not that big of a deal, I admit. Nonetheless, it somewhat irks me. Would Apple allow me to trade it in for another one because of that? I'm still debating whether or not it's worth the trouble.
I think it's highly unlikely they'll let you. AFAIK, there's no "official" policy, but you generally need a lot more than one dead pixel.
There's no official return policy? That seems kind of odd to me.
Not for dead pixels. Again - as far as I know.
depends if in your country their is a restocking fee, in the uk i think their is a 14 day return policy so you can return it "because you dont like it" and try your luck on a new one.
or you could just deal with it or try rubbing it out, which works 80% of the time.
yes there is a pixel policy.
Many Apple products use active-matrix LCD panels, including the iMac (Flat Panel), iBook, recent PowerBook computers, Apple Cinema displays, and iPod models with a color display. In addition to being slim and light, active-matrix LCD technology provides customers with many visual performance advantages when compared to traditional cathode-ray tube- (CRT) based displays, such as increased brightness, sharpness, and contrast ratio.
Active-matrix LCD technology uses rows and columns of addressable locations (pixels) that render text and images on screen. Each pixel location has three separate subpixels (red, green and blue) that allow the image to be rendered in full color. Each subpixel has a corresponding transistor responsible for turning the subpixel on or off.
There are typically millions of these subpixels on an LCD display. For example, the LCD panel used in the Apple Cinema HD display is made up of 2.3 million pixels and 6.9 million red, green, and blue subpixels. Occasionally, a transistor does not work perfectly, which may result in the affected subpixel being turned on (bright) or turned off (dark). With the millions of subpixels on a display, it is quite possible to have a low number of faulty transistors on an LCD. Therefore, a certain number of subpixel anomalies is considered acceptable. Rejecting all but perfect LCD panels would significantly increase the retail price for products using LCD displays. These factors apply to all manufacturers using LCD technology--not just Apple products.
Rubbing it out? Like... just rubbing the screen where the dead pixel is?
Ah, thanks. I guess they probably won't take it back, then. Oh well.
In the UK that depends on whether you bought it in a store or at a distance. If you bought it in a store it is assumed that you would take the time to check that the product is what you want so this rule does not apply.
If you buy over the phone or internet where you cannot check the product out then you have 14 days to check it out and return it if it is not what you want.
Typically if you just bought it Apple will do the return. Especially if you bought it at an Apple store. I've noticed the apple store likes to make all their customers leave happy. I'd try a in store exchange if you bought it at an apple store. Most of the employees don't care. If the computers is only a few days old they just label it as DOA and give you a new one. Seriously it's that easy. It gets tricker when the computer is out of it's return date because then they have to do the work on the monitor and are forced to follow apple's dead pixel policy.
it depends on you and your determination. I have known people who have returned displays and laptops with one dead pixel. They were incredibly insistent but polite. IF you keep talking, eventually they may well grease your wheels.
I don't know about that. I've owned/serviced many (20+) LCD's with dead/stuck pixels. Not one has been resolved using this rubbing method.
Just to clarify a little, I ordered it from apple.com. I didn't physically buy it in a store.
my ibook came with 5 dodgy pixels, i rubbed 4 of them out one has a stuck green bit at the bottom but it's barely noticeable.
also i got rid of two red stuck pixels on my sisters ibook using the method, thankfully i have none on any other lcd's i own.
This is from another thread...it helps with stuck pixels. just play the video over the effected area.
Also, if you exchange it, since it is technically under spec, you run the risk of getting one with 2, 3, 4 + dead pixels.
Well, thanks everybody. I'll probably just keep it and try out that pixelmurder file and see if it helps. Does anybody here have any experience using it? Maybe some tips? I've read some comments from people about it that say that it actually made things worse for them.


Makes a secure hold and professional finish on crimp beads and crimp tubes.
Manufacturer: Darice. Quantity: 1 per pack. Size: 5" long. Color: Black. Composition: Metal. Makes a secure hold and professional finish on crimp beads and crimp tubes.


It looks like we still got a few of these GE PLC around. A 90-30 and a Series one. Hoping someone familiar with these can help me with these questions below. The longer term plan is to replace them with AB.
1. From searching, it looks like Series One is the same as Automation Direct DL 305 but 90-30 is a different thing completely, correct?
For the 90-30 PLC, we found a copy of VersaPro 2.03 CD, we got it installed in trial mode in XP virtual machine but can't get comm going even after check for everything. We were able to open an offiline file though.
2. I also found a Proficy Machine Editon V5.00 CD today. Can I run this on Win 7?
3. Does VersaPro or Proficy ME work like RSLogix where the offline file reflect the comm setting the PLC port is set to? It seems weird that both port1 and port2 are set to "RTU" in the offline program.
The 90-30 is a model 363, there are 3 comm port I see, one DB15 on the Power supply, one DB15 on the CPU, and a RJ11 on the CPU. both DB15 are wired up and goes "somewhere".
4. The RJ11 is open, can I use the RJ11 to communicate? The manual shows a cable IC693CBL316 which I can make.
-I emailed GE about the price to upgrade the Proficy ME to the latest edition, haven't heard back yet.


text:
Brand Collective
Back to all client stories Ergobaby's client story
Ergobaby is headquartered in Los Angeles, California and is currently sold in more than 700 retailers in the United States, and in over 50 countries internationally.
Founded in 2003 by Karin Frost in Maui, Hawaii
Sold in more than 50 countries worldwide
700+ retail locations
Ergobaby already had a Digital Asset Management (DAM) system in place, but their distributors, and their own creative team, weren't happy with the usability or customer service it was providing.
The structure of the system was hard to work with. Our distributors would log in and then be logged off automatically. This led to a lot of calls from our partners thinking they were locked out of the software. Basically, our old DAM created more work for us, and when we would call customer service to resolve the issue, they would never get back to us.
In this case study, we sat down with Ergobaby's Creative Director Perry Hampton to discuss when Ergobaby knew it was time for a new platform, how they approached the selection process, and how Brandfolder has been a better fit for their team.
Ergobaby's checklist for a new DAM:
A beautiful front end
Onboarding was very streamlined and simple. Brandfolder has a really friendly sales and customer service team. They're easy to get a hold of, and quick to respond to any questions we have.
- Perry Hampton, Creative Director at Ergobaby
Ergobaby knew a DAM platform was crucial to their marketing strategy early on. It didn't take them long, however, to realize that the DAM they had chosen was not the right fit for their business. "We were unhappy with the platform, as well as the customer service, so we knew it was time for a switch," said Perry Hampton, Creative Director at Ergobaby.
Hampton and his team also found the platform outdated and clunky. "It was hard for our users to find files. You had to click through so many folders to get to the right file!"
And so, Ergobaby began searching for a new DAM. "I came across the site G2 Crowd and read all of the reviews and comparisons with other DAM companies. Brandfolder seemed like the best fit for our needs." Of the Brandfolder platform, Hampton says his team really loves how easy it is to use. Being able to search for files quickly and efficiently is a huge priority.
It really comes down to the simplicity of finding assets quickly and efficiently. It also helps that it's more visual. You see what you need right away, instead of seeing a tiny thumbnail and not quite knowing what you're getting.
Hampton says distributors are their most active users. "We don't provide dedicated support to them, so the DAM needs to be easy to use and intuitive. We don't have time to be fielding support requests." Ergobaby's Brandfolder ensures their distributors, from small shops to large box stores, have the correct and updated assets they need to advertise and sell Ergobaby products online and in their brick-and-mortar retail stores. Distributors get high-quality, creative-approved, 360-degree product photos of each item Ergobaby sells. They also gain access to curated lifestyle images, merchandising information, and usage guidelines for the Ergobaby brand and the products they're promoting.
Their sales team also shares assets stored in the Ergobaby Brandfolder with retailers. And Hampton's creative team uses Brandfolder to store videos, lifestyle photos, product photography, and other brand assets, including their company overview and brand book. They also use it to house seasonal advertising and social media campaign collateral, as well as training guides for new mothers. "It has definitely saved our executives frustration fielding calls from exasperated distributors. It really comes down to the simplicity of finding assets quickly and efficiently. It also helps that it's more visual. You see what you need right away, instead of seeing a tiny thumbnail and not quite knowing what you're getting."
When asked about adoption rates among the Ergobaby team, Hampton says, "We have around 80% of our staff using Brandfolder. The other 20% don't really need access to our DAM. Everyone internally really likes the system — or else they wouldn't have signed up for it!"
Interested in learning how Brandfolder can help you manage, distribute and analyze your brand assets?
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Nautical is an AkzoNobel brand. AkzoNobel is one of the largest global paints and coatings companies and a major producer of specialty chemicals. We supply industries and consumers worldwide with innovative products and are passionate about developing sustainable answers for our customers. Our portfolio includes well known brands such as Dulux, Sikkens, International and Eka. Headquartered in Amsterdam, the Netherlands, AkzoNobel is a Global Fortune 500 company and is consistently ranked as one of the leaders on the Dow Jones Sustainability Indexes. With operations in more than 80 countries, our 55,000 people around the world are committed to excellence and delivering Tomorrow's Answers Today™.


The technology group Wärtsilä has been selected to provide its Wärtsilä 31 main engine, the propulsion machinery, and the auxiliary engines for a new state-of-the-art pelagic trawler under construction at a Norwegian shipyard. The vessel has been ordered by Research Fishing Co based in Lerwick, Shetland Islands. The order with Wärtsilä was booked in April 2017. There is an option for a second vessel.
In specifying the Wärtsilä solutions, the customer cited the need for the latest technologies in order to optimise the total efficiency of the vessel. The Wärtsilä 31 engine has been recognised by Guinness World Records as being the world's most efficient 4-stroke diesel engine. This will be the first 12-cylinder version of this engine ordered.
In addition, Wärtsilä will supply the gearbox, the controllable pitch propeller with the Wärtsilä ProTouch propulsion control system, as well as one 8-cylinder and one 6-cylinder Wärtsilä 20 auxiliary engines. Delivery of the Wärtsilä equipment is scheduled to commence in November 2017.
"The Wärtsilä 31 engine is in a class of its own regarding fuel efficiency and total cost of ownership. Its efficiency reduces exhaust emission levels, and provides extended intervals between service requirements. We are proud to have been selected to provide a complete package of solutions for this extremely modern fishing vessel," says Stefan Wiik, Vice President, Engines, Wärtsilä Marine Solutions.
"This vessel has been designed to meet our needs well into the future. For this reason the machinery onboard has to be the best available, and we believe that by selecting Wärtsilä this requirement is achieved," says Gary Williamson, Skipper & Co owner.
When delivered in the autumn of 2018, this 79.8 metre long Skipsteknisk designed vessel will operate in Scottish fishing grounds.
Caption: The new trawler being built for Reseach Fishing Co will be equipped with a Wärtsilä 31 main engine, Wärtsilä propulsion machinery and auxiliary engines.


A "Lesson with a Star" was given today, September 13, in Comprehensive School No. 31 in St. Petersburg. The first Olympic wrestling champion of Russia among women (London 2012), silver medalist of the Olympic Games 2016 in Rio Natalia Vorobieva met with schoolchildren to tell and show them a lot. This meeting and the whole "Lesson with a Star" project was initiated by FIAS President Vasily Shestakov, who visited the school together with the champion.
During the official part of the meeting, Vasiliy Shestakov and Natalia Vorobieva answered the children's questions in the school's assembly hall. Representatives of the younger generation were interested in the details of the Olympic performances, asked about the most important qualities that a high-level athlete must have, asked about freestyle wrestling and SAMBO, found out where the Olympic medals were kept and whether those were really made of precious metals.
Next came the practical exercises. Natalia Vorobieva dressed in a typical training uniform and gave a champion master class for the children on a wrestling mat. It is worth noting that local students really know how to fight, because they are actively engaged in SAMBO. Vasileostrovsky district of St. Petersburg, along with Nevsky, is the most active participant of all parts of the city in the "SAMBO in St. Petersburg Schools" project. Five schools of the district have SAMBO groups, and school No. 31 won the first place this year at the regional tournament.
The "Lesson with a Star" will definitely help further enhance the pupils' leadership in school district competitions, and will give the children impetus and motivation for the sport. It is also possible that by the time they finish school, SAMBO wrestlers around the world will already be preparing to fight for Olympic medals.


An illustrated handbook to the terracotta garden pottery of Archibald Knox and Mary Seton Watts that decorated Arts and Crafts gardens in Britain and America. Jardinières, sundials, fountains and garden seats ornamented with symbolic Celtic Revival and Art Nouveau motifs, were produced for Liberty & Co by Carter & Co, and for the Potters' Arts Guild in Surrey and Inverness-shire. Liberty's announced in 1904, 'Never before, in this country at least, has the Garden Pot been treated as an item, per se, of decorative skill'.
Watts Chapel at Compton, Surrey, is one of the most remarkable Art Nouveau buildings in Britain. Designed on Ruskinian principles by the craftswoman Mary Seton Watts, wife of the Victorian artist G.F. Watts, its profuse terracotta symbolism interwoven within Celtic knotwork evokes the art of ancient civilisations worldwide. The exuberant interior is ablaze with jewel-like polychrome gesso decoration. Comparing a symbol to a magic key, Mary Watts observed that on the one hand it was just a piece of quaintly wrought iron or modelled clay, on the other, it unlocked a door into a world of enchantment.
20 Charlwood Road, London SW15 1PE.


text:
\section{Introduction}

Let $\mathscr{X}$ denote a bounded subset of $\mathbb{R}^k$, such as the unit ball $B_1$ with respect to the $\ell^2$-norm,
and $\mathscr{A}$ denote a quantization alphabet (a finite subset of $\mathbb{R}$), such as an arithmetic
progression. Given a finite frame, i.e., a list of $m$ vectors
$F:=\{f_1,\dots,f_m\}$ that span $\mathbb{R}^k$,
we consider
\begin{equation}\label{synthesis-dist}
 \mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X}) := \sup_{x \in \mathscr{X}} \inf_{q \in \mathscr{A}^m} \left \| x - \sum_{i=1}^m q_i f_i \right \|_2
\end{equation}
which represents the smallest error of approximation that can be attained uniformly over $\mathscr{X}$ by
considering linear combinations of elements of $F$ with coefficients in $\mathscr{A}$. We will call this
quantity the {\em synthesis distortion} associated with $(F,\mathscr{A},\mathscr{X})$ (hence the subscript `s').
Understanding the nature of
the synthesis distortion and finding quantization algorithms (i.e., maps
$x \in \mathscr{X} \mapsto q \in \mathscr{A}^m$) that achieve or even
approximate this quantity is generally an open problem, usually referred to as
the ``Synthesis Problem'' in the theory of quantization of frame representations
\cite{powell2013quantization}. The synthesis distortion can also be phrased as a generalized
linear discrepancy \cite{chazelle2002discrepancy,MathAtoD,matousek_discrepancy}.

Consider any other frame $E := \{e_1,\dots,e_m\}$ in $\mathbb{R}^k$ that is dual to $F$, i.e.
\begin{equation}\label{dual-frame}
\sum_{i=1}^m \langle x,e_i \rangle f_i = x \mbox{ for all } x \in \mathbb{R}^k.
\end{equation}
In this paper we will adopt a matrix-based notation, and
write $F$ for the matrix whose columns are $(f_i)_1^m$ and $E$ for the matrix whose
{\em rows} are $(e^\top_i)_1^m$. (Depending on the role it assumes,
$E$ will be called the analysis frame or the sampling operator, and similarly
$F$ will be called the synthesis frame or the reconstruction operator.)
It is then evident that \eqref{dual-frame} is equivalent to $FE = I$
and also that duality of frames is a symmetric relation.
We define the {\em analysis distortion} associated with $(E,\mathscr{A},\mathscr{X})$ by
\begin{equation}\label{analysis-dist}
 \sD_{\mathrm{a}}(E,\mathscr{A},\mathscr{X}) := \inf \{\mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X}) : FE= I \}.
\end{equation}

The analysis distortion has the following interpretation: Given an analysis frame $E$,
the encoder chooses a quantization map $Q:\mathbb{R}^m \to \mathscr{A}^m$ and the decoder chooses
a synthesis frame $F$ that is dual to $E$. The encoder maps each $x \in \mathscr{X}$ (or equivalently,
the measurement vector $y := Ex$) to its quantized version $q := Q(Ex))$, and using $q$,
the decoder recovers an approximation to $x$ in the form of $\hat x := Fq$.
The optimal choice of $Q$ and $F$ (dual to $E$) yields the analysis distortion. We will refer to this optimization problem as the ``Analysis Problem''. (Also see \cite{powell2013quantization}.)

There are some remarks to be made regarding the definition of the analysis distortion and
its above interpretation. First, the assumption that
the reconstruction must be ``linear'' is made for practical reasons. Non-linear methods (such as
{\em consistent reconstruction} \cite{thao1996lower,goyal1998quantized}) have also been considered
in the literature for the above type of encoders, but these methods generally come with higher
computational cost. Second, the assumption that $F$ be dual to $E$ is made so that
the reconstruction is guaranteed to be exact in the high-resolution quantization limit, or simply when
there is no quantization. It should be noted that
this assumption is also meaningful in order
to make sense of how the choice of $E$ might influence the reconstruction accuracy: without any
assumptions on $F$ or $Q$, $F$ can be chosen optimally (in the sense
of minimizing the synthesis distortion for $\mathscr{X}$) and $Q$ can be chosen to produce the corresponding
optimally quantized coefficients, thereby erasing the influence of $E$.
To make this precise, let us define
\begin{equation} \label{DEQFX}
 \mathscr{D}(E,Q,F,\mathscr{X}) := \sup_{x \in \mathscr{X}} \| x - F Q (Ex) \|_2
\end{equation}
for any pair of frames $(E,F)$ (not necessarily dual) and any
quantization map $Q:\mathbb{R}^m \to \mathscr{A}^m$.
The infimum in \eqref{synthesis-dist} is achieved for every $x$, so there is an optimal
$q^*(x) \in \mathscr{A}^m$ that yields
$$ \mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X}) = \sup_{x\in \mathscr{X}} \| x - F q^*(x) \|_2.$$
Since $E$ is injective, a map $Q^*:\mathbb{R}^m \to \mathscr{A}^m$ can be found such that $Q^*(Ex) = q^*(x)$, yielding
$$ \mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X}) = \mathscr{D}(E,Q^*,F,\mathscr{X}) =  \inf_Q  ~\mathscr{D}(E,Q,F,\mathscr{X}). $$
This relation is valid for all $E$ and $F$. Fixing $E$ and minimizing over its dual frames $F$ now
results in the interpretation of the analysis distortion given in the previous paragraph:
\begin{equation} \label{DaEAXeq}
\mathscr{D}_{\mathrm{a}}(E,\mathscr{A},\mathscr{X}) = \inf_{(Q,F) : FE = I}  ~\mathscr{D}(E,Q,F,\mathscr{X}).
\end{equation}
Finally, it should be noted that in the setup of this paper
we do not allow the decoder to pick a different linear map $F$
depending on the quantized vector $q$ that it receives.
While such an adaptive method could be implemented in practice, it is not much different from (and
sometimes equivalent to) using an arbitrary non-linear decoder.

The distortion performance of any given encoder-decoder scheme (adaptive or not) that
uses at most $N$ codewords for $\mathscr{X}$ is constrained by the
entropic lower bound $\varepsilon(N,\mathscr{X})$ which is defined to
be the smallest $\varepsilon>0$ for which there exists an $\varepsilon$-net for $\mathscr{X}$ of cardinality $N$.
Then we have
\begin{equation}\label{dist-relations-1}
\mathscr{D}(E,Q,F,\mathscr{X}) \geq \mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X})
\geq \mathscr{D}_{\mathrm{a}}(E,\mathscr{A},\mathscr{X}) \geq \varepsilon(|\mathscr{A}|^m,\mathscr{X})
\end{equation}
where the second inequality assumes that $FE=I$.
The entropic lower bound depends on $\mathscr{A}$ only through its cardinality, so it will not necessarily be
useful without a suitable matching between $\mathscr{A}$ and $\mathscr{X}$. For example, if
the convex hull of $F(\mathscr{A}^m)$ is small compared to $\mathscr{X}$, then
$\mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X})$ will be artificially large due to empty regions in $\mathscr{X}$.
For the rest of this paper, we will fix $\mathscr{X} = B_1$. With slight abuse of notation, define
\begin{equation} \label{DsFL}
\mathscr{D}_{\mathrm{s}}(F,L) := \inf_{\mathscr{A} : |\mathscr{A}| = L} \mathscr{D}_{\mathrm{s}}(F,\mathscr{A},B_1),
\end{equation}
and
\begin{equation} \label{DaEL}
\mathscr{D}_{\mathrm{a}}(E,L) := \inf_{\mathscr{A} : |\mathscr{A}| = L} \mathscr{D}_{\mathrm{a}}(E,\mathscr{A},B_1).
\end{equation}

With the simple volumetric bound $\varepsilon(N,B_1) \geq N^{-1/k}$, the relations in
\eqref{dist-relations-1} yield
\begin{equation}\label{dist-relations-2}
\mathscr{D}_{\mathrm{s}}(F,L) \geq \sD_{\mathrm{a}}(E,L) \geq L^{-m/k}.
\end{equation}

One of the main results of this paper is a matching upper bound for the analysis distortion for Gaussian random frames. We call $E$ a {\em standard Gaussian random frame} if $(E_{i,j})$ are independent standard Gaussian random variables.

\begin{theorem} \label{T:MAIN}
There exist absolute positive constants $c_1$ and $c_2$ such that for any scalar
$\eta \in (0,1/2]$, if $k \geq \frac{8}{\eta}$ and $\frac{m}{k} \geq \frac{c_1}{\eta^2} \log \frac{m}{k}$, then
with probability at least $1 - \exp(- c_2 \eta^2 m)$,
a standard Gaussian random frame $E$ of $m$ vectors in $\mathbb{R}^k$ satisfies
\begin{equation}\label{main-thm-bound}
 \sD_{\mathrm{a}}(E,L) \leq \sqrt{k} L^{-(1-\eta)m/k}.
\end{equation}
for all $L \geq 2$.
\end{theorem}

The proof of this theorem is constructive and algorithmic. Given $\eta$, we will, for each $L$,
provide a quantization alphabet $\mathscr{A}$ which is an arithmetic progression of size $L$, and an algorithmic
quantization map $Q:\mathbb{R}^m \to \mathscr{A}^m$ such that with high probability on $E$,
there exists a certain dual
frame $F$, determined by $E$ and $Q$ and computable by a fast, low-complexity algorithm,
which yields $\mathscr{D}(E,Q,F,B_1) \lesssim (m/k)^{3/2} k^{1/2}  L^{-(1-\eta)m/k}$. The chosen $Q$ is a variation on vector-valued $\beta$-expansions and $F$ is an alternative dual of $E$ determined by $Q$. Suitably restricting the range of $m$ and $k$ and
summing over the failure probability for each $L$ will then allow us to guarantee the distortion bound \eqref{main-thm-bound} uniformly over $L$.

Quantization of finite frames by sigma-delta modulation goes back to \cite{benedetto2006sigma,BPY2}.
Alternative duals for frames were introduced in the context of sigma-delta quantization in \cite{alternative2010,blum2010sobolev}, and more recently have been effectively used for Gaussian random
frames in \cite{gunturk2010sobolev} and more generally for sub-Gaussian random frames in \cite{krahmer2013subGaussian}. For these frames,
an $r$-th order sigma-delta quantization scheme coupled with an associated Sobolev dual can achieve the bound
$\sD_{\mathrm{a}}(E,L) \lesssim L^{-1} (cr)^{2r} (m/k)^{-r}$ with high
probability for some constant $c$.
Optimizing this bound for $r$ yields $\sD_{\mathrm{a}}(E,L) \lesssim L^{-1}e^{-\tilde c\sqrt{m/k}}$ for another
constant $\tilde c$ which is sub-optimal in terms of its dependence on both the alphabet size $L$ and the oversampling factor $m/k$ and $L$. In contrast,
our result achieves near-optimal dependence on both parameters: Not only is the dependence on $m/k$
exponential, but the rate can also be made arbitrarily close to its maximum value $\log_2 L$.
To the best of our knowledge, this is the first result of this kind.\footnote{In the case of conventional sigma-delta modulation, similar exponential error bounds that have been obtained previously are not compatible with the results of this paper: The results of \cite{gunturk2003one} and \cite{deift2011optimal} are for a fixed frame-like system, but using a different norm in infinite dimensions, and the dependence on $L$ is unavailable. The results in \cite{derpich2008}, obtained in yet another functional setting for sigma-delta modulation, come close to being optimal, however these results were obtained under modeling assumptions on the quantization noise and circuit stability, and therefore are also not compatible with this paper.}

The paper is organized as follows: In Section \ref{section-distributed-quantization}, we review the notion of general noise-shaping quantizers and their stability as well as alternative duals of frames based on noise shaping. In this section we introduce a number of new notions, namely, $V$-duals and $V$-condensation of frames and the concept of {\em distributed noise shaping}. In Section \ref{section-beta-duals}, we introduce the special case of {\em beta duals} and derive a general bound for the analysis distortion of an arbitrary frame using beta duals. As an example, we show in this section that beta duals are optimal for harmonic semicircle frames. In Section \ref{gaussian_frames}, we analyze the analysis distortion of Gaussian frames and prove Theorem \ref{T:MAIN} as well as some variations of it.

\section{Quantization via distributed noise shaping} \label{section-distributed-quantization}

\subsection{Noise-shaping quantizers} \label{noise-shaping}

Let $\mathscr{A}$ be a quantization alphabet and $J$ be a compact interval in $\mathbb{R}$.
Let $h = (h_j)_{j\geq0}$ be a given sequence, finite or infinite, and $h_0=1$.
By a noise-shaping quantizer with the transfer sequence $h$, we
mean any map $Q:J^m \to \mathscr{A}^m$ where $q := Q(y)$ satisfies
$$ y - q = h*u$$
and $\|u\|_\infty \leq C$ for some constant $C$ which is independent of $m$.
Here $h*u$ refers to the convolution of $h$ and $u$ defined by
$(h*u)_n := \sum h_j u_{n-j}$ where it is assumed that $u_n := 0$ for $n\leq 0$.

Noise-shaping refers to the fact that
the ``quantization noise'' $y-q$ cannot be arbitrary.
Even though the operator
$H:u \mapsto h*u$ is invertible on $\mathbb{R}^m$; the requirement that
$\|u\|_\infty$ must be controlled uniformly in $m$ imposes restrictions on what $q$ can be
for a given $y$.

The above formulation amounts to saying that $H$ is an $m\times m$
lower triangular Toeplitz matrix defined by $h$, in particular with unit diagonal.
Let us relax the notion of a noise-shaping quantizer and assume that $H$ is any given $m\times m$
lower triangular matrix with unit diagonal. We will refer to $H$ as the noise transfer operator and
keep the noise-shaping relation
\begin{equation} \label{y-q-H-u}
  y - q = Hu,
\end{equation}
but since $H$ is more general and $m$ is therefore static, we can no longer insist that
$\|u\|_\infty$ is controlled in the same manner. While we will mostly stick to convolution-based
noise-shaping as defined at the beginning, the above relaxation is conceptually and algebraically
simpler.

Noise-shaping quantizers do not exist unconditionally. However,
under certain suitable assumptions on $H$ and $\mathscr{A}$, they exist and
can be implemented via recursive algorithms. The simplest
is the greedy quantizer whose general formulation is given below:

\begin{proposition}\label{P:greedy}
Let $\mathscr{A} := \mathscr{A}_{L,\delta}$ denote the arithmetic progression in $\mathbb{R}$ which is of length $L$, spacing $2\delta$, and symmetric about $0$.
Assume that $H = I - \tilde H$, where $\tilde H$ is strictly lower triangular, and $\mu \geq 0$ such that
$\|\tilde H\|_{\infty \to \infty} + \mu/\delta \leq L$. Then
there exists a quantization map $Q_{H,\mathscr{A}}:[-\mu,\mu]^m \to \mathscr{A}^m$ implementable by a recursive
algorithm such that $q:=Q_{H,\mathscr{A}}(y)$ satisfies \eqref{y-q-H-u}
and $\|u\|_\infty \leq \delta$.
\end{proposition}
\begin{proof}
Since the endpoints of $\mathscr{A}$ are $\pm (L-1)\delta$,
we have $|w - \mathrm{round}_\mathscr{A}(w)| \leq \delta$ for all $|w| \leq L\delta$.
For $n=1,\dots,m$, we set
$$ q_n := \mathrm{round}_\mathscr{A}\left (y_n + \sum_{j=1}^{n-1} \tilde H_{n,n-j} u_{n-j} \right)$$
and
$$ u_n := y_n + \sum_{j=1}^{n-1} \tilde H_{n,n-j} u_{n-j} -q_n .$$
The proof now follows by induction. Setting $w=y_1$, we obtain $|u_1| \leq \delta$.
Assuming that $|u_j|\leq \delta$ for $j=1,\dots,n-1$, we obtain
$$\left |y_n + \sum_{j=1}^{n-1} \tilde H_{n,n-j} u_{n-j} \right|
\leq \mu + \delta \|\tilde H\|_{\infty \to \infty} \leq L\delta.$$
Hence $|u_n| \leq \delta$.
\end{proof}
\begin{remark}
In the special case $Hu = h*u$ where $h_0 = 1$, note that $\|\tilde H\|_{\infty \to \infty} = \|h\|_1 - 1$.
\end{remark}

\subsection{Alternative duals of frames for noise shaping} \label{V-duals}
Let $E$ be a frame and $y=Ex$ be the frame measurements of a given signal $x$. Assume that
we quantize $y$ using a noise-shaping quantizer with transfer operator $H$.
Any left-inverse (dual) $F$ of $E$ gives
\begin{equation}\label{error_FHu}
x - Fq = F(y - q) = FHu.
\end{equation}

As explained in the introduction, the choice of $F$ is to be made based upon $E$ and the quantizer
$Q_{H,\mathscr{A}}$, but not $u$ which is unavailable to the decoder.
We are interested in bounding $\|x - Fq\|_2$, but the best a priori bound for $u$ is
in the $\infty$-norm. This suggests that we employ the bound
\begin{equation}\label{error_bound_FHu}
 \|x - Fq \|_2 \leq \|FH\|_{\infty \to 2} \|u \|_\infty.
\end{equation}
With this bound, the natural objective would be to minimize $\|FH\|_{\infty \to 2}$ over all duals of $E$.
However, an analytical solution for this problem is not available to us. If instead we minimize
$\|FH\|_\mathrm{2 \to 2}$, then the solution is known explicitly, and given by
$F = (H^{-1}E)^\dagger H^{-1}$,
where $A^\dagger := (A^\ast A)^{-1} A^\ast$ is the pseudoinverse \cite{blum2010sobolev,gunturk2010sobolev}.

More generally, for any $p \times m$ matrix $V$ such that $VE$ is also a frame for $\mathbb{R}^k$
(i.e. of rank $k$),
\begin{equation}\label{E:vdual}
F_V := (VE)^\dagger V
\end{equation}
is a dual of $E$, which we will call the {\em $V$-dual} of $E$. (For the case $V = H^{-1}$,
the resulting dual was called the $H$-dual in \cite{gunturk2010sobolev} and denoted by $F_H$.
With our more general definition, this dual more accurately becomes the $H^{-1}$-dual
and denoted by $F_{H^{-1}}$.)
The particular case $V = D^{-r}$, called the ``Sobolev dual'',
was introduced in \cite{blum2010sobolev} and studied for random $E$ in \cite{gunturk2010sobolev}.

With a $V$-dual, we have $F_V H = (VE)^\dagger VH$. Given that $\| (VE)^\dagger \|_{2\to 2}
 = 1/\sigma_{\mathrm{min}}( VE)$, we employ the initial bound
$$\|F_V H\|_{\infty \to 2} \leq \frac{\| VH\|_{\infty \to 2} }{\sigma_{\mathrm{min}}( VE) }. $$
Note that this bound is still valid when $VE$ fails to be a frame, since then we have $\sigma_{\mathrm{min}}(VE)=0$.

Let us compare the following two routes for bounding $\| VH\|_{\infty \to 2}$:
\begin{itemize}
 \item[(i)]  $\| VH\|_{\infty \to 2} \leq \sqrt{m}\| VH\|_{2 \to 2} $. Since
$\| VH  \|_\mathrm{2 \to 2}/\sigma_{\mathrm{min}}( VE) \geq \|F_V H\|_{2\to2} \geq 1/\sigma_{\mathrm{min}}(H^{-1}E)$, it follows
that the minimum value of $\| VH  \|_\mathrm{2 \to 2}/\sigma_{\mathrm{min}}( VE)$ is attained when $V = H^{-1}$.
Hence, this route does not offer a new bound.
 \item[(ii)]  $\| VH\|_{\infty \to 2} \leq \sqrt{p}\| VH\|_{\infty \to \infty} $. The minimization of
$\sqrt{p} \| VH  \|_\mathrm{\infty \to \infty}/\sigma_{\mathrm{min}}( VE)$ opens up
the possibility of a smaller error bound because $V = H^{-1}$ readily achieves the same bound
as in (i).\footnote{The following example shows that the minimum value can be
strictly smaller: Let
$H = \left[\begin{matrix}1&0\\-1&1\end{matrix}\right]$, $E =\left[\begin{matrix}1\\1\end{matrix}\right] $
for which $\sqrt{m}/\sigma_{\mathrm{min}}(H^{-1}E) = \sqrt{2/5}$. Meanwhile,
$V = \left[\begin{matrix}1&1\end{matrix}\right]$ yields
$\sqrt{p}\|VH\|_{\infty \to \infty}/\sigma_{\mathrm{min}}(VE) = 1/2$.}
Let $v_1,\dots,v_p$ denote the rows of $V$. Note that
$$ \| VH  \|_{\infty \to \infty} =  \max_{1 \leq n \leq p} \| v_n H \|_1.$$
Hence an effective strategy for designing $V$ is based on choosing $v_1,\dots,v_p$ near (but not
necessarily equal to) the
bottom $p$ left singular vectors of $H$. We will exploit this principle in the next section.
\end{itemize}

Note that different choices of $V$ can yield the
same $V$-dual. In particular, given any $V$, the $k \times m$ matrix
$ \tilde V:= (W^*W)^{-1/2}W^*V $ yields $F_{\tilde V} = F_V$, where $W := VE$.
This $\tilde V$ is not particularly useful for computational purposes, but it shows that
$p$ can be chosen as small as $k$ without sacrificing performance.
In general, when $p < m$, we will call $VE$ the {\em $V$-condensation} of $E$.

\subsection{Distributed noise shaping} \label{distributed}

By a {\em distributed} noise-shaping quantization scheme, we will mean a noise-shaping quantizer defined by
a block-diagonal noise transfer matrix $H$, and for any given analysis frame $E$, an associated $V$-dual where
$V$ is also block-diagonal with matching sized blocks. More specifically, we set
$$H=\left[\begin{matrix} H_1  & & \\ & \ddots & \\ & & H_l \end{matrix}\right]
\quad\text{and}\quad
V=\left[\begin{matrix} V_1  & & \\ & \ddots & \\ & & V_l \end{matrix}\right]$$
where $H_i\in\mathbb{R}^{m_i\times m_i}$ and $V_i\in\mathbb{R}^{p_i\times m_i}$ with $\sum_i m_i=m$ and $\sum_i p_i=p$.
We further decompose $E$, $y$, $q$ and $u$ as
$$
E = \left[\begin{matrix}E_1 \\ \vdots \\ E_l \end{matrix}\right],\quad
y = \left[\begin{matrix}y_1 \\ \vdots \\ y_l \end{matrix}\right],\quad
q = \left[\begin{matrix}q_1 \\ \vdots \\ q_l \end{matrix}\right]
\quad\text{and}\quad
u = \left[\begin{matrix}u_1 \\ \vdots \\ u_l \end{matrix}\right],$$
where $E_i \in \mathbb{R}^{m_i \times k}$, and $y_i$, $q_i$, $u_i \in \mathbb{R}^{m_i}$, $i \in [l]:=\{1,\dots,l\}$.
Note that the $E_i$ may or may not be individual frames in $\mathbb{R}^k$. With this notation, we have
$l$ individual quantizers that run in parallel:
\begin{equation}\label{y-q-H-u-i}
E_i x - q_i = H_i u_i,\qquad i \in [l].
\end{equation}
Let $v_{i,j}$ denote the $j$th row of $V_i$, $i \in [l]$, $j\in[p_i]$. Then we have
$$\| VH  \|_{\infty \to \infty} =  \max_{i \in [l]} \max_{j \in [p_i]} \| v_{i,j} H_i \|_1.$$

Hence, it follows that the reconstruction error obeys the bound

$$ \|x - F_V q\|_2 \leq \frac{\sqrt{p} \|u \|_\infty}{\sigma_{\mathrm{min}}(VE)}
 \max_{i \in [l]} \max_{j \in [p_i]} \|v_{i,j} H_i \|_1.$$

\begin{remark}
For conceptual clarity, we may assume that $l \not= 1$ so that distributed noise shaping is genuinely a special case of noise shaping as described in Section \ref{V-duals}. However, we will not need to turn this into an explicit assumption because we will only consider $l \geq k$, and most often, $l > k$.
\end{remark}

\begin{remark}
Our formalism for noise-shaping quantization and alternate-dual reconstruction assumes
that the frame vectors and the measurements are processed in the given (specified) order.
For distributed noise shaping, we assume that these vectors are collected into groups according to the same ordering.
\end{remark}

\section{Beta duals}\label{section-beta-duals}

We now propose a specific distributed noise-shaping quantization scheme and corresponding
alternative duals for reconstruction which will form the basis for the main results of this paper.

For any $\beta > 1$, let $h^\beta$ be the (length-$2$)
sequence given by $h^\beta_0 = 1$ and $h^\beta_1 = -\beta$.

Given $k \leq l \leq m$, let $\mbox{\boldmath$m$}:=(m_1,\dots,m_l)$ and $\mbox{\boldmath$\beta$}:=(\beta_1,\dots,\beta_l)$ be such that
$\sum m_i = m$ and $\beta_i > 1$, $i \in [l]$. For each $i$, we set $H_i$
equal to the convolution operator $u \mapsto h^{\beta_i}*u$, $u \in \mathbb{R}^{m_i}$, as defined in
Section \ref{noise-shaping}.
Note that each $H_i$ is an $m_i \times m_i$ bidiagonal matrix with
diagonal entries equal to $1$ and the subdiagonal entries equal to $-\beta_i$.

We set $p_i = 1$ and
$V_i =  v_{i,1} := [\beta_i^{-1} ~~ \beta_i^{-2}~\cdots~~\beta_i^{-m_i}]$, $i \in [l]$.
Let $V =: V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}$ and $H$ be as in Section \ref{distributed}. Note that $p=l$. It follows that
$V_i H_i = [0~\cdots~0~~\beta_i^{-m_i}]$,
$\|V_i H_i\|_{\infty \to \infty} = \beta_i^{-m_i}$, and therefore
$\|VH\|_{\infty \to \infty} = \beta_*^{-m_*}$, where $\beta_* := \min_i \beta_i$ and $m_* := \min_i m_i$.
With the setup and the notation of Section \ref{distributed},
it follows that
\begin{equation}
\|x - F_{V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}} q\|_2 \leq
\frac{\sqrt{l} \|u\|_\infty}{\sigma_{\mathrm{min}}(V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}E)} \beta_*^{-m_*}.
\label{E:bdest}
\end{equation}

Note that this particular choice of $V_i = v_{i,1}$ is derived from
$\beta$-expansions of real numbers. In fact, solving the
difference equations (\ref{y-q-H-u-i}) for $q_i$ and recovering with
$v_{i,1}$ gives a truncated $\beta_i$-expansion of $v_{i,1}E_ix$.

We will refer to
$V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}E$ as the {\em $(\mbox{\boldmath$\beta$},\mbox{\boldmath$m$})$-condensation} of $E$, and
the corresponding $V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}$-dual $F_{V_{\mbox{\scriptsize \boldmath$\beta$},\mbox{\scriptsize \boldmath$m$}}}$ as the
{\em $(\mbox{\boldmath$\beta$},\mbox{\boldmath$m$})$-dual} of $E$. We will use the term {\em beta dual} when implicitly referring to
any such dual.

In a typical application all the $\beta_i$ would be set equal to some $\beta = \beta_*$ and the $m_i$
(approximately) equal to $m/l$. To denote this choice, we will replace the vector pair $(\mbox{\boldmath$\beta$}, \mbox{\boldmath$m$})$
by the scalar triplet $(\beta, m, l)$, ignoring the exact values of the $m_i$ as
long as $m_* \geq \lfloor m/l \rfloor$.

\subsection{Bounding the analysis distortion via beta duals}
\label{betadualsteps}

For any integer $L \geq 2$ and scalar $\mu > 0$, let
\begin{equation}\label{S_muL}
 S_{\mu,L}:= \{(\beta,\delta) : \beta > 1, ~\delta > 0, ~\beta + \mu/\delta \leq L \}.
\end{equation}
This set corresponds to the admissible set of $\beta$ and $\delta$ values such that the greedy
quantization rule described in Proposition \ref{P:greedy} using $\mathscr{A}_{L,\delta}$ and a noise
shaping operator $H$ as defined above
(with all $\beta_i = \beta$) is stable with $\|u\|_\infty \leq \delta$ uniformly for
all inputs bounded by $\mu$. Note that the best $\ell_\infty$-norm
bound for $y = Ex$ over all $x \in B_1$ is equal to $\| E \|_{2 \to \infty}$ and that
$S_{\mu,L}$ decreases with $\mu$.
Combining \eqref{E:bdest} with \eqref{DEQFX}, \eqref{DaEAXeq} and \eqref{DaEL}, we
immediately obtain the following result:
\begin{proposition} \label{analysis_dist}
Let $E$ be an $m \times k$ frame and $\mu \geq \| E \|_{2 \to \infty}$. For any integer $L \geq 2$, we have
\begin{equation}\label{DaEL_bound1}
 \sD_{\mathrm{a}}(E,L) \leq \inf_{\underset{l \geq k}{(\beta, \delta) \in S_{\mu,L}}} ~~
 \frac{\delta \sqrt{l} \beta^{-\lfloor m/l\rfloor}}{\sigma_{\mathrm{min}}(V_{\beta,m,l}E)}.
\end{equation}
\end{proposition}
In order to find good upper bounds for $\sD_{\mathrm{a}}(E,L)$ via this proposition, we will
do the following:
\begin{enumerate}
\item[(1)] Find an upper bound $\mu$ for $\|E\|_{2\to \infty} =
\displaystyle \max_{1 \leq i \leq m} \|e_i\|_2$.
\item[(2)] Given any $\beta > 1$ and $l\geq k$, find a lower bound on
$\sigma_{\mathrm{min}}(V_{\beta,m,l}E)$.
\item[(3)] Find optimal (or near-optimal) values of $\beta$, $\delta$, and $l$
for the resulting effective upper bound replacing \eqref{DaEL_bound1}.
\end{enumerate}

We will primarily carry out this procedure for random frames.
However, before doing that in Section \ref{gaussian_frames}, we will
discuss in Section \ref{harm_semi} a specific family of (deterministic) frames, namely the harmonic semicircle frames
in $\mathbb{R}^2$.

\subsection{Example: the analysis distortion for the harmonic semicircle frames}
\label{harm_semi}

Harmonic semicircle frames  are obtained from harmonic frames by removing some of their symmetry (see \cite{framepath2007} for the general
definition). In
$\mathbb{R}^2$, the harmonic semicircle frame of size $m$ is defined by the $m\times 2$ matrix
$E:=E_{\mathrm{hsc},m}$ given by
$$
E_{\mathrm{hsc},m}:= \left[\begin{matrix}
\cos \pi/m & \sin \pi/m \\
\cos 2\pi/m & \sin 2\pi/m \\
\vdots & \vdots \\
\cos \pi & \sin \pi
\end{matrix}\right]
$$
We set $l=2$, split $E$ into $E_1$ and $E_2$ as in Section \ref{distributed}, and for simplicity,
assume that $m$ is even. (The odd case can be handled with a slight alteration.)

For any $\beta > 1$, consider $V_{\beta,m,2}E$, which is given by the $2 \times 2$ matrix
$$\left[\begin{matrix} v_\beta E_1 \\ v_\beta E_2 \end{matrix}\right] $$
where $v_\beta := V_1 = V_2 = [\beta^{-1} \cdots \beta^{-m/2} ]$.
Note that each row of $E_2$ is a 90 degree rotation of the corresponding row of $E_1$.
Therefore $ v_\beta E_2 $ is also a
90 degree rotation of $ v_\beta E_1 $ which implies that
$\sigma_{\mathrm{min}} (V_{\beta,m,2}E) = \|v_\beta E_1 \|_2 = : C_{\beta,m}$. An exact
expression for $C_{\beta,m}$ can be found easily, namely,
$$ C_{\beta,m} = \frac{(1+\beta^{-m})^{1/2}}{|\beta-e^{\pi i /m}|}.$$
It is easily seen that $C_{\beta,m} \geq \beta^{-1}$ for all $m$ (with equality if and only if $m=2$).

Since all frame vectors are of unit norm, we can set $\mu=1$. Finally, we
choose $\beta = mL/(1 + m)$ and $\delta = (1+m)/L$, which are near optimal values
as suggested by Proposition \ref{analysis_dist}. With these values, it follows that
\[ \sD_{\mathrm{a}}(E_{\mathrm{hsc},m},L) < \sqrt{2e} m L^{-m/2}. \]

In Figure \ref{fig:bdukcondtwo} we illustrate how the $(\beta,m,2)$-dual for $m=12$ and $\beta = 1.6$.

\begin{figure}[htp]
\centering
\includegraphics[scale=0.5]{hsc2}
\caption{Harmonic semicircle frame using circles and $(\beta,m,l)$-dual frame with stars,
$\beta = 1.6$, $m=12$, $l=2$.}
\label{fig:bdukcondtwo}
\end{figure}

\section{Analysis distortion of Gaussian frames} \label{gaussian_frames}
In this section, we assume that $E$ is an $m \times k$ random matrix with entries drawn independently from
$\mathcal{N}(0,1)$. As before, $L$ stands for a given alphabet size. To bound the analysis distortion $\sD_{\mathrm{a}}(E,L)$, we will again follow the three step plan outlined in section \ref{betadualsteps} with a slight modification: Note that the entries of the $l \times k$ matrix $V_{\beta, m, l} E$ are mutually independent and the entries of the $i$th row are identically distributed according to $\mathcal{N}(0,\sigma^2_{m_i})$, where $$\sigma^2_n := \sum_{j=1}^{n} \beta^{-2j}$$
and the $m_i$ satisfy $\sum m_i = m$ and $m_i \geq \lfloor m/l \rfloor$ for all $i$. Our analysis of $\sigma_{\mathrm{min}}(V_{\beta, m, l} E)$ would be simpler if all the $\sigma_{m_i}$ were equal, which holds if and only if $m$ is a multiple of $l$. Since $l$ is a parameter which will be optimized later, this condition cannot be enforced in advance. To circumvent this issue, we define for each $l$, the submatrix $\tilde E_l$ of $E$ formed by the first $\tilde m$ rows of $E$ where $\tilde m := \lfloor m/l \rfloor l$. Provided that $\tilde E_l$ is a frame for $\mathbb{R}^k$ (which holds almost surely if $l\geq k$), the definition of the analysis distortion $\sD_{\mathrm{a}}(E,\mathscr{A},\mathscr{X})$ given by \eqref{analysis-dist} implies that
$$ \sD_{\mathrm{a}}(E,\mathscr{A},\mathscr{X})  \leq \sD_{\mathrm{a}}(\tilde E_l,\mathscr{A},\mathscr{X}), $$
because for any $\tilde F$ such that $\tilde F \tilde E_l = I$, the zero-padded $k \times m$ matrix $F:= [\tilde F ~ 0]$ satisfies $FE = I$ and
$$\mathscr{D}_{\mathrm{s}}(F,\mathscr{A},\mathscr{X}) = \mathscr{D}_{\mathrm{s}}(\tilde F,\mathscr{A},\mathscr{X}). $$
Hence we replace \eqref{DaEL_bound1} with
\begin{equation}\label{DaEL_bound2}
 \sD_{\mathrm{a}}(E,L) \leq \inf_{l \geq k} ~~\sD_{\mathrm{a}}(\tilde E_l,L)
\end{equation}
where
\begin{equation}\label{DaEL_bound3}
\sD_{\mathrm{a}}(\tilde E_l,L) \leq \inf_{(\beta, \delta) \in S_{\mu,L}} ~~
 \frac{\delta \sqrt{l} \beta^{-\lfloor m/l\rfloor}}{\sigma_{\mathrm{min}}(V_{\beta,\tilde m,l} \tilde E_l)}.
\end{equation}

\subsection{Upper bound for $\|E\|_{2\to \infty}$}
 Note that because the Gaussian distribution has unbounded support, there is no absolute bound on $\|E\|_{2\to \infty}$ that is applicable almost surely,
so we employ concentration bounds which hold with high probability.
Let
\begin{equation}\label{E:gaussopnormbdM}
\mathcal{M}:=\left \{\|E\|_{2\to \infty} \leq \mu := 4\sqrt{m}\right \}
\end{equation}
The following well-known result will be suitable for this purpose:
\begin{proposition}[{\cite[Theorem II.13]{ds2001}}{\cite{ds2003}}]\label{ds2001}
 Let $m \geq k$ and $E$ be an $m \times k$ random matrix whose entries are i.i.d. standard Gaussian variables. For any $t >0$,
$$ \mathbb{P} \left(\left\{\sigma_\mathrm{max}(E) \geq \sqrt{m}+\sqrt{k} + t \right \} \right) < e^{-t^2/2}.$$
\end{proposition}
Setting $t = 2\sqrt{m}$ and noting that $\sigma_\mathrm{max}(E) = \|E\|_{2\to 2} \geq \|E\|_{2\to\infty}$,
we have
\begin{equation}
\mathbb{P}(\mathcal{M}^c) \leq \mathbb{P}\left (\left \{\|E\|_{2\to 2} > 4\sqrt{m}\right \} \right ) < e^{-2m}.
\end{equation}

\subsection{Lower bound for $\sigma_{\mathrm{min}}(V_{\beta,\tilde m,l}\tilde E_l)$} Again, we employ concentration bounds. For any $\gamma > 0$, we are interested in the event
\begin{equation}\label{Egamma}
 \mathcal{E}(\gamma) :=  \mathcal{E}(\gamma,\beta,l) :=\left\{ \sigma_{\mathrm{min}}(V_{\beta,\tilde m,l}\tilde E_l) \geq \gamma \right\}.
\end{equation}
Note that the entries of the $l\times k$ matrix $V_{\beta, \tilde m, l} \tilde E_l$ are now independent and
identically distributed, with each entry having distribution $\mathcal{N}(0,\sigma^2_{\lfloor m/l \rfloor})$.
It will suffice to use standard methods and results on the concentration
of the minimum singular value of Gaussian matrices. The nature of these results differ depending on whether $l=k$ or $l > k$ and the optimal value of $l$ depends on the desired level of probabilistic assurance.

For $l=k$, we use the following result:
\begin{proposition}[{\cite[Theorem 3.1]{rudelson2010non}}, {\cite{edelman1988eigenvalues}}]\label{P:squaregaussminsv}
Let $\Omega$ be a $k \times k$ random matrix with entries drawn
independently from $\mathcal{N}(0,1)$. Then for any $\varepsilon > 0$,
\[ \mathbb{P}\left(\left\{ \sigma_{\mathrm{min}}(\Omega) \leq \frac{\varepsilon}{\sqrt{k}}\right\}\right) \leq \varepsilon. \]
\end{proposition}

For $l>k$, there are standard estimates concentration estimates (e.g. \cite{ds2001, rudelson2009smallest, vershynin2010introduction}) which take the lower end of the spectrum $\sqrt{l}-\sqrt{k}$ as a reference point.
In contrast, the following result concerns the probability of the smallest singular value being near zero.
\begin{theorem}\label{T:svmingaussrect}
Let $l > k$ and $\Omega$ be an $l \times k$ random matrix whose entries are drawn independently from $\mathcal{N}(0,1)$. Then for any $0 < \varepsilon < 1$,
\[ \mathbb{P}\left(\left\{ \sigma_{\mathrm{min}}(\Omega) \leq \varepsilon\sqrt{l}/2\right\}\right) \leq
\left (10+8\sqrt{\log \varepsilon^{-1}}\right)^k  e^{l/2} \varepsilon^{l-k}.
 \]
\end{theorem}
For a proof, see Appendix \ref{appendix}.
Note that there exists $\varepsilon_0 := \varepsilon_0(k,l) < 1$ such that this probability bound is not viable (i.e., exceeds $1$) for $\varepsilon > \varepsilon_0$. This may turn out be the case in Theorem \ref{thm_lgtrk} (see Section \ref{lbiggerk} below) if neither $m/l$ nor $L$ is sufficiently large. However, the stronger assumptions in Corollary \ref{cor} and Theorem \ref{T:MAIN} rule out this possibility.

\subsection{Bounding the analysis distortion via optimal choice of $\beta$ and $\delta$}

\subsubsection{Choosing $l=k$}

Let $\eta \in (0,1)$ be arbitrary. We employ Proposition \ref{P:squaregaussminsv} with
$\varepsilon = \beta^{-\eta\lfloor m/k \rfloor}$ which corresponds to $\gamma = \frac{\varepsilon}{\sqrt{k}}\sigma_{\lfloor m/k \rfloor}$ in \eqref{Egamma}. With the simple bound $\sigma_{\lfloor m/k \rfloor} \geq \beta^{-1} \geq L^{-1}$, the analysis distortion bound (\ref{DaEL_bound3}) yields
\begin{equation} \label{E:bdgreedyP2}
\sD_{\mathrm{a}}(\tilde E_k,L) \leq \inf_{(\beta, \delta) \in S_{\mu,L}}  k L \delta \beta^{-(1-\eta)\lfloor m/k \rfloor}
\end{equation}
for all $E$ in the event $\mathcal{E}(\gamma)\cap\mathcal{M}$, which occurs with probability at least $1-\beta^{-\eta \lfloor m/k \rfloor} - e^{-2m}$. Note that $\mu = 4\sqrt{m}$.

A straightforward calculation shows that for any $\alpha > 1/(L-1)$, the minimum of $\delta \beta^{-\alpha}$ over the set $S_{\mu,L}$ is equal to $\mu \alpha \left( \frac{1+\alpha}{\alpha L}\right)^{1+\alpha}$, which is attained at $\beta = L\alpha/(1+\alpha)>1$ and $\delta=\mu(1+\alpha)/L$.

We use these values of $\beta$ and $\delta$ in the bound \eqref{E:bdgreedyP2} with $\alpha = (1-\eta)\lfloor m/k \rfloor $ and $\mu = 4\sqrt{m}$. Noting that $(1+1/\alpha)^\alpha < e$, we have proved

\begin{theorem}\label{thm_leqk}
Let $m \geq k$ and $L \geq 2$, except assume $L > 2$ if $m<2k$. Suppose
$E$ is an $m \times k$ random matrix with entries drawn independently
from $\mathcal{N}(0,1)$. For any $0< \eta <1-\frac{1}{\lfloor m/k \rfloor (L-1)}$, the analysis distortion for $E$ satisfies the bound
\[ \sD_{\mathrm{a}}(E,L) <  8e m^{3/2} L^{-(1-\eta)\lfloor m/k \rfloor} \]
with probability at least $1-L^{-\eta \lfloor m/k \rfloor}e^{\eta/(1-\eta)} - e^{-2m}$.
\end{theorem}

\begin{remark}
The most favorable choice of $\eta$ depends on $m$, $k$, $L$ as well as the probabilistic assurance that is desired. For example, as $\lambda:=m/k \to\infty$, $\eta=\lambda^{-1/2}$ yields
$\sD_{\mathrm{a}}(E,L) =  O(m^{3/2} L^{-\lambda(1+o(1))})$ with probability $1-o(1)$.
\end{remark}

\subsubsection{Choosing $l>k$}\label{lbiggerk}

Again, let $\eta \in (0,1)$. We now employ Theorem \ref{T:svmingaussrect} with $\varepsilon = \beta^{-\eta\lfloor m/l \rfloor}$ which corresponds to $\gamma = \varepsilon \sigma_{\lfloor m/l \rfloor} \sqrt{l} / 2$.
Again using $\sigma_{\lfloor m/l \rfloor} \geq L^{-1}$,  the analysis distortion bound (\ref{DaEL_bound3}) becomes
\begin{equation} \label{E:bdgreedyP3}
\sD_{\mathrm{a}}(\tilde E_l,L) \leq \inf_{(\beta, \delta) \in S_{\mu,L}} 2L\delta\beta^{-(1-\eta)\lfloor m/l \rfloor }
\end{equation}
for all $E$ in the event $\mathcal{E}(\gamma,\beta,l)\cap\mathcal{M}$, which occurs with probability
at least
$$1 - e^{l/2}(6+4 \sqrt{ \eta \lfloor m/l \rfloor \log \beta })^k \beta^{-\eta \lfloor m/l \rfloor(l-k)}
-e^{-2m}.$$

Motivated by the same calculation as in
the $l=k$ case, we choose $\alpha = (1-\eta)\lfloor m/l \rfloor $,
$\beta = L\alpha/(1+\alpha)$ and $\delta = \mu(1+\alpha)/L$, which now shows:
\begin{theorem}\label{thm_lgtrk}
Let $m \geq l > k$ and $L \geq 2$, except assume $L > 2$ if $m<2l$. Suppose
$E$ is an $m \times k$ random matrix with entries drawn independently
from $\mathcal{N}(0,1)$. For any $0< \eta <1-\frac{1}{\lfloor m/l \rfloor (L-1)}$, the analysis distortion for $E$ satisfies the bound
\begin{equation}\label{analysis_dist_2}
\mathscr{D}_{\mathrm{a}}(E,L) < 16em^{3/2}l^{-1} L^{-(1-\eta)\lfloor m/l \rfloor}
\end{equation}
with probability at least
\[ 1 - e^{l/2}(10+8 \sqrt{ \eta \lfloor m/l\rfloor \log L })^k e^{\frac{\eta}{1-\eta} (l-k)}L^{-\eta \lfloor m/l \rfloor (l-k)}-e^{-2m}.\]
\end{theorem}

In particular, this result implies that the failure probability can be arranged to be exponentially small in $m$ while ensuring that the distortion bound is still of the same order as in Theorem \ref{thm_leqk}. Indeed, let $l = k+ \lfloor \theta k \rfloor$ for some $\theta \geq 1/k$ (so that $l > k$).
With this value of $l$, we have
\begin{equation}\label{analysis_dist_3}
\mathscr{D}_{\mathrm{a}}(E,L) < 16em^{3/2}k^{-1} L^{-(1-\eta)\lfloor \frac{m}{(1+\theta)k} \rfloor}
\end{equation}
for all $E$ in the event $\mathcal{E}(\gamma,\beta,l)\cap\mathcal{M}$,
where the probability of $\mathcal{E}(\gamma,\beta,l)^c$ is bounded by
$$\exp\left\{k \left( \frac{1}{2}(1+\theta) + c_0 +\frac{1}{2}  \log\frac{m}{k} + \frac{1}{2} \log \log L  + \theta \frac{\eta}{1-\eta} \right)
-\eta\left \lfloor \frac{m}{(1+\theta)k} \right \rfloor  \lfloor \theta k \rfloor  \log L \right \}$$
with $c_0 := \log(21)$. (In deriving this simplification, we have used that
$k < l \leq (1+\theta)k$ and $10+8 \sqrt{ \eta \lfloor m/l\rfloor \log L } \leq 21\sqrt{(m/k)\log L } $.)
If $k\geq 8$, $2/k \leq \theta \leq 1/4$ and
$m/k \geq 5/2$,  we have
$$ \left \lfloor \frac{m}{(1+\theta)k} \right \rfloor  \lfloor \theta k \rfloor \geq \frac{m\theta}{4(1+\theta)} \geq\frac{m\theta}{5} $$
so that the probability of $\mathcal{E}(\gamma,\beta,l)^c$ is further bounded by
$$\exp\left\{
k \left( \frac{1}{2}(1+\theta) + c_0 +\frac{1}{2}  \log\frac{m}{k} + \frac{1}{2} \log \log L  + \theta \frac{\eta}{1-\eta} \right)
-\frac{1}{5}\eta \theta m \log L \right \}.$$

Let $\theta \in (0,1/4]$ and $\eta \in (0,1/2]$ be fixed. There exists an absolute constant $c_1 > 0$ such that if
$\frac{m}{k} \geq \frac{c_1 }{\eta\theta} \log \frac{m}{k}$ and $k \geq 2/\theta$, we have
$$
k \left( \frac{1}{2}(1+\theta) + c_0 +\frac{1}{2}  \log\frac{m}{k} + \frac{1}{2} \log \log L  + \theta \frac{\eta}{1-\eta} \right)
\leq \frac{1}{10}\eta\theta m \log L$$
so that
\begin{equation}
 \mathbb{P}(\mathcal{E}(\gamma,\beta,l)^c) \leq L^{-\frac{1}{10}\eta \theta m}.
\end{equation}
(It can be checked that a sufficiently large value of $c_1$ implies a correspondingly large lower bound on $m/k$, which in turn guarantees that  $1-\frac{1}{\lfloor m/l \rfloor (L-1)} > \eta$.) Note that for fixed values of $\eta$ and $\theta$, the event $\mathcal{E}(\gamma,\beta,l)$ depends on $L$ through $\gamma$ and $\beta$. Since $m \geq 20/\eta\theta$ (by readjusting the value of $c_1$ if necessary), it follows that
\begin{equation}
 \mathbb{P}\left (\bigcup_{L=2}^\infty \mathcal{E}(\gamma,\beta,l)^c \right ) \leq
\sum_{L=2}^\infty L^{-\frac{1}{10}\eta \theta m} \leq
3\cdot2^{-\frac{1}{10}\eta \theta m}.
\end{equation}
Note also that if $m/k \geq (1+\theta)/\theta^2$, then
$$ \left \lfloor \frac{m}{(1+\theta)k} \right \rfloor \geq
\frac{m}{(1+\theta)k} - 1 \geq (1-\theta)\frac{m}{k}$$
so that the error bound \eqref{analysis_dist_3} can be simplified to
$$ \mathscr{D}_{\mathrm{a}}(E,L) < 16em^{3/2}k^{-1} L^{-(1-\eta-\theta)\frac{m}{k}}. $$

Choosing $\theta = \eta \in (0,1/4)$, replacing $\eta$ by $\eta/2$, we obtain the following corollary:
\begin{corollary}\label{cor}
There exists an absolute constant $c_1 > 0$ such that if $\eta \in (0,1/2)$, and
$m$ and $k$ are such that $k \geq 4/\eta$ and $\frac{m}{k} \geq \frac{c_1}{\eta^2} \log \frac{m}{k}$, an $m \times k$ random matrix $E$ with entries drawn independently
from $\mathcal{N}(0,1)$ satisfies
\[ \mathscr{D}_{\mathrm{a}}(E,L) < 16em^{3/2}k^{-1} L^{-(1-\eta) \frac{m}{k}} \]
for all $L \geq 2$ with probability at least $1 -3\cdot 2^{-\frac{1}{40}\eta^2 m} - e^{-2m}$.
\end{corollary}

\paragraph{Proof of Theorem \ref{T:MAIN}.}
We make the following observations. First, increasing the value of $c_1$ in Corollary \ref{cor} if necessary, the assumption
$\frac{m}{k} \geq \frac{c_1}{\eta^2} \log \frac{m}{k}$
implies that $\frac{m}{k}  \geq \frac{c_1}{\eta^2}$, and therefore
$$  16e \left (\frac{m}{k} \right)^{3/2} <  e^{5+1.5 \log(m/k)} < L^{c_3\eta^2 m/k},$$
where $c_3 \leq 13/c_1$. Again increasing the value of $c_1$ if necessary,
we can assume that $c_3\leq 4$ so that $1-\eta - c_3  \eta^2 \geq 1-2\eta$. Next, we can restrict $\eta \in (0,1/4]$, replace $\eta$ by $\eta/2$ as before, and replace $c_1$ with $4c_1$. This yields the error bound given in \eqref{main-thm-bound}. For the probability guarantee,
note that $m \geq c_1/\eta^3$. Once again increasing the value of $c_1$ if necessary, we can find an absolute constant $c_2>0$ such that $3\cdot 2^{-\frac{1}{160}\eta^2 m} + e^{-2m}
\leq e^{-c_2 \eta^2 m}$ for all admissible $m$, hence Theorem \ref{T:MAIN} follows.

\section*{Acknowledgements}

The authors would like thank Thao Nguyen, Rayan Saab and \"Ozg\"ur Y\i lmaz for useful conversations.

redpajama_set_name: RedPajamaArXiv


Bolton to light up the skies for the Queen – Bolton Live!
Bolton Council will be putting on a firework display and lighting beacons to celebrate the 90th birthday of Her Majesty the Queen.
The council will be organising a spectacular firework display on Spa Road Playing Fields at Queens Park on Thursday April 21 at 8.30pm to mark the historic occasion.
The official viewing area for the display will be the playing field off Spa Road, however the display is an all aerial spectacular and so the display will be visible from locations right across the borough.
Parking is available at town centre car parks for those attending on the night.
The fireworks will include the colours of the Union Flag — red, white and blue — in a patriotic tribute to Queen Elizabeth II.
Councillor Carole Swarbrick, Mayor of Bolton said: "We hope that residents of Bolton will join us to celebrate the Queen's birthday and either watch one of the beacons being switched on, or view the fantastic fireworks.
The council will also respond to a request from the Queen's Pageant Master by lighting two beacons — one in Horrocks Wood car park off Scout Road and another at USN Bolton Arena. The beacons will be lit at 8.30pm.


text:
So You Want to be Successful on Instagram? Part II
This post is the second in the "So You Want to be Successful on Instagram?" series, which explores Instagram as the ultimate visual platform. Kike Calvo is a National Geographic Creative photographer whose travels you can follow on Instagram.
After gaining over 300 million users in around three years, Instagram has become the undisputable social network for visual people. It places photo sharing and video sharing at your fingertips, no matter where you are in the world. High visibility on Instagram is becoming a coveted commodity among artists who are already mastering the platform as a mean to build a following for their work. However, many are just discovering this app. They are asking themselves a variety of questions: how do you get more Instagram followers? What is the secret behind accounts with thousands of followers? How do you find your niche as a visual provider in a network with more than 20 billion images and counting?
Read rules 1-4 in Part I
Photo © KIKE CALVO
Cross market using your other platforms: By using other platforms you can drive traffic to your Instagram account. It is important to remember too, that you can set your account so you will be able to save time and automatically share your content on Facebook, Twitter or any another social media outlets you may use. "We share many of our Instagram photos on other platforms, especially on Facebook — that's a richer user experience, I think," says social photo editor for @nytimes Kerri MacDonald. "Sometimes I'll post photos directly from our photographers' feeds to Facebook and encourage people to follow them to see more from a story. There is so much that can be done, and I think it's wise to experiment with cross-collaboration in as many ways as possible."
"Only occasionally I get that sophisticated cross marketing with other platforms," says National Geographic photographer Jim Richardson (@jimrichardsonng) of his own strategies. "I know I should but I don't very much. I do use IFTTT to send my Instagram posts to Twitter and Flickr. It's not ideal because the formatting is different. But better than nothing, I hope." Though it may not be essential to post on all your social media channels, it can be a useful way to gain exposure with as many people as possible.
Snorkeling in Granito de Oro Island. Coiba National Park. Photo © KIKE CALVO / National Geographic Creative
Understand hashtags: Explore the hashtags related to your type of account, interests or industry. However, remember not to overdo it. Like metadata embedded in our pictures, hashtags act as keywords. They help people connect with their communities or people they admire. "I learned that hashtags are extremely important for Instagram users who are not currently following you to find you," advises National Geographic Creative photographer Mike Teiss. "Insert some of the strongest hashtags, much like they're keywords. But remember: there is a fine line between that and overdoing it, putting too many hashtags so it looks "spammy" and not just putting the basics for people to find the photo."
Websta, a site that allows you to see statistics for your instagram account, provides a list of the top hashtags:
#love (894,241,625 posts)
#instagood (407,569,165 posts)
#me (366,957,537 posts)
#tbt (343,476,966 posts)
#cute (329,285,399 posts)
#follow (327,830,890 posts)
#followme (316,202,640 posts)
#photooftheday (309,918,269 posts)
Lilac Breasted Roller bird (Coracias caudate). Lake Nakuru (Kenya) Photo © KIKE CALVO / National Geographic Creative
"Jessica Anderson, a social strategy editor, does most of our thinking about hashtags," says MacDonald. "We're trying to use them sparingly. I think we are still working on the best practices, but at this point we have been using no more than five per post. We're trying to use them in places where they'll provide value for our followers. Hashtags with massive numbers don't always do that. Neither do hashtags that bring you to a large collection of disparate images. We have a few of our own: #nytweekender and for our weekend projects; #nytassignment, for photographers who are posting work from assignments, for instance. We're working on a few others."
"I tag if there is an obvious location that I want people there to see, like #scotland if I think people in Scotland will be interested," says Richardson. "I have a hashtag of #islandobsession for the series I do on Scottish islands. And I try to be transparent about whether the picture is from the archive or is new work on my iPhone. I resist hashtagging that I feel would be manipulation. Just dumping in popular hashtags strikes me as tacky."
"At National Geographic Travel we have been successful in sharing and featuring user-generated images through the #natgeotravelpic hashtag (which currently has over 257K tags)," says associate photography producer for @natgeotravel Tyler Metcalfe. "This gives everyone a chance to provide their perspective on travel, which benefits everyone."
Ballerina dancing in the historical district Pelourinho in Salvador de Bahia. © KIKE CALVO for World of Dances.
Interact with other users: Like on most social media, the more you engage, the more activity you receive on your account. This is based on the premise that you are engaging the right way and following the rules of the platform in question. Be kind to other people, and leave appreciative comments under their images, if you truly like them. Based on a survey conducted by Pew Research Center, "most notably, 53% of young adults ages 18-29 now use (Instagram), compared with 37% who did so in 2013. Besides young adults, women are particularly likely to be on Instagram, along with Hispanics and African-Americans, and those who live in urban or suburban environments."
"Many photographers are getting known and developing careers from Instagram," says Lens blog co-editor and senior staff photographer James Estrin. "Take Ruddy Roye for example. But he works at it almost full-time. A few photo editors have told me they find people through Instagram."
"My job running @nytimes is still very new," says MacDonald. "We launched two and a half months ago and this weekend we reached 358k followers. As the account grows, I'm trying to get a handle on how, and when, to interact with our followers. I'm still trying to sort out how to find the right balance of imagery for the account — I want to be sure to show difficult images, but many people don't come to Instagram to see these kinds of things. It's interesting to see how our followers react to some of the stories we cover."
"Our followers hold our National Geographic brand to extremely high standards and expect to see the very best from our photographers," says Metcalfe. " It is great to be held to such high standards, no matter the platform we are publishing on."
Lion (Panthera leo). Masai Mara National Park (Kenya). Photo © KIKE CALVO
Engage your followers. Ask questions as part of your captions. Speak loudly about things you care about. Interact with your users. Make them feel like part of a community. Engagement is the secret to success in most social media platforms. Some brands like GoPro include user-generated content, giving the spotlight to their customers. People are curious by nature. By sharing information about the photographs you are posting, you are sharing dreams, thoughts, experiences, achievements and tiny pieces of your life. Many brands are now posting "behind the scenes" images of their work. Why shouldn't you?
"But like any audience building exercise, you need to determine who your audience is and how to create engaging content for them," says co-founder of PhotoShelter Allen Murabayashi. "The average consumer isn't going to hire a professional photographer, so it's a false premise to say "you must build a huge audience." On the other hand, photographers with 100k+ followers have real revenue opportunities that aren't available to photographers with smaller followings."
"Many of the digital agencies are increasingly looking at engagement rather than raw followers nowadays. Engagement is a percentage of likes and comments," adds Murabayashi. "And is a better indicator of how influential a photographer is with their audience. I think looking at a multi-contributor feed like @natgeo and seeing which photos get lots of engagement is pretty instructive."
Nang Keo traditional dance. Amantaka. Luang Prabang (Laos). Photo © KIKE CALVO
"Instagram is very easy to use compared to Facebook and Twitter," says co-author of The Art of Social Media and social media strategist Peg Fitzpatrick. " It's 100% mobile, which is where a huge percentage of online activity comes from these days. 90% of Instagrammers are under the age of 35 – it's a great place to hit up a younger millennial audience. 58% of Instagrammers use Instagram daily compared with 23% of Pinterest users."
"Instagram reveals very quickly what kinds of pictures people really like," Richardson says. "I'd already seen some of that because I have my own gallery and we offer hundreds of photographs in cards in our shop. You see very quickly what people really gravitate to and what doesn't interest them. It's not always what I would have suspected. Specifically, photojournalism (particularly gritty news images) is much less popular than many of us thought. Animals are golden, partly because they are culturally agnostic. Viewers gravitate to images that celebrate their own cultures, however much we like to think we are lovers of world cultures."
Experiment and learn from your mistakes. Set realistic goals. Keep in mind the importance of measuring your results. There are some tools out there to get this job done. Success, like everything in life, is relative. Bringing more followers to your account can bring some personal satisfaction or even monetary rewards with time, but never lose perspective of things. Have fun.Teiss shared some of his lessons: "Be persistent but be patient, you will gain followers with time. Hold back your average material and only post your best. Post daily, but no more than once a day. Use hashtags to get new followers. Have fun doing it and respond to your followers answering their questions in comments. It is impossible to answer every single one but you can try."
Leopard (Panthera pardus). Masai Mara National Park (Kenya). Photo © KIKE CALVO
"I think that people assume that Instagram has to be polished," says MacDonald. "Some of the most beautiful work I've seen there is more spontaneous — it's work that really shows that the photographer is experimenting, and learning. That's one of the most fun things about Instagram, I think. It's something I want to show more on the @nytimes account, when possible."
MacDonald adds, "I've been working with a few photographers who are new to Instagram and they're all having a lot of fun with it. Generally, I think people often begin by posting a little too much. Instagram isn't a place to publish 12 images at once. It's a place to tell stories a little more slowly — despite its "instant" nature. I also see people going crazy with the hashtags. I think captions (and images, of course!) are definitely a personal choice, but I'm not a huge fan of accounts using too many hashtags and not enough personality. Finally, I think you really need to interact with your followers. We're trying to do this as well."
"Is it too late to become successful on Instagram?" I ask myself. "There was a bit of a gold rush in the early days because the discovery tools were very static," admits Murabayashi. "But I think it's still possible to grow an audience with compelling content and a few key press hits."
"I don't think it is too late to gain followers," says Richardson. "Photographers are building followers all the time. What may be happening is that followers are getting more discriminating. I think there was a time when you could get a lot of followers just because people were trying to figure out who to follow. But they weren't necessarily followers who really stuck around or really cared. You can see that, I think, in some photographers who have vast followings but haven't gained a lot of followers in the last year, and who don't get likes in proportion to the number of followers. That tells me that they have a lot of inactive followers in the rolls. I think anyone seeking to engage an Instagram photographer for work, such as commercial jobs, should be paying a lot of attention to who those followers are and how engaged they are. To be blunt, are they real?"
A desert safari is an exhilarating way of discovering Dubai's natural beauties (United Arab Emirates) Photo © KIKE CALVO
As you probably have figured out by now, brands are looking to spread their brands to broad audiences, but think wisely.
"While advertisers are coming forward to pay for the eyeballs photographers have collected in their followings, I think photographers must be extremely careful when they then share those followers for commercial purposes," concludes Richardson. "Photographers must realize that using their followers for an advertiser or sponsors project may cannibalize their following. If so they must consider the consequences and either decline or accept the loss and charge the client accordingly."
"We are seeing more and more clients who want to leverage the reach of any social media influencers," adds marketing manager for @natgeocreative Ashley Thomas. "And of course, this includes photographers on Instagram with a strong feed of content and healthy followings."
We live in a visual world. Instagram lets you be part of it. Instagram also helps you strengthen your relationships in a fun way. "I think the beauty of the platform is the way in which it encourages even professional photographers to see things a little bit differently, and creatively," says MacDonald. "Instagram gives photographers so much power. I really believe that the captions can be just as powerful as the images; everyone agrees with me on that point."
"A lot of the images you see on people's Instagram feeds are not going to change the world," adds MacDonald." They are just a more personal way of looking at it, and I love that."
Any opinions, advice, statements, or other information that constitutes part of the content are copyrighted and property of their respective owners.
Follow Kike Calvo @kikeo. Follow National Geographic Creative @natgeocreative.
So You Want to be Successful on Instagram? Part I
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Instagram Marketing On Fire: How Smart Companies and Entrepreneurs Ignite Their Instagram Followers and Turn Them Into Repeat Customers
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Tags Kike Calvo
Award-winning photographer, journalist, and author Kike Calvo (pronounced key-keh) specializes in culture and environment. He has been on assignment in more than 90 countries, working on stories ranging from belugas in the Arctic to traditional Hmong costumes in Laos. Kike is pioneering in using small unmanned aerial systems to produce aerial photography as art, and as a tool for research and conservation. He is also known for his iconic photographic project, World of Dances, on the intersection of dance, nature, and architecture. His work has been published in National Geographic, New York Times, Time, The Wall Street Journal, The Washington Post, New York Magazine, Rolling Stone, and Vanity Fair, among others. Kike teaches photography workshops and has been a guest lecturer at leading institutions like the School of Visual Arts and Yale University. He is a regular contributor to National Geographic blog Voices. He has authored nine books, including Drones for Conservation; So You Want to Create Maps Using Drones?; Staten Island: A Visual Journey to the Lighthouse at the End of the World; and Habitats, with forewords by David Doubilet and Jean-Michel Cousteau. Kike's images have been exhibited around the world, and are represented by the National Geographic Image Collection. Kike was born in Spain and is based in New York. When he is not on assignment, he is making gazpacho following his grandmother's Andalusian recipe. You can travel to Colombia with Kike: www.colombiaphotoexpeditions.com
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Home Latest News US Restrictions Impact Chinese Markets
US Restrictions Impact Chinese Markets
The UK's leading 100 stock index outperformed all major indices last week, recovering 2.27% from Tuesday's lows to finish the week up 1.56%. Strong performance towards the end of the week may have been aided by the Bank of England's (BoE) latest policy meeting, as rates remained unchanged at 0.1% while the BoE raised its growth forecasts to 7.25% this year, the highest in more than seven decades. Although this is up from the 5% projection made back in February, Andrew Bailey, the central bank's Governor, ensured the move was not taken out of context as the growth forecast purely stems from the economy shrinking in the last year. He said: "[The strong recovery] takes us back by the end of this year to the level of output we had at the end of 2019... two years of output growth have been lost." The BoE also announced a reduction in the pace of quantitative easing yet stressed this does not signal a change in stance of its monetary policy.
Chinese equities took another hit last week after geopolitical relationships withered on two fronts. First, China indefinitely suspended all activity with Australia under a China-Australia Strategic Economic Dialogue, as some Australian Commonwealth Government officials were accused by China's National Development and Reform Commission (NDRC) on Thursday of "launching a series of measures to disrupt the normal exchanges and cooperation between China and Australia out of Cold War mindset and ideological discrimination." The relationship between the two Asia-Pacific countries has gradually deteriorated since Australia became the first country to ban Chinese tech company Huawei from its 5G network. Essentially, the suspension of the agreement will ban exports of Australian coal with copper concentrate imports to China also dropping off. Australia's commodity sector is therefore expected to take a hit, but its largest export in iron ore is likely to remain protected from the current political tensions. As a result, the Australian Dollar initially dropped but has recovered somewhat. Chinese equities were not so fortunate, however. The CSI 300 index was down 2% on Friday morning, no doubt also impacted by a report revealing US President Joe Biden's administration is set to maintain its pressure on China by preserving former President Donald Trump's limitations on US investments in Chinese companies.
US markets had a mixed week. The Dow Jones Industrial index hit a record high while the Nasdaq and S&P 500 both fell at the start of the week to slightly recover going into the weekend. As of Friday morning, the Dow Jones was up 2.07%, the Nasdaq was down 2.01%, and the S&P 500 dropped by 0.08%. An inspiring weekly jobless claim figure helped to provide some insight on the US recovery, but questions have begun to surface as to whether there are enough workers to fuel economic growth. Some companies in the US, such as Chipotle and MGM resorts, have stated they are struggling to employ or attract workers for their businesses. Stimulus cheques and generous unemployment benefits could be a contributing factor to what is unfolding to be a "Job Paradox" in the US, especially for companies in the leisure and hospitality sectors, according to one Senior Economist at Bank of America Corp.
In the UK, one of Rotherham's largest employers, Liberty Steel, may be set to secure a £200m lifeline after its previous lender Greensill Capital went bust in early March this year. Almost all work at the Rotherham site has ground to a halt with 3,000 workers across 11 different sites at risk of losing their jobs. It is clear Sanjeev Gupta, the CEO of steel and mining conglomerate Gupta Family Group (GFG), is trying hard to keep the company alive. He recently contacted the UK Government for £170m in support but was rejected. Business Secretary Kwasi Kwarteng told the BBC he wanted to give the firm "time to find finance".
Thankfully, that appears to be a possibility as California-based investment firm White Oak Global Advisors are in the preliminary stage of an agreement, but this first needs the approval of GFG's creditors including Credit Suisse and Tata Steel. If the loan does go ahead, a degree of job security will return to Liberty Steel employees. However, Sanjeev Gupta has also appointed four new directors to lead a committee that will be given the freedom to sell underperforming parts of the company. This could potentially lead to some UK plants being sold off to pay back the debt, creating further fear within the already declining UK steel industry.
Please note that investments and income arising from them can fall as well as rise in value. This communication is for information only and does not constitute a recommendation to buy or sell the shares of the investments mentioned.
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