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Granite Channel Chief Charlie Pagliazzo Named a Channel Futures Networking & Connectivity Leader
Inaugural List Honors 20 Leaders Who Are Determining the Future of the Telecom Channel
"This award recognizes Granite Channels' commitment to providing our partners with the training and support needed to deliver the technology solutions their clients look to them to provide. Partners are (our) customers, and empowering them to leverage all that Granite offers is what drives (us)."
QUINCY, Mass. (PRWEB) November 10, 2022
Granite Telecommunications, a provider of communications service and technology solutions to multilocation businesses and government agencies, announced today that the company's Vice President of Channels, Charlie Pagliazzo, is among the 20 honorees on the inaugural Channel Futures Networking & Connectivity Channel Leaders list. The list recognizes these individuals as leaders who will determine the channel's future as they redefine the partner-supplier relationship.
Channel Futures selected the honorees based on their companies' market share, growth potential, partner network strength, partner program scope, and the individual's impact on the partner ecosystem.
Pagliazzo, a 29-year telecom veteran, heads up sales, support and program management for Granite Channels. Over the past 19 years, Pagliazzo established Granite Channels' nationwide presence and grew the partner program to more than $420 million in annual revenue. Channel Futures recognized Pagliazzo's leadership in advocating for indirect sales at Granite and contributing to the company's philanthropic initiatives. Previously, Pagliazzo was named to the 2020 Channel Futures Top Gun 51 and 2016 Channel Partners Circle of Excellence lists.
"I'm honored to be included on the Channel Futures Channel Leaders list for networking and connectivity," said Pagliazzo. "This award recognizes Granite Channels' commitment to providing our partners with the training and support needed to deliver the technology solutions their clients look to them to provide. Partners are Granite Channels' customers, and empowering them to leverage all that Granite offers is what drives our team."
"As the industry faces one of its most tumultuous periods, it is essential for channel partners to connect with the leaders of today's strategic technology providers. These individuals are the ones who are guiding their organizations and partner ecosystems into the future with innovative thinking and industry-defining channel programs. Channel Futures has identified the individuals who are fueling a new wave of growth and solutions partners can bring to customers." said Robert DeMarzo, Vice President, Content, Informa Tech Channels.
About Channel Futures
Channel Futures is a media and events destination for the information and communication technologies (ICT) channel community. We provide information, perspective and connection for the entire channel ecosystem, including solution providers (SPs), managed service providers (MSPs), managed security service providers (MSSPs), cloud service providers (CSPs), value-added resellers (VARs) and distributors, technology solutions brokerages, subagents and agents, as well as leading technology vendor partners and communication providers.
Our properties include many awards programs such as the Channel Futures MSP 501, a list of the most influential and fastest-growing providers of managed services in the technology industry; Channel Partners events, which delivers unparalleled in-person events, including Channel Partners Conference & Expo, Channel Partners Leadership Summit, Women's Leadership Summit, the MSP Summit and Channel Partners Europe; and a DEI Community Group, our initiative to educate, support and promote diversity, equity and inclusion (DE&I) in the ICT channel industry. Channel Futures is where the world meets the channel; we are leading Channel Partners forward. More information is available at channelfutures.com.
Channel Futures is part of Informa Tech, a market-leading B2B information provider with depth and specialization in ICT sector. Every year, we welcome 14,000+ subscribers to our research, more than 4 million unique monthly visitors to our digital communities, 18,200+ students to our training programs and 225,000 delegates to our events.
About Granite
Granite delivers advanced communications and technology solutions to businesses and government agencies throughout the United States and Canada. The $1.8 billion company serves more than two-thirds of Fortune 100 companies and has 1.75 million voice and data lines under management, supporting more than 650,000 locations. Founded in 2002, Granite has grown to be one of the largest competitive telecommunications carriers in the US by simplifying sourcing and management of voice, data and cellular service with a single point of contact and consolidated invoicing for all locations nationwide. Today, Granite supports customers with a wide range of services, including access, UCaaS, mobile voice and data, and MSP solutions for SD-WAN, monitoring and network management. Granite employs more than 2,250 people at its headquarters in Quincy, Mass. and 11 regional offices nationwide. For more information, visit http://www.granitenet.com.
Jeremy Robison


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Dallas Smith – Some Things Never Change
Multi-Platinum award winning country star Dallas Smith has announced the details for his long-awaited return to the road with the 2022 Some Things Never Change Tour with James Barker Band and special guest Meghan Patrick plus emerging country artists Shawn Austin, JoJo Mason, Kelly Prescott, and Manny Blu . The Tour will kick off May 24 in Brandon, MB, and head to over 20 cities including St. John's on June 14, 2022 at the Mary Brown's Centre.
Tickets range in price from $45.94 to $103.44 (plus fees and service charges). Tickets will be available beginning at 12 noon on Friday, January 7, 2022 at Mary Brown's Centre Box Office, 709 576-7657 and online at mbcentre.ca.
VIP Tickets are also available for $390.94 (plus fees and service charges) and include:
Some Things Never Change VIP Experience
One (1) standing ticket in the GA PIT
Invitation to an exclusive VIP experience featuring pre-show performances from Dallas Smith, James Barker Band, and guests!
A Q&A session with Dallas Smith, James Barker Band, and guests
VIP-exclusive Dallas Smith gift item
An autographed Dallas Smith Some Things Never Change Tour poster
Official VIP Experience laminate
Tour merchandise shopping before door time
Dedicated entrance (where available)
On site VIP host
NHL Legends put the Christmas Kettle on in Support of the Salvation Army!
NHL Legends will put the Christmas Kettle on, in Support of the Salvation Army, during the NHL ALUMNI GAME game on THURSDAY, DECEMBER 14TH at Mile One Centre.
Boston Bruins LEGENDS lineup includes: Raymond Bourgue, Rick Middleton, Terry O'Reilly & Ken Linsman
Montreal Alumni LEGENDS lineup includes: John LeClair, Stephane Richer, Chris Nilan & Steve Penny
Doors open at 6:30pm / Puck Drops at 7:30pm
Tickets available at Mile One Centre box office, 709.576.7657 and www.mileonecentre.com.
Tickets $25.00 for students with I.D and $35.00 for an adult.

redpajama_set_name: RedPajamaCommonCrawl


Brazil's fascinating for everyone, from eight year olds watching Robinho do crazy things with a ball, to 28 year olds watching Gisele while doing crazy things with their own...Robinhos. Now making Brazil fascinating for your eyes, Chilli Beans.
CB is Brazil's largest sunglass company, just now making its US debut, with fresh drops coming in each week to a small, LCD-TV-enhanced, mirror-laden shop in Santa Monica Place -- a safe choice as Melrose Place is full of Shooters. Just released shades include a futuristic, bright red pair; a gold-rimmed, aviator-style number w/ thin black lines running down each side; and an uber-Euro-looking pair with thick earpieces and a one-piece lens that's all yellow, so buy them, and expect Coldplay to write a song for you, and all the things you do. They've also got other accessories, like an extensive selection of watches, with options like wide-circled stainless steel analogs, as well as fitted logo caps, striped fedoras, and plaid drivers caps, though it remains unclear why your golf club needs a cap, especially when its got that sweet Snoopy cover.
To celebrate their LA launch, they've also got a few limited-edition, store-only pairs boasting a graphic of the ferris wheel on Santa Monica Pier, which is also a fascinating place, but be sure and keep your wallet in your front pocket, to protect against getting robbinho'd.


A gorgeous story spanning four generations of a Canadian family, this page-turner begins in the early 20th century when James Piper marries Materia, a teenage, Lebanese girl from a wealthy family. When the couple give birth to Kathleen, a dangerous obsession develops for which many family members will pay the consequences.
As the characters travel from Cape Breton to New York City, the novel explores family dynamics and the role of hazardous secrets.


For my thoughts on privacy in general, take a look at my privacy statement on my other blog. In short, I hate modern analytics and tracking, and I will never track you in any way. This means that you actually have to contact me if you want to give feedback. I won't be watching from behind the scenes.
I reserve the right to advertise in the future, but if I do, it will be in a way that does not track you.


Whales and dolphins join the ranks of the free in Costa Rica, following a decree signed last weekend by President Oscar Arias that protects these marine mammals in all of Costa Rica's ocean territory.
The decree forbids the "pursuit, capture, injury, netting or commercialization" of all whale and dolphin species on the country's expansive, 640,000 square kilometer marine territory, the largest in Central America.
Arias waxed poetic about his move, likening it to the country's now half-century old decision to do away with the army.
"Today, we have another peace agreement to sign, and another military force to abolish: We must come to peace with our environment,"he said in a speech in Puerto Jiménez, overlooking well-known whale and dolphin-viewing hotspot Golfo Dulce, in the Southern Zone.
Environment Minister Roberto Dobles seconded Arias' commitment, stating that the country had proven once again that it was at the "forefront" of biodiversity protection worldwide.
While the decree makes the policy official, Costa Rica has traditionally never had a whale hunt, unlike nations such as Norway, Japan and Iceland, and dolphins have rarely been the target of commercial fishers here.
Earlier this year,Dobles voted against reinstating a limited hunt worldwide, despite the pressure from the Japanese, at a meeting of the International Whaling Commission (IWC) in Alaska.
Costa Rica can put another notch in its belt of global recognition for greenness after scoring high on a new environmental performance list released by Yale and Columbia universities.
While Switzerland and the wealthy Scandinavian countries dominate, Costa Rica placed fifth overall, came first in Latin America and placed well among industrialized nations on the first official Environmental Performance Index released Wednesday. After Switzerland came Sweden, Norway, Finland and Costa Rica.
"Costa Rica is proud to be a global leader in environmental conservation," said President Oscar Arias.
In Latin America, Colombia came second after Costa Rica, finishing ninth.
The United States came 39th among the index's 149 countries, with the United Kingdom ranked 14th, Canada 12th and Australia 46th, according to the Web site http://epi.yale.edu/home.
In Costa Rica, a separate report also brought good news to the environment. The country has cut its imports of ozone depleting gases by 71.5%, the Environment Ministry said. In 2004, the country imported 105 tons of chlorofluorocarbons (CFCs), while in 2007, the amount was just 30 tons, according to newswire ACAN-EFE.
Industries have also reduced the purchase of harmful agricultural chemicals such as methyl bromide, typically used by melon growers to control a wide variety of pests. Imports of that chemical dropped from 1,070 tons in 1999 to 400 last year, thanks to a national phase-out program.


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Bank Consolidation: Regulators Always Have the Power to Pull the Plug
By Adam M Zaretsky
Between 1990 and 2002, the total number of banks and thrifts in the United States fell from 15,131 institutions to 9,336. The foremost cause of the decline was a significant relaxation of U.S. bank branching laws, which allowed mergers and acquisitions that previously had been prohibited. Such mergers and acquisitions cannot take place, however, without the approval of a federal banking regulatory agency. (See table.) These banking regulatory agencies scrutinize proposed transactions to ensure that they do not violate any of the U.S. antitrust statutes.
The practice of antitrust analysis of bank mergers and acquisitions dates back to 1963, when the U.S. Supreme Court held that commercial banking, like other industries in the United States, is subject to the Sherman Antitrust Act of 1890 and the Clayton Act of 1914.1 In its opinion, the court noted that the test for anticompetitive behavior is whether the effect of a bank merger "may be substantially to lessen competition … in any line of commerce in any section of the country."
Foundations of Antitrust
To apply this test, the court defined the "line of commerce" for the banking industry as the cluster of products and services-demand deposits, trust administration and extension of various types of credit, for example-that banks uniquely provide to their customers. In other words, the court determined that the products and services denoted by the term "commercial banking" compose a distinct line of commerce.
To define "section of the country"—that is, the relevant geographical market—the court looked to where the effect of a merger on competition would be "direct and immediate." For banking, this effect occurs in the customers' local communities because individuals and firms typically conduct the bulk of their banking transactions at banks with local offices.
A bank regulatory agency's final step is to determine whether the effect of the merger "may be substantially to lessen competition." In its ruling, the Supreme Court recognized that the answer to this question involved not only the immediate effects of a merger on competition, but also its anticipated future effects.2 Such a prediction relies on the structure of the relevant market-that is, market concentration, the market shares of individual banks and number of market competitors. Banking antitrust is based on the assumption that the structure of a market influences how firms in that market will act, which, in turn, affects the firms' overall performance (otherwise known as the structure-conduct-performance hypothesis). In other words, the merger's effect on these measures of structure, particularly market concentration, is thought to be a reliable gauge of whether the merger will lessen competition substantially. Therefore, a proposed merger that increases market concentration considerably would probably fail this test, and the federal regulator would not approve it. The federal regulator might approve it, however, if other evidence exists to mitigate the proposal's anticompetitive effects on market structure. That said, the Department of Justice could challenge the decision and possibly sue to prevent the merger.
Antitrust in Action: the Guidelines
To minimize the chances that a decision will be challenged and to align the antitrust analyses of the federal regulators, the Justice Department has periodically issued guidelines that define the circumstances under which an application is likely to exceed its antitrust standards and, therefore, warrant closer scrutiny. The federal banking regulators use these guidelines to help them identify proposals that are likely to raise concerns about adverse effects of mergers on competition.
The Justice Department's antitrust standards identify potentially anticompetitive mergers in terms of prescribed levels, and changes in levels, of a commonly used measure of market concentration, the Herfindahl-Hirschman Index (HHI). HHI is calculated by squaring each bank's share of deposits in a market and then adding these squared shares. The index number can range from zero (a perfectly competitive market) to 10,000 (a pure monopoly). For example, the perfectly competitive market would consist of many firms, each with about the same market share. As the number of firms in this market increases, each firm's share decreases, until it approaches the limit of zero. The square of zero is zero; so, the sum of those squares is still zero. The pure monopoly market would have only one firm that controls 100 percent of the market. The square of 100 is 10,000.
According to the guidelines, a market can be broadly characterized as unconcentrated if the HHI is less than 1,000 points, as moderately concentrated if the HHI is between 1,000 and 1,800, and as highly concentrated if the HHI is above 1,800.3 These thresholds apply not only to banking, but to all industries in the United States. The Justice Department distinguishes banking from other industries, however, by allowing it more latitude for increases in HHI. That is, the department normally will not challenge a bank merger or acquisition unless the resulting increase in HHI is at least 200 points and the post-merger market HHI is at least 1,800 (highly concentrated). The additional cushion afforded the banking industry accounts for the competition banks now face from thrifts, credit unions and other providers of financial services. In fact, thrifts-that is, savings and loan associations and savings banks-so resemble banks today in their financial service offerings that their deposits are commonly included in antitrust analyses at 50 percent weight.4
Deposits at credit unions are rarely included in banking antitrust analyses. Being membership organizations, credit unions offer their financial services only to their members, and these services are usually quite limited when compared with those offered by banks and thrifts. As such, credit unions do not necessarily compete in the same product market as banks and thrifts.5 In certain cases, however, credit union deposits might be included in the analysis of a specific market (at fractional weighting) if substantial evidence supports their inclusion. One piece of such evidence would be that the share of deposits at credit unions in the market area greatly exceeded the national average. On top of that, a particular credit union should have liberal membership rules (typically, at least 70 percent of market residents must be eligible for membership) and offices that are easily accessible to local residents.
Antitrust in Action: Beyond the Guidelines
Determining the change in HHI and its post-merger level is not the end of the story. If these numbers were to fall outside of Justice Department thresholds, the merger or acquisition would not automatically be denied. Such an outcome would indicate only that regulators consider the concentration of the market to be high enough to enable the firms in the market to keep prices above the competitive level for a significant period of time. Such a case would require that a more-detailed economic analysis be conducted before a decision could be made. This analysis would seek to determine whether other factors, such as potential competition and economic conditions of the market, could mitigate the anticompetitive structural effect of the merger and, thereby, suggest that the HHI does not tell the whole story. An applicant might avoid the more-detailed analysis, however, if it were to choose or agree to divest to a third party some of its offices in the affected markets to get those markets' competitive structures to fall within guidelines.6
Having a post-merger HHI and an increase in HHI that exceed the department's thresholds is not the only reason an application might receive closer scrutiny. A bank that would end up controlling more than 35 percent of the deposits in a particular market after a merger or acquisition would also trigger a more in-depth examination by the Federal Reserve, even if the HHI indicates no significant change in market concentration.7 In such a case, the Fed's antitrust analysis would focus on whether any factors might mitigate the anticompetitive effects of the merger. One such mitigating factor could be that recent economic growth in the market has been strong enough to indicate that it is attractive for entry by other banks or thrifts.8
These procedures are an effective and consistent method by which federal banking regulators ensure that bank mergers and acquisitions do not "substantially ... lessen competition … in any line of commerce in any section of the country." They have been questioned and tested many times, particularly since changes in the law have allowed many more types of financial institutions to offer similar products.9 In the end, those tests continue to show that these procedures are still the best ones to use in these antitrust analyses.
Federal Banking Regulators
Regulates
Office of the Comptroller of the Currency (OCC) Commerical banks with national charters
Federal Reserve System (Fed) Bank-holding companies and state-chartered commercial banks that are Fed members
Federal Deposit Insurance Corp. (FDIC) State-chartered commercial banks that are not Fed members
Office of Thrift Supervision (OTS) Thrifts
Department of Justice (DOJ) Oversees all
United States v. Philadelphia National Bank, 374 U.S. 321 (1963). [back to text]
This is how the court interpreted Congress' directive to "arrest anticompetitive tendencies in their 'incipiency.' " [back to text]
Department of Justice, Antitrust Division, and the Federal Trade Commission, 1992 Horizontal Merger Guidelines (revised April 8, 1997). www.justice.gov/atr/public/guidelines/hmg.htm [back to text]
In some cases, deposits at thrifts may be weighted more or less than 50 percent, depending on the level of activity a particular thrift has in a region's commercial lending market. In addition, deposits of thrift subsidiaries of commercial banking organizations are included in the HHI calculation at 100 percent. [back to text]
The types of services credit unions offer are broadening, though. For more information about the evolution of credit unions, see Emmons and Schmid (2003). [back to text]
For more information about divestitures, see Webb (2001). [back to text]
See Department of Justice, "Bank Merger Competitive Review-Introduction and Overview" (1995), current as of September 2000. www.justice.gov/atr/public/guidelines/6472.htm [back to text]
For other examples of mitigating factors, see Holder (1993). [back to text]
For summaries of many of the tests that have been performed recently, see Gilbert and Zaretsky (2003). [back to text]
Emmons, William R. and Schmid, Frank A. "Credit Unions Make Friends-But Not with Bankers." Federal Reserve Bank of St. Louis The Regional Economist, October 2003, pp. 4-9.
Gilbert, R. Alton and Zaretsky, Adam M. "Banking Antitrust: Are the Assumptions Still Valid?" Federal Reserve Bank of St. Louis Review, November/December 2003, Vol. 85, No. 6, pp. 29-52.
Holder, Christopher L. "The Use of Mitigating Factors in Bank Mergers and Acquisitions: A Decade of Antitrust at the Fed." Federal Reserve Bank of Atlanta Economic Review, March/April 1993, Vol. 78, No. 2, pp. 32-44.
Webb, Robert L. "Divestiture: A Prescription for Healthy Competition." Federal Reserve Bank of St. Louis The Regional Economist, January 2001, pp. 10-11.


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johnny seven
Guest writer Stephen A. Nelson is a freelance journalist who grew up in Toronto back when, as he says "they still had a Santa Claus at the Eaton's and Simpson's department stores, where magical window displays with expensive moving toys gave joy to many." Here Stephen shares a true, right from the heart, childhood Christmas Story, and adds some of his own favourite media Christmas Stories.
THE MANY FACES OF FATHER CHRISTMAS
A Real Christmas Story from the Kid Who Lived It!
by Stephen A. Nelson (reprinted)
The Salvation Army has been in the news; especially one Christmas, when it was discovered that a Grinch – masquerading as Santa Claus – stole more than $2 million worth of donated toys that were meant to go to needy kids in the Toronto area. The stolen toys were recovered and Mean Mr. Grinch was arrested and charged with stealing Christmas. It's sad and shocking, but at the same time a reminder that – while Scrooges and Grinches may be real – so is Someone Else. The Salvation Army proceeds this year with the 128th annual Christmas Kettle Campaign, a mission to aid millions of people living in poverty.
My Christmas Story
My Christmas story takes place not in the 1940s, but in the 1960s, shortly after we'd arrived in Canada from England. The Beatles were more popular than Jesus and it was a very good time to be English in Canada. Especially if Canadians thought you sounded "just like The Beatles."
We moved into a neighbourhood in West Toronto. Our family of six was living in a small two-bedroom flat above a restaurant on Dundas Street West now known as The Junction. We did not have much, in fact, we had very little. But we had the two most valuable things in the world: family and friends.
"Family" included my Uncle Terry, my mother's beloved brother. The Nelson family had arrived in Canada on the unforgettable night of the Great Northeast Blackout in November of that year — that night everything from Niagara Falls to New York plunged into complete darkness. That was the night Uncle Terry had driven all night to get us from Montreal to our new home in Toronto.
"Family" also included the people at the Salvation Army in West Toronto. That was the great thing about the Sally Ann then. We had travelled to the other side of the world, but the Salvation Army still felt like home. It was more than a church, more than a charity. It was family.
Foremost among that Salvation Army family was a man we called Uncle Harry. He virtually had adopted my dad when my dad first arrived in Canada. And for as long as he lived, Uncle Harry was like a father and a big brother to my Dad. And until his dying day, Uncle Harry was a real uncle and a granddad to us kids.
When we arrived in Toronto, winter was already upon us. More snow than we had ever seen turned Toronto into a Winter Wonderland. And before we knew it, it was Christmas. It was a beautiful glorious Christmas, around which the entire year of being a kid revolved. Christmas, time for Christmas carols and Father Christmas — or as Canadians called him, "Santa Claus."
I was a true believer in Father Christmas. So although there were many department-store Santas, I knew there was only one true Father Christmas. He was at the North Pole now, making his toys. But I had met him at the Salvation Army Christmas party and I knew he was real. I knew because he knew me and he knew me by name.
But I was worried: Even though I'd seen Father Christmas, I hadn't told him what I really wanted. And with all the excitement of moving to a new country, I hadn't had time to write to him! "How will Father Christmas know where we are? How will he know what I want?" I wondered.
Mum, another true believer, assured me that Father Christmas would know and that he would find us.
"But we don't have a chimney in our flat! How will Father Christmas get into our house on Christmas Eve?"
Mum assured me that Father Christmas had a magic chimney he could use to get into houses that had no chimney.
Well if Mum says so, it's got to be true.
But I was still worried.
I was like Ralphie from the character who played in A Christmas Story. A movie favorite of mine that centres around a young, nine-year-old, blue-eyed, blond-haired boy name Ralph "Ralphie" Parker (Peter Billingsley). The first time I saw the film, I immediately turned to my friend, pointed at the kid on the screen with big glasses and said, "That's me!"
But Ralphie is an Everyman, and the film is the tale of his quest to realize his heart's desire and secure his holy grail of Christmas presents — a Daisy, Red Ryder, Carbine-Action , 200-shot Range Model BB Rifle. He was trying to relay the message to the world that there was one thing I wanted more than anything else in the world. This was my heart's desire. This was my holy grail.
It wasn't a Red Ryder BB Gun. No, in 1965 it was a Johnny Seven O.M.A. (One Man Army) — a toy gun that was seven guns in one.
In these cyber-days of Halo, WarCraft and Assassin, such toy weapons are either politically incorrect or passé. Forbidden or forgotten. But in 1965, the Johnny Seven was "the bomb."
It was a Red Ryder BB Rifle on steroids. Ralphie's Red Ryder was a steel-blue beauty that fired BBs and had "a compass in the stock and this thing that tells time." It was everything cowboy hero Red Ryder would need to fend off the evil Black Bart, rustlers, and other Bad Guys. But a Johnny Seven One Man Army was a plastic mini-arsenal that fired an anti-tank rocket; shot an armour-piercing shell; launched an anti-bunker missile; shot 10 bullets as a rifle; made a rat-a-tat-tat sound as a Tommy gun; and had a pistol that detached and worked as a cap gun with a very loud "bang!" In short, it was everything that G. I. Joe would ever need to fight the Viet Cong. And since it was three-feet long, it was literally the yardstick against which all other boys' toys were measured. It was the talking Malibu Barbie of boys' toys. It was the perfect present.
Like Ralphie, I knew exactly what I wanted. And like Ralphie, I was worried that I wasn't going to get it. After all, if Father Christmas couldn't find me, how could he give one to me? And if Father Christmas couldn't deliver, who could?
On Christmas Eve, there were no signs of any Christmas presents in our small flat. I don't even remember seeing a Christmas tree.
Still, milk and cookies were placed out in a dish with care, in hopes that Father Christmas soon would be there. There were no visions of sugar plums dancing in my head. I was dreaming of a Johnny Seven. I went to bed hoping and praying that Father Christmas would indeed find us. But I feared he would not.
On Christmas morning, I woke up to find a sign of hope. A red Christmas stocking had been hung up with care, a sign Father Christmas had really been there.
When out from the living room there was such a clatter, I sprang down hallway and said "What's the matter?"
I turned into the living room and beheld a sight more wonderful than anything I had ever seen before or since. Where there had been darkness before, there was now the most wondrous light. Where there had been emptiness, there was now a cornucopia of Christmas presence. An Aladdin's cave of magical things that I had never seen before in my life, including something called a "toboggan." I had no idea what it was for, but I thought it was amazing.
In a room full of children's treasures, I almost missed the Holy Grail itself. But there it was, in the middle of the living room floor, set like a jewel in the crown: my Johnny Seven. It was perfect.
I promptly fired off all seven guns in rapid order and managed to do it without shooting anybody's eye out!
I was overwhelmed with joy at getting my Johnny Seven. But I was even more happy that Father Christmas had found us and he had delivered. For years to come, this was proof to me that Father Christmas was real. And later, when my non-believing school friends said, "There's no such thing as Santa; it's just your parents," I knew they were wrong.
Over the years in Canada, we would spend many Christmases at my Uncle Terry's and we would learn many things about Father Christmas. We learned that, here in Canada, Father Christmas went by many other names: Saint Nicholas, Saint Nick, Sinter Klaas, Santa Claus, Santa, Kris Kringle, or even Père Noël.
Years later, I found out that, in our home, Father Christmas had two other names: Sometimes we called him "Dad": other times we called him "Uncle Terry."
But at our other home, the Salvation Army, we learned that the most important person at Christmas was not Father Christmas, but that other person who also comes at Christmas and who is also known by many names.
And at the Salvation Army, we learned that the real Father Christmas had yet another name. There we called him "Uncle Harry."
Welcome Christmas, while we stand
heart to heart and hand in hand…
Christmas Day will always be
Just as long as we have we
My Favourite Media Christmas Stories
Stephen A. Nelson
Everybody has a favourite TV holiday special at this time of the year.
For me, there are two classic Christmas programs that are especially powerful, even after more than 40 years of repeated viewing.
The first and still the best is A Charlie Brown Christmas, featuring the Peanuts gangand the coolest of all Christmas soundtracks by Vince Guaraldi. A very close second is How the Grinch Stole Christmas starring Boris (The Grinch) Karloff and the coolest of all Christmas songs, You're A Mean One, Mr. Grinch .
And when it comes to Christmas movies on TV, you still can't beat the 1951 black-and-white classic A Christmas Carol, starring Alistair (Scrooge) Sim and three very spooky Christmas spirits. Mind you, Jim Carrey. The Muppets, and even Doctor Who have all done a great job of bringing this story to life.
What these all have in common is the theme of redemption. That and the radical, almost heretical message expressed by the Grinch: the idea that "Maybe Christmas… doesn't come from a store. Maybe Christmas, perhaps, means a little bit more!"
So maybe it's odd that my other holiday favourite is the 1983 movie A Christmas Story. Odd because, at first glance, the movie seems to say that Christmas does come from a store — and that happiness is a new gun.
A Christmas Story starring Darren (Kolchalk: The Night Stalker) McGavin is a classic based on the book In God We Trust, All Others Pay Cash by Jean Shepherd. A sleeper hit when it was first released, the film has become a perennial favourite on TV. One American cable station shows it all day on Christmas Day.
The film, narrated by the author himself, takes place in a mythical mid-western US town. But a lot of it was shot in Toronto. In fact, in a part of Toronto that looks a lot like the Toronto I grew up in. It even has the classic red-and-yellow TTC streetcars that Mike Filey loves.
It also has all the elements of my first Christmas in Canada: the Salvation Army band playing Christmas carols in the frosty air; the Santa Claus parades, the department store Santas. And most magical of all, the Christmas windows displays at stores like downtown Toronto's Eaton's and Simpson's, where half-frozen kids press their noses up against the frosty glass to get a closer look of at the electronic panoramas of mechanized magic.
Perhaps that's why this film stays with me, because it feels like my city and my childhood. I feel I could have easily run into the characters in this film. In fact, I think I did.
But more than that, I think it stays with me because — in many ways — A Christmas Story is my story: "except for the name and a few other changes, when you talk about me, the story's the same one."
Ralphie, for most of the film, is scheming to get his mitts on one of these beautiful, steel-blue pieces of pre-adolescent weaponry- a Daisy, Red Ryder, Carbine-Action , 200-shot Range Model BB Rifle… For him, it is not only the Holy Grail; in his hands it will become Excalibur.
So our hero does everything he can to persuade every adult he knows — his teacher, his parents, even Santa — that a firearm is the perfect present. The response is always discouraging and always the same: "You'll shoot your eye out!" And once even Santa Claus rains on his parade, Ralphie knows the game is over and he'll probably end up with a football ("not a very good present") for Christmas.
Needless to say, our hero is more than a little discouraged come Christmas Eve. By Christmas morning, when all the presents have already been opened, he has despaired of ever achieving the Holy Grail.
But wait! Somehow Santa delivers a last-minute Christmas miracle and Ralphie gets his BB gun — his faith in Santa and in Christmas is fully restored. That night he goes to sleep with his holy grail in his hands and all is right with the world. It is a very merry Christmas.
Donations to the 2018 Christmas Campaign can also be made at SalvationArmy.ca, by calling 1-800-SAL-ARMY (725-2769), via mail to The Salvation Army, 2 Overlea Blvd, Toronto, ON M4H 1P4.
Donors can also text HOPE to 45678 from most mobile carriers in Canada. A $5 donation will be added to their monthly mobile bill.
Stephen A. Nelson has a master's degree in Theological Studies from Lutheran Theological Seminary at the University of Saskatchewan. In his spare time, he plays old rock 'n' roll at the local jam nights, sings in church, and enjoys his Jasper mountain paradise. Write Stephen Nelson at http://ca.linkedin.com/in/stephenanelson or stephena.nelson@gmail.com or
Posted in Christmas, Daily Blogs
Tagged charlie brown, Eaton's, Emotions, Heart, johnny seven, merry christmas, moving toys, mr grinch, my story, sally ann, salvation army, Simpson's, stealing christmas Stephen A Nelson


text:
For the Japanese standards committee, see Japanese Industrial Standards Committee.
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1 April 1993 (1993-04-01)
not-for-profit company
To enable people in higher education, further education and skills in the UK to perform at the forefront of international practice by exploiting fully the possibilities of modern digital empowerment, content and connectivity
Bristol, London, Manchester and Harwell, Oxfordshire
Region served
Dr Paul Feldman[1]
www.jisc.ac.uk
Jisc (formerly the Joint Information Systems Committee) is a United Kingdom not-for-profit company[2] whose role is to support post-16 and higher education, and research, by providing relevant and useful advice, digital resources and network and technology services, while researching and developing new technologies and ways of working. It is funded by a combination of the UK further and higher education funding bodies, and individual higher education institutions.[3]
Jisc's vision is for the UK to be the most digitally-advanced higher education, further education and research nation in the world.
Services offered include:
Providing UK universities and colleges with shared digital infrastructure and services, such as the superfast JANET network
Operating shared digital infrastructure and services, including:
The Archives Hub, a catalogue of the collections of academic and other institutions
SUNCAT, a catalogue of serial publications
Negotiating sector-wide deals, such as the Jisc Collections licenses for academic journals as well as deals with IT vendors and commercial publishers
Providing expert and trusted advice on digital technology for education and research, built from over 30 years' experience.
Jisc was established on 1 April 1993 under the terms of letters of guidance from the Secretaries of State to the newly established Higher Education Funding Councils for England, Scotland and Wales, inviting them to establish a Joint Committee to deal with networking and specialist information services. JISC was to provide national vision and leadership for the benefit of the entire Higher Education sector. The organisation inherited the functions of the Information Systems Committee (ISC) and the Computer Board, which had served universities. An initial challenge was to support a much larger community of institutions, including the ex-polytechnics and higher education colleges.
Four sub-committees initially supporting the JISC committee: their remits covered Networking; Awareness, Liaison and Training; Electronic Information and Technology Applications activities.
In 1995, the Northern Ireland Department of Education became a full partner in JISC. The organisation expanded again in 1999 when the further education funding bodies became funding partners. This expansion prompted a restructuring and a new set of committees:
JISC Committee for Authentication and Security (JCAS) November 1999 – January 2002. The work of this committee was taken over by JCN2 and JCIE3.
JISC Committee for Electronic Information (JCEI) November 1999 – January 2002. The work of this committee was taken over by JCIE3 and JCCS4.
JISC Committee for Integrated Environments for Learners (JCIEL) November 1999 – January 2002. The work of this committee was taken over by JCLT5.
JISC Committee for Awareness, Liaison and Training (JCALT) The work of this committee was taken over by JOS6.
JISC Committee on Networking (JCN2).
The expansion also raised wider concerns about JISC's governance, and Professor Sir Brian Follett was appointed to carry out an independent enquiry. His report, published in November 2000, concluded that "JISC is perceived as a UK success story, providing a network of world-class standard and a range of excellent services. Importantly, it evolves continuously and is an excellent example of collaboration between the community and the funding bodies". However, Follett made various recommendations for reform, most of which were accepted by the funding bodies.
A new structure was therefore put in place from December 2001, consisting of a JISC Board, advised by a steering committee made up of senior officers from each funding body. Six sub-committees fell under two main heads: strategy and policy committees, which aim to ensure that the needs of specific communities are met (in the fields of research, learning and teaching, and management); and functional committees, concentrating on specific areas of work (networking, information environment, and content acquisition).
In January 2005, the Freedom of Information Act 2000 came into full effect, giving the public a general right to access all types of recorded information held by public bodies. JISC took an important lead in raising the Act's profile in the academic community and published its own FOI Publication Scheme.
In the spring of 2005, the Learning and Skills Council (LSC) announced a reduction in its funding contribution to JISC. The LSC and JISC Executive negotiated a package of services that the LSC would continue to contribute towards in the future.
Also in 2005, a number of strategies and policies for UK Higher and Further Education were published, reaffirming the importance of ICT: JISC was seen as taking an important role in supporting and implementing these strategies.
In the same year, JISC signed agreements with several international partners:
SURF Foundation in the Netherlands, cementing the considerable areas of co-operation between the two organisations;
Australian Department for Education, Science and Training (DEST) supporting the e-Framework initiative which aims to develop a service-oriented approach to the development and integration of IT systems;
Deutsche Forschungsgemeinschaft (Germany), Denmark's Electronic Research Library (DEFF) and SURF to create the Knowledge exchange which aims to increase the return on investment by the individual organisations in ICT infrastructure, services and projects.
Under the Government Spending Review 2004, JISC was awarded additional funding of £81 million for the period April 2006 to March 2009, and was able to launch a range of new programmes to support the work of the higher education and research community. Funding for SuperJANET5 was provided by all JISC's funding partners. Other activities were funded by the Higher Education Funding Councils for England and Wales.
A review of JISC's work, commissioned by the Higher Education Funding Council for England under the chairmanship of Professor Sir Alan Wilson, was published in February 2011. One of the principal findings of the review was that there is no comparable body to JISC within the UK, and internationally its reputation is outstanding as a strategic leader and partner. The Review concluded that, "JISC is an invaluable national resource which has evolved in response to increasing demands over 20 years". However, the Review also recommended that, due to the breadth and complexity of JISC's activity, its structure, processes, projects, programmes and governance need to be simplified and reshaped: "[i]n an era of financial constraint, it is necessary to refocus activities around clearer priorities, and to ensure JISC operates with a sustainable financial model".[4]
Ronald Urwick Cooke, then recently retired as Vice-Chancellor of the University of York, was appointed Chair of JISC in 2004, and served until 2008. He was succeeded by Timothy O'Shea, Principal and Vice-Chancellor of the University of Edinburgh, who was appointed with effect from 1 January 2009. Martin Hall, Vice-Chancellor of the University of Salford served as Chair until 2015 when he was replaced by David Maguire, Vice-Chancellor of the University of Greenwich. From February 2012 the Chief Executive was Martyn Harrow, who was succeeded by Paul Feldman in October 2015.[1]
^ a b "Appointment of new Jisc chief executive". Jisc. Retrieved 21 October 2015.
^ "Company and charity details". Jisc. Jisc. Retrieved 3 May 2016.
^ "How we are funded". Jisc. Retrieved 21 October 2015.
^ HEFCE. Feb 2011. Review of the Joint Information Systems Committee Accessed: 31 March 2012
Jisc website


text:
Q: Modeling Quantum Aspects with Probability This is a question I've had a while about quantum theory.
Many times when I look at books and equations about this subject matter I see that the use many concepts in probability. (Correct me if Im wrong) Like how the movement of atom can be modeled using probability distributions. So I was wondering when physics takes this approach to this subject matter is it saying that the movement of an atom is actually random or that the details of explaining may be to cumbersome and be approximated with great accuracy just by using these distributions?

A: The quantum world is really random. There are no local hidden variables.

A:
So I was wondering when physics takes this approach to this subject matter is it saying that the movement of an atom is actually random or that the details of explaining may be to cumbersome and be approximated with great accuracy just by using these distributions?

Depends on the physicist. Some like to think there is fundamental randomness in nature, some think that unpredictability is due to our poor knowledge. It is hard to prove one or either point of view right.
This is an old argument that is not necessarily associated with quantum theory only. Similar problems plague theory of probability.

redpajama_set_name: RedPajamaStackExchange


text:
Peter Sturgeon (born February 12, 1954) is a Canadian retired professional ice hockey player. He was drafted by the Boston Bruins with the 36th overall pick in the 1974 NHL Entry Draft. He went on to play in six games in the National Hockey League, all with the Colorado Rockies.

He is the father of musician Adam Sturgeon of Status/Non-Status and OMBIIGIZI.

Career statistics

Regular season and playoffs

References

External links

Hockey Draft Central

1954 births
Living people
Baltimore Clippers players
Boston Bruins draft picks
Broome Dusters players
Canadian ice hockey left wingers
Colorado Rockies (NHL) players
Columbus Owls players
Fort Worth Texans players
Grand Rapids Owls players
Ice hockey people from Yukon
Kitchener Rangers players
New England Whalers draft picks
Phoenix Roadrunners (PHL) players
Rochester Americans players
Sportspeople from Whitehorse

redpajama_set_name: RedPajamaWikipedia


text:
ACCEPTED

#### According to
Index Fungorum

#### Published in
in Bailey, Queensland Agric. J. 5: 41 (1899)

#### Original name
Lecidea subcaerulea Stirt.

### Remarks
null

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction\label{Section: Intro}}

The aim of this note is to give several applications of the recently
introduced calculus of thick distributions in several variables \cite{YE2},
generalizing the thick distributions of one variable \cite{EF07}. The thick
distributional calculus allows us to study problems where a finite number of
special points are present; it is the distributional version of the analysis
of Blanchet and Faye \cite{BlanchetFaye}, who employed the concepts of
Hadamard finite parts as developed by Sellier \cite{Sellier3} to study
dynamics of point particles in high post-Newtonian approximations of general
relativity. We give a short summary of the theory of thick distributions in
Section \ref{Section: Thick distributions}.

Our first application, given in Section \ref{Section; The distribution 1}, is
the computation of the distributional derivatives of homogeneous distributions
in $\mathbb{R}^{n}$ by first computing the thick distributional derivatives
and then projecting onto the space of standard distributions. Our analysis
makes several delicate points quite clear.

Next, in Section \ref{Section; Bowen}, we consider an application to point
source fields. In \cite{Bowen}, Bowen computed the derivative of the
distribution
\begin{equation}
g_{j_{1},\ldots,j_{k}}\left(  \mathbf{x}\right)  =\frac{n_{j_{1}}\cdots
n_{j_{k}}}{r^{2}}\,, \label{IN.1
\end{equation}
of $\mathcal{D}^{\prime}\left(  \mathbb{R}^{3}\right)  ,$ where $r=\left\vert
\mathbf{x}\right\vert $ and $\mathbf{n=}\left(  n_{i}\right)  $ is the unit
normal vector to a sphere centered at the origin, that is, $n_{i}=x_{i}/r.$
His result can be written as\footnote{Following the notation introduced by the
late Professor Farassat \cite{Farassat}, we shall denote distributional
derivatives with an overbar.
\begin{equation}
\frac{\overline{\partial}}{\partial x_{i}}g_{j_{1},\ldots,j_{k}}=\left\{
\sum_{q=1}^{k}\delta_{ij_{q}}\frac{n_{j_{1}}\cdots n_{j_{k}}}{n_{j_{q}
}-\left(  k+2\right)  n_{i}n_{j_{1}}\cdots n_{j_{k}}\right\}  \frac{1}{r^{3
}+A\delta\left(  \mathbf{x}\right)  \,, \label{IN.2
\end{equation}
where $n_{i}n_{j_{1}}\cdots n_{j_{k}}=n_{1}^{a}n_{2}^{b}n_{3}^{c},$ and $A=0$
if $a,$ $b,$ or $c$ is odd, whil
\begin{equation}
A=\frac{2\Gamma\left(  \left(  a+1\right)  /2\right)  \Gamma\left(  \left(
b+1\right)  /2\right)  \Gamma\left(  \left(  c+1\right)  /2\right)
{\Gamma\left(  \left(  a+b+c+3\right)  /2\right)  }\,, \label{IN.3
\end{equation}
if the three exponents are even. Interestingly, he observes that if one tries
to compute this formula by induction, employing the product rule for
derivatives, the result obtained is \emph{wrong.} In this article we show that
one can actually apply the product rule in the space of thick distributions,
obtaining (\ref{IN.2}) by induction; furthermore, our analysis shows
\emph{why} the wrong result is obtained when applying the product rule in
\cite{Bowen}.

Finally in Section \ref{Section: Higher order derivatives} we show how the
thick distributional calculus allows one to avoid mistakes in the computation
of higher order derivatives of thick distributions of order $0.$

\section{Thick distributions\label{Section: Thick distributions}}

We now recall the basic ideas of the thick distributional calculus \cite{YE2}.
If $\mathbf{a}$ is a fixed point of $\mathbb{R}^{n},$ then the space of test
functions with a thick point at $\mathbf{x=a}$ is defined as
follows.\smallskip

\begin{definition}
\label{Definition Test}Let $\mathcal{D}_{\ast,\mathbf{a}}\left(
\mathbb{R}^{n}\right)  $ denote the vector space of all smooth functions
$\phi$ defined in $\mathbb{R}^{n}\setminus\left\{  \mathbf{a}\right\}  ,$ with
support of the form $K\setminus\left\{  \mathbf{a}\right\}  ,$ where $K$ is
compact in $\mathbb{R}^{n},$ that admit a strong asymptotic expansion of the
form
\begin{equation}
\phi\left(  \mathbf{a}+\mathbf{x}\right)  =\phi\left(  \mathbf{a
+r\mathbf{w}\right)  \sim\underset{j=m}{\overset{\infty}{\sum}}a_{j}\left(
\mathbf{w}\right)  r^{j},\ \ \ \text{as }\mathbf{x}\rightarrow\mathbf{0}\,,
\label{Def.1
\end{equation}
where $m$ is an integer (positive or negative), and where the $a_{j}$ are
smooth functions of $\mathbf{w},$ that is, $a_{j}\in\mathcal{D}\left(
\mathbb{S}\right)  .$ The subspace $\mathcal{D}_{\ast,\mathbf{a}}^{\left[
m\right]  }\left(  \mathbb{R}^{n}\right)  $ consists of those test functions
$\phi$ whose expansion (\ref{Def.1}) begins at $m.$ For a fixed compact $K$
whose interior contains $\mathbf{a},$ $\mathcal{D}_{\ast,\mathbf{a}}^{\left[
m;K\right]  }\left(  \mathbb{R}^{n}\right)  $ is the subspace formed by those
test functions of $\mathcal{D}_{\ast,\mathbf{a}}^{\left[  m\right]  }\left(
\mathbb{R}^{n}\right)  $ that vanish in $\mathbb{R}^{n}\setminus K.$\smallskip
\end{definition}

Observe that we require the asymptotic development of $\phi\left(
\mathbf{x}\right)  $ as $\mathbf{x}\rightarrow\mathbf{a}$ to be
\textquotedblleft strong\textquotedblright. This means \cite[Chapter
1]{GreenBook}\ that for any differentiation operator $\left(  \partial
/\partial\mathbf{x}\right)  ^{\mathbf{p}}=\left(  \partial^{p_{1}
...\partial^{p_{n}}\right)  /\partial x_{1}^{p_{1}}...\partial x_{n}^{p_{n}},$
the asymptotic development of $\left(  \partial/\partial\mathbf{x}\right)
^{\mathbf{p}}\phi\left(  \mathbf{x}\right)  $ as $\mathbf{x}\rightarrow
\mathbf{a}$ exists and is equal to the term-by-term differentiation of
$\sum_{j=m}^{\infty}a_{j}\left(  \mathbf{w}\right)  r^{j}.$ Observe that
saying that the expansion exists as $\mathbf{x}\rightarrow\mathbf{0}$ is the
same as saying that it exits as $r\rightarrow0,$ uniformly with respect to
$\mathbf{w}.$

We call $\mathcal{D}_{\ast,\mathbf{a}}\left(  \mathbb{R}^{n}\right)  $ the
space of test functions on $\mathbb{R}^{n}$ with a thick point located
at\textbf{\ }$\mathbf{x}=\mathbf{a}.$ We denote $\mathcal{D}_{\ast,\mathbf{0
}\left(  \mathbb{R}^{n}\right)  $ as $\mathcal{D}_{\ast}\left(  \mathbb{R
^{n}\right)  .$

The topology of the space of thick test functions is constructed as
follows.\smallskip

\begin{definition}
\label{DefinitionTopology}Let $m$ be a fixed integer and $K$ a compact subset
of $\mathbb{R}^{n}$ whose interior contains $\mathbf{a}.$ The topology of
$\mathcal{D}_{\ast,\mathbf{a}}^{\left[  m;K\right]  }\left(  \mathbb{R
^{n}\right)  $ is given by the seminorms $\left\{  \left\Vert
\ \ \ \ \right\Vert _{q,s}\right\}  _{q>m,s\geq0}$ defined as
\begin{equation}
\left\vert \left\vert \phi\right\vert \right\vert _{q,s}=\sup_{\mathbf{x-a}\in
K}\sup_{\left\vert \mathbf{p}\right\vert \leq s}\frac{\left\vert
\displaystyle\frac{\partial^{\mathbf{p}}\phi}{\partial\mathbf{x}}\left(
\mathbf{a+x}\right)  -\underset{j=m-\left\vert \mathbf{p}\right\vert
}{\overset{q-1}{\sum}}a_{j,\mathbf{p}}\left(  \mathbf{w}\right)
r^{j}\right\vert }{r^{q}}\,, \label{def 2
\end{equation}
where $\mathbf{x}=r\mathbf{w}$ an
\begin{equation}
\displaystyle\frac{\partial^{\mathbf{p}}\phi}{\partial\mathbf{x}}\left(
\mathbf{a+x}\right)  \sim\underset{j=m-\left\vert \mathbf{p}\right\vert
}{\overset{\infty}{\sum}}a_{j,\mathbf{p}}\left(  \mathbf{w}\right)  r^{j}.
\label{def 2b
\end{equation}

The topology of $\mathcal{D}_{\ast,\mathbf{a}}^{\left[  m\right]  }\left(
\mathbb{R}^{n}\right)  $ is the inductive limit topology of the $\mathcal{D
_{\ast,\mathbf{a}}^{\left[  m;K\right]  }\left(  \mathbb{R}^{n}\right)  $ as
$K\nearrow\infty.$ The topology of $\mathcal{D}_{\ast,\mathbf{a}}\left(
\mathbb{R}^{n}\right)  $ is the inductive limit topology of the $\mathcal{D
_{\ast,\mathbf{a}}^{\left[  m\right]  }\left(  \mathbb{R}^{n}\right)  $ as
$m\searrow-\infty.$\smallskip
\end{definition}

A sequence $\left\{  \phi_{l}\right\}  _{l=0}^{\infty}$ in $\mathcal{D
_{\ast,\mathbf{a}}\left(  \mathbb{R}^{n}\right)  $ converges to $\psi$ if and
only there exists $l_{0}\geq0,$ an integer $m,$ and a compact set $K$ with
$\mathbf{a}$ in its interior, such that $\phi_{l}\in\mathcal{D}_{\ast
,\mathbf{a}}^{\left[  m;K\right]  }\left(  \mathbb{R}^{n}\right)  $ for $l\geq
l_{0}$ and $\left\vert \left\vert \psi-\phi_{l}\right\vert \right\vert
_{q,s}\rightarrow0$ as $l\rightarrow\infty$ if $q>m,s\geq0.$ Notice that if
$\left\{  \phi_{l}\right\}  _{l=0}^{\infty}$ converges to $\psi$ in
$\mathcal{D}_{\ast,\mathbf{a}}\left(  \mathbb{R}^{n}\right)  $ then $\phi_{l}$
and the corresponding derivatives converge uniformly to $\psi$ and its
derivatives in any set of the form $\mathbb{R}^{n}\setminus B,$ where $B$ is a
ball with center at $\mathbf{a};$ in fact, $r^{\left\vert \mathbf{p
\right\vert -m}\left(  \partial/\partial\mathbf{x}\right)  ^{\mathbf{p}
\phi_{l}$ converges uniformly to $r^{\left\vert \mathbf{p}\right\vert
-m}\left(  \partial/\partial\mathbf{x}\right)  ^{\mathbf{p}}\psi$ over all
$\mathbb{R}^{n}.$ Furthermore, if $\left\{  a_{j}^{l}\right\}  $ are the
coefficients of the expansion of $\phi_{l}$ and $\left\{  b_{j}\right\}  $ are
those for $\psi,$ then $a_{j}^{l}\rightarrow b_{j}$ in the space
$\mathcal{D}\left(  \mathbb{S}\right)  $ for each $j\geq m.$

We can now consider distributions in a space with one thick point, the
\textquotedblleft thick distributions.\textquotedblright\smallskip

\begin{definition}
The space of distributions on $\mathbb{R}^{n}$ with a thick point at
$\mathbf{x=a}$ is the dual space of $\mathcal{D}_{\ast,\mathbf{a}}\left(
\mathbb{R}^{n}\right)  .$ We denote it $\mathcal{D}_{\ast,\mathbf{a
}^{^{\prime}}\left(  \mathbb{R}^{n}\right)  ,$ or just as $\mathcal{D}_{\ast
}^{^{\prime}}\left(  \mathbb{R}^{n}\right)  $ when $\mathbf{a=0}.$\smallskip
\end{definition}

Observe that $\mathcal{D}\left(  \mathbb{R}^{n}\right)  ,$\ the space of
standard test functions, is a closed subspace of $\mathcal{D}_{\ast
,\mathbf{a}}\left(  \mathbb{R}^{n}\right)  ;$ we denote b
\begin{equation}
i:\mathcal{D}\left(  \mathbb{R}^{n}\right)  \rightarrow\mathcal{D
_{\ast,\mathbf{a}}\left(  \mathbb{R}^{n}\right)  \,, \label{Test.3
\end{equation}
the inclusion map and b
\begin{equation}
\Pi:\mathcal{D}_{\ast,\mathbf{a}}^{\prime}\left(  \mathbb{R}^{n}\right)
\rightarrow\mathcal{D}^{\prime}\left(  \mathbb{R}^{n}\right)  \,,
\label{Dis.1
\end{equation}
the projection operator, dual of the inclusion (\ref{Test.3}).

The derivatives of thick distributions are defined in much the same way as the
usual distributional derivatives, that is, by duality.\smallskip

\begin{definition}
\label{Definition: Derivaative}If $f\in\mathcal{D}_{\ast,\mathbf{a}}^{\prime
}\left(  \mathbb{R}^{n}\right)  $ then its thick distributional derivative
$\partial^{\ast}f/\partial x_{j}$ is defined a
\begin{equation}
\left\langle \frac{\partial^{\ast}f}{\partial x_{j}},\phi\right\rangle
=-\left\langle f,\frac{\partial\phi}{\partial x_{j}}\right\rangle
\,,\ \ \ \ \ \phi\in\mathcal{D}_{\ast,\mathbf{a}}\left(  \mathbb{R
^{n}\right)  \,. \label{Der.1
\end{equation}

\end{definition}

We denote by $\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $ the space of
smooth functions in $\mathbb{R}^{n}\setminus\left\{  \mathbf{a}\right\}  $
that have a strong asymptotic expansion of the form (\ref{Def.1});
alternatively, $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $\ if
$\psi=\psi_{1}+\psi_{2},$ where $\psi_{1}\in\mathcal{E}\left(  \mathbb{R
^{n}\right)  $\footnote{In general $\mathcal{E}\left(  U\right)  $ is the
space of all smooth functions in the open set $U.$} and where $\psi_{2
\in\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  .$ The space
$\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $ is the space of
\emph{multipliers} of $\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  $ and
of $\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{n}\right)  .$ Furthermore
\cite{YE2}, the product rule for derivatives holds
\begin{equation}
\frac{\partial^{\ast}\left(  \psi f\right)  }{\partial x_{j}}=\frac
{\partial\psi}{\partial x_{j}}f+\psi\frac{\partial^{\ast}f}{\partial x_{j}}\,,
\label{Der.3
\end{equation}
if $f$ is a thick distribution and $\psi$ is a multiplier. Notice that
$\partial\psi/\partial x_{j}$ is the \emph{ordinary} derivative in
(\ref{Der.3}).

Let $g\left(  \mathbf{w}\right)  $ is a distribution in $\mathbb{S}.$ The
thick delta function of degree $q,$ denoted as $g\delta_{\ast}^{\left[
q\right]  },$ or as $g\left(  \mathbf{w}\right)  \delta_{\ast}^{\left[
q\right]  },$ acts on a thick test function $\phi\left(  \mathbf{x}\right)  $
as
\begin{equation}
\left\langle g\delta_{\ast}^{\left[  q\right]  },\phi\right\rangle
_{\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{n}\right)  \times
\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  }=\frac{1}{C}\left\langle
g\left(  \mathbf{w}\right)  ,a_{q}\left(  \mathbf{w}\right)  \right\rangle
_{\mathcal{D}^{\prime}\left(  \mathbb{S}\right)  \times\mathcal{D}\left(
\mathbb{S}\right)  }\,, \label{Dis.15
\end{equation}
where $\phi\left(  r\mathbf{w}\right)  \sim\sum_{j=m}^{\infty}a_{j}\left(
\mathbf{w}\right)  r^{j},$ as $r\rightarrow0^{+},$ and where
\begin{equation}
C=\frac{2\pi^{n/2}}{\Gamma\left(  n/2\right)  }\,, \label{Area
\end{equation}
is the surface area of the unit sphere $\mathbb{S}$ of $\mathbb{R}^{n}.$ If
$g$ is locally integrable function in $\mathbb{S},$ then
\begin{equation}
\left\langle g\delta_{\ast}^{\left[  q\right]  },\phi\right\rangle
_{\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{n}\right)  \times
\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  }=\frac{1}{C}\int
_{\mathbb{S}}g\left(  \mathbf{w}\right)  a_{q}\left(  \mathbf{w}\right)
\,\mathrm{d}\sigma\left(  \mathbf{w}\right)  \,. \label{Dis.16
\end{equation}
Thick deltas of order $0$ are called just thick deltas, and we shall use the
notation $g\delta_{\ast}$ instead of $g\delta_{\ast}^{\left[  0\right]  }.$

Let $g\in\mathcal{D}^{\prime}\left(  \mathbb{S}\right)  .$ The
\begin{equation}
\frac{\partial^{\ast}}{\partial x_{j}}\left(  g\delta_{\ast}^{\left[
q\right]  }\right)  =\left(  \frac{\delta g}{\delta x_{j}}-\left(  q+n\right)
n_{j}g\right)  \delta_{\ast}^{\left[  q+1\right]  }. \label{DD.1
\end{equation}
Here $\delta g/\delta x_{j}$ is the $\delta-$derivative of $g$ \cite{EK1,
EK2}; in general the $\delta-$derivatives can be applied to functions and
distributions defined only on a smooth hypersurface $\Sigma$ of $\mathbb{R
^{n}.$ Suppose now that the surface is $\mathbb{S},$ the unit sphere in
$\mathbb{R}^{n}$ and let $f$ be a smooth function defined in $\mathbb{S},$
that is, $f\left(  \mathbf{w}\right)  $ is defined if $\mathbf{w}\in
\mathbb{R}^{n}$ satisfies $\left\vert \mathbf{w}\right\vert =1.$ Observe that
the expressions $\partial f/\partial x_{j}$ are not defined and, likewise, if
$\mathbf{w}=\left(  w_{j}\right)  _{1\leq j\leq n}$ the expressions $\partial
f/\partial w_{j}$ do not make sense either; the derivatives that are always
defined and that one should consider are the $\delta f/\delta x_{j},$ $1\leq
j\leq n.$ Let $F_{0}$\ be the extension of $f$\ to $\mathbb{R}^{n
\setminus\left\{  \mathbf{0}\right\}  $ that is homogeneous of degree $0,$
namely, $F_{0}\left(  \mathbf{x}\right)  =f\left(  \mathbf{x}/r\right)  $
where $r=\left\vert \mathbf{x}\right\vert .$ Then \cite{YE2
\begin{equation}
\frac{\delta f}{\delta x_{j}}=\left.  \frac{\partial F_{0}}{\partial x_{j
}\right\vert _{\mathbb{S}}\,. \label{Su.4
\end{equation}
Also, if we use polar coordinates, $\mathbf{x}=r\mathbf{w},$ so that
$F_{0}\left(  \mathbf{x}\right)  =f\left(  \mathbf{w}\right)  ,$ then
$\partial F_{0}/\partial x_{j}$ is homogeneous of degree $-1,$ and actually
$\partial F_{0}/\partial x_{j}=r^{-1}\delta f/\delta x_{j}$ if $\mathbf{x\neq
0.}$

The matrix $\mathbf{\mu=}\left(  \mu_{ij}\right)  _{1\leq i,j\leq n},$ where
$\mu_{ij}=\delta n_{i}/\delta x_{j},$ plays an important role in the study of
distributions on a surface $\Sigma.$ If $\Sigma=\mathbb{S}$ then $\mu
_{ij}=\delta n_{i}/\delta x_{j}=\delta_{ij}-n_{i}n_{j}.$ Observe that
$\mu_{ij}=\mu_{ji},$ an identity that holds in any surface.

The differential operators $\delta f/\delta x_{j}$\ are initially defined if
$f$ is a smooth function defined on $\Sigma,$ but we can also define them when
$f$ is a distribution. We can do this if we use the fact that smooth functions
are dense in the space of distributions on $\Sigma.$

\section{The thick distribution $\mathcal{P}f\left(  1\right)
\label{Section; The distribution 1}}

Let us consider one of the simplest functions, namely, the function $1,$
defined in $\mathbb{R}^{n}.$ Naturally this function is locally integrable,
and thus it defines a regular distribution, also denoted as $1,$ and the
ordinary derivatives and the distributional derivatives both coincide and give
the value $0.$ On the other hand, $1$ does not automatically give an element
of $\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{n}\right)  $ since if
$\phi\in\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  $ the integral
$\int_{\mathbb{R}^{n}}\phi\left(  \mathbf{x}\right)  \,\mathrm{d}\mathbf{x}$
could be divergent, and thus we consider the \emph{spherical} finite
part\footnote{If instead of removing balls of radius $\varepsilon,$ solids of
other shapes are removed one obtains a different thick distribution.} thick
distribution $\mathcal{P}f\left(  1\right)  $ given a
\begin{equation}
\left\langle \mathcal{P}f\left(  1\right)  ,\phi\right\rangle =\mathrm{F.p.
\int_{\mathbb{R}^{n}}\phi\left(  \mathbf{x}\right)  \,\mathrm{d
\mathbf{x=}\mathrm{F.p.}\lim_{\varepsilon\rightarrow0^{+}}\int_{\left\vert
\mathbf{x}\right\vert \geq\varepsilon}\phi\left(  \mathbf{x}\right)
\,\mathrm{d}\mathbf{x}\,. \label{T1.1
\end{equation}
The derivatives of $\mathcal{P}f\left(  1\right)  $ do not vanish, since
actually we have the following formula \cite{YE2}.\smallskip

\begin{lemma}
\label{Lemma T1.1}In $\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R
^{n}\right)  ,$
\begin{equation}
\frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(  1\right)
\right)  =Cn_{i}\delta_{\ast}^{\left[  -n+1\right]  }, \label{T1.1p
\end{equation}
where $C$ is given by (\ref{Area}).
\end{lemma}

\begin{proof}
One can find a proof of a more general statement in \cite{YE2}, but in this
simpler case the proof can be written as follows
\begin{align*}
\left\langle \frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(
1\right)  \right)  ,\phi\right\rangle  &  =-\left\langle \mathcal{P}f\left(
1\right)  ,\frac{\partial\phi}{\partial x_{i}}\right\rangle \\
&  =-\mathrm{F.p.}\lim_{\varepsilon\rightarrow0^{+}}\int_{\left\vert
\mathbf{x}\right\vert \geq\varepsilon}\frac{\partial\phi}{\partial x_{i
}\,\mathrm{d}\mathbf{x}\\
&  =\mathrm{F.p.}\lim_{\varepsilon\rightarrow0^{+}}\int_{\varepsilon
\mathbb{S}^{n-1}}n_{i}\phi\,\mathrm{d}\sigma\,,
\end{align*}
so that if $\phi\in\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  $ has the
expansion $\phi\left(  \mathbf{x}\right)  \sim\sum_{j=m}^{\infty}a_{j}\left(
\mathbf{w}\right)  r^{j},$ as $\mathbf{x}\rightarrow\mathbf{0},$ the
\[
\int_{\varepsilon\mathbb{S}^{n-1}}n_{i}\phi\,\mathrm{d}\sigma\sim\sum
_{j=m}^{\infty}\left(  \int_{\mathbb{S}}n_{i}a_{j}\left(  \mathbf{w}\right)
\,\mathrm{d}\sigma\left(  \mathbf{w}\right)  \right)  \varepsilon^{n-1+j},
\]
as $\varepsilon\rightarrow0^{+}.$ The finite part of the limit is equal to the
coefficient of $\varepsilon^{0},$ thu
\begin{align*}
\mathrm{F.p.}\underset{\varepsilon\rightarrow0}{\lim}\int_{\varepsilon
\mathbb{S}^{n-1}}n_{i}\phi\,\mathrm{d}\sigma &  =\int_{\mathbb{S}}n_{i
a_{1-n}\left(  \mathbf{w}\right)  \,\mathrm{d}\sigma\left(  \mathbf{w}\right)
\\
&  =\left\langle Cn_{i}\delta_{\ast}^{\left[  1-n\right]  },\phi\right\rangle
\,,
\end{align*}
as required.\smallskip
\end{proof}

If $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $ is a multiplier
of $\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  ,$ then we define, in a
similar way, the thick distribution $\mathcal{P}f\left(  \psi\right)
\in\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{n}\right)  ,$ and we clearly
have the useful formul
\begin{equation}
\mathcal{P}f\left(  \psi\right)  =\psi\mathcal{P}f\left(  1\right)  \,,
\label{T1.2
\end{equation}
which immediately gives the thick distributional derivative of $\mathcal{P
f\left(  \psi\right)  $ a
\[
\frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(  \psi\right)
\right)  =\frac{\partial\psi}{\partial x_{i}}\mathcal{P}f\left(  1\right)
+\psi\frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(
1\right)  \right)  \,,
\]
so that we obtain the ensuing formula.\smallskip

\begin{proposition}
\label{Prop. T1.1}If $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)
$ the
\begin{equation}
\frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(  \psi\right)
\right)  =\mathcal{P}f\left(  \frac{\partial\psi}{\partial x_{i}}\right)
+Cn_{i}\psi\delta_{\ast}^{\left[  1-n\right]  }. \label{T1.3
\end{equation}

\end{proposition}

Notice that, in general, the term $Cn_{i}\psi\delta_{\ast}^{\left[
1-n\right]  }$ is \emph{not} a thick delta of order $1-n.$ Indeed, let us now
consider the case when $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R
^{n}\right)  $ is homogeneous of order $k\in\mathbb{Z}.$ Then $\psi\left(
\mathbf{x}\right)  =r^{k}\psi_{0}\left(  \mathbf{x}\right)  ,$ where $\psi
_{0}$ is homogeneous of order $0.$ Since $r^{k}\delta_{\ast}^{\left[
q\right]  }=\delta_{\ast}^{\left[  q-k\right]  }$\cite[Eqn. (5.16)]{YE2} we
obtain the following particular case of (\ref{T1.3}), where now the term
$Cn_{i}\psi_{0}\delta_{\ast}^{\left[  1-n-k\right]  }$ \emph{is} a thick delta
of order $1-n-k.$\smallskip

\begin{proposition}
\label{Prop.T1p}If $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $
is homogeneous of order $k\in\mathbb{Z},$ the
\begin{equation}
\frac{\partial^{\ast}}{\partial x_{i}}\left(  \mathcal{P}f\left(  \psi\right)
\right)  =\mathcal{P}f\left(  \frac{\partial\psi}{\partial x_{i}}\right)
+Cn_{i}\psi_{0}\delta_{\ast}^{\left[  1-n-k\right]  }\,, \label{T1.4
\end{equation}
where $\psi_{0}\left(  \mathbf{x}\right)  =\left\vert \mathbf{x}\right\vert
^{-k}\psi\left(  \mathbf{x}\right)  .$\smallskip
\end{proposition}

If we now apply the projection $\Pi$ onto the usual distribution space
$\mathcal{D}^{\prime}\left(  \mathbb{R}^{n}\right)  ,$ we obtain the formula
for the distributional derivatives of homogeneous distributions. Observe first
that if $k>-n$ then $\psi$ is integrable at the origin, and thus $\psi$ is a
regular distribution and $\Pi\left(  \mathcal{P}f\left(  \psi\right)  \right)
=\psi.$ If $k\leq-n$ then $\Pi\left(  \mathcal{P}f\left(  \psi\right)
\right)  =\mathcal{P}f\left(  \psi\right)  ,$ since in that case the integral
$\int_{\mathbb{R}^{n}}\psi\left(  \mathbf{x}\right)  \phi\left(
\mathbf{x}\right)  \,\mathrm{d}\mathbf{x}$ would be divergent, in general, if
$\phi\in\mathcal{D}\left(  \mathbb{R}^{n}\right)  .$ A particularly
interesting case is when $k=-n,$ since if $\psi$ is homogeneous of degree $-n$
an
\begin{equation}
\int_{\mathbb{S}}\psi\left(  \mathbf{w}\right)  \,\mathrm{d}\sigma\left(
\mathbf{w}\right)  =0\,, \label{T1.4p
\end{equation}
then the \emph{principal value} of the integra
\begin{equation}
\mathrm{p.v.}\int_{\mathbb{R}^{n}}\psi\left(  \mathbf{x}\right)  \phi\left(
\mathbf{x}\right)  \,\mathrm{d}\mathbf{x=}\lim_{\varepsilon\rightarrow0^{+
}\int_{\left\vert \mathbf{x}\right\vert \geq\varepsilon}\psi\left(
\mathbf{x}\right)  \phi\left(  \mathbf{x}\right)  \,\mathrm{d}\mathbf{x}\,,
\label{T1.4p2
\end{equation}
actually exists for each $\phi\in\mathcal{D}\left(  \mathbb{R}^{n}\right)  ,$
so that $\mathcal{P}f\left(  \psi\right)  =\mathrm{p.v.}\left(  \psi\right)
,$ the principal value distribution\footnote{Let $\Sigma$ be a closed surface
in $\mathbb{R}^{n}$ that encloses the origin. We describe $\Sigma$ by an
equation of the form $g(\mathbf{x})=1,$ where $g(\mathbf{x})$ is continuous in
$\mathbb{R}^{n}\setminus\left\{  \mathbf{0}\right\}  $ and homogeneous of
degree $1.$ Then $\left\langle \mathcal{R}_{\Sigma}\left(  \psi(\mathbf{x
)\right)  ,\phi(\mathbf{x})\right\rangle =\underset{\varepsilon\rightarrow
0}{\lim}\int_{g(\mathbf{x}\dot{)}\geqslant\varepsilon}\psi(\mathbf{x
)\phi(\mathbf{x})\,\mathrm{d}\mathbf{x}\,,$ defines another regularization of
$\psi,$ but in general $\mathcal{R}_{\Sigma}\left(  \psi(\mathbf{x})\right)
\neq\mathrm{p.v.}\left(  \psi\left(  \mathbf{x}\right)  \right)  $ \cite{YE},
a fact observed by Farassat \cite{Farassat}, who indicated its importance in
numerical computations, and studied by several authors \cite{VH1, YE}.}.
Condition (\ref{T1.4p}) holds whenever $\psi=\partial\xi/\partial x_{j}$ for
some $\xi$ homogeneous of order $-n+1.$\smallskip

\begin{proposition}
\label{Prop. T1.2}Let $\psi$ be homogeneous of order $k\in\mathbb{Z}$ in
$\mathbb{R}^{n}\setminus\left\{  \mathbf{0}\right\}  .$ Then, in
$\mathcal{D}^{\prime}\left(  \mathbb{R}^{n}\right)  $ the distributional
derivative $\overline{\partial}\psi/\partial x_{i}$ is given as follows
\begin{equation}
\frac{\overline{\partial}\psi}{\partial x_{i}}=\frac{\partial\psi}{\partial
x_{i}}\,,\ \ \ \ \ k>1-n\,, \label{T1.5
\end{equation}
equality of regular distributions
\begin{equation}
\frac{\overline{\partial}\psi}{\partial x_{i}}=\mathrm{p.v.}\left(
\frac{\partial\psi}{\partial x_{i}}\right)  +A\delta\left(  \mathbf{x}\right)
\,,\ \ \ \ k=1-n\,, \label{T1.6
\end{equation}
where $A=\int_{\mathbb{S}}n_{i}\psi_{0}\left(  \mathbf{w}\right)
\,\mathrm{d}\sigma\left(  \mathbf{w}\right)  =\left\langle \psi_{0
,n_{i}\right\rangle _{\mathcal{D}^{\prime}\left(  \mathbb{S}\right)
\times\mathcal{D}\left(  \mathbb{S}\right)  },$ whil
\begin{equation}
\frac{\overline{\partial}\psi}{\partial x_{i}}=\mathcal{P}f\left(
\frac{\partial\psi}{\partial x_{i}}\right)  +D\left(  \mathbf{x}\right)
\,,\ \ \ \ k<1-n\,, \label{T1.7
\end{equation}
where $D\left(  \mathbf{x}\right)  $ is a homogeneous distribution of order
$k-1$ concentrated at the origin and given b
\begin{equation}
D\left(  \mathbf{x}\right)  =\left(  -1\right)  ^{-k-n+1}\sum_{j_{1
+\cdots+j_{n}=-k-n+1}\frac{\left\langle n_{i}\psi_{0},\mathbf{w}^{\left(
j_{1},\ldots,j_{n}\right)  }\right\rangle }{j_{1}!\cdots j_{n}!
\mathbf{D}^{\left(  j_{1},\ldots,j_{n}\right)  }\delta\left(  \mathbf{x
\right)  \,. \label{T1.8
\end{equation}

\end{proposition}

\begin{proof}
It follows from (\ref{T1.3}) if we observe \cite[Prop. 4.7]{YE2} that if
$g\in\mathcal{D}^{\prime}\left(  \mathbb{S}\right)  $ the
\begin{equation}
\Pi\left(  g\delta_{\ast}^{\left[  q\right]  }\right)  =\frac{\left(
-1\right)  ^{q}}{C}\sum_{j_{1}+\cdots+j_{n}=q}\frac{\left\langle g\left(
\mathbf{w}\right)  ,\mathbf{w}^{\left(  j_{1},\ldots,j_{n}\right)
}\right\rangle }{j_{1}!\cdots j_{n}!}\mathbf{D}^{\left(  j_{1},\ldots
,j_{n}\right)  }\delta\left(  \mathbf{x}\right)  \,, \label{T1.9
\end{equation}
and, in particular,
\begin{equation}
\Pi\left(  g\delta_{\ast}\right)  =\frac{1}{C}\left\langle g\left(
\mathbf{w}\right)  ,1\right\rangle \delta\left(  \mathbf{x}\right)  \,,
\label{T1.10
\end{equation}
if $q=0.$\smallskip
\end{proof}

Our next task is to compute the second order thick derivatives of homogeneous
distributions. Indeed, if $\psi$ is homogeneous of degree $k$ then we can
iterate the formula (\ref{T1.4}) to obtai
\begin{align}
&  \frac{\partial^{\ast2}}{\partial x_{i}\partial x_{j}}\left(  \mathcal{P
f\left(  \psi\right)  \right)  =\frac{\partial^{\ast}}{\partial x_{i}}\left(
\mathcal{P}f\left(  \frac{\partial\psi}{\partial x_{j}}\right)  +Cn_{j
\psi_{0}\delta_{\ast}^{\left[  1-n-k\right]  }\right) \label{T1.11}\\
&  \ \ \ \ \ \ \ \ \ =\mathcal{P}f\left(  \frac{\partial^{2}\psi}{\partial
x_{i}\partial x_{j}}\right)  +Cn_{i}\xi_{0}\delta_{\ast}^{\left[
2-n-k\right]  }+\frac{\partial^{\ast}}{\partial x_{i}}\left(  Cn_{j}\psi
_{0}\delta_{\ast}^{\left[  1-n-k\right]  }\right)  \,,\nonumber
\end{align}
where $\xi=\partial\psi/\partial x_{j}$ is homogeneous of degree $k-1$ and
$\xi_{0}\left(  \mathbf{x}\right)  =\left\vert \mathbf{x}\right\vert ^{1-k
\xi\left(  \mathbf{x}\right)  $ is the associated homogeneous of degree $0$
function. Use of (\ref{DD.1}) and (\ref{Su.5}) allows us to writ
\begin{align}
&  \frac{\partial^{\ast}}{\partial x_{i}}\left(  Cn_{j}\psi_{0}\delta_{\ast
}^{\left[  1-n-k\right]  }\right)  =C\left(  \frac{\delta}{\delta x_{i
}\left(  n_{j}\psi_{0}\right)  +\left(  k-1\right)  n_{i}n_{j}\psi_{0}\right)
\delta_{\ast}^{\left[  2-n-k\right]  }\label{T1.12}\\
&  \ \ \ \ \ =C\left(  \left(  \delta_{ij}-n_{i}n_{j}\right)  \psi_{0
+n_{j}\frac{\delta\psi_{0}}{\delta x_{i}}+\left(  k-1\right)  n_{i}n_{j
\psi_{0}\right)  \delta_{\ast}^{\left[  2-n-k\right]  }\nonumber\\
&  \ \ \ \ \ =C\left(  \left(  \delta_{ij}+\left(  k-2\right)  n_{i
n_{j}\right)  \psi_{0}+n_{j}\frac{\delta\psi_{0}}{\delta x_{i}}\right)
\delta_{\ast}^{\left[  2-n-k\right]  }\,,\nonumber
\end{align}
while the equation $\psi=r^{k}\psi_{0}$ yields $\partial\psi/\partial
x_{j}=r^{k-1}\{kn_{j}\psi_{0}+\delta\psi_{0}/\delta x_{j}\},$ so tha
\begin{equation}
\xi_{0}=kn_{j}\psi_{0}+\frac{\delta\psi_{0}}{\delta x_{j}}\,. \label{T1.13
\end{equation}
Collecting terms we thus obtain the following formula.\smallskip

\begin{proposition}
\label{Prop. T1. 4}If $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)
$ is homogeneous of order $k\in\mathbb{Z},$ the
\begin{align}
&  \frac{\partial^{\ast2}}{\partial x_{i}\partial x_{j}}\left(  \mathcal{P
f\left(  \psi\right)  \right)  =\mathcal{P}f\left(  \frac{\partial^{2}\psi
}{\partial x_{i}\partial x_{j}}\right) \label{T1.14}\\
&  \ \ \ \ \ +C\left(  \left(  \delta_{ij}+2\left(  k-1\right)  n_{i
n_{j}\right)  \psi_{0}+n_{j}\frac{\delta\psi_{0}}{\delta x_{i}}+n_{i
\frac{\delta\psi_{0}}{\delta x_{j}}\right)  \delta_{\ast}^{\left[
2-n-k\right]  }\,.\nonumber
\end{align}
where $\psi_{0}\left(  \mathbf{x}\right)  =\left\vert \mathbf{x}\right\vert
^{-k}\psi\left(  \mathbf{x}\right)  .$\smallskip
\end{proposition}

Projection onto $\mathcal{D}^{\prime}\left(  \mathbb{R}^{n}\right)  $ of
(\ref{T1.14}) gives the formula for the distributional derivatives
$\overline{\partial}^{2}/\partial x_{i}\partial x_{j}(\mathcal{P}f\left(
\psi\right)  )$ if $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)  $
is homogeneous of order $k\in\mathbb{Z}.$ In case $k=2-n$ we obtain the
following formula.\smallskip

\begin{proposition}
\label{Prop. T1.5}If $\psi\in\mathcal{E}_{\ast}\left(  \mathbb{R}^{n}\right)
$ is homogeneous of order $2-n,$ the
\begin{equation}
\frac{\overline{\partial}^{2}}{\partial x_{i}\partial x_{j}}\left(
\psi\right)  =\mathrm{p.v.}\left(  \frac{\partial^{2}\psi}{\partial
x_{i}\partial x_{j}}\right)  +B\delta\left(  \mathbf{x}\right)  \,,
\label{T1.15
\end{equation}
wher
\begin{equation}
B=\left\langle \psi_{0},2n_{i}n_{j}-\delta_{ij}\right\rangle _{\mathcal{D
^{\prime}\left(  \mathbb{S}\right)  \times\mathcal{D}\left(  \mathbb{S
\right)  }\,. \label{T1.16
\end{equation}

\end{proposition}

\begin{proof}
If we apply the operator $\Pi$ to (\ref{T1.14}) and employ (\ref{T1.10}) we
obtain (\ref{T1.15}) wit
\[
B=\left\langle \left(  \delta_{ij}+2\left(  k-1\right)  n_{i}n_{j}\right)
\psi_{0}+n_{j}\frac{\delta\psi_{0}}{\delta x_{j}}+n_{i}\frac{\delta\psi_{0
}{\delta x_{j}},1\right\rangle _{\mathcal{D}^{\prime}\left(  \mathbb{S
\right)  \times\mathcal{D}\left(  \mathbb{S}\right)  }.
\]
But \cite[(2.6)]{YE2} yield
\begin{equation}
\left\langle n_{j}\frac{\delta\psi_{0}}{\delta x_{j}},1\right\rangle
_{\mathcal{D}^{\prime}\left(  \mathbb{S}\right)  \times\mathcal{D}\left(
\mathbb{S}\right)  }=\left\langle \psi_{0},n\,n_{i}n_{j}-\delta_{ij
\right\rangle _{\mathcal{D}^{\prime}\left(  \mathbb{S}\right)  \times
\mathcal{D}\left(  \mathbb{S}\right)  }, \label{T1.17
\end{equation}
and (\ref{T1.16}) follows since $k=2-n.$\smallskip
\end{proof}

We would like to observe that while $\psi_{0}$ has been supposed smooth, a
continuity argument immediately gives that $\psi_{0}$ could be any
distribution of $\mathcal{D}^{\prime}\left(  \mathbb{R}^{n}\setminus\left\{
\mathbf{0}\right\}  \right)  $ that is homogeneous of degree $0.$

\section{Bowen's formula\label{Section; Bowen}}

If we apply formula (\ref{T1.6}) to the function $\psi=n_{j_{1}}\cdots
n_{j_{k}}/r^{2},$ which is homogeneous of degree $-2$ in $\mathbb{R}^{3}$ we
obtain at once that
\begin{align}
&  \frac{\overline{\partial}}{\partial x_{i}}\left(  \frac{n_{j_{1}}\cdots
n_{j_{k}}}{r^{2}}\right)  =\label{B.1}\\
&  \mathrm{p.v.}\left(  \left\{  \sum_{q=1}^{k}\delta_{ij_{q}}\frac{n_{j_{1
}\cdots n_{j_{k}}}{n_{j_{q}}}-\left(  k+2\right)  n_{i}n_{j_{1}}\cdots
n_{j_{k}}\right\}  \frac{1}{r^{3}}\right)  +A\delta\left(  \mathbf{x}\right)
\,,\nonumber
\end{align}
where
\begin{equation}
A=\int_{\mathbb{S}}n_{i}n_{j_{1}}\cdots n_{j_{k}}\,\mathrm{d}\sigma\left(
\mathbf{w}\right)  \,. \label{B.2
\end{equation}
This integral was computed in \cite[(3.13)]{EK85b}, the result being
\begin{equation}
A=\frac{2\Gamma\left(  \left(  a+1\right)  /2\right)  \Gamma\left(  \left(
b+1\right)  /2\right)  \Gamma\left(  \left(  c+1\right)  /2\right)
{\Gamma\left(  \left(  a+b+c+3\right)  /2\right)  }\,, \label{B.3
\end{equation}
if $n_{i}n_{j_{1}}\cdots n_{j_{k}}=n_{1}^{a}n_{2}^{b}n_{3}^{c},$ and $a,$ $b,$
or $c$ are even, while $A=0$ if any exponent is odd. Bowen \cite[Eqn.
(A5)]{Bowen} also computes the integral, and obtains a different but
equivalent expression; in particular, his formula for $k=3$ reads a
\begin{equation}
A=\frac{4\pi}{15}\left(  \delta_{ij_{1}}\delta_{j_{2}j_{3}}+\delta_{ij_{2
}\delta_{j_{1}j_{3}}+\delta_{ij_{3}}\delta_{j_{1}j_{2}}\right)  \,,
\label{B.3p
\end{equation}
so that (\ref{B.3}) or (\ref{B.3p}) would yield that if $\left(  a,b,c\right)
$ is a permutation of $\left(  2,2,0\right)  $ then $A=4\pi/15$ while if a
permutation of $\left(  4,0,0\right)  $ then $A=4\pi/5.$

Our main aim is to point out why the product rule for derivatives, as employed
in \cite{Bowen} does not produce the correct result. Indeed, if we use
\cite[Eqn. (16)]{Bowen} written as\footnote{We shall employ our notation, not
the original one of \cite{Bowen}.
\begin{equation}
\frac{\overline{\partial}}{\partial x_{i}}\left(  \frac{n_{j_{1}}}{r^{2
}\right)  =\mathrm{p.v.}\left(  \frac{\delta_{ij_{1}}-3n_{i}n_{j_{i}}}{r^{3
}\right)  +\frac{4\pi}{3}\delta_{ij_{1}}\delta\left(  \mathbf{x}\right)  \,,
\label{B.4
\end{equation}
and then try to proceed as in \cite[Eqn. (18)]{Bowen}
\begin{equation}
\frac{\overline{\partial}}{\partial x_{i}}\left(  \frac{n_{j_{1}}n_{j_{2
}n_{j_{3}}}{r^{2}}\right)  \text{\textquotedblleft\textquestiondown
=\text{?\textquotedblright\ }n_{j_{1}}n_{j_{2}}\frac{\overline{\partial
}{\partial x_{i}}\left(  \frac{n_{j_{3}}}{r^{2}}\right)  +\frac{n_{j_{3}
}{r^{2}}\frac{\overline{\partial}}{\partial x_{i}}\left(  n_{j_{1}}n_{j_{2
}\right)  \,. \label{B.5
\end{equation}
Thus (\ref{B.4}) and the formul
\begin{equation}
\frac{\overline{\partial}}{\partial x_{i}}\left(  n_{j_{1}}n_{j_{2}}\right)
=\frac{\delta_{ij_{1}}n_{j_{2}}+\delta_{ij_{2}}n_{j_{1}}-2n_{i}n_{j_{1
}n_{j_{2}}}{r}\,, \label{B.6
\end{equation}
giv
\begin{equation}
n_{j_{1}}n_{j_{2}}\frac{\overline{\partial}}{\partial x_{i}}\left(
\frac{n_{j_{3}}}{r^{2}}\right)  +\frac{n_{j_{3}}}{r^{2}}\frac{\overline
{\partial}}{\partial x_{i}}\left(  n_{j_{1}}n_{j_{2}}\right)
=\text{\textquotedblleft Normal\textquotedblright}+\text{\textquotedblleft
Src\textquotedblright}\,, \label{B.7
\end{equation}
wher
\begin{equation}
\text{\textquotedblleft Normal\textquotedblright}=\mathrm{p.v.}\left(
\frac{\delta_{ij_{1}}n_{j_{2}}n_{j_{3}}+\delta_{ij_{2}}n_{j_{1}}n_{j_{3
}+\delta_{ij_{3}}n_{j_{1}}n_{j_{2}}-5n_{i}n_{j_{1}}n_{j_{2}}n_{j_{3}}}{r^{3
}\right)  \,, \label{B.8
\end{equation}
coincides with the first term of (\ref{B.1}) while
\begin{equation}
\text{\textquotedblleft Src\textquotedblright}=\frac{4\pi}{3}\delta_{ij_{3
}n_{j_{1}}n_{j_{2}}\delta\left(  \mathbf{x}\right)  \,. \label{B.9
\end{equation}
The right hand side of (\ref{B.9}) is not a well defined distribution, of
course, but Bowen suggested that we treat it as what we now call the
projection of a thick distribution, that is, as
\begin{equation}
\text{\textquotedblleft Src\textquotedblright}=\Pi\left(  \frac{4\pi}{3
\delta_{ij_{3}}n_{j_{1}}n_{j_{2}}\delta_{\ast}\right)  =\frac{4\pi}{9
\delta_{ij_{3}}\delta_{j_{1}j_{2}}\delta\left(  \mathbf{x}\right)  \,,
\label{B.10
\end{equation}
since $\Pi\left(  n_{j_{1}}n_{j_{2}}\delta_{\ast}\right)  =\left(  1/3\right)
\delta_{j_{1}j_{2}}\delta\left(  \mathbf{x}\right)  $ \cite[Example 5.10
{YE2}. In order to compare with (\ref{B.1}) and (\ref{B.3p}) we observe that
by symmetry the same result would be obtained if $j_{3}$ and $j_{1},$ or
$j_{3}$ and $j_{2},$ are exchanged, so that if in the term \textquotedblleft
Src\textquotedblright\ we do these exchanges, add the results and divide by
$3,$ we would get
\begin{equation}
\text{\textquotedblleft SrcSym\textquotedblright}=\frac{4\pi}{27}\left(
\delta_{ij_{1}}\delta_{j_{2}j_{3}}+\delta_{ij_{2}}\delta_{j_{1}j_{3}
+\delta_{ij_{3}}\delta_{j_{1}j_{2}}\right)  \delta\left(  \mathbf{x}\right)
\,, \label{B.11
\end{equation}
and thus the symmetric version of the (\ref{B.7}) is \textquotedblleft
Normal\textquotedblright$+$\textquotedblleft SrcSym\textquotedblright, which
of course is different from (\ref{B.1}) since the coefficient in (\ref{B.3p})
is $4\pi/15,$ while that in (\ref{B.11}) is $4\pi/27.$ Therefore, the relation
\textquotedblleft\textquestiondown $=$?\textquotedblright\ in (\ref{B.5})
cannot be replaced by $=.$

Hence the product rule for derivatives fails in this case. \emph{The question
is why? }Indeed, when computing the right side of (\ref{B.5}), that is, the
left side of (\ref{B.7}), we found just one irregular product, namely
$n_{j_{1}}n_{j_{2}}\delta\left(  \mathbf{x}\right)  ,$ but using the average
value $\left(  1/3\right)  \delta_{j_{1}j_{2}}\delta\left(  \mathbf{x}\right)
$ seems quite reasonable.

In order to see what went wrong let us compute $\overline{\partial}/\partial
x_{i}\left(  n_{j_{1}}n_{j_{2}}n_{j_{3}}/r^{2}\right)  $ by computing the
thick derivative $\partial^{\ast}/\partial x_{i}\mathcal{P}f\left(  n_{j_{1
}n_{j_{2}}n_{j_{3}}/r^{2}\right)  ,$ applying the product rule for thick
derivatives, and then taking the projection $\pi$ of this. We have
\begin{align*}
\frac{\partial^{\ast}}{\partial x_{i}}\mathcal{P}f\left(  \frac{n_{j_{1
}n_{j_{2}}n_{j_{3}}}{r^{2}}\right)   &  =\frac{\partial^{\ast}}{\partial
x_{i}}\left[  n_{j_{1}}n_{j_{2}}\mathcal{P}f\left(  \frac{n_{j_{3}}}{r^{2
}\right)  \right] \\
&  =n_{j_{1}}n_{j_{2}}\frac{\partial^{\ast}}{\partial x_{i}}\mathcal{P
f\left(  \frac{n_{j_{3}}}{r^{2}}\right)  +\frac{\partial\left(  n_{j_{1
}n_{j_{2}}\right)  }{\partial x_{i}}\mathcal{P}f\left(  \frac{n_{j_{3}}
{r^{2}}\right)  \,,
\end{align*}
and taking (\ref{T1.4}) into account, we obtain
\begin{align*}
&  n_{j_{1}}n_{j_{2}}\left\{  \mathcal{P}f\left(  \frac{\delta_{ij_{3}
-3n_{i}n_{j_{3}}}{r^{3}}\right)  +4\pi n_{j_{3}}n_{i}\delta_{\ast}\right\} \\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{\delta_{ij_{1
}n_{j_{2}}+\delta_{ij_{2}}n_{j_{1}}-2n_{i}n_{j_{1}}n_{j_{2}}}{r
\mathcal{P}f\left(  \frac{n_{j_{3}}}{r^{2}}\right)  \,,
\end{align*}
that is, $\partial^{\ast}/\partial x_{i}\mathcal{P}f\left(  n_{j_{1}}n_{j_{2
}n_{j_{3}}/r^{2}\right)  $ equal
\begin{align}
&  \mathcal{P}f\left(  \frac{\delta_{ij_{1}}n_{j_{2}}n_{j_{3}}+\delta_{ij_{2
}n_{j_{1}}n_{j_{3}}+\delta_{ij_{3}}n_{j_{1}}n_{j_{2}}-5n_{i}n_{j_{1}}n_{j_{2
}n_{j_{3}}}{r^{3}}\right) \label{B.12}\\
&  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +4\pi n_{j_{1}}n_{j_{2
}n_{j_{3}}n_{i}\delta_{\ast}\,.\nonumber
\end{align}
Applying the projection operator $\Pi$ we obtain that the $\mathcal{P}f$
becomes a $\mathrm{p.v.}$, so that the term \textquotedblleft
Normal\textquotedblright\ given by (\ref{B.8}) is obtained, while
(\ref{T1.10}) yields that the projection of thick delta is exactly
$A\delta\left(  \mathbf{x}\right)  $ where $A=\int_{\mathbb{S}}n_{i}n_{j_{1
}n_{j_{2}}n_{j_{3}}\,\mathrm{d}\sigma\left(  \mathbf{w}\right)  ,$ that is,
the \emph{correct} term
\[
\frac{4\pi}{15}\left(  \delta_{ij_{1}}\delta_{j_{2}j_{3}}+\delta_{ij_{2
}\delta_{j_{1}j_{3}}+\delta_{ij_{3}}\delta_{j_{1}j_{2}}\right)  \delta\left(
\mathbf{x}\right)  \,.
\]
The reason we now obtain the correct result is while it is true that
$\Pi\left(  n_{j_{1}}n_{j_{2}}\delta_{\ast}\right)  =\left(  1/3\right)
\delta_{j_{1}j_{2}}\delta\left(  \mathbf{x}\right)  $ and that $\Pi\left(
n_{j_{3}}n_{i}\delta_{\ast}\right)  =\left(  1/3\right)  \delta_{ij_{3}
\delta\left(  \mathbf{x}\right)  ,$ it is \emph{not} true that the projection
$\Pi\left(  4\pi n_{j_{1}}n_{j_{2}}n_{j_{3}}n_{i}\delta_{\ast}\right)  $ can
be obtained as $4\pi\left(  1/3\right)  \delta_{ij_{3}}\Pi\left(  n_{j_{1
}n_{j_{2}}\delta_{\ast}\right)  $ nor as $4\pi\left(  1/3\right)
\delta_{j_{1}j_{2}}\Pi\left(  n_{j_{3}}n_{i}\delta_{\ast}\right)  ,$ and
actually not even the symmetrization of such results, given by (\ref{B.11}),
works. Put in simple terms, it is not true that the average of a product is
the product of the averages!

One can, alternatively, compute $\partial^{\ast}/\partial x_{i}\mathcal{P
f\left(  n_{j_{1}}n_{j_{2}}n_{j_{3}}/r^{2}\right)  $ a
\begin{equation}
\frac{\partial}{\partial x_{i}}\left(  \frac{n_{j_{3}}}{r^{2}}\right)
\mathcal{P}f\left(  n_{j_{1}}n_{j_{2}}\right)  +\left(  \frac{n_{j_{3}}
{r^{2}}\right)  \frac{\partial^{\ast}}{\partial x_{i}}\mathcal{P}f\left(
n_{j_{1}}n_{j_{2}}\right)  \,, \label{B.13
\end{equation}
sinc
\begin{equation}
\frac{\partial^{\ast}}{\partial x_{i}}\mathcal{P}f\left(  n_{j_{1}}n_{j_{2
}\right)  =\mathcal{P}f\left(  \frac{\delta_{ij_{1}}n_{j_{2}}+\delta_{ij_{2
}n_{j_{1}}-2n_{i}n_{j_{1}}n_{j_{2}}}{r}\right)  +4\pi n_{j_{1}}n_{j_{2}
n_{i}\delta_{\ast}^{\left[  -2\right]  }\,. \label{B.14
\end{equation}
Here the thick delta term in (\ref{B.13})\ is $4\pi\left(  n_{j_{3}
/r^{2}\right)  n_{j_{1}}n_{j_{2}}n_{i}\delta_{\ast}^{\left[  -2\right]  },$
which becomes, as it should, $4\pi n_{j_{1}}n_{j_{2}}n_{j_{3}}n_{i
\delta_{\ast}.$

Complications in the use of the product rule for derivatives in one variable
were considered in \cite{EF07} when analysing the formula \cite{Vibet
\begin{equation}
\frac{d}{dx}\left(  H^{n}\left(  x\right)  \right)  =nH^{n-1}\left(  x\right)
\delta\left(  x\right)  \,, \label{B.15
\end{equation}
\ where $H$ is the Heaviside function; see also \cite{Paskusz}.

\section{Higher order derivatives\label{Section: Higher order derivatives}}

We now consider the computation of higher order derivatives in the space
$\left(  \mathcal{D}_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{n}\right)
\right)  ^{\prime}.$ If $f\in\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R
^{n}\right)  $ then, of course, the thick derivative $\partial^{\ast
}f/\partial x_{i}$ is defined by duality, that is,
\begin{equation}
\left\langle \frac{\partial^{\ast}f}{\partial x_{i}},\phi\right\rangle
=-\left\langle f,\frac{\partial\phi}{\partial x_{i}}\right\rangle \,,
\label{h.1
\end{equation}
for $\phi\in\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  .$ Suppose now
that $\mathcal{A}$ is a subspace of $\mathcal{D}_{\ast}\left(  \mathbb{R
^{n}\right)  $ that has a topology such that the imbedding $i:\mathcal{A
\hookrightarrow\mathcal{D}_{\ast}\left(  \mathbb{R}^{n}\right)  $ is
continuous; then the transpose $i^{T}:\mathcal{D}_{\ast}^{\prime}\left(
\mathbb{R}^{n}\right)  \rightarrow\mathcal{A}^{\prime}$ is just the
restriction operator $\Pi_{\mathcal{A}}.$ If $\mathcal{A}$ is closed under the
differentiation operators\footnote{The space $\mathcal{A}^{\prime}$ would be a
space of (thick) distributions in the sense of Zemanian \cite{zem}.}, then we
can also define the derivative of any $f\in\mathcal{A}^{\prime},$ say
$\partial_{\mathcal{A}}f/\partial x_{i},$ by employing (\ref{h.1}) for
$\phi\in\mathcal{A}.$ The
\begin{equation}
\Pi_{\mathcal{A}}\left(  \frac{\partial^{\ast}f}{\partial x_{i}}\right)
=\frac{\partial_{\mathcal{A}}}{\partial x_{i}}\left(  \Pi_{\mathcal{A}}\left(
f\right)  \right)  \,, \label{h.2
\end{equation}
for any thick distribution $f\in\mathcal{D}_{\ast}^{\prime}\left(
\mathbb{R}^{n}\right)  .$ In the particular case when $\mathcal{A
=\mathcal{D}\left(  \mathbb{R}^{n}\right)  $ then $\partial_{\mathcal{A
}f/\partial x_{i}=\overline{\partial}f/\partial x_{i},$ the usual
distributional derivative, and thus (\ref{h.2}) becomes \cite[Eqn.
(5.22)]{YE2}
\begin{equation}
\Pi\left(  \frac{\partial^{\ast}f}{\partial x_{i}}\right)  =\frac
{\overline{\partial}\Pi\left(  f\right)  }{\partial x_{i}}\,. \label{h.3
\end{equation}
What this means is that one can use thick distributional derivatives to
compute $\partial_{\mathcal{A}}f/\partial x_{i},$ as we have already done to
compute distributional derivatives.

When $\mathcal{A}$ is not closed under the differentiation operators then
$\partial_{\mathcal{A}}f/\partial x_{i}$ cannot be defined by (\ref{h.1}) if
$f\in\mathcal{A}^{\prime}$ since in general $\partial\phi/\partial x_{i}$ does
not belong to $\mathcal{A}$ and thus the right side of (\ref{h.1}) is not
defined. However, if $f\in\mathcal{A}^{\prime}$ has a \emph{canonical}
extension $\widetilde{f}\in\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R
^{n}\right)  $ then we could define $\partial_{\mathcal{A}}f/\partial x_{i}$
as $\Pi_{\mathcal{A}}\left(  \partial^{\ast}\widetilde{f}/\partial
x_{i}\right)  .$ This applies, in particular when $\mathcal{A}=\mathcal{D
_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{n}\right)  :$ if $f\in\left(
\mathcal{D}_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{n}\right)  \right)
^{\prime}$ then $\partial_{0}^{\ast}f/\partial x_{i}=\partial_{\mathcal{A
}f/\partial x_{i}$ cannot be defined, in general, but if $f$ has a canonical
extension $\widetilde{f}\in\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R
^{n}\right)  $\ then $\partial_{0}^{\ast}f/\partial x_{i}$ is understood as
$\Pi_{\mathcal{D}_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{n}\right)
}\left(  \partial^{\ast}\widetilde{f}/\partial x_{i}\right)  .$

Our aim is to point out that, in general, if $P=RS$ is the product of two
differential operators with constant coefficients, then while, with obvious
notations, $P^{\ast}=R^{\ast}S^{\ast},$ $P_{\mathcal{A}}=R_{\mathcal{A
}S_{\mathcal{A}},$ if $\mathcal{A}$ is closed under differential operators,
and $\overline{P}=\overline{R}\,\overline{S},$ it is \emph{not} true that
$P_{0}^{\ast}=R_{0}^{\ast}S_{0}^{\ast}.$ Therefore the space $\left(
\mathcal{D}_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{n}\right)  \right)
^{\prime}$ is not a convenient framework to generalize distributions to thick
distributions; the whole $\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R
^{n}\right)  $ is needed if we want a theory that includes the possibility of
differentiation.\smallskip

\begin{example}
\label{Example h.1}Let us consider the second order derivatives of the
distribution $\mathcal{P}f\left(  1\right)  .$ Formula (\ref{T1.14}) yield
\begin{equation}
\frac{\partial^{\ast2}}{\partial x_{i}\partial x_{j}}\left(  \mathcal{P
f\left(  1\right)  \right)  =C\left(  \delta_{ij}-2n_{i}n_{j}\right)
\delta_{\ast}^{\left[  -n+2\right]  }. \label{H.4
\end{equation}
In particular, in $\mathbb{R}^{2},$ $\partial^{\ast2}/\partial x_{i}\partial
x_{j}\left(  \mathcal{P}f\left(  1\right)  \right)  =2\pi\left(  \delta
_{ij}-2n_{i}n_{j}\right)  \delta_{\ast}.$ If we consider the function $1$ as
an element of $\left(  \mathcal{D}_{\ast}^{\left[  0\right]  }\left(
\mathbb{R}^{2}\right)  \right)  ^{\prime}$ then it has the canonical extension
$\mathcal{P}f\left(  1\right)  \in\mathcal{D}_{\ast}^{\prime}\left(
\mathbb{R}^{2}\right)  $ and s
\[
\frac{\partial_{0}^{\ast}\left(  1\right)  }{\partial x_{j}}=\Pi
_{\mathcal{D}_{\ast}^{\left[  0\right]  }\left(  \mathbb{R}^{2}\right)
}\left(  2\pi n_{j}\delta_{\ast}^{\left[  -1\right]  }\right)  =0\,,
\]
and consequently
\begin{equation}
\frac{\partial_{0}^{\ast}}{\partial x_{i}}\left(  \frac{\partial_{0}^{\ast
}\left(  1\right)  }{\partial x_{j}}\right)  =\frac{\partial_{0}^{\ast
}{\partial x_{i}}\left(  0\right)  =0\neq2\pi\left(  \delta_{ij}-2n_{i
n_{j}\right)  \delta_{\ast}=\frac{\partial^{\ast2}\left(  1\right)  }{\partial
x_{i}\partial x_{j}}\,. \label{h.5
\end{equation}
Observe that $\Pi\left(  2\pi\left(  \delta_{ij}-2n_{i}n_{j}\right)
\delta_{\ast}\right)  =0,$ but observe also that this means very
little.\smallskip
\end{example}

\begin{example}
\label{Example h.2}It was obtained in \cite[Thm. 7.6]{YE2} that in
$\mathcal{D}_{\ast}^{\prime}\left(  \mathbb{R}^{3}\right)  $
\begin{equation}
\frac{\partial^{\ast2}\mathcal{P}f\left(  r^{-1}\right)  }{\partial
x_{i}\partial x_{j}}=\left(  3x_{i}x_{j}-\delta_{ij}r^{2}\right)
\mathcal{P}f\left(  r^{-5}\right)  +4\pi\left(  \delta_{ij}-4n_{i
n_{j}\right)  \delta_{\ast}\,. \label{PD.9
\end{equation}
Since $\Pi\left(  n_{i}n_{j}\delta_{\ast}\right)  =\left(  1/3\right)
\delta_{ij}\delta\left(  \mathbf{x}\right)  $ in $\mathbb{R}^{3},$ this yields
the well known formula of Frahm \cite{Frahm
\begin{equation}
\frac{\overline{\partial}^{2}}{\partial x_{i}\partial x_{j}}\left(  \frac
{1}{r}\right)  =\mathrm{p.v.}\left(  \frac{3x_{i}x_{j}-r^{2}\delta_{ij}
{r^{5}}\right)  -\left(  \frac{4\pi}{3}\right)  \delta_{ij}\delta\left(
\mathbf{x}\right)  \,. \label{PD.10
\end{equation}
We also immediately obtain that
\begin{equation}
\frac{\partial_{0}^{\ast2}\mathcal{P}f\left(  r^{-1}\right)  }{\partial
x_{i}\partial x_{j}}=\mathcal{P}f\left(  \frac{3x_{i}x_{j}-r^{2}\delta_{ij
}{r^{5}}\right)  +4\pi\left(  \delta_{ij}-4n_{i}n_{j}\right)  \delta_{\ast}\,,
\label{h.6
\end{equation}
a formula that can also be proved by other methods \cite{YE3}. On the other
hand, in \cite{Franklin} one can find the computation o
\begin{equation}
\frac{\partial_{0}^{\ast}}{\partial x_{i}}\left(  \frac{\partial_{0}^{\ast
}{\partial x_{j}}\left(  \frac{1}{r}\right)  \right)  =\mathcal{P}f\left(
\frac{3x_{i}x_{j}-r^{2}\delta_{ij}}{r^{5}}\right)  -4\pi n_{i}n_{j
\delta_{\ast}\,. \label{h.7
\end{equation}
The fact that $\displaystyle\frac{\partial_{0}^{\ast}}{\partial x_{i}}\left(
\frac{\partial_{0}^{\ast}}{\partial x_{j}}\right)  \neq\frac{\partial
_{0}^{\ast2}}{\partial x_{i}\partial x_{j}}$ is obvious in the Example
\ref{Example h.1}, but it is harder to see it in cases like this
one\footnote{That the two results are different is overlooked in
\cite{Franklin}.}. Observe that the projection of both $4\pi\left(
\delta_{ij}-4n_{i}n_{j}\right)  \delta_{\ast}$ and of $-4\pi n_{i}n_{j
\delta_{\ast}$ onto $\mathcal{D}^{\prime}\left(  \mathbb{R}^{3}\right)  $\ is
given by $-\left(  4\pi/3\right)  \delta_{ij}\delta\left(  \mathbf{x}\right)
,$ but this does not mean that they are equal; observe also that one needs the
finite part in (\ref{h.6}) and in (\ref{h.7}) since the principal value, as
used in (\ref{PD.10}), exists in $\mathcal{D}^{\prime}\left(  \mathbb{R
^{3}\right)  $\ but not in $\left(  \mathcal{D}_{\ast}^{\left[  0\right]
}\left(  \mathbb{R}^{3}\right)  \right)  ^{\prime}.$\smallskip
\end{example}

.

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