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CHAPTER ONE
GENERAL INTRODUCTION
1.1 Background to the study
A major development issue facing many developing countries has been the need to reduce the scale and depth of poverty among the growing population. Chandy and Gerts (2011) estimate that there were about 878.2 million people living below the poverty line in the year 2010. Of this number over 700 million live in rural areas. African has a 369.9 million population or proportion of poor people the elimination and reduction of poverty is a key concern of development thinkers and practitioner (Coyle, 2007; Ifrpi, 2008).
The beginning of the 1970s saw attention general towards improving the wellbeing of the rural poor who formed the majority of the population in developing countries, many governments and international and local agencies shifted their attention and channelled their resources towards the development of the rural area.
This idea was motivated by the intention of reducing the levels of unemployment, increasing access to public goods and services by the development of rural population and more particularly, lowering poverty and overcoming income inequalities in most developing and least-developed countries according to brad show (2006), the explanation of poverty may be structural, personal, social or economic according to (Burgess &Pande, 2002).
In the fight against poverty, it is believed that the introduction of banks in rural areas enhances the livelihood of the rural dwellers. It is assumed that intervention will change human behaviours and practices in a way that will lead to the achievement of the desired outcome.
According to Van Santen (2010), financial services for the poor have also been proven to be a powerful instrument for reducing poverty, enabling poor people to build assets and increase incomes, and reducing vulnerability to economic stress and shocks.
Microfinance is not a new concept. It dates back to the 19th century when money lenders were formally performing the role of now financial institutions. Generally, a large segment of the world’s population is still underprivileged as such; this fosters a great demand for the services of micro-financial institutions for rural development. These institutions extend loans to applicants who typically belong to the lowest group of people in society.
There have been government policies on the role of microfinance in the rural development process for more than four decades. In the 1960s and 1970s, the policies focused on the provision of agricultural technologies aimed at improving farmers’ incomes and feed the nations. Later the focus broadened to include credit provision to the rural population engaged in other enterprises.
Presently, the international development agenda is dominated by the millennium goals with poverty eradication heading the list of goals and with microfinance strongly linked to these goals. In the highly globalized present-day world, microfinance is undoubtedly an essential tool in facing poverty in the 3rd world countries by which inequality between the developed and the underdeveloped world could be reduced.
Since independence, the government of Cameroon has embarked on several attempts aimed at promoting agricultural development in the country, especially in rural areas. In 1961, she brought forth the policy of the “Green revolution”, which was aimed at encouraging the development of agriculture (Simarski 1992).
Other efforts included the setting up of agencies like the National Fund for Rural Development (FONADER) and other rural agricultural extension programs. Despite all these attempts, much is still needed to boost this sector, particularly in rural areas, since it is very vital in the economic life of the state.
A recent development in this sector has been the increasing involvement of NGOs and microfinance institutions in the process of enhancing the development of small and medium-sized enterprises particularly at the rural level.
The question now is why microfinance at this point in time? The notion of credit unions was brought to the country by Rev Father Anthony Jansen and this was as a result of complaints that were coming up from the farmers and inhabitants in the locality in which he was living.
Among the difficulties faced by these people, was the fact that most of them save their money by hiding it in some parts of the house in which case ants and rats often eat them up. Besides, some farmers sold their crops before harvest due to a lack of storage facilities.
It was then that the first credit union was formed so that farmers could have a bit of financial power to afford better seedlings (www.camccul.org). How then are these microfinance institutions of significance to the sustainable development of the country particularly rural areas?
Even though the government promotes rural development, it is still important to know how micro-financial institutions influence development in rural areas. This work will be concerned with the role which the Mbiame Cooperative Credit Union Ltd (MBIACCUL) plays in the rural development of Mbiame.
1.2 Statement of the Problem
Micro Finance Institutions could be a powerful strategy or instrument among several others, for Development in general and Rural Development in particular in developing countries. Although many developing countries, such as African have scored relative successes in using microfinance institutions as an instrument for development in general, and rural development in particular, it has not been so for many other developing countries.
Most of the micro-finance programs operated in these countries have left the so-called beneficiaries in debts. In a similar vein, a most organization involved in providing micro-finance service, including government institution, co-operatives and Nongovernmental Organizations (NGOs) have in most cases performed very poorly.
High rates of non-repayment of loans by clients have on several occasions led to the collapse of micro-finance institutions. Notwithstanding this, micro-finance has continued to gain popularity among rural developers as a visible tool for improving rural agricultural practice and the diversification of economic activities of smallholder farming householders.
Lack of adequate loan funds, inadequate institutional capacities, poor coordination, little or no participation of the beneficiaries in the planning of micro-finances programs, lack of effective training programs for both beneficiaries and operators of the programs are some of the reasons behind the ineffectiveness of micro-finance as a strategy for Rural Development in developing countries.
It is just recent that MFIs gained recognition thanks to the noble price winner Yunus Mohammed (2003) of the Grameen bank. According to him, MFIs are financial intermediaries which are aimed at providing services such as accepting deposit savings, microcredit facilities, money transfer and other businesses.
It is generally known that the poor cannot borrow from banks because they do not have what is required to be granted a loan. The lack of financial power is a contributing factor to most societal problems. MFIs target the poor who are considered risky but their repayment rate tends to be positive as compared to regular commercial banks (Zellar and Sharma,1998). These services will go a long way to raise income levels and standards of living for rural people.
Besides the development of these institutions has gone away with the various transactions taking place in society that are related to the development of the world economy as a whole. Thus in most developing countries such as Cameroon, the development of MFIs have had a significant impact on the development of the rural economy.
However, the problem here is that there is no clear distinction of the extent to which MFIs help rural development. Therefore, studies relating to the impact of micro-financial institutions on rural development would explain and indicate in quantitative terms the impact of loans on the growth of the GDP in rural areas.
1.3 Research questions
In order to enable us to determine the impact of MFIs in the development of rural areas, the following research questions are formulated:
- What are the types of loans offered by MBIACCUL?
- What are the effects of the various types of loans offered by MBIACCUL on the growth of agriculture, small and medium-sized enterprises, education, employment in Mbiame?
1.4 Objectives of the study
The main objective of this study is to examine the contributions of MFIs to the development of rural areas with the case study of Mbiame Cooperative Credit Union Ltd (MBIACCUL). The specific objectives include:
- To identify the types of loans offered by MBIACCUL.
- To examine the effects of the various loan types on agriculture, small and medium-sized businesses, employment and education. | https://www.researchkey.net/the-role-of-financial-institutions-on-the-development-of-rural-areas-case-of-mbiame-cooperative-credit-union-ltd/ |
FIN500 New Mexico Bond Yield Asset Probability & Issues of Debit Problems Paper
No more than two pages, double spaced, explaination of equations used, 12 point. 1 inch margins.
Question 1:
Expected yield – You own a 6% bond maturing in two years and priced at 88%. Suppose that there is a 9% chance that at maturity the bond will default and you will receive only 41% of the promised payment.
What is the bond’s promised yield to maturity?
Question 2:
The following table shows some financial data for two companies:
A B
Total assets $1,587.1 $1,600.7
EBITDA –53 77
Net income + interest -73 31
Total liabilities 744.0 1,467.1
- Calculate the probability of default for the two companies.
- Which company has the higher probability?
Question 3:
Refer to the following information:
Amount issued $400 million
Offered Issued at a price of 101.50% plus accrued interest (proceeds to company 101.300%) through First Boston Corporation.
Interest 9.25% per annum, payable February 15 and August 15.
- Suppose the debenture was issued on September 1, 1992, at 101.50%. How much would you have to pay to buy one bond delivered on September 15? Don’t forget to include accrued interest.
- What is the amount of the first interest payment?
Question 4:
ABC Corp. is prohibited from issuing more senior debt unless net tangible assets exceed 150% of senior debt. Currently, the company has outstanding $100 million of senior debt and has net tangible assets of $201 million. How much more senior debt can ABC Corp. issue?
Question 5:
IMO Microsystems’ 12% convertible is about to mature. The conversion ratio is 34. Assume a face value of $1,000.
- What is the conversion price?
- The stock price is $54. What is the conversion value? | https://academicwritersden.com/fin500-new-mexico-bond-yield-asset-probability-issues-of-debit-problems-paper/ |
Daniel Ratner, Ira Pohl
This paper describes two new algorithms, Joint and LPA*, which can be used to solve difficult combinatorial problems heuristically. The algorithms find reasonably short solution paths and are very fast. The algorithms work in polynomial time in the length of the solution. The algorithms have been benchmarked on the 15-puzzle, whose generalization has recently been shown to be NP hard, and outperform other known methods within this context. | https://aiide.org/Library/AAAI/1986/aaai86-028.php |
Remember back when you had to grab a giant dictionary every time you wanted to find or translate one word?
Back when you had to say the alphabet from A all the way to that first letter of the word you were looking up…only to never find it or to take forever doing so?
We’re all happy those days are behind us, and that we now enjoy the luxury of Google Translate and the internet.
Still, learning new words in a foreign language like Arabic is difficult if you lack proper systems and strategic learning methods.
So, what’s a strategic way to learn new basic Arabic words for beginners?
According to statistics, learning 1000 words covers 85.5% of conversation in a given language.
Technically, this means you’ll be able to speak Arabic fluently if you learn the 1000 most used words.
Makes it sound way easier, right?
Below are 200 words to get you started.
Let’s dig in!Table of Contents
1. Pronouns
Personal Pronouns
To start composing basic sentences in Arabic, you’ll probably need to master personal pronouns. Here’s a list of the most essential personal pronouns in Arabic. | https://www.arabicpod101.com/blog/2021/09/23/arabic-beginner-words/ |
- Our Ratchet straps are designed to handle most types of flatbed tie-down requirements. All ratchet straps are tagged with their working load limit to meet DOT regulations and WSTDA recommended standards.
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These straps are meant to be hooked onto the trailer bed,
I would not recommend bolting them. The shortest strap
that I would carry would be 30' long. | https://www.myteeproducts.com/2-in-x-30-ft-ratchet-straps-with-wire-hook.html |
Provinces are octagonal chunks of land separated by impenetrable barriers called Borders, created by Jusicial after the death of Kathe to contain the wildly different magics and species from crossing and destroying the universe. There are 111 in total, 110 after the destruction of Darholm.
Shonova is the Province of Dreams, known for being in permanent twilight and granting it's residents powers over dreams.
A province full of treasures and magical artifacts, nullified by the area's anti-magic properties. | https://www.worldanvil.com/w/kollark-amevelloblue/c/provinces-category |
Q:
R plotCI mis-assigned colors plotted for series
In R v2.14.0 x64 on Windows 7, I am using the plotCI function in the gplots library, and trying to set the colour of each plot based on data within a data frame using:
plotCI(
x = data[1:2,3],
ui = data[1:2,5],
li = data[1:2,4],
col=data[1:2,6],
lty = 1, pch=20, xaxt ="n", xlim = c(1,42), ylim = c(0,100),
gap = 0 )
The plot occurs correctly except for the colour of the plotted points, which are mis-assigned to the wrong series (the colours are consistent within series however).
I have a data frame of structure (first 7 rows only):
size qim X1 lower upper color
1 1000 1 100.0000 99.6000 100.0000 blue
2 1000 2 99.8000 99.4000 100.0000 blue
3 1000 3 98.2000 96.6000 99.2000 blue
4 1000 4 62.7000 58.8000 65.7000 blue
5 1000 5 10.4000 9.0000 12.5000 blue
6 1000 6 3.9000 2.9000 4.9000 blue
7 5000 1 99.9000 99.4000 100.0000 red
I sort the data frame using:
data <- data.unsorted[with(data,order(qim,size)),]
The sort appears to have happened correctly, with the resultant data frame:
size qim X1 lower upper color
1 1000 1 100.0000 99.6000 100.0000 blue
7 5000 1 99.9000 99.4000 100.0000 red
13 10000 1 99.7000 99.4000 99.9000 green
19 40909 1 98.5000 98.5000 98.5000 black
25 152228 1 98.1000 98.1000 98.1000 black
31 241707 1 98.9000 98.9000 98.9000 black
37 434844 1 97.4000 97.4000 97.4000 black
In the resultant plot, the first line is plotted as red, and the second line is plotted as blue (reversed).
Is there something I'm doing wrong, or is there some other explanation for this?
A:
There's a factor/character confusion going on here. The color variable is being read into R as a factor, so its underlying numeric values are assigned according to the alphabetic order of the values: "black"=1, "blue"=2, (probably) "green"=3, "red"=4. Then colors are being mapped according to R's default palette: 1=black, 2=red, 3=green, 4=blue. This leads to the (admittedly seemingly bizarre) correspondence: "black"=black, "blue"=red, "green"=green, "red"=blue (!!). The fix is actually pretty easy: just use as.character around your color variable.
data.unsorted <- read.table(textConnection(
header=TRUE)
library(gplots)
data <- data.unsorted[with(data,order(qim,size)),]
I will also point out that a with statement makes it much easier to read the code and make sure that you're getting the right columns:
with(data[1:2,],
plotCI(x=X1,li=lower,ui=upper,
col=as.character(color),
lty=1,pch=20,
xaxt="n",
xlim=c(1,42),ylim=c(0,100),
gap=0))
| |
A WELSH champion was yesterday named one of the best young chefs in the world after striking gold in New Zealand.
Gary Griffiths, of the Castle Hotel, Conwy, was one of 24 competitors in the World Junior Chefs' Challenge at the World Association of Cooks' Societies Congress in Auckland.
The 20-year-old junior sous chef said: "It's been an amazing experience. I wasn't expecting a gold medal and I'm thrilled to finish third in the world,"
Each competitor was given a mystery box of ingredients from which they had to devise a menu and cook a three-course meal for six.
His starter was pan-fried red snapper, salmon tortellini and mustard-scented leeks with a tomato and cream sauce.
The main course was roasted venison with fennel, pear and date chutney, garlic braised potato, beetroot and thyme jus.
Dessert was baked chocolate fondant with rhubarb parfait and cinnamon palmiers.
Gary had plenty of support in New Zealand including Peter Jackson, chairman of the Welsh Culinary Association. | https://www.dailypost.co.uk/news/north-wales-news/young-chef-wins-world-accolade-2897167 |
By Narjas Zatat for Independent.
Archaeologists believe they have solved one of the history’s most puzzling questions – how the ancient Egyptians transported over 170,000 tons of limestone to build the Great Pyramid at Giza.
New findings at the site on the outskirts of Cairo have revealed purpose-built boats were used to transport the huge stones.
The findings shed new light on how King Khufu’s tomb, built over 4,000 years ago in about 2550 BC, was built
Archeologists have long known that some rock had been extracted eight miles from Giza in a place called Tura, while granite was quarried from over 500 miles away.
The way in which these materials were transported, however, has long been a source of disagreement amongst academics.
A group of archaeologists working at the Giza pyramid complex – an archaeological site – have unearthed an ancient papyrus scroll, remains of a boat and a network of waterways at the site of the pyramid, providing new evidence that points to how the oldest of the Seven Wonders of the Ancient World was built.
Pierre Tale, who spent four years painstakingly deciphering the papyrus written by an overseer working on the pyramid’s construction, told Channel 4 in the new documentary Egypt’s Great Pyramid: The New Evidence: “Since the very day of the discovery it was quite evident that we have the oldest papyrus ever found in the world.”
The document was apparently written by a man called Merer who was in charge of 40 elite sailors. Archaeologists discovered that thousands of trained workers used boats to navigate canals dug along the River Nile for the purposes of transporting limestone.
The boats were held together by thick, twisted ropes, some of which have survived and were found in good condition.
After collecting the materials, workers would bring them to an inland port a few meters from the base of the pyramid. In total, some 2.3 million blocks of stone were shipped across the land over the course of two decades.
American archaeologist Mark Lehner, who has over 30 years experience excavating in Egypt, said: “We’ve outlined the central canal basin, which we think was the primary delivery area to the foot of the Giza Plateau.”
Source: Independent.
New proof shows how the Egyptians transported 2½-ton blocks for 500 miles.
Blocks of limestone and granite built the tomb of Pharaoh Khufu in 2,600 BC.
170,000 tons carried along the Nile in wooden boats held together by ropes.
Boats used purpose-built channels and canals to inland port yards from site.
By Claudia Joseph For The Mail.
The detailed archaeological material shows that thousands of skilled workers transported 170,000 tons of limestone along the Nile in wooden boats held together by ropes, through a specially constructed system of canals to an inland port just yards from the base of the pyramid.
A scroll of ancient papyrus has also been found in the seaport Wadi Al-Jarf which has given a new insight into the role boats played in the pyramid’s construction.
Written by Merer, an overseer in charge of a team of 40 elite workmen, it is the only first-hand account of the construction of the Great Pyramid and describes in detail how limestone casing stones were shipped downstream from Tura to Giza.
In his diary, Merer also describes how his crew was involved in the transformation of the landscape, opening giant dykes to divert water from the Nile and channel it to the pyramid through man-made canals.
Although it has long been known that the granite from the pyramid’s internal chambers was quarried in Aswan, 533 miles south of Giza, and the limestone casing stones came from Tura, eight miles away, archaeologists disagreed over how they were transported.
Now archaeologist Mark Lehner, a leading expert in the field, has uncovered evidence of a lost waterway beneath the dusty Giza plateau. ‘We’ve outlined the central canal basin which we think was the primary delivery area to the foot of the Giza Plateau,’ he said.
The new discoveries are revealed in tonight’s Channel 4 documentary Egypt’s Great Pyramid: The New Evidence, which also includes another team of archaeologists who have unearthed a ceremonial boat designed for Khufu to command in the afterlife, which gives new insights into the construction of vessels at the time.
A team of specialists restored the wooden planks before scanning them with a 3D laser to work out how they were assembled. They discovered that they were sewn together with loops of rope. | https://treasuresegypt.com/tag/great-pyramids/ |
Is this a Novel I See Before Me? James Robertson’s And the Land Lay Still
Not for the first time, we have Walter Scott to thank, or perhaps to blame. Conventionally, the historical novel as a European phenomenon – and it was certainly phenomenal – is claimed as the Shirra’s invention, his great insight. Its emergence gave rise to one of the great paradoxes: how can the past be fiction, but also “true”? And does the burden of history have anything to do with the fact that Scott is no longer widely read?
Nor is he often emulated. By that I mean very few in these parts have set out consciously to create the grand, definitive, historical statement through stories. Robert Louis Stevenson said, near the end, that he felt sure he could match the Shirra given – no luck there, poor soul – the chance. RLS was proud of his knowledge of Scotland’s history. But was such a book ever likely? Nothing in Stevenson’s art suggests a writer suited to the creation of a Covenanting War and Peace.
Where else to look? Sunset Song might come to mind, but it lacks the impersonality, the multiplicity of perspectives, that generally attend historical fiction. William McIlvanney’s Docherty is assuredly a novel-in-history, but it does not, save allusively, aim to tell “a nation’s story”. Class is another matter: one of the book’s explicit aims was to restore common folk to the accepted historical record.
But that version is tenacious. Scottish history has received most attention, welcome or not, in popular fiction. There they all are: those kings, queens, heroes and villains. Great fun, too. Yet distant, impossibly different, from anything recognised as the ordinary, modern sense of “Scotland’s story”. To put it otherwise: we have nothing to approach Gore Vidal’s (sometimes) great series of fictions based around the rise – and possible fall – of the American republic and empire.
There is another issue. Scotland’s history is itself disputed ground. When even a popular TV history can cause “controversy” because professors dislike its emphases and style, the idea of a common story becomes fanciful. Then you are reminded to ask why TV should even bother with such a project. One answer: because we Scots are astonishingly ignorant of simple facts, thanks mostly to the shameful failures of the education system. Three centuries of Union, of rewriting, Balmoralising and tartanry have hardly helped.
The fact remains that the greatest number of us have a strange, disjointed sense of what happened, and why it might matter. Scottish history is received almost as a series of parables with no obvious – nor continuing – connections between them. Anyone who “did” history in our schools knows the joke. What happened? Er, Picts, Bannockburn, Bonnie Prince Charlie, and some stuff about an industrial revolution.
History is a problem for any novelist, in any case. You can see as much in the greatest narrative cannonball of them all, War and Peace. Tolstoy did an immense amount of research, as any proper voice-of-the-nation would. No detail was too small, no battlefield tour too dull. But what was he then supposed to do with this immense quantity of stuff? Stick it in the novel, page after page, of course.
Newer translations give a better idea of the greatness of War and Peace, but still they fail to erase a tiny, lingering question: is this a history book, or a fiction based in history? Why does Tolstoy seem to veer from one to the other? More to the point, can they be reconciled?
It becomes a question of licence. By what right does a novelist mess around with historical fact? But if a writer is not messing around, or making fictive art, why bother with a novel? Nothing prevented Tolstoy from writing a very large piece of historical non-fiction. Would humanity, symbolism, the mythic dimension and sheer drama have thereby been lost? The better historians struggle constantly with that little difficulty.
Some reviewers of Anglo-American fiction nevertheless bemoan two facts, as though they are connected. First, they note that a great many successful contemporary novels seem to take refuge in history, as though in the playground, at the expense of the contemporary world. David Mitchell, born to play, is one of the prominent examples of the moment. So is this why, hacks further ask, that a book such as Alan Hollinghurst’s The Line of Beauty, set slap in the middle of the Thatcherite Eighties, is a rare thing? Where are the state-of-the-nation novels?
All of this is a long preamble to saying that James Robertson’s And the Land Lay Still is very long, in places very fine, and afflicted too often by the need to lay out the facts and headlines of recent history. Implicit in its creation, nevertheless, is a teasing question: can Scotland hope for one of those state-of-the-nation epics when the nation of Scotland remains a submerged and stateless entity?
The novel aims to be our contemporary. Which is to say it carries a large cast – sometimes too large – through the social and political changes of the recent past, from the mid-1960s to the present. Its backdrop is the re-emergence of that strange beast, national identity, in the years when we all began to learn to ask what it means to be a Scot. It is framed – a nice virtual pun – by a photographic exhibition being organised by a son in honour of his dead father. Had Gordon Williams not written a novel named From Scenes Like These, Robertson might have been spoiled for a choice of titles.
Michael Pendreich, that son in his father’s shadow, is the book’s hub. Through him, the double notion of images and history – a half a century of both – will be transmitted. A few will dismiss the novel, no doubt, as “the Nationalist version”, given the clear identification of homeland with home rule. But Robertson, to his credit, is smarter than that: his finest creation is a Tory MP with certain tastes and, in the end, a great dignity.
The book makes a brave attempt, too, to marry social realism with the version too often, and wrongly, called magical. There are italicised pages – some work, some do not – that strive to give a sense of the land itself, its mythic variety, its hold on the imagination. There is also the figure, for an example, of a haunted wanderer escaping into the map of Scotland, yet never seeming to reach a destination.
But there are, too, all the clichés of recent Scottish political history, the stereotypical attitudes and arguments of what passes for “debate”. Activists, artists, journalists and politicians contend. Which is fine: they did and do. But the book would have been enhanced, I think, by a clearer acknowledgement that for much of the time ordinary people did not, in fact, give a toss about Scot-land’s great upheavals, its referendums and its personality issues. Life is commonplace, but the historical novel needs, or is thought to need, more.
Exposition is the death of fiction. Too often Robertson finds it necessary for one character or another to “bring us up to date”. Failing that, the author himself wades in with the facts he holds to be pertinent. The decision is arguable, often enough, but on no occasion does it benefit the prose. Thus: “The poll tax – or community charge, as it was officially known – was born of the Scottish rates revaluation of early 1980s. When property owners saw what their new bills were likely to be, they howled, and the Scottish Tories, anxious to appease their own natural supporters.…”
The reader and the book could live safely, I think, without such passages. Raise the poll tax rebellion by all means, but each time Robertson surrenders to history in this fashion – as though worried that his structure lacks an armature – he supplies passages that read like paraphrases of the many “books, magazines, journals and other documents” he acknowledges dutifully at the novel’s end.
It counts as a technical problem, but a large problem. In his better pages, Rob-ertson meditates on history, using it as the point at which art can begin. At his worst, he offers the textbook version, as though journalism and academic history are no different from lived experience. Fiction is more than a recitation of what went on, according to the press – a baffled press, as it happened – in the 1979 referendum. So a fissure arises within the novel.
Take, in contrast, a lovely few pages on Edinburgh in the early 1970s. Plenty of facts, each of them exact, are here, but there’s more: the atmosphere, the sense of the place, without tedious documentary validation.
There follow several pages on the brief, miserable existence of the Scottish Labour Party. Perhaps in a big, capacious novel such details add texture, but details – and barely remembered details – they remain. Nor do they, though such is the clear intention, develop character much. History in this sense seems like a very long haul.
Better by far are Robertson’s attempts to depict one of those morose – and here vengeful – British government spooks we used to believe were hanging around. More human is the woman journalist brutalised and degraded. More truthful – more like the truth of the times – is Michael Pendreich’s liberation as a gay man. And the death of that traumatised, haunted wanderer – “You ate the stones, and the sea faded, and the land faded…” – seems to point to a novel that might have been written. A better novel, perhaps.
History is as slippery as fiction. But in fiction there is, persistently, the bigger story. | http://www.scottishreviewofbooks.org/2010/08/is-this-a-novel-i-see-before-me-james-robertsons-and-the-land-lay-still/ |
Between 1650 and 1750, four Catholic churches were the best solar observatories in the world. Built to fix an unquestionable date for Easter, they also housed instruments that threw light on the disputed geometry of the solar system, and so, within sight of the altar, subverted Church doctrine about the order of the universe.
A tale of politically canny astronomers and cardinals with a taste for mathematics, The Sun in the Church tells how these observatories came to be, how they worked, and what they accomplished. It describes Galileo's political overreaching, his subsequent trial for heresy, and his slow and steady rehabilitation in the eyes of the Catholic Church. And it offers an enlightening perspective on astronomy, Church history, and religious architecture, as well as an analysis of measurements testing the limits of attainable accuracy, undertaken with rudimentary means and extraordinary zeal. Above all, the book illuminates the niches protected and financed by the Catholic Church in which science and mathematics thrived.
Superbly written, The Sun in the Church provides a magnificent corrective to long-standing oversimplified accounts of the hostility between science and religion.
Publisher: Harvard University Press
ISBN: 9780674005365
Number of pages: 384
Weight: 748 g
Dimensions: 248 x 171 x 24 mm
MEDIA REVIEWS
Dr. Heilbron reveals the ubiquity of the solar observatories, which heretofore were little known among scholars. And he shows that the church was not necessarily seeking knowledge for knowledge's sake, a traditional aim of pure science. Rather, like many patrons, it wanted something practical in return for its investments: mainly the improvement of the calendar so church officials could more accurately establish the date of Easter. -- William J. Broad * New York Times *
A book both elegant and learned, exploring the installation of vast (but often easily overlooked) astronomical instruments in major churches by authorities sometimes thought, wrongly, to have opposed astronomical research. * New York Times Book Review *
In this elegant work, Heilbron recounts how in the seventeenth and eighteenth centuries the Roman Catholic Church fashioned several of its major cathedrals into precision instruments for studying the motions of the sun. The aim was to determine the time between vernal equinoxes, so that the dates for Easter could be forecast accurately...Heilbron, upending common views of the Church's relationship to science after it condemned Galileo, shows that Rome handsomely supported astronomical studies, accepting the Copernican hypothesis as a fiction convenient for calculation. * New Yorker *
Heilbron's book tells of the struggle to determine dates more accurately, including a little-known aspect of the history of the calendar--the use of churches as giant sundials to make astronomical measurements. -- Kate Noble * Time *
The historical perception of post-Renaissance Italian astronomy has become so over-charged with the Roman Catholic Church's condemnation of Galileo in 1633 that it is commonly assumed that no significant science took place south of the Alps until the 19th century. But, as John Heilbron's learned, elegant and finely phrased book reminds us, this was not the case...Though Heilbron supplies all the necessary geometry to demonstrate how the meridianae [(a solar measuring instrument)] were constructed and used within the great architectural masterpieces into which they were incorporated, his book is arranged and illustrated in such a way that non-mathematical persons can enjoy it. -- Allan Chapman * Times Higher Education Supplement *
John Heilbron's book does tell a gripping story and with a splendid literary flair...By subtly inserting critical comments, the author evaluates the interactions of science in its gestation with the culture of those centuries and the repercussions that these interactions have has down to our own times. And so it becomes a story about people, and Heilbron tells it in a masterfully human way. -- George V. Coyne * Nature *
In The Sun in the Church, historian John Heilbron argues convincingly that long-held interpretations [in astronomy] are too simplistic and must be revised...Heilbron tells an important story, one that is not so much neglected as unknown among historians of science. Even in histories of astronomy, there is usually only a passing reference to it. -- Albert Van Helden * Science *
The spectacle of the image of the sun projected on meridian lines in several of the great Italian cathedrals is captured in the beautiful color plates highlighting this book...This excellent book explains the difficulties posed by the inconvenient lengths of the lunar month and solar year, and discusses how observations of the solar image crossing a precisely aligned mark could solve the problem...The book is well written. -- D. E. Hogg * Choice *
Heilbron chronicles the ironic relationship between astronomy and the Catholic Church as it seeks the means to determine [the date for Easter]. This is the story of politically astute astronomers and cardinals who have to reconcile church doctrine with Galileo's universe...The text is filled with fine detail and is richly illustrated. An erudite and scholarly work. -- James Olson * Library Journal *
J. L. Heilbron depicts the unusual intersection of architecture, science, ecclesiastical and civil history, mathematics and philosophy that led the church to construct the buildings only a few years after it martyred Galileo. Erudite, accessible and wryly humorous, Heilbron's engaging book is a first-rate work of science history. * Publishers Weekly *
A fascinating history of astronomy that shows, as no other work has done so well, what happened to Italian science after Galileo's trial. An astonishing display of erudition and linguistic control, with a wealth of fine details, this is a major history that carves out a unique territory. -- Owen Gingerich, Harvard University
The innumerate reader will learn much from Heilbron's book, and may come away with a different appreciation of the stars above us. -- Ingrid D. Rowland * New York Review of Books *
He tells his story in rich detail, reconstructing characters and circumstances with ironic verve. His theme is the meridian lines (meridianae) laid down in the marble floors of cathedrals for quantifying the sun's annual motion... Heilbron's book is a treasure trove of fascinating information. -- Curtis Wilson * Isis *
This excellent book adds a welcome complexity to the historiography of astronomy in the years after Galileo's abjuration allegedly brought Italian astronomy to its knees Heilbron's book also reinterprets the relations of science and religion in the shadow of the Galileo affair. The novelty of his argument is neither that religion can stimulate astronomy nor that ecclesiastical patronage encouraged learning It is rather that the Church signally fertilized astronomy in an era when most historians portray the two as antagonists [one] will appreciate the witty prose of the argument and the elegant design of this important book. -- Michael H. Shank * Renaissance Quarterly *
The Sun in the Church: Cathedrals as Solar Observatories is a historical, well-documented, scholarly book concerned both with the use of churches in Italy during the 16th and 18th centuries to obtain observations of the sun for calendric and scientific purposes and with the relationship between the Church of Rome and the heliocentric views of many of the scientists of those times. -- Arnold M. Heiser * Science Books and Films *
Heilbron combines the history of astronomy, mathematics, architecture, patronage, and religion to tell a story that very much alters the common picture of the progress in astronomy in the early modern period and the place of the Catholic Church in that history. The story is well told, and the mathematics is given in a way that could discourage only the most innumerate. -- Sheila J. Rabin * The Sixteenth Century Journal *
J. L. Heilbron's remarkable book draws our attention to church users of a very different kind: early modern astronomers measuring the solar path to correct the shift of the ancient Julian calendar The Sun in the Church tells their history in detail, alongside an exceptionally comprehensive and clear account of medieval and early modern astronomy The Sun in the Church is an illuminous book, possibly as durable as the meridianae it celebrates. -- Sergio Sanabria * Technology and Culture *
This book offers a different kind of travel guide for the 'mathematical tourist,' providing an itinerary of Italian cities and churches in which to find meridians, analemmas, armillary spheres and gnomons. These are good reminders of the role of the church in the history of science and testify to the fact that everything applied to the church, even the most apparently ornamental, served a didactic purpose. -- Paul A. Calter and Kim Williams * Nexus Network Journal *
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This master thesis project has focused on the development of an open-domain generative model for chatbots. In particular, we target one of the under-studied aspects in the existing chatbots, which is the emotion modelling of the natural language. The motivation of introducing emotions in a chatbot is to improve the communication quality by making the chatbot more human-like — the chatbot acts as an artificial companion that is able to generate emotional responses. Such a functionality is accomplished by following an emotion model which enables us to customize emotional content in generated sentences.
The proposed model as shown in Fig. 1 is associated with two types of text representations: affective representation (SenticNet ) and semantic representa-tion (GloVe ). It follows a two-fold process: first, we represent the input-output pairs by the concepts and their emotion scores which are provided by the Sen-ticNet. Concepts are terms that contain conceptual and affective information of the text; emotion scores are two-dimensional values that correspond to the valence and arousal levels of the emotion. Given the emotion scores of the texts, the emotion model utilizes a fully connected neural network to learn the emotion relations between the input and output in a common conversation. Second, we apply a Seq2Seq (Sequence-to-Sequence) model as a basic structure for text generation, and modify the model to integrate the emotion relations learned by the emotion model. The modified Seq2Seq model takes word vectors from GloVe to represent the input, and combines the GloVe representation with the intended emotion score of the output generated by the emotion model. The rest of the network remains the same with the basic Seq2Seq model which applies recurrent neural networks with long short-term memory (LSTM) cells as used in .
2 Y. Zheng et al.
Fig. 1. The modified Seq2Seq model with emotion components, where X denotes the input, Affect(X) and GloVe(X) are the affective and semantic representations of X respectively, Y is the generated response, and Affect(Y) is the affective representation of the output produced by the emotion model given the current Affect(X).
using the movie-dialogs extracted from the corpus randomly, the proposed model is able to offer a finer-tuned emotion status while training with movie-dialogs from a specific category. Such a selection of the categories can be manipulated by the end-user as well in order to enrich the functionality of a chatbot, and to lengthen the conversation by increasing the user’s interest. In this thesis project, we have performed a preliminary test for integrating emotion component with Seq2Seq model, and comprehensive comparisons are planned for the next stage due to the time constraints. For future research, such a chatbot can be extended to provide a companionship that personalizes the user experience as in .
References
1. Cambria, E., Poria, S., Bajpai, R. and Schuller, B.: SenticNet 4: A semantic resource for sentiment analysis based on conceptual primitives. In: Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers, pp. 2666-2677.
2. Pennington, J., Socher, R., and Manning, C.: GloVe: Global vectors for word rep-resentation. In: Proceedings of the Conference on Empirical Methods in Natural Language Processing (EMNLP), 2014, pp. 1532-1543.
3. Sutskever, I., Vinyals, O., and Le, Q. V.: Sequence to sequence learning with neural networks. In: Advances in Neural Information Processing Systems, 2014, pp. 3104-3112.
4. Danescu-Niculescu-Mizil, C., and Lee, L.: Chameleons in imagined conversations: A new approach to understanding coordination of linguistic style in dialogs. In: Proceedings of the Workshop on Cognitive Modeling and Computational Linguistics, ACL, 2011. | https://123dok.net/document/yng9jlkl-approach-incorporate-emotions-chatbot-seq-seq-model.html |
During the 1970's, National Semiconductor sold many calculators under the Novus brand name. Some of the high-end models, including the Novus 4525, were programmable.
These Novus calculators, like the famous classics from Hewlett-Packard, were RPN models; binary operators were entered in postfix notation. The Novus 4525 has a four-level stack. The similarities with HP end here, however. Whereas HP calculators have legendary accuracy, the Novus 4525 is notoriously inaccurate: for instance, 1 arc tan yields 45.000654 instead of 45. In my collection, only the Sinclair Cambridge Programmable is worse in this regard. The manual (thanks, Martin!) reasons that this inaccuracy exists "because extra guard digits and rounding techniques are not employed [...] in order to simplify the technical design of your calculator." But, then, how come even the Russians were able to make calculators that used no guard digits, yet were able to deliver results to 8 digits of accuracy with an error typically no more than ±1 in the last decimal position?
The programming model of the Novus 4525 is rather unusual. Programming is started by moving the slide switch to the Load position and pushing the start button. However, there's no visual indication (other than the position of the slide switch) that the calculator is in program mode. There are no program steps, no key codes, operations are executed just like they would normally with results appearing on the display, except that the keystrokes are also invisibly recorded. Only a very limited editing facility is provided: the del key can be used to erase the most recent program step. When you switch to the Load setting, the program counter is positioned at the end of the program, so it is possible to backspace (erasing steps) or add program steps to the existing program. The halt and skip keys make it possible to enter multiple programs.
According to the manual, the program capacity of the calculator is 100 program steps. The example shown below, however, is 102 steps in length, no matter how I count it; or perhaps 103 if the start key counts as an extra step. Adding a single step triggers an error condition, so this really must be the maximum size of program memory. There's no mention in the manual of merged program steps, but in any case, I cannot see how merged steps would save exactly 2 (or 3) steps from the length of this program. Which leads me to believe that the actual program capacity of this machine may be 102 steps, not 100.
The calculator has no conditional branch capability; indeed, no control transfer capability of any kind. Programs are merely keystroke sequences, executed in fixed order. You cannot use an error as a halting condition either; program execution happily continues after a calculation results in an error such as an infinite result.
Needless to say, the lack of a conditional branching capability makes it impossible to implement algorithms that require iteration, so for instance, a factorial program is not possible on this unit. (The manual does suggest an iteration technique, but it requires the user to press a button every time the loop is restarted.) As a further limitation, the calculator has only one memory register. With all these restrictions, it is no small surprise (and a great testament to the superiority of the RPN model) that a Gamma function implementation is possible on the Novus 4525, with only a minor compromise in result accuracy.
The program presented here calculates the Gamma function to more than six digits of precision. This is actually better than the accuracy of built-in trigonometric functions.
There is no need to store any constants before using the program; simply key in the argument and hit the start button (making sure first that the slide switch is in the Run position.). | http://airy.rskey.org/CMS/index.php/7?manufacturer=National+Semiconductor&model=4525+Scientist+PR |
Last July 28, 2013, my partner and I are able to understand a person who Is considered as a deviant In this society. HIS sister let us enter Inside the life of his brother and shared to us the process of how he had become an "outsider" of this society. Julius Bella, our subject, Is a drug addict. In his case, It was clearly not Inherited or an Innate behavior, but It was the Influence of the people around him who drove Julius In the Intake of Illegal drugs.
Since Julius ND his family live In an unprivileged area where there were limited resources of good education, we assumed that Julius, in such a young age, was provided little knowledge about the effects of the drugs he took. Only later did he realize the consequences of his actions. Even if his friends' actions were irrational, Julius chose to conform. As Sash's research stated, many people are willing to negotiate their own judgments of right and wrong to avoid being considered as an outcast and different.
The theory of differential association introduced by Edwin Sutherland indicated that person's tendency toward conformity or deviance depends on the amount of contact with other who encourages or rejects conventional behavior (Twelfth Edition: Sociology; John J. Macaroni). In this case, the subject spent almost all of his time with his group of friends. Thus, in order to blend in with his friends, Julius had to agree and behave in compliance with his friends' definition of normal; moreover, motive for continued behavior evolves through participation in the behavior in the company of others (http://www. Ms. Du/?Keller/180/Theodore. HTML). Julius never gained to have a second chance in his life. Social control is the attempt by society to regulate people's thoughts and behavior (Twelfth Edition: Sociology; John J. Macaroni), and self-control Is under the category of social control. Social control's attempts to manage people's behaviors would not be achievable if the people in it do not have self-control. Every society has groups of people in charge to regulate peace and order In a society, such as policemen, traffic enforcers, lawmakers, and a lot more.
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In Julius' area, we presumed that they have fewer policemen than urban areas do, less focused by the government, and fewer people to look out for their behavior. In a small society with weak bonds of social control, the people living there are more likely free to deviate since there would be less chance that they will be restrained. According to social control theory, what causes people to use Illegal drugs Is the absence of social controls promoting conformity. On the other hand, lack of parental guidance Is one of he many causes of low self-control of deviant people.
Being neglectful parents could greatly Impact the life of their children since the young ones generally look up to their parents as the role model of their lives. So parents who fail to show care and control is caused by a factor that takes place very early in one's life, whereas social control can operate more or less throughout one's lifetime (http://higher. McGraw- hill. Com), but in a society, self-control and social control ought to have continuous balance in order to attain organization. | https://phdessay.com/case-study-on-deviance/ |
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DescriptionUKG is looking for a dedicated and driven Delivery Assurance Manager who will be accountable for ensuring the successful planning and execution of product development and product launch within the assigned group. The role requires a person who can build strong relationships, and is highly functional in a dynamic, challenging, and technical environment. You will be responsible for ensuring all programs successfully navigate from ideation through product launch including organizational readiness.
Primary/Essential Duties and Key Responsibilities:
Define and execute initiatives from design to release, identifying resources, and managing implementation
Clearly communicate goals, roles, responsibilities, and desired outcomes
Act as a cross-functional liaison to ensure relevant areas of the organization are engaged
Manage multiple internal and external, cross-functional and remote stakeholders
Track timely delivery against program objectives
Drive continuous improvement
Facilitate discussions and lead effective meetings
Lead without authority by building relationships
Monitor and report on initiative progress to leadership and stakeholders
Identify and manage risks and dependencies
QualificationsRequired Qualifications:
Strong detail orientation and ability to organize, both self and others
Demonstrated ability to collaborate to drive results and problem solve
Ability to effectively and concisely communicate status to leadership
Proven ability to work in a fast-paced, ambiguous, deadline-oriented environment
Understanding of software development life cycle model
Experience working with technical and non-technical stakeholders across multiple business units
Bachelor’s Degree
3-5 years’ experience successfully managing multiple projects concurrently
Preferred Qualifications:
Agile and/or Project Management Professional (PMP) certification preferred
Familiarity with agile software development
Experience managing projects to launch new products or services, with an understanding of how to achieve organizational and customer readiness
Ability to interpret data to track project success
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Corporate overviewHere at UKG, Our Purpose Is People. UKG combines the strength and innovation of Ultimate Software and Kronos, uniting two award-winning, employee-centered cultures. Our employees are an extraordinary group of talented, energetic, and innovative people who care about more than just work. We strive to create a culture of belonging and an employee experience that empowers our people. UKG has more than 13,000 employees around the globe and is known for its inclusive workplace culture. Ready to be inspired? Learn more at www.ukg.com/careers
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Ultimate Kronos Group is proud to be an equal opportunity employer and is committed to maintaining a diverse and inclusive work environment. All qualified applicants will receive considerations for employment without regard to race, color, religion, sex, age, disability, marital status, familial status, sexual orientation, pregnancy, genetic information, gender identity, gender expression, national origin, ancestry, citizenship status, veteran status, and any other legally protected status under federal, state, or local anti-discrimination laws.
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For individuals with disabilities that need additional assistance at any point in the application and interview process, please email [email protected] or please call 1 (978) 250 9800. | https://careers.ukg.com/careers/JobDetail/Delivery-Assurance-Manager/33477 |
Being bilingual or even multilingual should be seen as an asset and resource to schools and communities, rather than a deficit that needs to be remedied, said Maria Franquiz, a bilingual education expert and University of Utah professor who spoke at Baylor University on Tuesday.
Franquiz spent time before her lecture touring schools in the Waco area and speaking with educators about bilingual education.
Standards for bilingual education vary from state to state, with more than 30 states having English-only statutes that do not require schools to offer instruction in another language, Franquiz said. She worked in Texas for 12 years as a professor at the University of Texas in Austin and San Antonio before moving to Utah.
“When you come to this country and you don’t speak English, the goal, whether it’s a child or whether it’s a parent, is to learn English,” she said. “A lot of students get English-as-a-second-language education because there isn’t a local bilingual program. But in the case of Texas or the state of California, for example, two states that have an increasing enrollment of Spanish speakers in school, then bilingual education ought to be available.”
But foreign-language speakers do not want to lose their native language and should not be forced to, Franquiz said. In a globalized world, being bilingual is a gift, while being monolingual can limit the ability to communicate on a daily basis, she said.
“For future generations, to become bilingual or multilingual would be a huge asset, and yet we have traditionally thought of people coming to the United States who do not speak English as having a deficit,” Franquiz said. “That thinking ought to be in the past, given the changes demographically that are occurring in our nation, in general, but in certain states specifically, such as Texas.”
Most students, or 60 percent, in the Waco Independent School District are Hispanic or Latino, according to district information from the 2016-2017 school year. But fewer than 5 percent of students participated in a bilingual program, and about 14 percent received ESL education.
Waco ISD introduced a new dual language program at Brook Avenue Elementary School this past fall with 17 3-year-old students, said Grace Benson, assistant superintendent for elementary curriculum and instruction. They learn in both English and Spanish, promoting proficiency in both languages. The school district plans to expand the program next year with two classes: one for 3-year-olds and one for 4-year-olds. It also will add grade levels and campuses to the program as the inaugural class advances.
Research has shown dual language learners have a high rate of achievement and success if they stay with the program at least through elementary school, Franquiz said. Achievement rates only increase the longer students remain with the program, which is what many other countries encourage. Many countries require students to learn three languages before going to university, she said.
“They’re more likely to go to college,” Benson said. “They’re more likely to stay in school and not drop out.”
Killeen ISD bilingual program specialist Eileen Lebron de Benitez said the public school system is designed to remove students’ native language and have them become monolingual, which makes implementing bilingual programs difficult. Texas will have numerous uneducated students if the state does not help them nourish skills in both languages.
“It’s like fighting with a ghost,” she said.
Franquiz said she knows what it is like to learn English and forget her native language. She grew up in Puerto Rico before moving with her military family to North Carolina in third grade, which is where she learned English. She did not have an option for bilingual education but managed to master this new language. When the family moved to El Paso, Franquiz found that neither the English she had learned in North Carolina nor the Spanish she knew from Puerto Rico was adequate at school.
“I have been critiqued for not speaking the right English and not speaking the right Spanish, depending on where I am,” she said, adding she relearned Spanish in college.
Knowledge of various dialects in two languages gave Franquiz a wealth of understanding. Students in dual language programs can earn this knowledge, as well, but only if it is available to them, she said.
“That’s a repertoire of knowledge that you can get, if you open yourself up to the idea that these are resources that can help me communicate better with more people, which is what I think the purpose of education should be,” Franquiz said. “It is about learning different disciplinary knowledge, but it’s also about learning how to communicate that knowledge to different kinds of audiences.”
Dual language programs differ from ESL in that half the class in a dual language program speaks one language as their dominant language and the other half is dominant in the other language. Students do not just learn from the teacher, who helps them understand both languages, but from their peers, as well, Franquiz said.
“It’s a beautiful model,” she said. “It’s been around for at least three decades in different states and shows a highly successful rate if a child stays in dual language until fifth grade. They will be able to test as well in their second language as their English-only peers.”
Franquiz said the dual language model also promotes better multicultural understanding and does not segregate students dominant in any particular language. Students interact with a broader group, rather than only those who speak their native language.
Conversely, teachers in ESL programs give instruction only in English and do not use Spanish at all, Benson said. Dual language or bilingual programs use students’ native language as a vehicle to learn a second language, gradually transferring knowledge into the new language. It also helps students with career opportunities, because they are both bilingual and bi-literate.
“By law, we are supposed to provide remedies for children who do not speak English,” Franquiz said. “The goal is about valuing bilingualism and making bilingualism available, whether you’re an English speaker or a Spanish speaker. It’s not seen as a remedy for people who don’t speak English. When you come to this country and you don’t speak English, it is seen often as a problem instead of a resource, so you get specialized instruction, which is sometimes in a segregated school environment, until you have enough English to be integrated into the whole student body. When it’s seen as a resource, you’re a resource for peers, for teachers, for the community.”
And dual language or bilingual programs allow native speakers to retain that part of their culture, rather than pave over it with English. | https://wacotrib.com/news/education/expert-bilingual-students-more-likely-to-succeed-in-school-go/article_8e89bc13-f662-5110-b3de-889d411d9d0a.html |
Evolution of Adaptive Syndromes
The ant and plant characteristics listed in the previous section correlate with the ants' ability or inability to run on waxy stems and the presence or absence of wax crystals. The combinations of characters may thus be considered adaptive syndromes. Both ant and host plant traits may maximize the ecological benefits and minimize the ecological costs of each stem wax type. For the host plants, the evolutionary outcome may be governed by trade-offs between costs of producing morphological exclusion filters and costs of sustaining highly competitive ant colonies. Similar trade-offs may be associated with the ants' capacity of wax running.
Reconstructions of ancestral states in Macaranga ant-plants are ambiguous with regard to the question of whether species devoid of wax crystals have evolved from waxy ancestors [21,27]. However, it is likely that wax barriers have been lost once at the base of a large clade in the genus section Pachystemon. Thus, Macaranga host plants have possibly undergone a change of adaptive syndrome, followed by radiation.
Even though the existing phylogenetic information on the Macaranga-associated Crematogaster (Decacrema) species group is still uncertain [21,25], it suggests that wax running capacity represents the ancestral condition in this clade and that non-wax runners have originated through several independent losses . The evolutionary origin of wax running capacity may thus have occurred before the radiation of this plant-ant clade.
To summarize, the slippery wax barriers and the ants' capacity to run on them represent a mechanical factor that plays a central role in the biology of Macar-anga-ant associations and has multiple implications on traits of both host plants and ant partners. An essential prerequisite for clarifying the mechanisms of speciation and the extent of coevolution vs. coadaptation (i.e., the match of characters with independent origins) in this ant-plant system is a knowledge of the mechanical basis of wax barriers and of wax running behavior. A biomechanical analysis of wax running is necessary to identify traits and adaptations involved in the capacity to colonize waxy hosts. By mapping these traits on the phylogeny of Crematogaster (Decacrema) ants, it will be possible to study the evolution of this biomechanical capacity and to compare it with the evolutionary history of host plant adapations. | https://www.ecologycenter.us/biomechanics/evolution-of-adaptive-syndromes.html |
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Zebra Rock Image copyright of WA Museum This distinctive reddish-brown and white-banded sedimentary rock from the east Kimberley of Western Australia is called ‘zebra rock’ or ‘zebra stone’ Composed essentially of small particles of quartz and ‘sericite’ (fine-grained white mica), zebra rock also contains the minerals kaolinite ....
Banded Gneiss - interlayered with calc-silicate rock, metaconglomerate, amphibolite, sillimanite-mica schist, and granitic rock Biotite Granitic Gneiss (Middle Proterozoic) Biotite Granitic Gneiss (950-1250 my) - unconformity; contains paragneiss and granitic to ,...
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El molino vertical de la serie LM es un equipo avanzado de molienda en polvo el cual es diseñado en base de las tecnologías mundial que aplicamos ampliamente Es un equipo que se dedica a la trituración, secador, molienda en polvo, transportación y clasificación El Molino vertical se aplica en el campo de cemento, energia electrica, metalurgia, quimica, mineral etc Se usa para moler los ....
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The Metamorphic Rock Called "Mariposite" The name "mariposite" is also used for rocks The rocks contain enough particles of green mica to produce a green color These rocks are metamorphic, have been altered by hydrothermal activity, and they are usually thought to have a serpentinite protolith...
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Sep 17, 2018· A Michigan man made the discovery of a lifetime when he stumbled on glowing rocks on the beaches of Lake Superior The rocks, which he named ,...
Molino de Bolas Los molinos de bolas se utilizan principalmente en la trituración y rectificaciónSegún la necesidad de los clientes, el molino de bolas puede ser de diseño húmedo o seco Los molinos de bolas se han diseñado en tamaños estándar de los productos finales entre 0,074 mm y 0,4 mm de diámetro MTM Molino Trapezoidal...
Oct 10, 2019· Granite is an igneous rock that forms when magma cools relatively slowly underground It is usually composed primarily of the minerals quartz, feldspar, and mica When granite is subjected to intense heat and pressure, it changes into a metamorphic rock called gneiss Slate is another common metamorphic rock ,...
ultramafic rocks (1) The term "Wissahickon" was first used by Bascom (1902) in Pennsylvania, then Mathews (1904) in Maryland Southwick and Fisher (1967) recognized five formations within the Wissahickon: Lower Pelitic Schist, Boulder Gneiss, Metaconglomerate, Metagraywacke, ,...
Rock, cantera, mineral, piedra, minería, Carbón , Precio del equipo de trituracion de mineral de cobre de flotacion de cobre beneficio equipos precio mica mineral equipos de molienda en la India , silica, slate, soapstone, Sal planta de equipos exportacin de la Obtener precio; Mica ,...
Proceso De Molienda De Mica molienda de correo electronico mica molienda de correo electronico mica mar 17 blog dedicado a los materiales plásticos características usos fabricación procesos de transformación y reciclado, Saber más molienda plantas de mica planta movil de trituracion y cribado...
Folleto de todo el equipo Sepro - Sepro Mineral Systems CAPACIDAD DE MOLIENDA (t/h) , desbordamiento dentro del circuito de molienda , Concentración primaria de estaño, tántalo, tungsteno, cromo, cobalto , descarga del molino SAG Para operaciones de zarandas de enjuague y Leer Más Servicio En Línea...
Metallurgists & Mineral Processing Engineers For its extensive practical experience, 911 Metallurgist has a clear understanding of what successful mineral processing engineering is and how to go about achieving it Your goal is the production of a material that is marketable and returns you and your investors sustainable revenu...
Apr 24, 2017· Quartz ••• Nastya22/iStock/Getty Imag Gold is most often found in quartz rock When quartz is found in gold bearings areas, it is possible that gold will be found as well Quartz may be found as small stones in river beds or in large seams in hillsid The white color of quartz makes it easy to spot in many environments...
Knowing the fossil record lets a geoscientist place a particular fossiliferous rock layer into the scale of geologic time But the time scale given by fossils is only a relative scale, because it does not give the age of the rock in years, only its age relative to other layers Long after the relative time scale was worked out from fossils, geologists developed methods for finding the absolute ....
Folleto de todo el equipo Sepro - Sepro Mineral Systems CAPACIDAD DE MOLIENDA (t/h) , desbordamiento dentro del circuito de molienda , Concentración primaria de estaño, tántalo, tungsteno, cromo, cobalto , descarga del molino SAG Para operaciones de zarandas de enjuague y Leer Más Servicio En Línea...
Maquinaria De Molienda Superfina Equipos De Trituracin Y Equipos De Molienda Carretera Mal Diseada Cuesta $27 Mill Ms De Lo Pactado Compra En Lnea Harina De Almendra Bobs Red Mill Super Fina 453 G Haz Tus Compras Por Internet, Djate De Cuarzo Maquinaria De La Planta En Mxico Venta De La Bauxita Mica Esquisto Proceso De Toma De Polvo...
Oct 06, 2005· Crystals of several mineral species may be found in metamorphic rocks, depending on the original compositon of the parent rock and the physical and chemical conditions that prevailed during metamorphism Marine muds that had previously hardened to form a sedimentary rock called shale were converted to mica-rich rocks such as slate and schist...
Mica minerals are major rock forming minerals found in gneiss, schist and granite The mica group includes muscovite mica and biotite mica They usually form in layers of sediment on ocean floors Weathering of continental rocks breaks large and small chunks of rock off the larger older rocks...
Mica, which is essentially a compound of silicate minerals, is comprised of closely related materials that have perfect basal cleavage They belong to the category of principal rock-forming minerals and are found in various categories like sedimentary, igneous, and metamorphic...
An Overview of Rocks and Minerals, including igneous, sedimentary, and metamorphic rocks, how they form, what they may be composed of, and what physical prop....
Aventurine is an inexpensive and popular material for making tumbled stones in a rock tumbler If the mica particles are small, a smooth and lustrous finish can be produced Coarse mica particles tend to pluck out, giving the polished stones a pitted appearance Aluminum oxide, cerium oxide, and tin oxide will all produce a bright polish on ....
Our Earth is made mostly of rocks The rocks are composed of mineral grains combined in different ways and having various properti Minerals are naturally occurring chemical compounds in which atoms are arranged in three-dimensional patterns The kind of elements and their arrangements lead to a particular appearance and certain properties for each mineral...
Rock Type: metamorphic Composition: quartz, feldspar, mica Original Rock: granite, gabbro Environment: Gneiss forms at high temperatures and pressur The temperature needed is about 700°C and the pressure needs to be about 12-15 kilo bars, which is at a depth of about 40 km!... | https://www.bioedukacja.org.pl/anual/4219/molienda-mica-rock.html |
Testing the firewall - Introduction
This section of the chapter excerpt will introduce the concepts of auditing and testing firewalls. The introduction will provide an overview of the general concepts behind firewall compliance and focus more specifically on configuring both the operating system and the firewall.
Service provider takeaway: Regulatory and standards compliance can provide several challenges from both a business and a technical perspective. This section of the chapter excerpt from the book The IT Regulatory and Standards Compliance Handbook:: How to Survive Information Systems Audit and Assessments will focus on testing the firewall and dealing with complex compliance requirements.
Download the .pdf of the chapter here.
In this chapter we will introduce the concepts of auditing or testing firewalls.
First we need to define a firewall. A firewall is an application, device, system, or a group of systems that controls the flow of traffic between two networks based on a set of rules, protects systems from external (internet) as well as internal threats, separates a sensitive areas of a private network from less sensitive areas, encrypts internal and external networks that transmit sensitive data (when used as a VPN endpoint), or hides internal network addresses from external networks (network address translator). A firewall picks up where the border router leaves off and makes a much more thorough pass at filtering traffic. Firewalls come in different types, including static packet filters (for example Nortel Accelar router), statefull firewalls (for example Cisco PIX), and proxy firewalls (for example Secure Computing Sidewinder).
Similar to routers, a firewall uses various filtering technologies or methods to ensure security.These methods include packet filtering, statefull inspection, proxy or application gateway, and deep packet inspection. A firewall can use just one of these methods, or it can combine different methods to produce the most appropriate and robust configuration.
A good way to start to test a firewall is to gather information from individuals that have some responsibility for it. These people may be members of the audit team, system administrators, network administrators, members of the policy team, and information security personnel. The idea is to gather and collate each person's perceptions of what the firewall's functionally should be and what it is configured to provide for the network and systems. Obtain any existing firewall documentation and network diagrams to verify the information gathered from the interview. Ideally, the firewall is a control designed to reflect policy. This means that policy must be in place before the firewall is configured. Sadly, this is seldom the case.
After the information detailed above has been collected, the auditor can develop an understanding of the firewall architecture, and determine whether the firewall is configured to correctly segment networks and defend information. The next step is to evaluate the operating system (OS) configuration. This is the configuration of the firewall platform itself. All firewalls have an OS. Do not be fooled by vendor assertions that firewalls have an appliance. A firewall appliance typically will just have an OS that has been hardened. The appliance could in fact be running a scaled down version of Unix or, in some cases, be running a customized OS written by the firewall company, as in the case of the Cisco Adaptive Security Appliance (ASA). Firewalls and routers are all software driven; all they do is make it more difficult to see the code.
Next it is important to ensure that system administration follows best practice: user management, patch updates, change control, and configuration backups. If the firewall is not patched it will eventually be compromised. Just because it is a security device, it is not automatically secure. Finally, it is necessary to validate that the firewall rulebase matches the organizational policy.
Testing the firewall should be coordinated with testing the other components of the organization's defense-in-depth methodology. The organization should not rely only on a single line of defense; if it does, raise a red flag. Firewalls are not the panacea for all security ills. They mainly slow attackers and log activity.
The overall result of the testing or audit of the firewall would be the identification of any security vulnerabilities, as well as an assessment of whether the firewall is fulfilling its function in relation to the security policy of the company. Assess whether the setup, configuration, and operation of the firewall are secured sufficiently to protect the information or services that the firewall is intended to guard, considering the risks that were identified and the likelihood of occurrence. The Center for Internet Security provides benchmarks for several specific brands of firewalls devices. The benchmarks (available at www.cisecurity.org) greatly aid in developing an audit program for firewalls. These benchmarks are the source of our checklist frameworks.
OS Configuration
When auditing the firewall, the auditor must look at the platform or the OS on which the firewall is running. An auditor needs to check on whether the OS on which the fi rewall is installed is stripped to contain only the minimum functionalities or services that are required to provide the functions it runs. The firewall should be an isolated system dedicated to one purpose only, which is filtering traffic based on defined rules. The less complex the installation, the simpler its administration will be. Fewer features equates to less patching and fewer vulnerabilities.
To verify this, commands can be used for determining what services and ports are available to the OS.
Many operating systems have a number of built-in tools that may be used to determine which ports are listening. Some examples are listed here with more in the chapters associated with specific operating systems:
- UNIX : lsof --I, netstat --a, and ps --aef
- Windows the Service Microsoft Management Console (MMC), netstat --a and fport
When first determining the open ports and services, the firewall should be turned off (disabled or running with a policy that allows all traffic). This is done to test only the operating-system-specific ports and services. It is important to do this on a secure network and not connect the firewall to the Internet at this point. Remember, the firewall is a router in this mode.
In addition, the security settings and vulnerabilities of the OS that is installed should be analyzed. Every OS includes a set of security features and vulnerabilities, which varies from vendor to vendor and even between versions. For instance, the default security settings of the OS may not be modified during the installation and such settings may not meet the desired level of security that is consistent with the security policy. Some of the most common security settings that can be evaluated are the access rules, password rules, and logging rules. Other OS/version-specific settings and parameters should also be verified.
Centre for Internet Security also provides benchmarks for several OS. Those benchmarks (available at www.cisecurity.org) can greatly aid in determining whether the OS is configured based on the general industry best practices.
Firewall configuration
After looking at the firewall platform's OS, the next stage involves the validation of the firewall configuration. All firewalls have both a configuration and policy. These should not be confused.
The configuration is the set of base settings associated with the firewall software and installation. Changes to the configuration of the firewall will change its behavior, and, hence, how it processes in accordance with the policy.
Again the auditor must check on whether the firewall sits on an isolated system dedicated to one purpose only, which is filtering packets (and logging, of course). For instance, DNS, e-mail, or server load-balancing functions should not be installed on the same host or be processed by the firewall platform. The sole exception here is that load-balancing the firewall itself is a function of a high-availability firewall and is allowed.
Since the fundamental purpose of the firewall is to manage the flow of information between two networks, the auditor must look at how it serves such a function by looking at the firewall's configuration. We need to verify whether the traffic that the firewall allows to pass through is consistent with the security policy. Testing the rulebase is discussed in the latter part of the chapter, but critical things to look at are that:
- The access rules (authentication, authorization, and accounting) for the firewall are in line with the security policy and best practices
- Access to the firewall system for management and maintenance is provided using an encrypted channel
- Physical access to the device is restricted
- The firewall is configured to hide internal restricted DNS information from external networks
- The external firewall restricts incoming SNMP queries
- The firewall is configured as fail closed
- The firewall hides internal information from external sources
- The firewall is configured to deny all services, unless explicitly allowed
- All security-related patches are applied to the firewall system
- Configuration settings are properly backed up and accessible to authorized personnel only
Figure 11.1 illustrates an example of a firewall's standard policy rules. In this example, the standard policy rules detail the default settings that will be merged with the policy before being installed. Thus, the configuration and the policy when applied together make the rules that are enforced at the firewall.
The IT Regulatory and Standards Compliance Handbook: How to Survive Information Systems Audit and Assessments
Introduction
Working with firewall builder
System administration
Packet flow from all networks
Validated firewalls
Creating your checklist and Summary
About the book
The IT Regulatory and Standards Compliance Handbook: How to Survive Information Systems Audit and Assessments provides detailed methodology of several techincally based and professional IT audit skills that lead to compliance. Purchase the book from Syngress Publishing.
Printed with permission from Syngress, a division of Elsevier. Copyright 2008. "The IT Regulatory and Standards Compliance Handbook: How to Survive Information Systems Audit and Assessments" by Craig S. Wright. For more information about this title and other similar books, please visit www.elsevierdirect.com. | https://searchitchannel.techtarget.com/tip/Testing-the-firewall-Introduction |
Bike Reviews
There’s a lively debate surrounding the use of headphones while riding. Some cyclists say that closing your ears to the sound of traffic is a potential danger, while others maintain that listening to music helps motivate them through a workout. I don’t want to choose sides in that debate, but I do know one thing: listening to music while warming up for a race on my trainer gives me a great sense of peace, helping to focus on the race ahead. These earphones, available in 10 colors, use an over-the-ear design that’s very comfortable, while a foam padding blocks out ambient sounds, letting me focus on my pre-race efforts, and environmental distractions. Sound quality is very good across the whole aural range, and 3.5mm stereo plug means these are compatible with nearly any media device.—Michael Yozell
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New insights on water- and nitrogen uptake in deep roots
As part of her Ph.D.-studies at Dept. of Plant and Environmental Sciences at University of Copenhagen, Guanying Chen has studied deep root and water- and nitrogen uptake during the course of the Deep Root project. She has carried out studies in the root towers facilities, focusing on the crops chicory and winter rapeseed. This led to some interesting and quite surprising findings.
Chicory can use water and nitrogen down to 3,5 meters soil depth. This was one of the main findings in Guanying Chen’s studies which she have just finished as part of the Deep Frontier project. The potential for uptake of water and nitrogen are different, though: The study showed that the deep roots took up nitrogen more rapidly than water. This difference was quite surprising, because nitrogen- and water uptake normally are considered to follow each other closely, but this was not the case in this study:
Compared with nitrogen, root water uptake have a tendency to decrease with increased depth and less roots. Only a part of the available water was taken from deep layers. On the other hand, roots were able to take up nearly all nitrogen available in the deep soil layers.
There can be several reasons for the observed difference in water and nitrogen absorption rates. Firstly, due to several reasons, water uptake was higher in the upper soil layers than in the lower ones, and along a single root, the radial flux was higher in the proximal segments. These radial and axial resistances, may inhibit root water uptake from subsoil but no evidence have been shown for similar inhibition of nitrate uptake. Secondly, the water supply from the topsoil may have been sufficient to supply most of the water demand by the plants with repeated irrigation to the topsoil, while the nitrogen demand by the plants exceeded the topsoil supply, leading them to deplete all available soil layers.
New knowledge - improved models
The new insights can be used to more accurately predict nutrition uptake in deeper soil layers through modelling.
Guanying Chen explains: “We hope our findings can help modellers to better simulate the uptake. The models we have now are not perfect and need to be developed, e.g. we found water and nitrogen uptake below two meters, and almost no models can simulate that, so we hope our findings can help to more precisely calibrate the models.”
Previously, other studies have shown that more than 30 percent of water uptake in crops happens below 1 meters’ depth.
To follow up, Guanying Chen made further studies in which the fertilization level and the water level were changed to see how deep roots were performing under water stress and with different nitrogen applications. These studies were made using winter rapeseed, a crop that has been widely cultivated all over the world. With roots growing more than 2 meter depth, winter rapeseed can efficiently take water and nitrogen from deep soil layers, thus reduce the risk of nitrogen leaching. Compared with high nitrogen application rate, applying less nitrogen was able to enhance deep (> 1.5 m) nitrogen uptake efficiency; while more nitrogen application enhanced deep water uptake as plants grew bigger and required more water. Under drought stress, the total water uptake decreased and more water was taken by deep roots; deep nitrogen uptake was not significantly affected by water status. In conclusion, deep roots play an important role on water and nitrogen uptake when plants were exposed to water and N deficiency. These findings have refreshed the knowledge of deep roots, although they present a small proportion in soil, deep roots contribute a lot to resources uptake, especially under stresses. | https://projects.au.dk/deepfrontier/news-archive/show/artikel/new-insights-on-water-and-nitrogen-uptake-in-deep-roots |
- Spring 2020 BFA Exhibition
This is the online Spring 2020 BFA Exhibition for the Department of Art, Design, and Visual Studies at Boise State University in Boise, Idaho. The sixteen artists featured in the exhibition are candidates for the Bachelor of Fine Arts degree in Art Education, Illustration, and Visual Art. The exhibition would have been held at the Blue Galleries located in the Center for the Visual Arts, but due to Covid 19, our student exhibitions were moved online.
The virtual show was completed as part of the Art 490 BFA Exhibition course with Gallery Director and Lecturer Kirsten Furlong. Please feel free to contact Kirsten with any questions about the class, exhibition, the Blue Galleries or our department at [email protected].
The Blue Galleries are closed through summer 2020. Follow us on Facebook, Instagram, or via our website for information on our reopening.
The Department of Art, Design, and Visual Studies at Boise State University is a vibrant center of excellence in art and design that educates and prepares students to be innovative, creative problem solvers, critical thinkers and globally-engaged citizens. Working within diverse cultural and visual contexts, we value inclusivity and experimentation, encourage risk-taking, and through our research and practice advance knowledge in our field. The faculty embraces traditional and contemporary approaches toward materials, forms, techniques, and ideas, while always seeking the refinements, innovations, and inversions that allow art to convey meaning and agency. We share with our students the physical and intellectual experience of art and craft, and assist them with acquiring the skills and knowledge to practice as artists, art teachers, art theorists and historians, designers, and illustrators. Learn more about our programs and facilities in this video. | https://spring2020bfaexhibition.com/contact.html |
Sri Lanka’s northern seas are considered to have the highest density of crustaceans, which automatically attracts a larger school of fish. Given that these international waters are required to be shared with India and the Maldives, many of the fishermen in the region are in a constant battle to secure their catch for the day.
During a recent visit to the northern peninsula, The Sunday Morning met up with several fishing communities along the northern coast to inquire about how they are managing their trade, especially with international competition, to secure the biggest possible catch each day.
We were told that at least 60% of the population is dependent on fisheries, with the rest of them either opting to work in state or private institutions.
The geoposition of the peninsula allows the fishermen to exploit the various resources of aquaculture to add to their line of products they offer consumers.
“The fishermen here can choose to specialise in lagoon breeds, freshwater breeds, and coast and offshore breeds of aquaculture. The biggest threat we face is when the multiday trawlers from India cross into these international waters. If we have just three trawlers, they would arrive with 50 trawlers, and it is difficult for us to compete with their numbers and the illegal techniques they use to catch fish,” Amal from Mullaitivu explained.
Amal pointed out that most often the fishermen from India employ the bottom trawling method, which in the long run will not be sustainable for marine aquaculture as it easily destroys mariculture that is setup.
Furthermore, the Sri Lankan fishermen get outnumbered in international waters and also can’t compete against the various techniques used by all the multiday trawlers from India.
The struggle to secure their catch is partly due to the fact that fishermen in northern Sri Lanka still employ conventional fishing methods, while their counterparts all over the world use state-of-the-art techniques in this regard.
Knowledge expansion lacking
“The majority of the populations in the Mannar, Jaffna, and Mullaitivu Districts depend on fisheries. The lack of investments towards upgrading fishing techniques and knowledge expansion within the fishing communities is further challenged due to foreign fishermen. Their boats are equipped with the latest tracking and fish harvesting devices and when we have to share the catch with them in international waters, we are always at a loss,” Noel Peiris from Talaimannar told The Sunday Morning.
Peiris’s request is that the Government and other private entities put enough focus on uplifting the livelihoods of the fishing community, as there is a higher density of fishing products that can be harvested from the northern seas.
He also said that the Government should look at providing relief or concessions to the fishing communities, enabling them to upgrade their trade tools so they may be able to improve their market share.
Upgrading vessels and techniques
When contacted to inquire about the steps the Government of Sri Lanka (GoSL) is taking to improve the fisheries sector, Department of Fisheries Director General G.P.J. Kumara replied that last Wednesday (29), a special meeting had been called by the Presidential Task Force appointed to oversee the issues of the fisheries sector, where they had discussed matters at length.
“The fisheries issue in the northern seas has been long standing, and both the GoSL and the Indian Government are constantly discussing how both countries could fish in the shared space. We cannot blame only the multiday trawlers belonging to the Indian fishermen, as even our fishermen at times cross international waters and enter Indian territorial waters. So, fishermen in both countries are at fault,” Kumara elaborated.
However, in order to ensure that Sri Lankan fishermen stick to international waters when offshore fishing, the Department of Fisheries employs the Sri Lanka Navy to monitor them.
The presence of the naval surveillance teams somewhat keeps the situation under control as they are patrolling international waters from time to time, and are also able to apprehend any trawlers involved in illegal poaching or that use bottom trawling techniques.
At present, Sri Lanka has apprehended close to 150 such fishing vessels and has taken legal action against the fishermen and confiscated their fishing equipment in accordance with Sri Lankan law.
“If we have stronger laws and regulate any type of illegal fishing techniques that are currently in use, we should be able to have sufficient catch as well as streamline the methods of fishing employed by the Sri Lankan fishermen as well,” Kumara added.
The Navy conducts regular operations for the prevention of illegal fishing practices that adversely impact legitimate commercial fishers as well as marine ecosystems.
During 25-28 July, the Sri Lanka Navy managed to nab 15 persons engaged in illegal fishing and also seized the fishing gear belonging to them; the Navy took four unauthorised fishing nets, four dinghies, and fishing gear into custody, apart from the 15 accused.
The suspects together with the seized items were handed over to the Assistant Directorate of Fisheries in Mullaitivu, Fisheries Inspector of Kuchchaveli, and Assistant Directorate of Fisheries, Trincomalee through the Sri Lanka Coast Guard Regional Director of the Eastern Region for further investigation.
Kumara also affirmed that there is an issue with the vessels the Sri Lankan fishermen use at present, and said that in comparison to the vessels used by Indian fishermen, the trawlers we have are much smaller.
That provides them an advantage to gather more fish stocks than Sri Lankan fishermen.
Co-operation between civilians and Navy needed
Mannar Fishermen’s Association President Y. Baskaran, speaking to The Sunday Morning, revealed: “We have also lost a lot of fishing grounds due to the naval command that is set up in Mullikulum and Silavaturai, which hold the highest density of fry fish that attract all the commercial fish closer to the coastline.
“Our fishermen have to now venture into the deep sea using multiday trawlers, which is when we encounter the Indian fishing boats. We are discussing with the Sri Lanka Navy to allow fishermen to continue fishing activities within their naval command area under their surveillance.
“We hope that once the new parliament is established, there will be a considerable amount of co-operation between civilians and the Navy.”
Baskaran noted that recently, the Navy had allowed a handful of fishing families to resettle in Mullikulum, and they will be able to go to sea in the area in the coming weeks. | https://www.themorning.lk/northern-fishermen-caught-in-a-net-of-issues/ |
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It is hoped that the Gender Assessment of National Law-Making Mechanisms and Processes in selected Southeast Asian Countries: A CEDAW Perspective will serve Members of Parliament (MPs), legislators, parliamentary staff, women’s advocates and relevant stakeholders as a vital reference tool.
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This publication is intended to share an analytical framework for investigating plural legal systems from the gender perspective. It focuses on the broad spectrum of the legal orders, including those that are informal, not formally recognized, or not State sanctioned covering customary, indigenous, traditional and religious orders.
Strengthening the Capacity of the Thai Judiciary to Protect Survivors of Domestic Violence:Justice for Society’s “Little Dots”
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This study concluded that the promotion of gender equality and women’s access to justice required not only the enactment of new laws that were compliant with international standards such as the Convention on the Elimination of All Forms of Discrimination against Women (CEDAW) but that these laws should be implemented by a gender-sensitive administration of justice. This indeed is one of the State obligations under CEDAW which calls on states to “take all appropriate measures,...
Terminal report for Valuing the Social Cost of Migration across Four Countries in Asia: An Exploratory Study
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The aim of this study is to contribute to a better understanding of the political, economic and social dimensions and costs of migration and consequences on the family, community and society. The study analyzes the social outcomes of migration and identifies strategies in response to the issues of families left behind with further examination of the government policy framework on migration in Indonesia, Thailand and the Philippines to justify the deployment of hundreds of thousands of women and men to work abroad. The study also inquires about the effectiveness of government programmes in addressing issues related to migrants and migrant families. Lastly, the study looks into the following issues of displacement, adjustment and adaptation faced by individuals within the family in response to the problems brought about by the migration of a family member.
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UN Women has developed this publication to bring the key challenges faced by women migrant workers in the low wage sectors of the informal economy within the ASEAN. It is expected that the study will help key labour sending and receiving countries with the ASEAN in developing and implementing national and regional policies that will empower and legally protect women migrant workers. | https://asiapacific.unwomen.org/en/digital-library/publications?country=df874a4d2b154980aefbfade31f3616d&f%5B0%5D=country_publications%3A1710&f%5B1%5D=published_by%3A2215&f%5B2%5D=resource_type_publications%3A2108&f%5B3%5D=resource_type_publications%3A2117 |
When you send us updates, you become part of a UK-wide wildlife monitoring and conservation project. Although we aren't currently processing the data we collect, we may use it in future to improve understanding of the most effective ways to have a tangible positive impact on wildlife across the UK.
All of our Naturehood actions are based on scientific evidence, providing you with the best information we have to support wildlife. It's predicted that doing this as a community will make an even bigger difference by providing a large, well-stocked area to support thriving wildlife populations rather than just a few individuals.
When you add, replace or remove a wildlife-friendly feature you can complete an Action Log to keep us up to date. These take about 1 minute, and contribute directly to our research.
If you haven’t completed the Me & My Naturespace survey yet, please do so first! | https://naturehood.uk/survey-your-space |
In continuation of its determination to ensure job opportunities for Liberians, the President George Manneh Weah’s administration, on Thursday, 09 September, 2021, finalized terms for an amendment to ArcelorMittal Mineral Development Agreement (MDA).
“The feat was achieved following two years of intense and difficult negotiations,” statement made available to The SUMMIT by the Deputy information Minister for Press and Public Affairs, Jawarlah A. Tonpo said.
“The Liberian Government and ArcelorMittal will sign the 3rd Amendment to the MDA on Friday September 9, 2021.
“The 3rd Amendment was required to ensure completion of ArcelorMittal Phase 2 project. The Government has secured this $800 million expansion project to build a mine concentrate, expand capacity of the Rail and Port from the current production of 5 MTPA to 15MTPA within 3 years and a reservation to increase production to 30MTPA over the long term.”
The statement further explained that the Government also secured a total $65 million payment from Arcelormittal which includes $55million USD reservation fee for further of expansion rights for the project to 30MTPA and $10 million signing bonus.
“The Government has also secured additional contribution of $300k to $500k annually for projects to be implemented directly by affected communities in Bong, Grand Bassa, and Nimba.
The ArcelorMittal expansion project will triple Iron ore production in Liberia with a corresponding increase in Government revenue and job creation. This $800 million Phase Two project will result in direct spending into the Liberian economy of approximately $200 million per year. The project will create approximately 1,200 direct employment and 400 indirect employment for Liberians. During Construction, there will be approximately 1,500 jobs created for Liberians.
The Amended MDA has improved the local content provisions of the agreement to require that within 2 years, AML and the Government to development a framework to ensure Liberian owned SMEs can participate in the expansion project through the purchase of goods and services from Liberian owned SMEs.
The Government ensure the possibility of a multiuser railroad and port infrastructure to allow Guinean Mining Companies to access Liberia’s transportation infrastructure. The Amended MDA allows for expansion of the Railroad and Port to potentially allow an initial 30MTPA of Iron ore from Guinea to transit through Liberia, with Liberia receiving transit fees and greater employment for Liberians.
Critically important, the Government has finally secured terms creating a framework for multiuse of the existing Rail and Port asset for transport of Guinea ore through Liberia. Following difficult discussions, the amended MDA now provides a clear framework in which the Government can enter into access agreements with Mining Companies for transport of Iron ore through Liberia. | https://summitpostnews.com/2021/09/09/liberian-govt-strikes-huge-deal-more-citizens-to-get-job/ |
This month, in a special Lawyer Monthly feature we decided to take a look into the surprising reports around the gender pay gap. Even though most suspected a level of disparity, no one had really quite predicted the sheer gap between both genders and how there is still such a way to go until women are given the same opportunities as men.
The legal industry, of course, was not exempt from this. So, what is next? We know that it is a widespread problem, but what are law firms doing about the issue?
We spoke with two of the top 100 UK law firms on this report and the steps they are taking to tackling the gender gap problem.
The Associate Justice of the Supreme Court of the United US once stated: “Women belong in all places where decisions are being made… It shouldn’t be that women are the exception”; such a quote remains true in all areas, even the modern workplace.
Earlier this year, companies in Britain had to report details regarding the pay gap between genders and unsurprising for some, yet a shock to others, the results presented quite a disparity between men and women.
The figures had revealed that if you are a woman working at a large UK firm, you are on average more likely to be paid less than men, with 78% of companies paying men more. A contributing factor towards this is due to the fact men tend to be hired for higher paid roles and are paid higher bonuses than their female counterparts.
The legal industry also portrayed such bias and criticism. When reports began to sift through, the results did not seem so bad. The average gender pay gap for the 25 largest UK firms (by revenue) was at 20%. The firm’s median pay gaps were higher than the mean pay gaps; some of the highest mean gender pay gaps were: Lathan & Watkins at 39.1%, Wei Gotshal at 38.1% and Kirkland & Ellis at 33.2%. The lowest mean gap included Forsters at 2.3% and Irwin Mitchel at 12.8%.
These reports sparked controversy, however, as many law firms were criticised for excluding partners from their pay gap reporting; with the Equality Act 2010 Regulations stating that firms need to release the statistics of employee’s pay, many big firms excluded partners with their reason being due to the fact that equity partners are not ‘salaried employees’. Many expressed this was unfair as it kept secret the most senior, well-paid and mostly-male lawyers.
A level of transparency was needed and some top firms, such as Clifford Chance knew this. Laura King, the firm’s global Head of People and Talent stated: “Unless you’re transparent you don’t really understand what’s going on in your organisation. Excluding partners, we felt would not give us the opportunity to examine ourselves.”
This transparency had shown that Clifford Chance had reported a mean pay gap of 66.3%; other firms inclusive of partners, Freshfields Bruckhaus Deringer, reported a mean pay gap of 60.4%; Linklaters posted 60.3%; and, Slaughter and May reported 61.8%.
From this, some law firms buckled under the pressure and showcased the statistics of all employees.
WHAT CAUSES THE GENDER PAY GAP IN LAW?
What are the reasons behind this disparity and what can be done? We hear from two of the UK’s Top 100 law firms: Keystone Law and Holman Fenwick Willan (HFW)* who speak on what they are doing to tackle the gender pay gap.
Problem: Male Domination
There are several reasons up for debate here. One of the main reasons behind women being paid, on average, less than men, is the fact that there are more male partners than female.
From a study conducted by the SRA in August 2017, they found that women make up 48% of all lawyers in law firms (and 47% of the UK workforce). Seniority made more a difference, however, with women making up 59% of non-partner solicitors and the larger law firms (with 50 plus partners), showing women make up 29% of partners.
Even Slaughter & May had highlighted that their gap is widened significantly due to the all-female secretarial positions.
Solution:
A spokesperson at Keystone Law, one of the UK’s Top 100 law firms, stated: “Many firms claim to have a diversity plan in place but invariably it’s more talk than action. What really makes an impact is truly recognising talent and showcasing ability.”
Keystone Law is not a traditional law firm, having chosen to go against the grain by eradicating the partnership model, in favour of a flat structure for all fee earners who have almost all been partners at some of the country’s leading law firms. All Keystone lawyers come on board at the same level and hold the same title. Of the lawyers Keystone currently engages 40% of them, are female. This stands in stark contrast to 18% of partners at Top 10 firms, as cited by PwC’s Law Firm Survey 2017.
HFW said: “Ensuring higher levels of female representation at the highest levels of the industry, particularly among partnerships, has been high on the law firm agenda for years, but we need action, not words. More firms should set defined targets, as we have, and be more proactive.”
Steps they have already taken (as part of our gender equality action plan) include:
- Setting an ambitious target of having 30% female fixed-share partners by 2020
- Raising awareness of unconscious bias and the need for creating an inclusive environment;
- Setting defined performance and remuneration criteria to ensure that individual compensation is not susceptible to conscious or unconscious gender bias;
- Providing specialist training to help female lawyers return from maternity leave, which has been extremely well received;
- Providing training to all lawyers, such as networking and presentation skills training, which has been particularly popular with women;
- Appointing Diversity and Inclusion Champions in each of our 18 international offices that are tasked with raising awareness, supporting initiatives and calling out bias.
They expanded: “The percentage of female fixed-share partners at the firm has increased from 12% in 2015 to 26% today, while women account for 50% of our new Legal Director roles. Over 40% of the firm’s internal partner and legal director promotions over the last two years were women.”
At the lower end of the pay scale, there are significant issues of gender stereotyping. The vast majority of secretarial services staff – not just in the legal industry, but more generally – are women, for example.
Commenting on the matter, HFW said: “This is harder to resolve. A law firm would currently find it challenging to achieve gender equality in that role due to a lack of male applicants. It is possible that law firms could visit schools to provide students with a broader outlook on future careers and we can adapt how we speak about secretarial roles so that they are attractive to men as well as women. This will be a longer-term challenge.”
Problem: Not enough law students?
UK law student figures show that 67% of applicants in 2016/2017 were female. We asked HFW: Will this ultimately lead to a redressing of the balance, or do you think law schools have a continued responsibility to enrol more females?
“This is not a new phenomenon – there have been high levels of female lawyers coming into the profession for a relatively long time. It is important that law schools maintain a strong pipeline of male and female graduates, but this alone will not redress the balance.”
Solution:
Going back to problem number one, HFW suggested: “Law firms must take proactive steps to encourage and enable more female lawyers to step into more senior roles.”
Problem: Women have a parental responsibility
The imbalance of men and women in the legal sector is often said to be due to parental responsibilities, which traditionally lie with women. We ponder on the thought that law firms need to offer solutions that will help re-dress the balance and thus the perception of the sector, (i.e. flexible working hours, allowing male staff to take on more parental responsibilities and part-time working for male staff).
“Because of their perceived reliability and stability, husbands and fathers in the workplace are often more likely to be given pay increases and promotions. Unfortunately, the opposite still seems to be true for women with children, whether they are married or not! The sector therefore needs to concentrate its efforts on removing any stigma that is attached to mothers as well as fathers who require flexibility and agility in order to look after their families”, replied Keystone.
Solution:
With Keystone’s business model allowing lawyers to work on their own terms (including earnings), they state: “Many of the reasons women, in particular, cite for being driven out of the legal profession are absent from the Keystone model. There are no set billing targets, lawyers can work from any location that they choose and work the hours that best suit them. This works towards re-dressing the balance of parental responsibilities as both men and women can tailor their practice to suit them – without worrying about being penalised for requiring flexibility.”
“Interestingly the shift towards the desire for greater flexibility is increasingly stemming from millennials. According to a recent Deloitte report, up and coming lawyers are increasingly being inspired by the growing gig economy, realising that earning well doesn’t need to involve being chained to a desk during fixed hours. Not only that, these young lawyers are rebelling to demand a better balance — and influencing those more senior to them in the process!”
HFW have a similar solution at hand: “We offer flexible working hours to all fee-earners, male and female, and also offer extremely generous shared parental leave to male employees at a rate well above the statutory requirement. But the industry could do more, such as lobbying government to come up with practical and pragmatic solutions that will encourage a societal change to parental responsibilities. We are keen to work with others in the sector to progress this.”
Problem: Do clients prefer male lawyers?
It has been suggested that this is a client level problem, being that clients may prefer male representation. Is this true?
HFW commented: “We disagree. Gender diversity, and diversity more generally, is an issue that clients now take incredibly seriously. Clients are increasingly asking law firms for diversity and inclusion data as part of the pitch process.”
Solution:
If you don’t have a diverse team, the chances are that you won’t win the work. The problem is with the law firms, and we can achieve more by working together with our clients on this issue.
Problem: Are there enough female law students?
Short answer: Yes. UK law student figures show that 67% of applicants in 2016/2017 were female. Could this ultimately lead to a redressing of the balance, or do law schools have a continued responsibility to enrol more females?
HFW said: “It is important that law schools maintain a strong pipeline of male and female graduates, but this alone will not redress the balance.
Solution:
“Law firms must take proactive steps to encourage and enable more female lawyers to step into more senior roles.”
What Can Law Firms Do?
Since the reports, there has been a surge of motivation to ensure the gap is narrowed. Whether is it the attitudes towards women in the legal workplace needing to change, or the traditional working models for our contemporary lawyers, there will be a strong push towards equality, which will, without doubt, change the legal sphere in the future. | https://www.lawyer-monthly.com/2018/08/gender-pay-gap-why-is-the-legal-sector-failing/ |
Diana Eng is the director of Culture’s Bioprocess Alliance Management (BAM) team where she leverages her experience in fermentation and technology transfer to support Culture’s clients. We sat down with Diana to learn about the BAM team.
Mattie: What brought you into the world of bioprocessing?
Diana: I first entered the field as a research associate at Amyris. It was a small startup at the time and I was in the biology group where I supported strain engineering and strain screening. Because it was a small startup company, the different groups helped each other out when needed. We did pretty much everything ourselves, from making our own media and setting up and taking down bioreactors to running analytical samples on the HPLC and everything in-between. It was exciting! I had so many bright colleagues at Amyris who mentored us more junior scientists.
There was a close collaboration between Amyris’ strain engineering group and fermentation scientists which is what got me interested in fermentation – I was fascinated by it! Eventually one of the directors of the fermentation team invited me to join their group. On the fermentation team, I was a part of process development and scale-up and supported tech transfers to CMOs. I traveled to many CMOs around the country which was a great learning experience since I was involved in preparing the process documentation needed for tech transfers. I learned what data is essential to provide the CMO and what other information is helpful to share so they can successfully execute the work.
After 10 years at Amyris I made the move to Bolt Threads, another biotech startup. At a small company, there's a lot of room to influence and build out different team roles and projects, and to have more ownership and independence to lead tech transfer projects. At Bolt I did more tech transfers with more CMOs, scaled-up processes from bench scale to the 3000-liter level and higher, and I had more opportunities to lead those efforts. I participated in early CMO conversations to prepare acceptance criteria, discuss process phases, and review equipment and technical feasibility. Over my six years at Bolt, I went from being a fermentation scientist to the Associate Director of Fermentation. There was a lot of learning during those years!
Mattie: What brought you to Culture Biosciences?
Diana: I had been following Culture for a number of years and I was so intrigued with their concept! I see Culture’s model as the modern-day solution for biomanufacturing – to be able to bring together fermentation, bioprocessing, software, and hardware to create this platform where clients can design their experiments and monitor everything as it proceeds without having to be in the lab.
So, I was very excited when Culture reached out to me about heading up their Bioprocess Alliance Management team – affectionately known as the BAM team. They wanted someone who had worked on the customer side of the table and could build an effective team that would anticipate and meet client needs. I knew that my understanding of fermentation, the bioprocessing workflow, and what it takes to have a successful relationship with a CMO would make me relatable to Culture’s clients.
Mattie: Tell me about the BAM team and what it does.
Diana: The BAM team is at the center of the Culture–Client relationship. Each client has one BAM Lead for all of their projects. We’re here to support our clients as well as our in-house operations team which includes BD, operations, software, and hardware teams. We work closely with our clients from tech transfer through the investigational and data analysis stages and beyond. This level of collaboration is important to the success of the client’s work.
During tech transfer, we work with the client to understand their process, criteria, decision trees and action plan, red flags, and so on. Our tech transfer process and acceptance criteria framework ensure that client processes successfully scale down into Culture’s reactors before starting a guaranteed capacity contract. We also make sure the client understands our platform, how their technology will be implemented at Culture, and how they will have cloud-based access to their investigations and data.
Throughout the bioprocessing phase, we check in regularly with the client. I know from experience that radio silence is not good. Our goal is ultimately to facilitate communications through Culture’s Cloud Console where runs are entirely planned, scheduled, designed, and updated through our software interface. While we build that, we need to ensure seamless communication and have been integrating with other messaging platforms such as Microsoft Teams and Slack.
During the data analysis phase, we work with the client to make sure they are getting the type of data they need, ensure any process tweaks are compatible with our technology, and discuss other ideas they might have to execute the next experiment.
Mattie: What does a typical day look like for you?
Diana: Every day I check in with my team to see if and where I can help them. The BAM team also meets weekly to review all client projects and see how they are going. Some may need a bit more attention one week, a technical lead’s input may be needed, or brainstorming a solution might be helpful.
Depending on what phase my client projects are in, I may work on technology transfers, respond to calls or requests, meet with clients, work with the bioprocess engineering team to help resolve technical and operational issues, collaborate with the product and software team to develop features that are important for clients, and various other tasks. I check on the status of each of my client projects to keep track of action items that are in the hands of Culture scientists or the client. I also review each project’s Console to see what is happening and review any observations posted for the run. We are always striving to be as proactive and ahead of schedule as circumstances allow.
Client feedback is important to us, so I may have a client feedback session to see if there are other tools or features that we can provide, or if there is any aspect of the partnership that could be improved. The last thing we want is to find out six months too late that a client was unhappy about something that could have been resolved within a week!
And I think that's why I'm so excited to be in this role, trying to bring best practices to Culture and our clients. We really want to be an extension of the client’s lab or, if they don't have a lab, to be their lab.
Mattie: You’ve mentioned the Console a few times. Can you tell us about Culture’s Cloud Console and what it enables scientists and their teams to do?
Diana: Our Cloud Console is like a LIMS with a client-facing component. The Console gathers and organizes the experimental data for a client’s project, and the client can access it remotely as if it was in their own lab. It also captures the observations and notes of the experiment so the client can get further insights into their project.
Clients are able to interrogate their data in different ways right on the Console – they don’t have to wait for the batch record and then use Excel to evaluate it. This is important for our clients because each one has specific needs for data evaluation. Our software team is continuously adding functionality to the Console in response to client needs. The BAM team plays an important part in relaying client-specific needs to the Console team.
Mattie: How else is Culture making bioprocessing easier and faster for clients?
Diana: We work with a wide variety of clients. Some are small startups and others are large established companies. The startups usually have a scientist who designs their process and knows what is needed, but they don’t have the time or money to build out a fermentation team, set up an in-house bioreactor lab, etc. I've been there, and I know that process is really slow. So Culture makes it easy for them to get started on their experimental work sooner rather than later, and without huge capital investments.
And then we have other clients that are more established. They have their own lab and have a lot of strains they want to screen or development work across different projects, but they don't have the capacity or space to expand. Culture enables those clients to get more data faster and helps them continue to grow in scale.
Mattie: You’ve been on the client’s side of the table. As someone who could have been a client of Culture’s in your previous role as a fermentation scientist, how could Culture have allowed you to work differently? What could it have helped you achieve?
Diana: When I was doing fermentation, I would have loved to focus on designing and analyzing experiments rather than managing a bioreactor lab. That would have allowed me to make faster progress because the overall process would have been more efficient.
It takes time and money to build out a fermentation operation—to hire the right people for specific roles, then get the right equipment and commission it—I know there are many inefficiencies with that process. To have had Culture available and be able to say, “I want these runs and data next month” and get them would have been amazing!
This is just what biotech companies get when working with Culture. A great example of this is our client Sestina Bio where Mona Mirsiaghi, Sestina’s Director of Fermentation, works directly with Culture to run experiments instead of running an in-house bioreactor lab. She manages her experiments through Culture’s Cloud Console in much less time than would be required by an in-house lab.
Mattie: What does the day-to-day work look like for a scientist who works with Culture in contrast to that of a scientist who runs all their experiments internally?
Diana: Culture’s clients are able to focus their energies on experimental design and data analysis. They don’t have to operate their own bioreactors or even hire and manage an in-house team of operators. This doesn’t mean they have no input during their runs – they just use our Cloud Console to monitor their runs, make process adjustments, and evaluate their data. And they can do all of this from anywhere, so they have more flexibility in their schedule.
Also, when a client is ready to expand, they don’t have to worry about building new lab space, installing new bioreactors, etc. They just reach out to Culture to get their experiments up and running.
Mattie: What do you want your clients to expect when working with you?
Diana: An important thing for them to embrace is that this is a partnership in which both parties have ownership and responsibilities. Culture is here to help facilitate our clients’ vision and goals. I’d like them to think of Culture as their own bioreactor lab, but a better alternative: we run the lab for them and conduct the fermentation experiments so they don’t have to maintain banks of reactors or manage in-house teams.
Given that our clients are designing the experiments, we need them to be very plugged into the process. They do so mainly through Culture’s Cloud Console. Plus, each client has one BAM Lead who is their primary point of contact throughout the process, and a Technical Project Lead who works with them to translate their experimental design into our own records and then supports experiment execution. Through the Console, BAM Lead, and Technical Project Lead, our clients have the same level of awareness and control over their experiments that they would have with an in-house bioreactor lab – it’s just less labor-intensive for them.
Mattie: What does your ideal client look like?
Diana: I would say the ideal client, regardless of their size, is one that has a scientist or director who is very knowledgeable about bioprocessing and knows exactly what they want. That really helps with the collaborative nature of our work and enables them to complete their work without having to build or expand a lab. And after establishing a partnership with Culture, the company can continue to grow down the road and increase its scale and capacity.
Mattie: Based on conversations with our clients, what are the biggest challenges in biomanufacturing right now?
Diana: We recently compiled and reviewed our client feedback and one of the biggest challenges clients listed was having the capacity to do everything they want to do. They want to do more assays and investigations, work with different organisms and products, and look at different measurements in multiple ways. There are a lot of biotech companies out there that face the same challenge. This is the challenge that Culture is uniquely positioned to meet.
Our clients also reported wanting a software platform that will support all of the “more” they want to do. They see the value in our client Console, particularly the live run monitoring, and appreciate the data transparency and having all of their data at their fingertips. That makes them want even more flexibility for data analysis, plotting, interpretation, and reporting – all in real time and with the utmost security. These client needs are what drive Culture to continuously improve our cloud-based customer platform.
Mattie: What do you enjoy most about your work on the BAM team?
Diana: I enjoy interacting with clients and working with them to find the solutions they need. A lot of this is thanks to our platform which solves so many inefficiencies that I've seen in the industry. To hear a client say “Culture helped us move from here to here in such a short time” is rewarding!
It’s also wonderful that I’m able to focus on fermentation and connect with so many people in the industry, including peers that I worked with that are now at other companies.
Mattie: What are you most excited about for this year?
Diana: I'm excited to see the growth in our Cloud Console and the development work that we have planned for it because that's our main software and everything runs through it. There’s a lot to do, and we have a bunch of new hires in place, so I'm looking forward to seeing those capabilities get built out in the next year.
It’s also exciting to know that as word spreads of our clients’ experiences with Culture, more biotech companies will be open to investigating the concept. Once they do, they’ll learn that by working with us, they won’t be giving up control over their work – they’re actually getting the best possible execution of their own experiment designs. | https://blog.culturebiosciences.com/bam-team-partners-with-clients-from-scale-down-to-scale-up |
Experts at the Ramsar Convention, an international treaty for the protection of wetlands, identified “significant changes” due to human interference in the ecological characteristics of the area. They recommended, among other things, that Colombia enroll the wetlands in the Montreux Record, a register of seriously threatened wetlands requiring immediate attention.
Bogotá, Colombia—Following a visit to the Ciénaga Grande de Santa Marta wetlands in August of last year, a mission of international experts from the Ramsar Convention, an inter-governmental treaty for wetland protection, released a report recommending that the Colombian government include the area in the Montreux Record—a register of gravely threatened wetlands requiring immediate attention.
“Given the significant changes in the ecological characteristics of the Ciénaga Grande wetlands, we recommend including it in the Montreux Record,” said the report issued last week. These changes “require urgent action by the government of Colombia to maintain and restore the area’s ecological character, and to protect it in accordance with the objectives of the Convention,” the report said.
Among changes mentioned in the report are overexploitation and contamination of the wetlands’ waters, diminished fresh water due to increased sedimentation and obstruction of waterways, “huge loss” of mangrove forests caused by road and infrastructure projects that block water flow, and declining fish populations.
“Including Ciénaga Grande in the Montreux Record would allow the Ramsar Wetland Conservation Fund to provide economic assistance through grants. It would also allow Ramsar scientists to provide expert advice and recommendations on best practices for the recovery and conservation of the ecosystem,” explained Juan Pablo Sarmiento Erazo, a researcher from the Universidad del Norte.
In addition, the Ramsar report recommends two other solutions to the wetlands’ rapidly degrading condition: performing effective dredging based on new plans for water management and strengthening coordination among institutions that manage the site.
“The key is that the Colombian government should follow the Ramsar recommendations to the letter, implement improvements as soon as possible, and make necessary changes in the site’s management,” said Gladys Martínez, an attorney with AIDA. “The Montreux Record is far from being a blacklist. It’s an opportunity for governments to demonstrate responsible management of natural resources that demand urgent attention.”
Ramsar experts visited the site from Aug. 22–26, 2016, following a 2014 petition filed with the Ramsar Secretariat by AIDA, el Universidad del Norte, and the University of Florida. Scientist Sandra Vilardy at Universidad del Magdalena also contributed.
“We hope the government will make the report official,” Vilardy said. “The document mentions that it is imperative to re-establish aquatic balance in the wetlands, emphasizing the role that rivers play in feeding Ciénaga Grande.”
More information on Ciénaga Grande de Santa Marta is available here. | https://aida-americas.org/en/press/ramsar-secretariat-advises-colombia-add-cienaga-grande-to-list-of-world-s-most-threatened-wetlands |
1. Field of the Invention
This invention relates to a wind directing apparatus for directing air into the interior of a structure, such as a marine craft through an opening such as a hatch on the deck of the marine craft.
2. Description of the Prior Art
In boats, including sailing boats, power boats, and like marine craft, the interior thereof is frequently not air conditioned. Accordingly, ventilation of the interior of various types of boats which include living space below deck is frequently a problem, especially when the marine craft is not moving. In order to overcome such a problem, developments in the prior art include various types of ventilator or wind deflecting or directing assemblies which are specifically structured for use on a marine craft and which are intended to "capture" or more specifically, direct air from the exterior of the craft down through an opening, such as an open hatch, port, or the like, into the interior of the craft. Such prior art devices are evidenced in the following U.S. Patents. Bliemeister discloses a wind deflecting ventilator comprising an open frame attached to a fabric scoop like receptacle for turning the direction of moving air down an open hatch on a boat. A plurality of cords are attached to a supporting frame for securement to an open hatch or like structure. This device, however, is primarily used to direct wind from a single direction rather than be efficiently operable to direct air into the interior of the structure regardless of the direction of the breeze or wind.
Vail discloses a swiveling wind scoop for ventilating the enclosed interior area of a boat through a hatch opening and comprising a flexible sail for continuously directing air flow regardless of the direction of travel of the boat. A mast for rotatably supporting the sail above the hatch opening is fastened above the opening. A supporting or mounting base is strapped or otherwise secured about the open hatch for bracing and support of the structure.
The references to Beck, U.S. Pat. No. 3,741,100; Fuerst, U.S. Pat. No. 4,050,363; Knight, U.S. Pat. No. 3,013,483; and Hunt, U.S. Pat. No. 1,115,315 are all directed to a scoop or air directing type of structure which has a generally fixed or rigid structural configuration and which is designed to primarily direct air coming from a single direction into the interior of the boat or like structure on which it is mounted by means of a hole leading to such interior.
Other prior art devices are represented in the following U.S. Patents which are directed to the same problem as set forth above but which are not specifically directed for use with a marine craft or the like. None of these structures are specifically adaptable to be selectively and easily positionable in both an operative and stored position since the majority of such structures are made from a somewhat rigid material and are intended to provide a fixed mounting over some type of ventilating opening into the interior of various types of structures. Such patents include Murray, U.S. Pat. No. 4,730,552; Comte, U.S. Pat. No. 4,241,645; Burns, U.S. Pat. No. 4,111,106; McIntosh, U.S. Pat. No. 4,535,715; and Jalbert, U.S. Pat. No. 3,757,664.
While the structures disclosed above are generally representative of prior art attempts to solve the above-noted problem, none are specifically capable of efficiently providing a stable structure capable of directing a breeze or air coming from any direction into the interior of a boat through an open hatch while providing no moving parts and which is effectively self-supporting and also adaptable for the ventilating of other types of structures other than boats.
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Transcosmos has released Transpeech2.0, the upgraded version of its speech recognition solution designed for the contact center industry. Transpeech 2.0 adds Quality Control Platform, AI Defender, Emotion Analysis, and Dialogue Summary features.
The Quality Control Platform seeks to improve call quality and contact center efficiency. With it, supervisors can monitor, evaluate, and visualize agents' performance and then coach them to provide better service, taking into consideration their past performance results.
The AI Defender automatically detects at-risk agent behaviors, such as failing to deliver required notifications or using inappropriate language, and sends alerts to supervisors' desktops. | https://www.speechtechmag.com/Articles/News/Speech-Technology-News/Transcosmos-Releases-Transpeech2.0-143521.aspx |
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Description
Magnetic Resonance Navigation (MRN) relies on Magnetic Nanoparticles (MNPs) embedded in microcarriers or microrobots to allow the induction of a directional propelling force by 3D magnetic gradients. These magnetic gradients are superposed on a sufficiently high homogeneous magnetic field to achieve maximum propelling force through magnetization saturation of the MNP. As previously demonstrated by our group, such technique was successful at maintaining microcarriers along a planned trajectory in the blood vessels based on tracking information gathered using Magnetic Resonance Imaging (MRI) sequences from artifacts caused by the same MNPs. Besides propulsion and tracking, the same MNPs can be synthesized with characteristics that can allow for the diffusion of therapeutic cargo carried by these MR-navigable carriers through the Blood Brain Barrier (BBB) using localized hyperthermia without compromising the MRN capabilities. In the present study, an external heating apparatus was used to impose a regional heat shock on the skull of a living mouse model. The effect of heat on the permeability of the BBB was assessed using histological observation and tissue staining by Evans blue dye. Results show direct correlation between hyperthermia and BBB leakage as well as its recovery from thermal damage. Therefore, the proposed navigable agents could be suitable for controlled opening of the BBB by hyperthermia and selective brain drug delivery.
Questions and AnswersYou need to be logged in to be able to post here. | http://techtalks.tv/talks/towards-mr-navigable-nanorobotic-carriers-for-drug-delivery-into-the-brain/55216/ |
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Clinical Pharmacist Specialist, Infectious Diseases
Department of Pharmaceutical Services
University Hospital
Newark, New Jersey
Arun Mattappallil, PharmD, has disclosed that he has an ownership interest in Relief Therapeutics.
Clinical Assistant Professor
Department of Internal Medicine
University of Texas Southwestern
Infectious Diseases Clinical Pharmacy Specialist
Department of Pharmacy
Parkland Health & Hospital System
Dallas, Texas
Jessica K. Ortwine, PharmD, BCIDP, has no relevant conflicts of interest to report.
In this commentary, Arun Mattappallil, PharmD, and Jessica K. Ortwine, PharmD, BCIDP, answer questions about monoclonal antibody (mAb) therapies for the treatment of COVID-19 from a ProCE webinar titled “Emerging Insights on the Role of Monoclonal Antibodies in Patients With COVID-19.” Slides from the webinar are also available for self-study or to use in your noncommercial presentations.
Are there any clinical trials using mAb infusions in the inpatient setting?
Jessica K. Ortwine, PharmD, BCIDP:
The most promising inpatient trial results to date come from the RECOVERY trial, assessing the combination of casirivimab and imdevimab vs standard of care alone in adult and pediatric patients 12 years of age or older admitted to the hospital with suspected or confirmed COVID-19. The primary outcome was 28-day, all-cause mortality assessed first among patients who were seronegative at baseline, and Hornby and colleagues found a mortality benefit favoring casirivimab and imdevimab. However, when all patients (both seronegative and seropositive) were included in the analysis, no mortality benefit was observed.
Another recently published trial looked at adults with confirmed COVID-19 experiencing ≤10 days of symptoms either on low-flow or no supplemental oxygen at baseline. A primary clinical efficacy outcome was progression to mechanical ventilation or death among patients with high baseline viral loads, and the results were not statistically significant. However, Somersan-Karakaya and colleagues found a 56% relative risk reduction in 28-day, all-cause mortality and in progression to mechanical ventilation or death among patients who were seronegative at baseline. I believe the study results have been submitted to the FDA and will expand the current indications for use of this mAb combination.
Can mAb infusions be used to treat breakthrough infections?
Jessica K. Ortwine, PharmD, BCIDP:
Based on what we know at this time, the vaccination status of the patient should not affect the decision to use mAb therapy if the patient meets Emergency Use Authorization (EUA) criteria. The National Institutes of Health (NIH) COVID-19 Treatment Guidelines state that, assuming no logistical or supply constraints, prior vaccination against SARS-CoV-2 should not affect decisions regarding the use of monoclonal antibody treatment. A recent study out of the Mayo Clinic looked at fully vaccinated patients with breakthrough COVID-19 infections, of which 38% received treatment with mAb. The authors found that there was a significantly lower rate of hospitalization among the ambulatory patients with breakthrough infections who received mAb therapy.
Are screening tests suggested before the administration of mAb?
Arun Mattappallil, PharmD:
Currently, rapid serology (antibody) testing that can identify seronegative individuals in real time is not widely available. The NIH COVID-19 Treatment Guidelines and FDA EUAs for all currently available COVID-19 mAb do not emphasize the need for serology testing (within authorized use) to determine patient eligibility for therapy. Recently, Bierle and colleagues found that mAb therapy also prevented disease progression in high-risk, seropositive individuals (this population was vaccinated). However, the utility of preadministration serology testing remains unclear.
What is the route of administration and optimal temperature for casirivimab and imdevimab in limited-resource settings?
Arun Mattappallil, PharmD:
Subcutaneous administration of casirivimab and imdevimab is a reasonable alternative administration method for COVID-19 treatment or for postexposure prophylaxis. As specified in the EUA, casirivimab and imdevimab should equilibrate to room temperature before administration and be administered as 4 consecutive injections at different injection sites. The preadministration procedures are similar whether being used for COVID-19 treatment or postexposure prophylaxis. Use of the subcutaneous route of administration still requires significant logistical coordination and training to ensure success.
How do hospitals monitor patients during the postinfusion, 1-hour observation period? Should vitals signs be monitored, or is observation adequate?
Arun Mattappallil, PharmD:
All EUAs for anti–SARS-CoV-2 mAb have the following requirement: “Clinically monitor patients after injections and observe patients for at least 1 hour.” No details about the methods of clinical monitoring are included in the EUA, although these infusions must be administered in settings with immediate access to medications to treat severe infusion or hypersensitivity reactions and the ability to activate emergency medical systems. So healthcare facilities should develop and follow a clear protocol regarding patient monitoring to ensure uniform practice within their facility.
Anecdotally, I have heard of facilities implementing a post-mAb observation protocol similar to their post–COVID-19 vaccine monitoring protocol. The Department of Health and Human Services Monoclonal Antibody Clinical Implementation Guide recommends taking vital signs at patient intake, which provides a baseline indicator of clinical status should adverse events occur post infusion.
Have pharmacoeconomic analyses shown that mAb infusions for postexposure prophylaxis are cost-effective?
Jessica K. Ortwine, PharmD, BCIDP:
There have not been any pharmacoeconomic analyses performed to date and the products are currently provided by the drug companies themselves and allocated free of charge by the federal government. I think that cost-effectiveness will be determined once the allocation process is discontinued and healthcare facilities are required to start purchasing the agents on their own. We don’t yet know what the costs of these agents will be.
Arun Mattappallil, PharmD:
The only overt pharmacoeconomic benefit that we are familiar with is the reduction in hospitalization costs. In terms of the product itself, the cost remains unclear until it becomes available for commercial purchase. There will also be added costs for preparation, administration, and monitoring that will need to be factored into the equation, and these will be more institution specific.
Can patients have worsening symptoms after receiving mAb?
Jessica K. Ortwine, PharmD, BCIDP:
While mAb significantly decreases the risk for hospitalization or death, it does not remove all possibility of these outcomes occurring. I think the vast majority of clinical worsening seen in some patients who received mAb is related to worsening COVID-19 symptoms and not new issues caused by the infusion itself.
As an example, in the phase III trial for outpatient treatment of mild to moderate infection with combination casirivimab and imdevimab, 46 of the events leading to medical attention among patients who received antibody therapy were related to COVID-19 disease, and 7 were not.
In the COMET-ICE study, which assessed the efficacy and safety of sotrovimab in preventing mild to moderate COVID-19 progression to severe disease, no serious adverse events (fatal or otherwise) were deemed to be related to study treatment. In all 3 key phase III ambulatory treatment trials, infusion-related reactions and adverse events severe enough to necessitate infusion interruption or study withdrawal were low (≤1%).
The EUA Provider Fact Sheets do include a disclaimer that monoclonal antibodies may be associated with worse clinical outcomes when administered to hospitalized patients with COVID-19 requiring high-flow oxygen or mechanical ventilation.
Your Thoughts?
How have your patients with COVID-19 responded to mAb therapy? Join the conversation by posting in the comments section.
For more information on mAb therapies for COVID-19, see our program here. | https://www.clinicaloptions.com/infectious-disease/programs/2021/covid19-mabs/clinicalthought/ct/page-1 |
Fields marked with an asterisk (*) are required.
Organizations recognize the criticality and vulnerability to fraud or misconduct situations. With increasing complex business landscape, the spectrum of white collar crimes is highly difficult to uncover and in most cases results in substantial strain on the business of organizations.
Timely unveiling the puzzle of suspicion is highly imperative in fraud investigations across corporates to resolve complicated questions and gather credible evidence which can withstand scrutiny and validation.
We provide assistance to clients in investigating allegations of misconduct or impropriety and provide recommendations around risk mitigations.
Our risk management professionals offer insights gained from extensive experience in conducting high-profile fraud, corruption and misconduct investigations. The team has substantial experience in handling corporate investigations pertaining to financial irregularities, process and transactional frauds, asset misappropriation, financial reporting and misstatements, bribery/corruption misconduct and whistle-blower allegations.
Stay one step ahead in a rapidly changing world and build a sustainable future with us. | https://www.mbgcorp.com/in/forensic-investigation/ |
In these Conditions:
“The Company” means CNBB LTD. T/A VTR North;
“Contract” means any contract between the Company and the Client from time to time for the provision of Services; “The Client” means the person, firm or company using the services of the Company;
“Services” means the services forming the subject of the contract between the Company and the Client;
2. INCORPORATION OF CONDITIONS
These Conditions are the only terms or conditions on which the Company gives quotations, accepts bookings or orders or supplies the Services. These Conditions override any other representations, terms or conditions stipulated, referred to or implied by the Client or the Company, its servants or agents whether in any order or in any document or in any negotiation or discussion. No variation of these Conditions shall be effective unless expressly agreed in writing, signed by or on behalf of the Company.
3. BOOKING AND ORDERS
A Contract shall exist between the parties upon and only upon the Company accepting the Clients booking or order. Each booking order accepted by the Company in accordance with these Conditions constitutes a separate Contract between the parties that is subject to these Conditions.
4. CHARGES AND RATE CARD
Unless otherwise agreed by the Company in writing, the charges applicable are those appearing upon the Company’s published Rate Card current at the time of the Company’s acceptance of the booking or order. Charges are quoted in Pounds Sterling exclusive of VAT, which the Client shall be additionally liable to pay to the Company. The Company reserves the right to alter the terms of its Rate Card at any time without prior notice provided that unless an increase in charges is solely the result of an increase in the cost to the Company of
fulfilling the booking or order upon being notified of an increase in charges the Client shall be entitled to cancel the booking or order. To be effective, a cancellation in the circumstances of the last preceding sentence must be in writing and received by the Company within 24 hours of the increase in charges being communicated to the Client.
5. PAYMENT TERMS
Payment for the Services shall be due within 28 days of the date of the invoice (“the Due Date”), time to be of the essence. This provision shall be without prejudice to the Company’s right to require immediate payment of all outstanding invoices rendered to the Client where the Client is in default of the payment terms in this Clause in respect of one or more invoices. If payment is not received by the Due Date , the Company reserves the right to withdraw any special terms or conditions or discounts or rebates that have or are intended to apply to any current or future Contract. The Company reserves the right to require payment in part or full in advance of the date of performance of the Services. If the Client fails to pay any invoice or any part thereof by the Due Date then, without prejudice to any other right or remedy available to the Company, the Company shall be entitled to either cancel the Contract, suspend the provision of further services to the Client or Charge the Client interest on any overdue payments at the annual rate of 5% above the base rate from time to time of the Company’s bankers (both before and after judgement) and that such interest may be claimed by the Company from the date of invoices until the date of payment.
6. PERFORMANCE
Any dates, times or periods quoted by the Company for performance of the Services are estimates only and the Company shall not be liable for failure to meet such estimates or for any costs charges or expenses incurred as a consequence of such failure and accordingly, the Client shall not be entitled to refuse to accept Services merely because of such failure. Time for performance shall not be of the essence unless previously agreed with the Company in writing. The Company may employ subcontractors to perform part or parts of the Service.
7. CLIENT’S REPRESENTATIVE
If requested to do so by the Company, the Client shall appoint a representative who shall be available at all reasonable times to approve the Services and if so required by the Company to be in attendance during the performance of the Services and whose approval in such circumstances shall be final and binding on the Client.
8. CANCELLATIONS
In its absolute discretion, the Company may, at any time, permit cancellation of a booking or order and reserves the right to cancel any booking or order in the event of any breach of any of these Conditions by the Client provided that in the event of any such cancellation, as liquidated damages, the Client agrees to pay a percentage of the value of the cancelled booking or order according to the length of time between cancellation and the date estimated for performance.
100% Payable Less than 2 days.
50% Payable Less than 5 but more than 2 days
Save as provided above, the Client may not cancel any order or booking that has been accepted by the Company except with the express written permission of the Company, which the Company may provide in its absolute discretion.
9. RISK AND LIABILITY
Risk in the Client’s films, tapes or other materials will at all times remain with the Client. The Company cannot and will not insure against loss of or damage to the content of Client’s films or tapes and accordingly LIABILITY IN RESPECT OF SUCH LOSS OR DAMAGE EVEN WHEN THE RESULT OF NEGLIGENCE ON THE PART OF THE COMPANY, ITS SERVANTS, AGENTS OR SUB-CONTRACTORS IS HEREBY EXCLUDED. The Company shall have no liability for any indirect or consequential losses or expenses suffered by the Client, howsoever caused, including but not limited to loss of anticipated profits, goodwill, reputation, business receipts or contracts, or losses or expenses resulting from third party claims.
10. INTELLECTUAL PROPERTY
Subject as hereinafter provided the Client hereby indemnifies and will keep indemnified the Company against all actions, claims, demands, costs, charges and expenses arising from or incurred by reason of any infringement or alleged infringement of any Copyright or other intellectual property rights or any defamation or alleged defamation arising out of the processing or reproduction of the Client’s films or tapes by the Company, or any work carried out by the Company on the Client’s instructions.
11. PUBLICITY
Unless otherwise instructed by the Client in writing, the Company may use the Client’s name and brief details of the Services for the purposes of the Company’s advertising and promotion of its business. Following the broadcast or exhibition of the work resulting from the Services, the Company shall have the right to use any part of the work solely for use in its corporate advertising and showreels.
12. FORCE MAJEURE
The Company reserves the right to defer the date of performance of the Services or to cancel the Contract or reduce the level of the Services ordered by the Client (without liability to the Client) if it is prevented from or delayed in the performance of the Services due to circumstances beyond the reasonable control of the Company including, without limitation, acts of God, governmental actions, war or national emergency, acts of terrorism, protests, riot, civil commotion, fire, explosion, flood, epidemic, lock-outs, strikes or other labour disputes (whether or not relating to either party’s workforce), or restraints or delays affecting carriers or inability or delay in obtaining
supplies of adequate or suitable materials.
13. INSOLVENCY OF THE CLIENT
This clause applies if the Client makes any voluntary arrangement with its creditors or becomes subject to an administration order or (being an individual or firm) becomes bankrupt or (being a company) goes into liquidation (otherwise than for the purposes of amalgamation or reconstruction); or an encumbrancer takes possession, or a receiver, administrative receiver or administrator is appointed, of any of the property or assets of the Client; or the Client ceases, or threatens to cease, to carry on business; or the Company reasonably apprehends that any of the events mentioned above is about to occur in relation to the Client and notifies the Client accordingly, then, without prejudice to any other right or remedy available to the Company, the Company shall be entitled to cancel the Contract or suspend the performance of any further Services under the Contract without any liability to the Client, and if Services have been performed but not paid for payment shall become immediately due and payable notwithstanding any previous agreement or arrangement to the contrary.
14. CLIENT’S UNDERTAKINGS
The Client shall at all times indemnify the Company in respect of all loss or damage suffered by any person, firm, company or property and against all actions, claims, demands, costs, charges or expenses in connection therewith for which the Company may become liable in respect of the Services. The Client accepts full liability for and shall at all times indemnify the Company against all actions, claims, demands, costs, charges and expenses whatsoever arising out of any loss of damage to any person, firm or company by reason of deficiencies in the materials or data or the like supplied to the Company by the Client in connection with the carrying out of the Services.
15. MATERIALS STORAGE
If the Company shall agree, at its absolute discretion, to store or hold any master tape or other material for the Client, then it shall do so entirely at the risk of the Client and shall not be liable for any loss or damage to such tape or other material, whether caused by the negligence of the Company or its employees, or otherwise. The Company reserves the right to charge the Client for such storage or to return the tape or other material to the Client at the Client’s expense.
16. GENERAL LIEN
The Company shall have a general lien over any of the goods or chattels of the Client in the Company’s possession for any moneys whatsoever due from the Client to the Company. If any lien is not satisfied within 14 days of such moneys becoming due, the Company may, in its absolutely discretion, sell or make use of such goods or chattels as agents for the Client and apply the proceeds towards the moneys due and the expenses of the sale, and shall upon accounting to the Client for the balance (if any) remaining, be discharged from all liability in respect of such goods or chattels.
17. ASSIGNMENT
Each Contract between the Company and the Client is personal to the Client who shall not assign or charge the benefit thereof without the Company’s express written consent. The Company may assign the performance of the Contract or any part thereof to any third party without the consent of the Client.
18. NOTICES
Notices shall be made in writing and posted in a first-class pre-paid envelope to, in the case of communications to the Company: to its registered office or such address as shall be notified to the Client by the Company. In the case of communications to the Client: to the registered office of the Client (if a limited company) or (in any other case) to any address of the Client set out in the Contract or such other address as shall be notified to the Company by the Client. A notice shall be deemed to have been received forty-eight hours after posting (or ninetysix hours after posting where the notice is sent from and/or addressed to an address outside the United Kingdom)
and in proving such service it shall be sufficient to show that the envelope was properly addressed and put into the post.
19. PROPER LAW
The validity, construction and performance of each contract shall be governed by English Law and the parties hereby submit to the sole and exclusive jurisdiction of the courts of England and Wales for the resolution of all disputes arising under any Contract.
20. SEVERANCE
If any provision of these Conditions is found by any court, tribunal or administrative body of competent jurisdiction to be wholly or partly illegal, invalid, void, voidable, unenforceable or unreasonable, it shall to the extent of such illegality, invalidity, voidness, voidability, unenforceability or unreasonableness be deemed severable and the remaining provisions of the Contract and the remainder of such provision shall continue with full force and effect.
21. WAIVER
No waiver by the Company of any breach of contract by the Client shall be considered a waiver of any subsequent breach of the same or any other provision.
Privacy Notice (GDPR)
Contents
This Privacy Notice explains in detail the types of personal data we may collect about you when you interact with us. It also explains how we’ll store and handle that data and keep it safe.
Legal Processing
Data protection law states there are several legal reasons why a company may collect and process personal data.
Contract – We will process your personal data to fulfil an order, deliver a service, essentially comply with our contractual obligations.
Legal Compliance – If the law or government requires us to, we may need to collect and process your data. An example might be passing on details of fraud to law enforcement.
Legitimate Interest – In cases, we require personal data to pursue our legitimate interests in a way which might reasonably be expected as part of running our business. This does not impact your freedom or rights. Examples might be using your postal or email details to send direct marketing information on content, products and services that we think might interest you with the aim of raising awareness of our business with yours and you, as an employee of that business. We may also use your browsing or purchase history to send you or make available personalised offers.
When do we collect, add to or amend your personal data?
When you visit any of our websites or speak us on the phone.
When you purchase a product or service or enter into negotiations to do so
When you engage with us on social media.
When you contact us by any means with queries, complaints etc.
When you book any kind of appointment with us or book to attend an event, for example a meeting to discuss an opportunity.
When you choose to complete any surveys we send you.
Any individual may access personal data related to them, including opinions. So if your comment or review includes information about the Partner who provided that service, it may be passed on to them if requested.
When you’ve given a third party permission to share with us the information they hold about you.
We may collect data from publicly-available sources where you have given your consent for them to share information publicly such as online directories or LinkedIn.
What sort of personal data do we collect?
Contact details at the company which you work for such as email address and phone number
Details of your interactions with us through phone contact, email or online.
Details of your visits to our websites, and which site you came from to ours.
To deliver the best possible web experience, we collect technical information about your internet connection and browser as well as the country and telephone code where your computer is located, the web pages viewed during your visit, the advertisements you clicked on, and any search terms you entered.
Your social media username, if you interact with us through those channels, to help us respond to your comments, questions or feedback.
How and why do we use your personal data?
We want to give you the best possible experience. One way to achieve that is to get the richest picture we can of who you are by combining the data we have about you. We then use this to offer you content, products and services that are most likely to interest you.
The data privacy law allows this as part of our legitimate interest in understanding our customers and potential customer and providing the highest levels of service.
Of course, if you wish to change how we use your data, you’ll find details in the ‘What are my rights?’ section below.
Remember, if you choose not to share your personal data with us, or refuse certain contact permissions, we might not be able to provide the best experience.
Here’s how we’ll use your personal data and why:
To process any orders or hold negotiations that you make by using our sales team or sales agents. If we don’t collect your personal data during negotiations or ordering, we won’t be able to process your order and comply with our legal obligations. We would then keep those details for reasonable period afterwards in order to fulfil contractual obligations
To respond to your queries. Handling the information you sent enables us to respond. We may also keep a record of these to inform any future communication with us and to demonstrate how we communicated with you throughout. We do this on the basis of our contractual obligations to you, our legal obligations and our legitimate interests in providing you with the best service and understanding how we can improve our service based on your experience.
To protect our business and yours from fraud and other illegal activities. We’ll also monitor your browsing activity with us to quickly identify and resolve any problems and protect the integrity of our websites. We’ll do all of this as part of our legitimate interest.
To market our product and services to you. We will use your personal data, preferences and details of your transactions to keep you informed by email, web, telephone about relevant content, products and services including information, tailored offers, discounts, promotions, events, competitions and so on. We market on the basis of legitimate interest and in a way which might reasonably be expected as part of running our business. This does not impact your freedom or rights. Our marketing will follow best practice laid out by the ICO when dealing with corporate subscribers under the PECR.
Of course, you are free to opt out of hearing from us at any time.
To send you relevant, personalised communications by post or email in relation to updates, content, offers, services and products. We’ll do this on the basis of our legitimate business interest.
You are free to opt out of hearing from us at any time.
To send you communications required by law or which are necessary to inform you about our changes to the services we provide you. For example, updates to this Privacy Notice, marketing data misuse issues, and legally required information relating to your orders. These service messages will not include any promotional content and do not require prior consent when sent by email or text message. If we do not use your personal data for these purposes, we would be unable to comply with our legal obligations.
To develop, test and improve the systems, services and products we provide to you. We’ll do this on the basis of our legitimate business interests.
To comply with our contractual or legal obligations to share data with law enforcement.
To send you survey and feedback requests to help improve our services. These messages will not include any promotional content and do not require prior consent when sent by email or text message. We have a legitimate interest to do so as this helps make our products or services more relevant to you.
Of course, you are free to opt out of receiving these requests from us at any time
How is personal data protected?
We will treat your data with the utmost care and take all appropriate steps to protect it.
We secure access to our websites and apps using ‘https’ technology.
Access to your personal data is password-protected, and sensitive data is secured by SSL encryption.
We regularly monitor our system to identify ways to further strengthen security.
Website
What specific information is obtained?
Our website does not collect specific, detailed information about you as a visitor to our site.
Where information is gathered, we will inform you of this along with the explanation of why it is required. This information will be for purposes such as responding to an enquiry made and we will ask for this information in order for us to get in touch and provide the relevant information which has been requested from you.
What information is retained?
Cookies and tracking software are used on a temporary basis and expire when the site has been left. These are used for purposes such as website improvement. Google Analytics is also used for the same reason of website improvement and understanding how you use it. This information is not shared with anyone.
The personal information we collect and hold via Tracking software is the following: –
Browser used (Bing, Chrome, Internet Explorer etc.)
Location (based on IP address)
Gender
Age group
Site searches
Device(s) used
Frequency on Site
Session duration (time spent on the site)
Page(s) visited
Page session duration (time spent on specific page)
Web Surfing Behaviours
We currently use the following companies that may process your personal data through website usage. Facebook, Twitter, LinkedIn, Google, HubSpot, SharpSpring, Instiller.
Our Details
If you need to contact us then please find the relevant details on our contact page here
Customers / Enquirers
We hold information about customers/ enquirers as it is necessary for us to conduct our business activities.
What personal information is held?
The personal information we collect and hold is the following: –
Full Name
Company Name
Company Address
Company Website Address
Email Address
Phone Number
Initial Requirement
This information may be linked together with what has already been gathered (as stated in the “what information is retained?” section) via Tracking Software.
How is this information gathered?
We gather this information through lead generation forms on our website and tracking software as mentioned in the “what information is retained?” section, as well as over the phone when relevant.
How long will the data be kept for?
For a reasonable period of time after you have stopped being an active customer, we may continue to contact you with information relating to our products and services. We deem a reasonable period of time to be three years after an order has been completed, or a relevant license period has expired. After this time, the data we hold will either be deleted, anonymised or retained providing you have asked us to do so.
Who will the data be passed to?
This information will not be passed on to anyone other than authorised employees of CNBB Ltd. T/A VTR North and relevant processors, for example a company managing marketing on behalf of VTR North.
Additionally, if a legal matter were to arise, then we may need to provide this information for those purposes.
In this instance, we provide only the information required to perform the service
The data will only be used for that exact purpose
Partners are required to protect and respect your privacy
If we stop using a partners service, your data will be deleted or rendered anonymous.
Examples of third parties we work with are:
Direct marketing companies who help us manage electronic communications and telemarketing
Google/Facebook/LinkedIn to show products that might interest you. This is based on consent or your acceptance of cookies on our website.
Data Insight companies to ensure your details are up to date and accurate.
IT Companies who support our website and business systems
What is the reason for holding/using the information held?
While you are a customer of VTR North, the basis for processing your data is ‘contract’. When you are not or no longer an active customer of VTR North the basis is ‘legitimate interest’ as we would like you order products/services from us.
We may use your details for the purpose of creating a profile to gauge decision making and for direct marketing purposes. For example, using information held, we may create profiles based on the following: –
Email Address
Telephone Number
Geographic Location
Job Title/Job Role
Data Subject Rights (Subject Access Request, Erasure, Rectification)
You have the right to ask for information we hold for you (Subject Access Request) which will be provided free of charge and if requested, can be presented in a spreadsheet based format.
The data we hold will be corrected when we are notified of errors.
Information we hold for you will no longer be processed if requested to do so.
Information we hold for you will be deleted if requested to do so. It is possible we may contact you at a later date as we will hold no record of you.
You have the right to withdraw consent at any time.
You have the right to stop the use of your personal data for direct marketing through selected or all channels.
You have the right to stop processing your personal data under legitimate interest for individual reasons. We will comply unless we believe there is a legitimate overriding reason to continue processing.
To stop the use of personal data for direct marketing:
Click Unsubscribe Link in any email communication that we send you. We will then stop all emails
Write to:
Compliance Officer
Our Details
If you need to contact us then please find the relevant details on our contact page
To contact the regulator, please see below: | https://vtrnorth.co.uk/detail/terms-and-conditions |
A comprehensive range of 3-dimensional geometric shapes.
||There is no PDF available for this product.|
(MPN: RVFM)
A pack of 36 solids made from tough non-toxic foam. The set includes 13 shapes: cones, spheres, cubes, rectangular prisms, pyramids and many more. | https://www.rapidonline.com/Education/Mini-maths-shapes-71831 |
Tracker:
This bill has the status Introduced
Here are the steps for Status of Legislation:
- Introduced
Subject — Policy Area:
- Science, Technology, Communications
- View subjects
Summary: H.R.3038 — 113th Congress (2013-2014)All Information (Except Text)
There is one summary for H.R.3038. Bill summaries are authored by CRS.
Shown Here:
Introduced in House (08/02/2013)
Suborbital and Orbital Advancement and Regulatory Streamlining Act or SOARS Act - Amends commercial space launch licensing requirements.
Revises the definition of "launch services" to include activities involved in the preparation of a launch vehicle (as under current law) or element thereof, including space flight participant training for a launch.
Authorizes the Secretary of Transportation (DOT) to issue a single license or permit for flight of a launch or reentry vehicle, or element thereof, in support of a launch or reentry, even when the vehicle or element is not being launched or reentered. Requires the Secretary to ensure that all DOT regulations for a licensed or permitted launch or reentry are satisfied under a single license or permit.
Authorizes the issuance of an experimental permit for a particular reusable suborbital rocket (as under current law) or rocket design after a license has been issued for the launch or reentry of a rocket of that design. Declares that any permits already issued shall remain valid for research and development (R&D) and other specified purposes.
Directs the Secretary to establish, under the Office of Commercial Space Transportation of the Federal Aviation Administration (FAA), a demonstration project to evaluate the benefits of using experimental aircraft for both the direct and indirect support of commercial space launch and reentry activities. | https://www.congress.gov/bill/113th-congress/house-bill/3038 |
The School of Architecture of the University of Puerto Rico has been educating architects for 55 years within the island’s oldest public institution. With over 400-students and 50-Faculty, it was last accredited by the National Architectural Accrediting Board in 2014. It offers a Bachelor\’s in Environmental Design and Master’s of Architecture, currently taught in Spanish. The School is in process of adding other nonprofessional degrees in related fields.
The School’s mission is to educate social and ethically driven professionals capable of generating change through research, creative and critical-thought analysis, whose actions contribute to the profession’s evolution and creation of socioeconomically and environmentally sustainable environments and the protection of built patrimony.
Minimum Requirements: Professional architecture, master’s, or doctoral degree granted by a recognized and accredited institution; Teaching experience offering architectural design courses, focusing on emergent features of the architectural profession, at both undergraduate and/or graduate levels and in recognized and accredited institutions; Professional experience in architecture (design, construction documents development and/or construction management) for a minimum of five years; Professional license in architecture is preferred; Capability to direct master’s thesis and investigations; Capability for in-person and online teaching; Experience in research, creative, and architectural work; Commitment to pursue external funding; and Publications in candidate’s area of expertise.
Responsibilities: The tasks that correspond to this teaching position include architectural design courses and other in-person and online specialized courses; availability to offer weekend and evening courses, at undergraduate and graduate levels, with a demonstrated interest in emergent architectural production, research, and publications in their area of expertise; participation in academic committees, and administrative support as part of the contractual obligations. The position is scheduled to begin August 1, 2021.
Required Documents: Letter of intent; Updated Curriculum Vitae; Official professional and academic credentials of all degrees; Evidence of Master thesis supervision and courses offered at University level; Evidence of recent presentations at conferences, publications, and research work; Portfolio of professional and creative work; Teaching and Research philosophy statements; Statement of intention to pursue external funding opportunities; and Two recommendation letters. | https://www.acsa-arch.org/job/tenure-track-teaching-position/ |
A. COLLECTION OF INFORMATION
The type of Information submitted to or collected by NanoTech Coatings varies depending upon the nature of the activity and relationship with NanoTech Coatings.
Browsing the Website: When You browse the company's website, NanoTech Coatings may collect information regarding the domain and host from which You access the Internet, the Internet Protocol address of the computer or Internet Service Provider You are using, and anonymous site statistical data. This information is collected for the purpose of assessing the effectiveness of the company's website and for security reasons.
Inquiries: The website contains various forms, links to company e-mail addresses, and fax numbers that You may use to solicit information about the website, NanoTech Coatings services, and the company in general. When You complete and submit a form, send us an e-mail, send us a fax, or contact us by telephone, NanoTech Coatings may store the inquiries and their contents, including any personally identifiable information You may have provided. Any personally identifiable information You submit via an inquiry is collected only with Your knowledge and active participation.
B. USE AND DISCLOSURE OF INFORMATION
Usage and disclosure of Your Information varies based on Your relationship with NanoTech Coatings. NanoTech Coatings never sells any personally identifiable information and, except as expressly set forth below, never discloses personally identifiable information to any third parties.
Browsing the Website: Information collected while You're browsing our website may be used to analyze trends, administer the website, improve site performance, gather broad demographic information, and for security purposes. Such Information may be disclosed to third parties to provide any of the aforementioned activities on behalf of NanoTech Coatings.
Inquiries: If you submit an inquiry via online form, e-mail, fax, or telephone call, the information collected may be used by NanoTech Coatings to respond to Your inquiry or to contact You to inform You of NanoTech Coatings products. Information collected during the course of an inquiry will not be disclosed to any third party unless such third party has been contracted by NanoTech Coatings, with obligations of confidentiality, to contact You on our behalf.
Surveys and Questionnaires: If you submitted an inquiry to NanoTech Coatings or if you are a NanoTech Coatings Customer, periodically, NanoTech Coatings may use your Information to contact You, directly or through a third party vendor, to complete a survey or questionnaire. Responses to any such survey or questionnaire may be used for internal business analyses and may be disclosed in aggregate form without disclosing Your personally identifiable information.
Investigations and Proceedings: NanoTech Coatings may use Information to conduct internal investigations. NanoTech Coatings may disclose Information to cooperate with an investigation by law enforcement agencies or other governmental authorities and may also disclose Information in response to a subpoena, warrant, court order, or other comparable legal processes.
C. INFORMATION SECURITY
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CROSS REFERENCE TO RELATED APPLICATIONS
BACKGROUND
SUMMARY
DETAILED DESCRIPTION
This application is a non-provisional of and claims priority to U.S. Provisional Patent Application No. 62/291,467, filed on Feb. 4, 2016, the entire contents of which are hereby incorporated by reference.
This specification relates to recurrent neural network architectures.
Neural networks are machine learning models that employ one or more layers of nonlinear units to predict an output for a received input. Some neural networks include one or more hidden layers in addition to an output layer. The output of each hidden layer is used as input to the next layer in the network, i.e., the next hidden layer or the output layer. Each layer of the network generates an output from a received input in accordance with current values of a respective set of parameters. Some reinforcement learning systems select the action to be performed by the agent in response to receiving a given observation in accordance with an output of a neural network.
Some neural networks are recurrent neural networks. A recurrent neural network is a neural network that receives an input sequence and generates an output sequence from the input sequence. In particular, a recurrent neural network can use some or all of the internal state of the network from a previous time step in computing an output at a current time step.
In general, one innovative aspect of the subject matter described in this specification can be embodied in systems that include a recurrent neural network implemented by one or more computers, wherein the recurrent neural network is configured to receive a respective neural network input at each of a plurality of time steps and to generate a respective neural network output at each of the plurality of time steps, wherein the recurrent neural network includes an associative long short-term memory (LSTM) layer, wherein the associative LSTM layer is configured to maintain N copies of an internal state for the associative LSTM layer, N being an integer greater than one, and wherein the associative LSTM layer is further configured to, at each of the plurality of time steps, receive a layer input for the time step, update each of the N copies of the internal state using the layer input for the time step and a layer output generated by the associative LSTM layer for a preceding time step, and generate a layer output for the time step using the N updated copies of the internal state.
Other embodiments of this aspect include methods that perform the operations that associative LSTM layer is configured to perform. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of software, firmware, hardware, or any combination thereof installed on the system that in operation may cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.
Implementations can include one or more of the following optional features. Updating each of the N copies of the internal state includes determining a cell state update for the time step from the layer input at the time step and optionally the layer output for the preceding time step; determining, for each of the N copies of the internal state, a corresponding transformed input key from the layer input at the time step and the layer output for the preceding time step; and for each of the N copies of the internal state, determining the updated copy of the internal state from the copy of the internal state, the cell state update, and the corresponding transformed input key.
Determining, for each of the N copies of the internal state, a corresponding transformed input key from the layer input at the time step and the layer output for the preceding time step includes determining an input key from the layer input at the time step and the layer output for the preceding time step; and for each of the N copies of the internal state, determining the corresponding transformed input key for the copy by permuting the input key with a respective permutation matrix that is specific to the copy.
Updating each of the N copies of the internal state further includes determining an input gate from the layer input at the time step and the layer output for the preceding time step, and determining a forget gate from the layer input at the time step and the layer output for the preceding time step.
Determining the updated copy of the internal state from the copy of the internal state, the cell state update, and the corresponding transformed input key includes applying the forget gate to the copy of the internal state to generate an initial updated copy; applying the input gate to the cell state update to generate a final cell state update; applying the corresponding transformed input key to the final cell state update to generate a rotated cell state update; and combining the initial updated copy and the rotated cell state update to generate the updated copy of the internal state.
Generating the layer output for the time step includes determining, for each of the N copies of the internal state, a corresponding transformed output key from the layer input at the time step and the layer output for the preceding time step; modifying, for each of the N copies of the internal state, the updated copy of the internal state using the corresponding transformed output key; combining the N modified copies to generate a combined internal state for the time step; and determining the layer output from the combined internal state for the time step.
Combining the N modified copies includes determining the average of the N modified copies.
Determining, for each of the N copies of the internal state, a corresponding transformed output key from the layer input at the time step and the layer output for the preceding time step includes determining an output key from the layer input at the time step and the layer output for the preceding time step; and for each of the N copies of the internal state, determining the corresponding transformed output key for the copy by permuting the output key with a respective permutation matrix that is specific to the copy.
Generating the layer output for the time step further includes determining an output gate from the layer input at the time step and the layer output for the preceding time step, and wherein determining the layer output from the combined internal state for the time step includes applying an activation function to the combined internal state to determine an initial layer output; and applying the output gate to the initial layer output to determine the layer output for the time step.
The subject matter described in this specification can be implemented in particular embodiments so as to realize one or more of the following advantages. Recurrent neural network layers with long short-term memory (LSTM) architectures can be implemented with additional memory units to store internal state values and with capabilities to index the internal state memory. Internal state values maintained by LSTM layers will be less noisy and more reliable. Accuracy of LSTM computations that involve tracking multiple elements in input data will be enhanced. By storing multiple copies of the internal state for a time step, LSTM layers become more resilient in the face of internal failures or loss of internal state data. The time complexity of LSTM layers can be reduced to a linear order of growth dependent on the number of stored internal state copies. Collisions between the storage of two or more internal state copies can be mitigated by using internal state copies that were not involved in the collision or for whom the collision has been resolved. By storing more internal state data, LSTM layers can better detect long-term dependencies between their input and output data.
The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims.
Like reference numbers and designations in the various drawings indicate like elements.
FIG. 1
100
100
shows an example neural network system . The neural network system is an example of a system implemented as computer programs on one or more computers in one or more locations, in which the systems, components, and techniques described below can be implemented.
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The neural network system is a machine learning system that receives a respective neural network input at each of multiple time steps and generates a respective neural network output at each of the time steps. That is, at each of the multiple time steps, the neural network system receives a neural network input and processes the neural network input to generate a neural network output. For example, at a given time step, the neural network system can receive a neural network input and generate a network output .
100
The neural network system can store the generated neural network outputs in an output data repository or provide the neural network outputs for use for some other immediate purpose.
100
The neural network system can be configured to receive any kind of digital data input and to generate any kind of score or classification output based on the input.
100
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For example, if the inputs to the neural network system are images or features that have been extracted from images, the output generated by the neural network system for a given image may be scores for each of a set of object categories, with each score representing an estimated likelihood that the image contains an image of an object belonging to the category.
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As another example, if the inputs to the neural network system are Internet resources (e.g., web pages), documents, or portions of documents or features extracted from Internet resources, documents, or portions of documents, the output generated by the neural network system for a given Internet resource, document, or portion of a document may be a score for each of a set of topics, with each score representing an estimated likelihood that the Internet resource, document, or document portion is about the topic.
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As another example, if the inputs to the neural network system are features of a personalized recommendation for a user, e.g., features characterizing the context for the recommendation, e.g., features characterizing previous actions taken by the user, the output generated by the neural network system may be a score for each of a set of content items, with each score representing an estimated likelihood that the user will respond favorably to being recommended the content item. In some of these examples, the neural network system is part of a reinforcement learning system that provides content recommendations to users.
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As another example, if the input to the neural network system is text in one language, the output generated by the neural network system may be a score for each of a set of pieces of text in another language, with each score representing an estimated likelihood that the piece of text in the other language is a proper translation of the input text into the other language.
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As another example, if the input to the neural network system is features of a spoken utterance, the output generated by the neural network system may be a score for each of a set of pieces of text, each score representing an estimated likelihood that the piece of text is the correct transcription for the utterance.
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As another example if the inputs to the neural network system are images, the output generated by the neural network system may be a score for each of a set of pieces of text, each score representing an estimated likelihood that the piece of text is text that is present in the input image.
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In particular, the neural network system includes a recurrent neural network which, in turn, includes an associative long short-term memory (LSTM) layer .
110
The recurrent neural network is configured to, at each of the time steps, receive the neural network input and to process the neural network input to generate the neural network output at the time step.
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The recurrent neural network may include one or more neural network layers in addition to the associative LSTM layer . For instance, the recurrent neural network can include one or more conventional LSTM layers, one or more other associative LSTM layers, one or more conventional recurrent neural network layers, and/or one or more feedforward neural network layers. In some implementations, the recurrent neural network is a deep LSTM neural network, where the input to the associative LSTM layer is either the input to the recurrent neural network or the output of another LSTM layer of the recurrent neural network .
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The associative LSTM layer is configured to, at each of the time steps, receive a current layer input and to process the current layer input and an internal state to generate an updated internal state. The associative LSTM layer generates the updated internal state by updating the internal state in accordance with the layer input .
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The associative LSTM layer can generate an update to the current internal state based on the values of a number of LSTM gates associated with the layer . In some implementations, the LSTM gates include an input gate, a forget gate, and an output gate.
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FIG. 3
The associative LSTM layer can use the value of the LSTM gates to determine a hidden state for a time step. The associative LSTM layer maintains more than one copy of the internal state. At each time step, the associative LSTM layer updates each of the copies of the internal state using the layer input for the time step and the updated internal state generated by the associative LSTM layer for a preceding time step. Updating the copies of the internal state is described in greater detail below with reference to .
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FIG. 2
The associative LSTM layer also generates for each time step, a layer output for the time step using the updated copies of the internal state. Generating the layer output for the time step using the updated copies of the internal state is described in greater detail below with reference to .
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In some implementations, by maintaining more than one copy of the internal state for each time step, the neural network system can reduce the amount of noise associated with values of the internal state used for the associative LSTM layer computations. Each copy of the internal state is a different representation of the internal state. To reduce the amount of noise associated with a retrieved value of the internal state, the neural network system can use a measure of central tendency (e.g., an average) of the representations of the internal state stored in the multiple copies to determine the hidden state of the associative LSTM layer for each time step.
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Individual copies of the internal state of the associative LSTM layer may include noise. This may for instance be the case when the internal state tracks the occurrence of multiple elements in a sequence of layer input values. In such circumstances, a measure of the internal state derived from multiple copies of the internal state can correct some of the noise and thus be more reliable and accurate.
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For instance, if the internal state for a time step aggregates multiple constituent values where each constituent value estimates an occurrence of one class of filters in a sequence of layer input values for a particular time step, the neural network system can reduce the noise associated with representation of each constituent value in the aggregated internal state by maintaining more than one copy of the internal state and using a measure of the central tendency of the multiple copies of internal state to compute the measure of internal state used for associative LSTM layer computations.
FIG. 2
FIG. 1
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is a flow chart of an example process for generating a layer output for a time step. For convenience, the process will be described as being performed by an associative LSTM layer implemented by a system of one or more computers located in one or more locations. For example, an associative LSTM layer in a neural network system, e.g., the associative LSTM layer of neural network system of , appropriately programmed in accordance with this specification, can perform the process .
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The associative LSTM layer receives a layer input for the time step ().
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FIG. 4
The associative LSTM layer updates each of the N copies of the internal state using the layer input for the time step and a layer output generated by the associative LSTM layer for a preceding time step (). Updating each of the N copies of the internal state is described in greater detail below with reference to .
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FIG. 3
The associative LSTM layer generates a layer output for the time step using the N updated copies of the internal state (). Generating a layer output for the time step using the updated internal state copies is described in greater detail below with reference to .
FIG. 3
FIG. 1
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is a flow chart of an example process for generating a layer output using updated copies of an internal state. For convenience, the process will be described as being performed by an associative LSTM layer implemented by a system of one or more computers located in one or more locations. For example, an associative LSTM layer in a neural network system, e.g., the associative LSTM layer of neural network system of , appropriately programmed in accordance with this specification, can perform the process .
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The associative LSTM layer receives a layer input ().
320
The associative LSTM layer determines a respective transformed output key for each internal state copy from the layer input at the time step and the layer output for the preceding time step ().
That is, the associative LSTM layer maintains multiple copies of the internal state from time step to time step. When a layer output is received for a given time step, the associative LSTM layer determines a transformed output key for each of the copies from the layer input at the time step and the layer output at the preceding time step.
For each internal state copy, the transformed output key is configured to be used for retrieving an internal state of a time step from a data structure with a distributed representation that represents all of the internal state copies for the time step. When the internal state copies are stored in a distributed representation data structure, the contents of the internal state copies can no longer be retrieved by the location of those copies as those contents are distributed in all of the locations of the distributed representation data structure. To retrieve an internal state copy, the associative LSTM layer needs to apply an extraction operation to the data in the distributed data structure. The associative LSTM layer uses the transformed output key for each internal state copy to obtain the value of that internal state copy from the distributed representation data structure.
Examples of a distributed representation data structure include a complex vector generated based on Holographic Reduced Representation (HRR).
In some implementations, the associative LSTM layer determines an output key from the layer input at the time step and the layer output for the preceding time step and determines the corresponding transformed output key for the copy by permuting the output key with a respective permutation matrix that is specific to the internal state copy.
In some implementations, the associative LSTM layer performs the following operation during each time step:
{circumflex over (r)}
=W
x
+W
h
+b
o
xh
t
hh
t-1
h
o
t
t-1
h
xh
hh
t
t-1
where {circumflex over (r)}is an initial output key of the respective time step, xis the layer input for the time step, his the layer output for the preceding time step, bis the bias vector for the time step, and Wand Ware parameter matrices applied to xand hrespectively.
In some implementations, the associative LSTM layer applies a bound function on the initial key to generate the output key for the time step.
In some implementations, the bound function is a function that operates on a complex vector and restricts the modulus of each complex pair in the complex vector.
For instance, the bound function may entail the following operation on a complex vector h:
<math overflow="scroll"><mrow><mrow><mi>bound</mi><mo></mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo></mo><mrow><mo>(</mo><mi>h</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><msub><mi>h</mi><mi>real</mi></msub><mo></mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo></mo><mi>Ø</mi><mo></mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo></mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>h</mi><mi>imaginary</mi></msub><mo></mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo></mo><mi>Ø</mi><mo></mo><mstyle><mspace width="0.6em" height="0.6ex" /></mstyle><mo></mo><mi>d</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></math>
where is an elementwise division operation with a vector d that causes an elementwise normalization by the modulus of each complex number in h. In some implementations, the associative LSTM layer may determine the value of the elements of d using the following equation:
h
⊙h
+h
⊙h
real
real
imaginary
imaginary
max(1,√{square root over ()})
where ⊙ is an element-wise vector multiplication operation.
330
FIG. 3
The associative LSTM layer updates each copy of the internal state (). Updating copies of the internal states is described in greater detail below with reference to .
340
The associative LSTM layer modifies each updated copy of the internal state using the corresponding transformed output key ().
In some implementations, the associative LSTM layer performs a binding operation, such as a complex vector multiplication operation, between an updated copy of the internal state and the transformed output key corresponding to the updated copy to modify the updated copy.
350
The associative LSTM layer combines the modified copies of the internal state ().
In some implementations, the associative LSTM layer computes a measure of central tendency (e.g., an average) of the modified copies of the internal state to generate a combined internal state.
360
The associative LSTM layer determines the layer output from the combined internal state for the time step ().
In some implementations, the associative LSTM layer maintains an output gate whose value at each time step is determined based on the layer input for the current time step and the layer output from a preceding the time step. For instance, the associative LSTM layer may determine the value of the output gate for each time step based on processing the layer input for the current time step and the layer output from a preceding the time step and applying an activation function (e.g., a logistic sigmoid function) to the result of that processing.
In some of those implementations, the associative LSTM layer applies an activation function (e.g., a hyperbolic tangent function) to the combined internal state to determine an initial layer output and applies the output gate to the initial layer output to determine the layer output for the time step.
In addition to the output gate, the associative LSTM layer may include: (1) a forget gate whose values determine whether the associative LSTM layer should process or ignore the output of the associative LSTM layer for a preceding time step as a hidden state and (2) an input gate whose values determine what the associative LSTM layer should adopt as the hidden state if it decides to ignore the output of the preceding time step.
For instance, the associative LSTM layer may perform the following operation:
h
=g
h
c
t
o
t
⊙ tan ()
t
o
t
where his the layer output for a time step, gis the value of the output gate during the time step, cis the combined internal state for the time step, and tan h is a hyperbolic tangent function.
FIG. 4
FIG. 1
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400
120
100
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is a flow chart of an example process for updating copies of an internal state for a particular time step. For convenience, the process will be described as being performed by a system of one or more computers located in one or more locations. For example, an associative LSTM layer in a neural network system, e.g., the associative LSTM layer of neural network system of , appropriately programmed in accordance with this specification, can perform the process .
410
The associative LSTM layer determines a cell state update for the time step from the layer input at the time step and optionally the layer output for the preceding time step (). The cell state update is a value that the associative LSTM layer calculates based on a measure of proposed update to the internal state for the time step. The associative LSTM layer may generate the proposed update to the internal state for the time step from the layer input for the time step and the layer output for a preceding time step.
In some implementations, the associative LSTM layer performs the following operations to generate the proposed update:
û=W
x
+W
h
+b
xu
t
hu
t-1
u
t
t-1
u
xu
hu
t
t-1
where û is the proposed update to the internal state for a time step, xis the layer input for the time step, his the layer output for a preceding time step, bis the bias value for the time step used to generate the proposed update, and Wand Ware parameter values applied to xand hto generate the proposed update.
In some implementations, the associative LSTM layer applies the bound function to the proposed update to the internal state for a time step to generate the cell state update for the time step.
420
The associative LSTM layer determines a transformed input key for the time step from the layer input at the time step and optionally determines a corresponding transformed input key for each internal state copy from the layer input at the time step and the layer output for the preceding time step ().
In some implementations, the associative LSTM layer uses an insertion operation to insert an internal state copy to a distributed representation data structure that includes the contents of the internal state copies for a time step (as explained above). The transformed input key may be an operand of that insertion operation.
In some of those implementations, the associative LSTM layer determines the transformed input key based on the input key by applying a permutation matrix for the specific internal state copy to the input key.
In some implementations, the associative LSTM layer uses the same copy-specific permutation matrix to generate both the transformed output key and the transformed input key for the specific internal state copy.
The copy-specific permutation matrix may be a random permutation matrix with following form:
<math overflow="scroll"><mrow><mo> </mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>P</mi><mi>s</mi></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>P</mi><mi>s</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></math>
s
where Pis matrix with random values which is applied to both the real and the imaginary part of a complex input key vector and a complex output key vector.
In some implementations, the associative LSTM layer performs the following operation:
{circumflex over (r)}
=W
x
+W
h
+b
i
xh
t
hh
t-1
h
i
t
t-1
h
xh
hh
t
t-1
where {circumflex over (r)}is an initial input key of the respective time step, xis the layer input for the time step, his the layer output for the preceding time step, bis the bias value for the time step, and Wand Ware parameter values applied to xand hrespectively.
In some implementations, the associative LSTM layer applies the bound function to the initial values for the input key of a time step to generate the input key for the time step.
430
The associative LSTM layer determines each updated copy of the internal state from the copy of the internal state, the cell state update, and the corresponding transformed input key ().
In some implementations, the associative LSTM determines each updated copy of the internal state for a time step from the layer input for the time step and the values of multiple LSTM gates in the time step.
In some implementations, the associative LSTM layer applies the transformed input key for an internal state copy to the cell state update for the time step and combines the resulting value with a measure of the internal state copy to determine the updated internal state copy.
In some implementations, the associative LSTM layer performs the following operation:
c
=g
⊙c
+r
g
⊙u
s,t
f
s,t-1
i,s
i
{circle around (*)} ()
s,t
f
i,s
i
where cis the updated copy of an internal state, gis the forget gate of the associative LSTM layer for the time step, ris the transformed input key for the internal state copy, gis the input gate of the associative LSTM layer for a time step, u is the cell state update for the time step, {circle around (*)} is a complex vector multiplication operation, and ⊙ is an element-wise vector multiplication operation.
Embodiments of the subject matter and the functional operations described in this specification can be implemented in digital electronic circuitry, in tangibly-embodied computer software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Embodiments of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non transitory program carrier for execution by, or to control the operation of, data processing apparatus. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them. The computer storage medium is not, however, a propagated signal.
The term “data processing apparatus” encompasses all kinds of apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). The apparatus can also include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.
A computer program (which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.
As used in this specification, an “engine,” or “software engine,” refers to a software implemented input/output system that provides an output that is different from the input. An engine can be an encoded block of functionality, such as a library, a platform, a software development kit (“SDK”), or an object. Each engine can be implemented on any appropriate type of computing device, e.g., servers, mobile phones, tablet computers, notebook computers, music players, e-book readers, laptop or desktop computers, PDAs, smart phones, or other stationary or portable devices, that includes one or more processors and computer readable media. Additionally, two or more of the engines may be implemented on the same computing device, or on different computing devices.
The processes and logic flows described in this specification can be performed by one or more programmable computers executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).
Computers suitable for the execution of a computer program include, by way of example, can be based on general or special purpose microprocessors or both, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Moreover, a computer can be embedded in another device, e.g., a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a Global Positioning System (GPS) receiver, or a portable storage device, e.g., a universal serial bus (USB) flash drive, to name just a few.
Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.
To provide for interaction with a user, embodiments of the subject matter described in this specification can be implemented on a computer having a display device, e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user and a keyboard and a pointing device, e.g., a mouse or a trackball, by which the user can provide input to the computer. Other kinds of devices can be used to provide for interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback, e.g., visual feedback, auditory feedback, or tactile feedback; and input from the user can be received in any form, including acoustic, speech, or tactile input. In addition, a computer can interact with a user by sending documents to and receiving documents from a device that is used by the user; for example, by sending web pages to a web browser on a user's client device in response to requests received from the web browser.
Embodiments of the subject matter described in this specification can be implemented in a computing system that includes a back end component, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front end component, e.g., a client computer having a graphical user interface or a Web browser through which a user can interact with an implementation of the subject matter described in this specification, or any combination of one or more such back end, middleware, or front end components. The components of the system can be interconnected by any form or medium of digital data communication, e.g., a communication network. Examples of communication networks include a local area network (“LAN”) and a wide area network (“WAN”), e.g., the Internet.
The computing system can include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other.
While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.
Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.
Particular embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In certain implementations, multitasking and parallel processing may be advantageous.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
shows an example neural network system.
FIG. 2
is a flow chart of an example process for generating a layer output for a time step.
FIG. 3
is a flow chart of an example process for generating a layer output using updated copies of an internal state.
FIG. 4
is a flow chart of an example process for updating copies of an internal state for a particular time step. | |
by Erin Larsen
A team of engineers at Ohio State University are looking to nature to redesign windmills. In a recent issue of the Journal of Sound and Vibration, OSU researchers reported that they have discovered new information about how vibrations pass through trees when they sway in the wind. They believe that this research can be used to design new tools for harvesting wind energy that look less like windmills and more like mechanical trees.
These tree-like structures would aesthetically be very simple, just a trunk with a few branches. Electromechanical materials would convert random forces intro structural vibrations that generate electricity. This technology may prove most useful on the small scale when other renewable energy sources such as solar are not a good fit, says project leader Ryan Harne, assistant professor of mechanical and aerospace engineering at Ohio State. Potential applications of the technology would include powering sensors used to monitor structural integrity and health of civil infrastructure. Harne envisions these trees taking advantage of the abundant vibrational energy that surrounds us every day and feeding voltages to sensors on the underside of a bridge. The structural monitoring systems would be powered by the same vibrations they are monitoring making them self-sufficient.
Despite how relatively straightforward the idea is, until now, researchers have not made a concentrated effort to capture ambient vibrations. The main deterrent was the assumption that the vibrations generated by nature would not be consistent enough or at a low enough frequency to produce the oscillations useful for electrical energies. Harne used mathematical modeling to demonstrate that it is indeed possible for tree-like structures to control random, high frequency forces. Harne determined that internal resonance could be exploited to covert strong, external vibrations into large amplitude, low frequency vibrations capable of generating energy.
This research paves the path towards more innovative, self-sustaining energy systems.
Frost Gorder, Pam. “News Room614-292-5962.” Turning Good Vibrations into Energy. Ohio State University, 1 Feb. 2016. Web. 02 Feb. 2016. | https://energyvulture.com/2016/02/20/designing-energy-systems-to-be-more-like-trees/ |
It is a voluntary Chilean program, administered by the Energy Sustainability Agency, which seeks to certify and recognize the efforts made by freight transport companies in the field of sustainability and energy efficiency.
In addition, Giro Limpio Program certifies cargo generating companies that prefer Clean Turn certified carriers, thus contributing to reduce energy consumption and emissions of Greenhouse Gases (GHG) and other local pollutants that affect the health of people, reducing the environmental impact of the various value chains in our country.
Commitments
Progress Report
Certify with Giro Limpio Program the internal truck fleet and those of SQM collaborating companies
The company is part of this initiative that seeks to improve and strengthen energy efficiency in road freight transport, to reduce the consumption of polluting fuels and CO2e emissions, to contribute to more efficient, competitive and environmentally friendly production. . In this sense, SQM was certified in 2021 together with its collaborating companies in the load generator category, since it moves 50% through truck fleets that already have the seal by meeting the requirements established by the program. | https://www.sqm.com/en/certificacion/giro-limpio/ |
The Maverick Newsletter, Volume 4, Issue 7, July 1966
Description: Monthly newsletter published by the Maverick Region of the Porsche Club of America containing stories related to the organization or of interest to the group's members including news, upcoming and past events, other feature articles, and classified advertisements.
Date: July 1966
Creator: Porsche Club of America. Maverick Region. | https://texashistory.unt.edu/search/?q=&t=fulltext&fq=dc_type:text_journal&fq=str_title_serial:Maverick%20Newsletter&fq=str_month:07_jul |
Q:
If $\gcd(a+b,c)=1$ and $a+b+c$ divides $1-abc$, does it follow that $a\mid b$ or $a\mid c$ or $b\mid c$?
Is it true that: For any integers $(\mid a\mid, \mid b \mid, \mid c\mid) \geq 2$ such that $\gcd(a+b,c)=1$, if $a+b+c$ divides $1-abc$ then one of $a$, $b$, and $c$ is a multiple of another ?
A:
If you take $a=5, b=7, c=409$,
then $a+b+c=421$ divides $1-abc=-14314$ and $\text{gcd}(12, 409)=1$;
but neither of $a, b, c$ divides another.
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b 1. Field of the Invention
The present invention relates to speech processing. In particular, the present invention relates to an enhanced method for performing speech modeling and vector quantization in speech encoding applications.
2. Discussion of the Related Art
Linear Predictive Coding (LPC) techniques are widely used in speech encoding applications. In the prior art, to efficiently code LPC parameters into as few bits as possible, and to maintain a linear distortion performance over a wide range of values of LPC parameters, LPC parameters are sometimes represented in the frequency domain as line spectral frequencies (LPFs) using, for example, any of the methods disclosed in Chapter 4, entitled xe2x80x9cLPC PARAMETER QUANTIZATION USING LSFSxe2x80x9d, in Digital Speech Coding for Low Bit Rate Communication Systems by A. M. Kondoz, published by Wiley and Sons (1994) (xe2x80x9cDigital Speech Codingxe2x80x9d). The principle steps of one such method are illustrated by process 100 of FIG. 1. Under this method, at step 101, a set of coefficients is first estimated using linear prediction represented by a linear predictor model LP(n) of order l given by: LP ( n ) = ∑ i = 1 l α i s ( n - i )
where s(n) is value of the speech signal at time n, xcex1i is ith LPC coefficient such that the error e(n)=s(n)xe2x88x92LP(n) is minimized. In one instance, l is 10. Typically, in the encoding process, the LPC coefficients are extracted every update period, which can be a time period 20-30 milliseconds long.
Then, at step 102, from the xcex1i""s, two xc2xd-degree polynomials P(x) and Q(x) are constructed. Polynomials P(x) and Q(x) are given by the following: P ( x ) = ∑ i = 0 l / 2 a i x i Q ( x ) = ∑ i = 0 l / 2 b i x i
The coefficients ai and bi are each a function of the LPC coefficients xcex1i. The l roots of polynomials P(x) and Q(x) are a set of values ki (1xe2x89xa7kixe2x89xa7xe2x88x921), in which the odd indices ki""s (i.e., i=1, 3, 5, . . . ) are roots of polynomial P(x) and the even indices ki""s (i.e., i=2, 4, 6. . . ) are roots of polynomial Q(x), and ordered such that ki greater than ki+1. and are typically grouped into xc2xd xe2x80x9cline spectral pairsxe2x80x9d (LSPs), each LSP consisting of a pair (ki, ki+1). FIG. 3 shows an example of a 5th order polynomial P (x) having roots k1, k3, k5, k7 and k9.
LSPs are, however, non-linear parameters, which are not suitable for efficient quantization. In particular, if linear quantization steps are used, requisite resolution may not attained over some range of values, and wasteful for unnecessary resolution over some other range of values. Thus, at step 103, the LSPs are transformed into the frequency domain by taking the arc-cosine (i.e., cosxe2x88x921 ki) of each root ki. The resulting values of the transformation are referred to as xe2x80x9cline spectral frequenciesxe2x80x9d (xe2x80x9cLSFsxe2x80x9d).
At step 104, the LSFs are then quantized. In one instance, the LSFs are xe2x80x9cvector quantizedxe2x80x9d by using the LSF values to search a xe2x80x9ccode bookxe2x80x9d for an index which represents the set of quantized LSF values. For example, the 2-vector (cosxe2x88x921 k1, cosxe2x88x921 k3) can be used to search a 2-dimensional table in the code book. If 6 bits are allocated to represent such a pair, the 2-dimensional table has 64 entry corresponding to 64 pairs of selected possible values for (cosxe2x88x921 k1, cosxe2x88x921 k3). In one implementation, the index of the entry (xi, xj) for which the mean squared error (xixe2x88x92cosxe2x88x921 k1)2+(xj xe2x88x92cosxe2x88x921 k3)2 is minimum is selected to represent the 2-vector (cosxe2x88x921k1, cosxe2x88x921 k3). Higher dimensional tables are possible for vector quantizing a larger number of LSF values. For example, at three bits per root, a 3-dimensional table searchable by a 3-vector (cosxe2x88x921 k1, cosxe2x88x921 k3, cosxe2x88x921 k5) has 9-bit indices, or 512 entries. Of course, for the same per-root bit allocation (e.g., 3 bits per root), the storage requirements grow exponentially with the number of dimensions. In communication or storage applications, for example, the indices are transmitted or saved. At a later time, speech is synthesized or reconstructed (e.g., at the receiver side, or when replaying from storage) using a process that is substantially the reverse of process 100 discussed above.
In the method described above, finding the l roots of polynomials P(x) and Q(x) at step 102 is typically performed using numerical methods (e.g., Newton""s method) which can be computationally intensive. In one method, each root ki is found by evaluating P(k) or Q(k) for the trial values k between xe2x88x921 and 1, at increments of 0.0005. Such a method requires substantial amount computation which is undesirable in real-time applications.
The present invention provides a linear predictive speech encoding method which combines the quantization step with the search of roots of line spectral pair (LSP) polynomials. In one embodiment, according to one embodiment of the present invention, an indexed table having as entries quantized line spectral pair (LSP) values is created from a table of quantized line spectral frequencies (LSFs). Under a method of the present invention, during each update period, a set of LPC coefficients is computed to derive LSP polynomials P(x) and Q(x). However, instead of finding the roots of the polynomials P(x) and Q(x), polynomials P(x) and Q(x) are evaluated using the quantized LSP values of the indexed table. The approximate roots of the polynomials P(x) and Q(x) are selected from the entry of the indexed table whose quantized LSP values give the least error when used to evaluate polynomials P(x) and Q(x). The index of the selected entry of the table can be used to representing the approximate roots in the speech encoding application.
In one embodiment, the method selects the approximate roots by selecting such quantized LSP values that provide a least mean squared error in evaluating polynomials P(x) and Q(x). Further, under one method of the present invention, a step is taken to ensure that each selected LSP value corresponds to a designated root of the polynomials P(x) and Q(x). In one instance, the ensuring step is achieved by examining the direction of change in value of polynomial P(x) when successively decreasing LSP values for x are substituted into polynomial P(x). In one implementation, each of said polynomials P(x) and Q(x) is 5th-order.
According to another aspect of the present invention, a code book used in conjunction with the present invention can be organized as a number of multi-dimensional tables each representing vectors of quantized LSP values corresponding to multiple roots of the LSP polynomials. In one embodiment, the entries of each table of LSP values are arranged in a decreasing order of proposed LSP values in a designated root of the LSP polynomials.
Under the present invention, during run time, complex operations for searching the roots of the polynomials are avoided. Further, because the code book is prepared from an LSF code book, the desirable linear distortion performance of quantization in the LSF domain is preserved.
The present invention is better understood upon consideration of the detailed description below and the accompanying drawings.
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Do your students have trouble with elapsed time? It can be a tricky skill, but I have learned a few things that help my students to be successful!
1.) Read the problem once and then discuss or think about the story. Is it in order? What does the story tell you?
2.) Read the story a second time and think about SEE … S=Start time, E=Elapsed time, and E=End time. What does the story tell you and what does the question ask you?
3.) Make a number line that shows what you know.
4.) Use the number line to find what is missing. I use “mountains and hills”. The mountains are hours and the hills are minutes.
5.) For initial lessons, consider color coding. Try using green for start, red for end and any other color for elapsed. Eventually switch out to pencil, but refer back to the color coding for students who need that support.
For More Resources, Click the Photos Below: | https://learningwithmrskirk.com/2017/04/30/elapsed-time-success/ |
This allowed the resident a chance to overcome his concerns by talking to someone with the qualification to help efficiently and ability to maintain professionalism effectively. This outcome also permitted myself to maintain a boundary of professionalism and be aware of my own weaknesses and expand my own personal knowing to be able to deal with similar situations in the future in a better way.
Situation
Whilst on clinical placement, we were required to complete a Clinical Assessment Profile or CAP. During filling out a student objective sheet I became confused on whether the student objectives were supposed to be from a personal level of goals or had to be in accordance with the ANMC National Competency Standards for the Registered Nurse. As our facilitator was always available at the facility, I went to ask her if she knew what was expected of us and if so what were the specific requirements. Target
To use effective questioning skills to communicate my concerns to the facilitator about the assignment that was to be completed during clinical placement. Also to ensure the objectives I then chose were to the standards of a nursing student’s competencies. Activities
While speaking to the facilitator about my concerns I asked many open questions that allowed her to wholly answer my questions, for example, “What do we have to include as part of the objectives in this section?” Then when more specific answers were needed I asked a closed question that involved the answer being “yes” or “no”. I also used appropriate non verbal communication like eye contact, facial expressions and gestures when needed to portray my engagement. Results
As a result I achieved my target by having initiated and maintained an effective communication system of questioning and non verbal communication techniques with the facilitator. It also consequentially achieved my objectives that were at fault of my need for questioning and I completed my CAP booklet accurately.
Situation
During my placement a resident was having treatment on an ulcer wound to her lower left leg. The doctor prescribed a solution of neat vinegar to be sprayed each morning. This treatment was administered for 2 days, though by the third day the resident was complaining of severe pain during administration. The doctor was then asked to re-assess her and so changed the treatment to a solution of vinegar and water diluted to the ratio of 1:10. The nurse was not informed of this change. Target
To confer with the nursing staff, being part of the health care team, the change in treatment and allow the patient to continue with the correct management of wound care. Activities
As I was able to observe the wound management on a day to day basis, I noticed the current nurse who was administering the solution to the wound on the third day was not following the recent changes in the... | http://www.studymode.com/essays/Reflective-Essay-453470.html |
Atmospheric correction over case 2 waters with an iterative fitting algorithm: relative humidity effects.
In algorithms for the atmospheric correction of visible and near-IR satellite observations of the Earth's surface, it is generally assumed that the spectral variation of aerosol optical depth is characterized by an Angström power law or similar dependence. In an iterative fitting algorithm for atmospheric correction of ocean color imagery over case 2 waters, this assumption leads to an inability to retrieve the aerosol type and to the attribution to aerosol spectral variations of spectral effects actually caused by the water contents. An improvement to this algorithm is described in which the spectral variation of optical depth is calculated as a function of aerosol type and relative humidity, and an attempt is made to retrieve the relative humidity in addition to aerosol type. The aerosol is treated as a mixture of aerosol components (e.g., soot), rather than of aerosol types (e.g., urban). We demonstrate the improvement over the previous method by using simulated case 1 and case 2 sea-viewing wide field-of-view sensor data, although the retrieval of relative humidity was not successful.
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Jill Randall makes sculpture and installations, and has exhibited her work extensively in the UK and abroad. Much of the work is site-specific, and has included large-scale public art projects involving collaborations with architects and engineers, and Residencies in industry have increasingly become the starting-point and driver of the work. Group gallery exhibitions have included the Bluecoat Gallery, Liverpool, Cornerhouse, Manchester, Artists Space, New York, and “European Sculpture-Diversity and Difference in Practice”, Martini Arte Internazionale, Turin, Italy.Solo shows include ”Light Matter”, the Lowry, Salford. - a commission from the Lowry for new work produced from an Artist’s Residency at Magnesium Elektron, Salford, and “Secrets and Lives”, the Yard Gallery, Nottingham. Randall has been the recipient of many individual awards, including a Travel Scholarship to Barcelona from the McColl Arts Foundation, and a Royal Society of Arts “Art for Architecture” Award. She was Guest Artist-In-Residence at Claremont School of Art, Perth ,Western Australia in 1994, and major UK Residencies have included Grizedale Forest , Cumbria in 1999, and The Irwell Sculpture Trail in 2001. Randall has an important parallel practice working on major public art projects in collaboration with other professionals, 3 of which have won national awards. She is half of the “Dogs In Space” public art partnership with artist Alan Birch.Recent projects include” A13 Artescape” commission for Barking Town Centre, London, a Public Art Consultancy for Ordsall Hall, Salford ,“Valley of Stone “, Rossendale, Lancs, and a commission for “Hiddenplace”, Burnley.
Jill Randall is currently Programme Leader for B.A. Visual Arts Course as well as a practising artist, and her teaching has always been characterised by the “real-world” professional experience of the artist . Jill Randall has held a permanent post since 1997, and has over 20 years teaching experience, being promoted to 0.6 Senior Lecturer in August 2011, in recognition of the quality and volume of her research activities and the contribution these have made to teaching. She has held the post of Programme Leader since Jan 2012. Jill Randall is especially interested in professional development such as 'Live' placements for students, European Study Trips and Erasmus exchanges.
She was Level 5 co-ordinator from 2002-10, teaches across all 3 levels and has been involved in remodelling Level 5 of the course and curriculum development.
Site-specific sculpture and public art projects, Artists Residencies in industry using context as the starting-point and driver of the work.
Acting Associate Head (Research and Innovation) for the School of Art & Design for 1 Semester, 2011.
My research outputs are in the form of exhibitions, commissions, consultancies and publications, nationally and internationally. One of my current research projects is the subject of one of the School’s Impact Case Studies for the REF 2013, and its quality has been highly commended by the Pro VC for Research. I have won and realised many large scale public art commissions in the built environment collaborating with architects, landscape architects and engineers, 3 of which have won national awards, including an “Art for Architecture” Award from the Royal Society of Arts. Significant external funding and awards won in open competition include a Travel Scholarship from the McColl Arts Foundation, 4 Arts Council Awards, Artworks Wales Award, Foundation for Sport & The Arts and the National Lottery .
I have undertaken International residencies in Barcelona, and Perth ,Western Australia. Major commissions have included a Grizedale Forest Sculpture Residency, a commission for the Irwell Sculpture Trail, and the A13 Artescape Public Art Project in Barking, one of the largest public art projects in the world.
M.A. Fine Art (Sculpture).Manchester Polytechnic, 1982.
B.A. Hons Fine Art (Sculpture). Falmouth School of Art. 1981. Art History Prize for Dissertation.
Member, European Sculpture Network. 2005 -. Based Berlin, Germany.
Member, International Sculpture Centre. 1999- Based New Jersey, USA.
Visual Arts Representative, Rossendale Arts Alliance, Rossendale, Lancs. 2004-5.
Founder-member and Chair of SIGMA, Sculptors In Greater Manchester Association Studio Co-operative. 1983-2003.
Board of Trustees, A&E Artists In Education, Manchester. 2004-5.
Artists Space, New York, USA. 2002-.
2005- Member of European Sculpture Network.
2002- Member of Ixia - Public Art Forum.
1999 Director of Mart Network, Major Lottery-funded visual arts event Manchester .
*2011“Golden Venture”-Jill Randall- Artists Residency at Parys Mountain Copper Mine, Anglesey. To be published June 2011.
” A Sense of Place”, ISBN: 978-0-901507-04-4. Publ. University of Leicester 2008.
2005 “Jill Randall. Secrets and Lives -An Artist’s Diary.” ISBN 0 905634 75 6.
*1996 “Terra Incognita”. ISBN 0 9529470 0 5.
1992 “Artists Handbook: Across Europe”. Spain’ chapter, Commissioned by “AN Publications”. ISBN 0 907730 159.
Inclusion in book “Public Sculpture of Greater Manchester “ - Public Sculpture of Britain .Vol. 8. By Terry Wyke. Refs pages 275, 458,170,196-7,197 Full page photo of sculpture “Arresting Time” , Irwell Sculpture Trail, notes and biography. ISBN 0-85323-567-8. Published 2004.
European Sculpture Parks book. P195 “Guia de Europa Parques de Esculturas “,Fundacion NMAC Barcelona.
Private Collections in Spain, USA, Portugal, Australia, France, Greece, UK. | https://www.salford.ac.uk/arts-media/our-staff/arts-media-academics/jill-randall |
We drive inclusion and belonging to achieve innovation, equity, and excellence in the global community. We achieve this by focusing on three commitments highlighted below.
Our offices
Office of Diversity, Equity & Inclusion
Luis Sotelo, Vice President for Diversity, Equity, and Inclusion
402.826.8116
[email protected]
Leah Cech, DEI Program Coordinator
402.826.8118
[email protected]
Office of Religious and Spiritual Life
Religious and Spiritual Life provides resources and connections for those of all religious, secular, and spiritual identities among our students, faculty, and staff. We also recognize an individual's additional diverse identities that may play a role in how they seek spiritual growth and a religious community. Our office is here to engage with you on all religious, secular and spiritual questions and to help you make connections in our communities. We can also refer you to one-on-one connections in our communities or for one-on-one sessions with a Spiritual Director.
Leah Cech, Director
402.826.8118
[email protected]
Campus Advocacy, Prevention, and Education (CAPE) Project
The CAPE Project for Doane University will shift campus and community social norms to increase trauma-informed networks of support for survivors of interpersonal violence while decreasing incidents of sexual assault, dating/domestic violence, and stalking through inclusive prevention and education programs.
Get in touch with CAPE Project: [email protected]
Office of Veteran and Military Services
Assimilating to college life can be difficult, especially after serving in the armed services. Our Office of Veteran and Military Services serves as a resources to our military-connected students to ensure a smooth and successful transition.
Sarah McNeel, Director
402.467.9070
[email protected]
Nexus Center for Inclusive Excellence
The Nexus Center serves as the home of the DEI Division offices. We’d love for you to drop by and visit us on the lower level of Perry Campus Center. | https://www.doane.edu/student-life/diversity-equity-and-inclusion |
1. Field
Embodiments of the invention generally relate to the field of automated translation of natural-language texts using linguistic descriptions and various applications in such areas as automated abstracting, machine translation, natural language processing, control systems, information search (including on the Internet), semantic Web, computer-aided learning, expert systems, speech recognition/synthesis and others.
2. Related Art
The ability to understand, speak and write one or more languages is an integral part of human development to interact and communicate within a society. Various language analysis/synthesis approaches have been used to dissect a given language, analyze its linguistic structure in order to understand the meanings of a word, a sentence in the given language, extract information from the word, the sentence, and, if necessary, translate into another language or synthesize into another sentence, which expresses the same semantic meaning in some natural or artificial language.
Prior machine translation (MT) systems differ in the approaches and methods that they use and also in their abilities to recognize various complex language constructs and produce quality translation of texts from one language into another. According to their core principles, these systems can be divided into the following groups.
One of the traditional approaches is based on translation rules or transformation rules and is called Rule-Based MT (RBMT). This approach, however, is rather limited when it comes to working with complex language phenomena. In the recent years no significant breakthroughs have been achieved within this field. The best known systems of this type are SYSTRAN (SYSTRAN S. A., Paris, France), PROMT (PROMT OOO, Sankt Petersburg, Russian Federation) and ETAP-3 (Institute For Information Transmission Problems, Moscow, Russian Federation). The known RBMT systems, however, usually possess restricted syntactic models and simplified dictionary descriptions where language ambiguities are artificially removed.
Rule-based concept has evolved into Model-Based MT (MBMT) which is based on linguistic models. Implementing a MBMT system to produce quality translation demands considerable effort to create linguistic models and corresponding descriptions for specific languages. Evolution of MBMT systems is connected with developing complex language models on all levels of language descriptions. The need in today's modern world requires translation between many different languages. Creating such MBMT systems is only possible within a large-scale project to integrate the results of engineering and linguistic research.
Another traditional approach is Knowledge-Based MT (KBMT) which uses semantic descriptions. While the MBMT approach is based on knowledge about a language, the KBMT approach considers translation as a process of understanding based on real knowledge about the World. Presently, interest in Knowledge-Based Machine Translation (KBMT) has been waning.
Example-Based MT (EBMT) relates to machine translation systems using automated analysis of “examples”, which is very similar to Statistics-Based MT (SBMT). The best known systems of this type is Google-translator (Google, Inc., Mountain View, Calif., USA), as well as translation engines with language-specific rules-based elements, such as Microsoft Bing Translator (Microsoft, Inc., Redmond, Wash., USA) and Yahoo Babelfish (Yahoo! Inc., Sunnyvale, Calif., USA). In recent years, the SBMT approach has received a strong impetus from the following factors: appearance of Translation Memory (TM) systems and availability of powerful and relatively affordable bilingual electronic resources, such as TM databases created by corporations and translation agencies, electronic libraries, and specialized Internet corpora. The TM systems have demonstrated their practical efficiency when translating recurrent text fragments on the basis of minimal knowledge about languages such that researchers and developers are encouraged to try and create advanced and relatively exhaustive SBMT and HBMT (Hybrid Based MT) systems.
Most machine translation systems, both rule-based and statistics-based, concentrate on proper transfer of language information directly between a source sentence and an output sentence and usually do not require any full-fledged intermediary data structures to explicate the meaning of the sentence being translated. For example, a system based on linguistic models would know how to build thousands of syntactic variants of verb phrases-constituents. A system which is based on purely statistical approach would not know anything about the connections between these variants and would not be able to obtain a correct translation of one phrase on the basis of another. In addition, most-used probabilistic (statistic) approaches and statistics-based systems have a common drawback of taking no consideration of semantics. As a result, there is no guarantee that the translated (or generated) sentence has the same meaning as the original sentence.
Thus, even though some linguistic approaches have been proposed, most of them have not resulted in any useful algorithms or industrial applications because of poor performance in translating complete sentences. Complex sentences, which may express different shades of meaning, or the author's attitude and/or have different styles or genre, or which may be very long and contain various punctuation marks and other special symbols, have not been successfully generated/translated by prior art systems, language generation programs, or machine translation systems. It is especially difficult to translate or generate complex sentences, such as those found in technical texts, documentation, internet articles, journals, and the like and is yet to be done.
Accordingly, there are many ways to improve the methods and systems for translating natural language sentences between languages.
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Q:
How to determine $x$ and $y$ intercepts for $y = 4(x - 2)^2(x + 2)^3$
I need help to determine $x$ and $y$ intercepts for
$$
y = 4(x - 2)^2(x + 2)^3
$$
I guess my first question is, do I need to get the equation into
$$
ax^3 + bx^2 + cx + d
$$
form before starting?
A:
The $x$-intercepts are defined to be the points where $y=0$. Since what you have is factored, it is easy to see where $y=0$. If $y=0$ then one of the factors must be $0$. The $y$-intercept is where $x=0$. If it exists, it is unique. Just plug in $x=0$ and see what you get.
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Package 2-3: Measuring muffins
In this task your child will develop measurement skills through baking muffins or another recipe of your choice.
Week 3 - Package 3 - Year 5 and 6 Mathematics - Measuring muffins
Things you need
Have these things available so your child can complete this task.
Ideal
Ingredients for Blueberry Muffin recipe (see below)
Sieve, large bowl, wooden spoon, spoon, oven, 12 hole muffin tin, skewer ,wire rack
Measuring jug, weighing scales
Oven gloves
Paper and pencil
Counter, coins or other small set of countable objects such as Lego bricks
Back up
If your child has allergies or does not like blueberries use one of your own recipes for muffins or biscuits. The recipe should be in metric measurements – grams, millilitres
See your own recipe for requirements
Conversion measurements if needed
Before you start
This is a fun activity that is also an opportunity to spend some time being creative with your child. It is important to allow enough time for the practical activity so that it doesn’t become stressful. Also you may want to decide ahead of time who is going to be responsible for cleaning the dishes and who will clean the work space.
Remember this is a shared activity and you will be using a hot oven. If you prefer you could take charge of putting things in and taking things out of the oven.
Make sure you have all of the ingredients and equipment ready for your activity and a damp cloth or two for if things get messy.
What your child needs to know and do
Your child is going to help you bake muffins or another recipe of your choice that makes muffins or biscuits.
Blueberry Muffin Recipe
295g self-raising flour
90g salted butter
150g brown sugar
125g fresh blueberries
250ml milk
2 eggs, lightly beaten
Method
Preheat the oven to 180 degrees C (160 degrees fan forced) and grease a 12-hole muffin tin.
Sift the flour into the large bowl. Using fingertips, rub the butter into the flour until the mixture looks like fine breadcrumbs. Stir in the sugar.
Make a well in the centre of the flour mixture and pour in the blueberries, milk and lightly beaten eggs. Gently stir until just combined.
Carefully spoon the mixture into the greased tin.
Bake for 25 minutes or until a skewer inserted in centre of 1 muffin comes out clean.
Leave the muffins in the tin for 5 minutes before turning out onto a wire rack to cool.
What to do next
Before the muffins are shared and eaten, but after the washing up has been done ask your child to consider the following questions.
Imagine that the muffins have been baked for a family of four. How many would each person in the family get?
What other sized groups of people could you divide the muffins between and ensure that everyone got at least 1 muffin?
Ask your child if they would like to bake the muffins for their class at school when everyone is able to go back. The recipe gives the ingredients to make enough batter for 12 muffins. How many batches of the recipe would they have to make in order to make sure everyone in their class (and the teacher of course) gets at least one whole muffin?
Level 1: If you think this is going to be tricky for your child, stick to whole multiples, for example, they might double the recipe for 24 or triple for 36.
Ask your child then to rewrite the recipe so there is enough of each ingredient to make 2 or 3 batches.
On completion ask your child if this is a practical way of baking. Would all of the ingredients fit into the bowl? Do you have another muffin tin? Would it be better to bake the batches one after the other and leave the recipe as written?
Level 2: For students who are confident with decimals and fractions they could rewrite the recipe to make exactly 30 muffins.
On completion ask your child if this is a practical way of baking. Would all of the ingredients fit into the bowl? Do you have another muffin tin? Would it be better to bake the batches one after the other and leave the recipe as written?
Eat, and enjoy, the muffins.
Options for your child
Activity too hard?
You may need to support your child with reading the scale when measuring the ingredients.
Give your child 12 counters, coins or lego bricks to divide into as many equal groups as they can.
See option 1 above.
If your child has difficulty multiplying the amounts in the recipe by 2 or 3 ask if they can think of another way of doing it. If necessary a calculator can be used.
Activity too easy?
What would happen if there were 5 people in your family and everyone wanted their fair share? Draw a diagram of how you would share the muffins so that everyone got an equal share. This could take some time. What is the exact number of muffins that each person gets? This question could be answered as a fraction or as a decimal fraction. See option 2 above.
Extension/Additional activity
Counting the cost
Ask your child to draw up the following table on a piece of paper or on their device. Alternatively you could print out this table.
|Ingredient||Cost per item||Cost for 12 muffins||Cost for 48 muffins|
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Blueberries
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Milk
|
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Brown Sugar
|
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Salted Butter
|
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Eggs
|
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Self Raising Flour
|
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Total Cost
Explain to your child that if you didn’t have any of these ingredients in your pantry you would have to buy them as packaged. This would mean that the cost for 12 muffins would be the same as the cost for all of the ingredients. This means that working out the costs of baking isn’t as straightforward as it might seem.
Check that your child understands that 48 muffins would be four times the original number of muffins. For this column they will need to refer back to the original recipe and work out how much they will need to make 4 times the amount. Then they will need to refer back to the package size to work out the costing.
Was your child surprised by how little difference in cost there is for making 12 muffins and making 48 muffins? Can they explain why? | https://education.nsw.gov.au/teaching-and-learning/learning-from-home/learning-at-home/learning-packages/year-5-and-6-learning-packs/maths/pacckage-3-measuring-muffins |
Looking for stock on 256K 9S12 parts.
Does anyone know of a North American distributor with stock of 256K 9S12 parts? Can be A or D series.
I can't find any and the lead times are into November.
I don't think you're going to find any. We're having the same problem. We are in the process of modifying hardware/software so that we can use either 128/256/512 parts. The 128 and 512 parts are still readily available, for now. | https://community.nxp.com/thread/16501 |
Q:
How to merge other rows of data frame to the current row with Python/Pandas
I have a data frame that looks something like this:
A1 A2 A3 A4
1001 1002 1003 1004
5001 5002 5003 5004
7001 7002 7003 7004
I would like to merge the other rows to the current row to look like this. For Eg: For the first row the first four columns remain the same but the columns B1 to B4 are copy of 2nd row from A1 to A4 and C1 to C2 are copy of 3rd row from A1 to A4. Similar merging for the 2nd and 3rd row.
A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
1001 1002 1003 1004 5001 5002 5003 5004 7001 7002 7003 7004
5001 5002 5003 5004 7001 7002 7003 7004 1001 1002 1003 1004
7001 7002 7003 7004 1001 1002 1003 1004 5001 5002 5003 5004
I have tried multiple things like groupby, indexing, icol, loops etc but unable to get the desired result.
A:
If you have already created the other two dataframes, you can just follow the concat code. Else you can create samples of the same dataframe based on np.random.permutation like below:
>>df1
A1 A2 A3 A4
0 1001 1002 1003 1004
1 5001 5002 5003 5004
2 7001 7002 7003 7004
df2 = df1.iloc[np.random.permutation(len(df1))]
df2.columns=['B{}'.format(i) for i in range(1, len(df1.columns) + 1)]
>>df2
B1 B2 B3 B4
1 5001 5002 5003 5004
0 1001 1002 1003 1004
2 7001 7002 7003 7004
df3 = df2.iloc[np.random.permutation(len(df2))]
df3.columns=['C{}'.format(i) for i in range(1, len(df1.columns) + 1)]
>>df3
C1 C2 C3 C4
2 7001 7002 7003 7004
0 1001 1002 1003 1004
1 5001 5002 5003 5004
Once you have the dataframes ready, you can concat them on axis=1 like:
pd.concat([df1,df2,df3],axis=1)
A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
0 1001 1002 1003 1004 5001 5002 5003 5004 7001 7002 7003 7004
1 5001 5002 5003 5004 7001 7002 7003 7004 1001 1002 1003 1004
2 7001 7002 7003 7004 1001 1002 1003 1004 5001 5002 5003 5004
Note This process is on permutations so you can expect equal number of combinations which would not be same everytime the code is ran.
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How To Make A Wooden Frontgate Mailbox
Frontgate Mailbox – When you need a new mailbox, there are several different types in a large price range. However, making your own rustic mailbox can be simple and convenient. In less than an hour you can have a traditional mailbox made easy to make ready to use.
Drill a 1-foot (30 cm) deep hole large enough for a 4 by 4-inch (10 by 10 cm) post. Place the post inside the hole and fill it with soil. Pack the soil very well around the pole until it does not move when pushed. Hold two of the 1-foot (30 cm) boards 6 inches (15 cm) apart, each resting on the 1-inch (2.5 cm) width along its 1-foot (30 cm) side. Place another 1-foot (30 cm) board over the top of the other two and line up the edges to start forming the box. Place a nail through the top table and inside each side. Turn the box. Center the frontgate mailbox on the post and the nails through the bottom and inside the post to join it.
13 Photos Gallery of: How To Make A Wooden Frontgate Mailbox
Place the last 1-foot (30 cm) table on the top of the box and a nail through it to join them. Add the backside by placing one of the 8-inch boards in the back of the box. Keep the box lined up on all sides and put a nail to join it. Place your hinge in the center of the board 8 inches (20 cm) on the side 6 inches (15 cm). Screw the hinge on the table. Hold the door aligned with the box. Double check the frontgate mailbox placement of your door and attach it to the other side of the hinge in the base of the box.
Glue the handle by marking a point on the inside of the gate. Screw through the door from the inside until the tip comes out. Place the handle on the screw and continue until it does not move. Attach the lock of the magnetic door by screwing it on the inside along the top of the frontgate mailbox. Find the matching touch point inside the door and secure the other half of the lock at that point.
Add the frontgate mailbox flag according to the manufacturer’s instructions. Flags can be found at home improvement stores or online. Different flags include different frame parts, so choose the one you want to install. Note that most flags are placed on the right side of the mailbox near the front.
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Mean- The average. S- Sample Standard Deviation, if the list of values you entered is only a sample from the entire population, this is the best estimation for the It is free online. It counts the standard deviation, variance, mean, sum, and confidence interval approximations for given numbers. Enter a data set up to 5000 It can do all the basics like calculating quartiles, mean, median, mode, variance, standard deviation as well as the correlation coefficient. You can also do almost Calculate online the standard deviation of a set of values, the exact result is The standard deviation calculator supports numeric expressions but also literal. The average function allows to calculate the arithmetic mean of a series of values. getcalc.com's Standard Deviation (s) Calculator is an online statistics how many standard deviation, the whole elements of data distributed around the mean in Step 3: Square each deviation from mean. Squared negatives become positive. Step 4: Sum the squared deviations (Add up the numbers from step 3). Step 5: 11 Sep 2019 There are many A/B testing statistical significance calculators online; some test it using the standard mean and standard deviation approach.
Algorithms for calculating variance play a major role in computational statistics. A key difficulty This is particularly bad if the standard deviation is small relative to the mean. However For such an online algorithm, a recurrence relation is required between quantities from which the required statistics can be calculated in a
How to calculate standard deviation? Method of calculate standard deviation step by step. Find out the mean (µ) of the given data. Subtract the mean (µ) from each given value the result will be called the deviation from the mean. Take square of the each deviation of the mean. Find out the summation of the taken squares Free standard deviation calculator online: calculates the sample standard deviation and the population standard deviation based on a sample. Standard deviation for binomial data (proportion) based on sample size and event rate. The Standard Deviation Calculator is used to calculate the mean, variance, and standard deviation of a set of numbers. Standard Deviation. Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how precise your data is. The standard deviation is easier to relate to, compared to the variance, because the unit is the same as for the original values. A variance of 13 years² correspond to a standard deviation of approximately 3.61 years. Note that this does not mean that the average deviation from the mean is 3.61 years. The standard deviation, denoted by the greek letter sigma, \( \sigma \), is a measure of how much a set of numbers varies from the mean, \( \mu \). It is calculated using the following equation, which can look intimidating but can be broken up into smaller steps that are easier to understand. Standard Deviation. Standard deviation is used when it comes to statistics and probability theory. It is used in order to measure both variability and diversity and will show the precision of your data. When you calculate standard deviation, you are calculating the square root of its variance. Standard deviation calculator calculates the sample standard deviation from a sample `X : x_1, x_2, . . . , x_n`, using simple method. It’s an online Statistics and Probability tool requires a data set (set of real numbers or valuables). The result will describe the spread of dataset, i.e. how widely it is distributed about the sample mean.
A free on-line calculator that estimates sample sizes for a mean, interprets the results and creates visualizations and tables for Expected Standard Deviation.
A common estimator for σ is the sample standard deviation, typically denoted by s. It is worth noting that there exist many different equations for calculating sample standard deviation since unlike sample mean, sample standard deviation does not have any single estimator that is unbiased, efficient, and has a maximum likelihood. Standard Deviation Calculator Download App Standard deviation is a measure of spread of numbers in a set of data from its mean value. Use our online standard deviation calculator to find the mean, variance and arithmetic standard deviation of the given numbers. Code to add this calci to your website Used in various fields like analysis of a given data. Also, it is universally denoted by σ (Lowercase Sigma of Modern Greek alphabet). Manually determining the Standard Deviation of big data is a long process. The best way is to use the Online Standard Deviation Calculator with mean value, variance, and formula.
Definition: The standard deviation measures how close the set of data is to the mean value of the data set. If data set have high standard deviation than the
Free online standard deviation calculator and variance calculator with steps. The bulk of data (68%) lies within one standard deviation from the mean. Like this The Standard deviation calculator that helps in calculating the Mean, Variance and Standard Deviation for the given data sets. Solve your statistics problems Free standard deviation calculator online: calculates the sample standard to use stdev calculator, that also outputs variance, standard error of the mean (SEM) , Combined Mean and Combined Standard deviation calculator - Find Combined Mean and Combined Standard deviation from number of Observations, Average
Use this Mean and Standard Deviation Calculator to get step-by-step calculation of the sample mean, variance and standard deviation.
In short, the relative standard deviation is also known as RSD. The RSD is always measured in percentage. It can be done by simply dividing the standard deviation by mean and multiplying by 100. i.e with the formula, In the below online relative standard deviation calculator, enter the given set of data and then click calculate to find the output. | https://mt4forexwti.web.app/fexawisivibyl/online-standard-deviation-and-mean-calculator84.html |
Diffusion tensor imaging, a variant of MRI, detects the random movement of water through brain tissue at the microscopic level.
Using the technique, Giovanni Carlesimo, MD, PhD, of Tor Vergata University and the Santa Lucia Foundation in Rome, and colleagues found that mean diffusivity in the hippocampus, and not hippocampal volume, predicted performance on memory tests among healthy adults. Mean diffusity can be used to track pathologic changes in white and gray matter.
The findings, reported online in Neurology, "imply that memory deficits in healthy individuals have a biologic underpinning without apparent tissue loss," Norbert Schuff, PhD, of the Veterans Affairs Medical Center in San Francisco, wrote in an accompanying editorial.
Action Points
- Explain to interested patients that this study compared conventional MRI with a new MRI variant in their ability to predict changes in memory in a group of healthy individuals.
- Point out that this new MRI variant might better detect very early changes indicative of Alzheimer's disease, although it remains unclear whether any of the participants in this study will actually develop dementia.
"Alterations in the brain's microstructure may potentially provide new clues for a separation of normal aging from pathology," he said.
"However, unless prospective studies are conducted, the issue remains open whether mean diffusivity is better as an early predictor of Alzheimer disease than volume."
Previous studies have shown that mean diffusivity in the hippocampus is significantly greater in patients with amnestic mild cognitive impairment or Alzheimer disease than in healthy individuals.
Although none of the individuals in the current study met criteria for mild cognitive impairment or more severe cognitive deficits, "in a number of individuals beyond their 50s, high mean diffusion values in the hippocampal formation were accompanied by performance scores that were not truly pathologic but fell in the the lower portion of the normal range on tests of declarative memory," Carlesimo and colleagues noted in the paper.
"The relevant question here is whether these individuals represent a very early stage in the progression of Alzheimer disease, anterior to the development of an amnestic mild cognitive impairment, or whether, instead, they represent the lower portion of the normal distribution formed by aged individuals who will not convert to dementia."
They said a longitudinal study is needed to settle the question.
Carlesimo and colleagues recruited 76 healthy people ages 20 to 80 (mean age 50.6) to undergo MRI scans with diffusion tensor imaging.
They were also given a neuropsychological assessment that included tests of verbal and visuospatial memory.
All anatomic measures -- hippocampal mean diffusivity and fractional anisotropy, total brain volume, and hippocampal volume -- were significantly correlated with age in the overall group (P<0.05 for all).
Mean diffusivity values were fairly stable with increasing age up to 50, after which they began a steady rise.
In the overall cohort, high mean diffusivity values in both the left and right halves of the hippocampus were associated with low scores on the verbal memory test (P<0.05).
Broken down by age, however, there were no significant associations between any of the anatomic measures and performance on either memory test in those younger than 50.
In individuals 50 and older, high hippocampal mean diffusivity, but not hippocampal fractional anisotropy, hippocampal volume, or total brain volume, significantly predicted performance on both memory tests (P<0.05 for both).
The effect was confined to changes in the left half of the hippocampus.
In an interview, the editorialist Schuff said, "Potentially this new imaging variant, diffusion tensor imaging, may be a very sensitive marker to identify very early changes that potentially could be related to Alzheimer disease."
But he said future studies would have to establish the sensitivity and specificity of the measure before widespread clinical application.
In the short term, he said, diffusion tensor imaging might be used to power clinical trials of potential Alzheimer disease therapies targeted very early in the disease process, when treatment would be assumed to have the greatest benefit.
Disclaimer
The study was funded by the Italian Ministry of Health.
Carlesimo reported receiving research support from the Italian Ministry of Health. His co-authors reported conflicts of interest with the Italian Ministry of Health, Wyeth, and Novartis.
Schuff reported receiving travel expenses for educational activities not funded by industry and receiving research support from Synarc, the NIH, the U.S. Department of Defense, and the Michael J. Fox Foundation.
Primary Source
Neurology
Source Reference: Carlesimo G, et al "Hippocampal mean diffusivity and memory in healthy elderly individuals" Neurology 2010; 74: 194-200.
Secondary Source
Neurology
Source Reference: Schuff N "A new sensitive MRI marker for memory deficits in normal aging" Neurology 2010; 74: 188-89. | https://www.medpagetoday.com/Neurology/AlzheimersDisease/17818 |
After riding Space Mountain, we decided to head to Fantasyland. Mad Tea Party had a really short lineup, the sign said the queue was 15 minutes but it was actually only about 5 minutes.
As we were exiting the ride, Pooh and Tigger walked past us, so we decided to follow them to their meet and greet area. There were only a few people in line already, and I love those characters so we hopped in line for our first character meet of our trip!
After meeting Pooh and Tigger, it was time for our third fastpass of the day – Seven Dwarfs Mine Train. It was a cute ride for sure, but way too quick and I can’t imagine waiting in line for over an hour for it. I also realized way too late that I didn’t get my on-ride picture and video
Since we had now used up our three pre-selected Fastpasses, we headed to a kiosk to pick out a 4th one. We ended up in line behind people who were completely clueless, and had no idea what they were doing or what they wanted to select. The Cast Member was very patient, but you could tell they were getting a little frustrated too, since the people were looking at their phone, and would say they wanted a fastpass for a ride that wasn’t even in Magic Kingdom. Eventually we made it to the front and selected a 4th fastpass for Pirates of the Caribbean.
We checked wait times on the Disney app, and decided to head to Ariel’s Undersea Adventure. The wait time said 25 minutes, and it was pretty much exact.
The ride itself is basically the exact same as the one at Disney California Adventure, but the exterior is really impressive at WDW. I loved how it looked like you were entering a cave while waiting in the queue line. I also apologize for continuously comparing the rides in Florida and California, but I had been to Disneyland first..
We walked over to the Storybook Circus area to check on the wait for Dumbo – only 15 minutes so into another line we get!
From there, we hopped on the Railroad – with zero wait time. We rode it all the way around to Frontierland. We still had time to kill before our FP+ window opened for POTC, so off to the Enchanted Tiki Room! We didn’t have time to check it out when we were in California (either time as it so happens). We got into line and as we walked through the queue line the doors opened up – perfect timing once again! I have to say – seeing that once is enough. Our FP window was open by the time we left the Tiki Room; in less than 5 minutes we were on a boat off to see some Pirates. Once again, another ride which is way better at Disneyland. The ride ended, and Luke and I just looked at each other and said “that was it?!”. It’s way too short at WDW. | http://fallonkendra.com/2017/02/04/january-6-2016-magic-kingdom/ |
Q:
What is the biological significance of finding palindromes in DNA sequence?
I found a function called palindromes in Matlab that finds palindromes from DNA sequence. Now what is the biological intention behind incorporating this function? What the biological significance of finding palindrome in DNA sequences?
A:
My knowledge of biology is extremely limited, but this is what I know of palindromic sequences:
Palindromic sequences are their own reverse complements. I have seen many restriction sites be palindromic. Also, some Transcription Factor Binding sites are palindromic. The canonical E-box site, for example, can be expressed as CANNTG.
Also, these palindromes, when occurring in pairs with a sequence between the occurrences, can double back and bind to each other (when the DNA is transcribed to RNA, say), and this would result in the RNA stem and loop structure.
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However, the average paralegal hourly rate of pay for a paralegal in the United States is $21.97. This range is similar to the $25.44 per hour 2020 median hourly pay for paralegals and legal assistants presented by the U.S. Bureau of Labor Statistics.
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Paralegal – $100 to $200 per hour.
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How much does a Paralegal make? While ZipRecruiter is seeing hourly wages as high as $32.45 and as low as $11.30, the majority of Paralegal wages currently range between $17.79 (25th percentile) to $25.00 (75th percentile) across the United States.
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Many law firms have minimum billable hour requirements, somewhere between 1,800 and 2,200 hours per year for first-year associates, according to the National Law Review.
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Most law firms have their attorneys bill time in one-tenth hour increments, with the smallest time increment possible at 0.10-hour. … One hour “on the clock” breaks down into 10 six-minute standard billing increments, making the shortest time possible to perform a task six minutes.
Is Paralegal a good paying job?
$78,478. Known also as government legal assistants, government paralegals make an average of $78,478 per year and are typically employed by federal, state, and local government law offices. Typical job duties of a government paralegal include: researching legislation and regulations.
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Salary by Paralegal Type
There are various types of paralegals, and some have higher earning potential than others. According to the U.S. Bureau of Labor Statistics (BLS), the 2019 median pay for paralegals and legal assistants was $51,740 per year. The highest 10% earned more than $82,000.
How much will a lawyer charge to write a letter?
According to our database of legal fees, an attorney practicing on their own will charge anywhere between $750 and $1,200 to write and send a demand letter. A smaller law firm will charge anywhere from $1,000 to $1,500 for their services.
Is a paralegal a white collar job?
This is occurring in part because many men in blue-collar jobs found themselves on the unemployment line when the recession began in 2008. … But now, it refers to any class of job traditionally filled by women, like nurse, teacher, receptionist, paralegal, social worker and personal aide.
What is the difference between paralegal and legal assistant?
Paralegals are more involved with the actual technicalities of the law, whereas legal assistants undertake broader tasks. If you are looking for a more hands-on law career, becoming a paralegal may interest you more.
Are paralegals overworked?
Being a paralegal is stressful, and paralegal burnout is real. … Because of this, paralegals’ bosses—who are often stressed-out attorneys—can be demanding perfectionists who require an extremely high standard of work from their legal support staff. A mistake or omission can be incredibly costly—to the firm and attorney.
How do paralegals increase billable hours?
Here are five things you can do to make sure you’re maximizing your billable hours:
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abstract: 'The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in the sense that it only detects the equality of means and the traces of covariance matrices in the high-dimensional setup. We propose a new class of metrics which inherits the desirable properties of the energy distance and maximum mean discrepancy/(generalized) distance covariance and the Hilbert-Schmidt Independence Criterion in the low-dimensional setting and is capable of detecting the homogeneity of/completely characterizing independence between the low-dimensional marginal distributions in the high dimensional setup. We further propose t-tests based on the new metrics to perform high-dimensional two-sample testing/independence testing and study their asymptotic behavior under both high dimension low sample size (HDLSS) and high dimension medium sample size (HDMSS) setups. The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data. We demonstrate the superior power behavior of the proposed tests for homogeneity of distributions and independence via both simulated and real datasets.'
author:
- |
Shubhadeep Chakraborty\
Department of Statistics, Texas A&M University\
and\
Xianyang Zhang\
Department of Statistics, Texas A&M University
title: '**A New Framework for Distance and Kernel-based Metrics in High Dimensions**'
---
\#1
[*Keywords:*]{} Distance Covariance, Energy Distance, High Dimensionality, Hilbert-Schmidt Independence Criterion, Independence Test, Maximum Mean Discrepency, Two Sample Test, U-statistic.
Introduction
============
Nonparametric two-sample testing of homogeneity of distributions has been a classical problem in statistics, finding a plethora of applications in goodness-of-fit testing, clustering, change-point detection and so on. Some of the most traditional tools in this domain are Kolmogorov-Smirnov test, and Wald-Wolfowitz runs test, whose multivariate and multidimensional extensions have been studied by Darling(1957), David(1958) and Bickel(1969) among others. Friedman and Rafsky(1979) proposed a distribution-free multivariate generalization of the Wald-Wolfowitz runs test applicable for arbitrary but fixed dimensions. Schilling(1986) proposed another distribution-free test for multivariate two-sample problem based on $k$-nearest neighbor ($k$-NN) graphs. Maa et al.(1996) suggested a technique for reducing the dimensionality by examining the distribution of interpoint distances. In a recent novel work, Chen and Friedman(2017) proposed graph-based tests for moderate to high dimensional data and non-Euclidean data. The last two decades have seen an abundance of literature on distance and kernel-based tests for equality of distributions. Energy distance (first introduced by Székely (2002)) and maximum mean discrepancy or MMD (see Gretton et al.(2012)) have been widely studied in both the statistics and machine learning communities. Sejdinovic et al.(2013) provided a unifying framework establishing the equivalence between the (generalized) energy distance and MMD. Although there have been some very recent works to gain insight on the decaying power of the distance and kernel-based tests for high dimensional inference (see for example Ramdas et al.(2015a, 2015b), Kim et al. (2018) and Li (2018)), the behavior of these tests in the high dimensional setup is still a pretty unexplored area.
Measuring and testing for independence between two random vectors has been another fundamental problem in statistics, which has found applications in a wide variety of areas such as independent component analysis, feature selection, graphical modeling, causal inference, etc. There has been an enormous amount of literature on developing dependence metrics to quantify non-linear and non-monotone dependence in the low dimensional context. Gretton et al.(2005, 2007) introduced a kernel-based independence measure, namely the Hilbert-Schmidt Independence Criterion (HSIC). Bergsma and Dassios(2014) proposed a consistent test of independence of two ordinal random variables based on an extension of Kendall’s tau. Josse and Holmes(2014) suggested tests of independence based on the RV coefficient. Székely et al.(2007), in their seminal paper, introduced distance covariance (dCov) to characterize dependence between two random vectors of arbitrary dimensions. Lyons(2013) extended the notion of distance covariance from Euclidean spaces to arbitrary metric spaces. Sejdinovic et al.(2013) established the equivalence between HSIC and (generalized) distance covariance via the correspondence between positive definite kernels and semi-metrics of negative type. Over the last decade, the idea of distance covariance has been widely extended and analyzed in various ways; see for example Zhou(2012), Székely and Rizzo(2014), Wang et al.(2015), Shao and Zhang(2014), Huo and Székely(2016), Zhang et al.(2018), Edelmann et al.(2018) among many others. There have been some very recent literature which aims at generalizing distance covariance to quantify the joint dependence among more than two random vectors; see for example Matteson and Tsay(2017), Jin and Matteson(2017), Chakraborty and Zhang(2018), Böttcher(2017), Yao et al.(2018), etc. However, in the high dimensional setup, the literature is scarce, and the behavior of the widely used distance and kernel-based dependence metrics is not very well explored till date. Székely and Rizzo(2013) proposed a distance correlation based t-test to test for independence in high dimensions. In a very recent work, Zhu et al.(2018) showed that in the high dimension low sample size (HDLSS) setting, i.e., when the dimensions grow while the sample size is held fixed, the sample distance covariance can only measure the component-wise [*linear dependence*]{} between the two vectors. As a consequence, the distance correlation based t-test proposed by Székely et al.(2013) for independence between two high dimensional random vectors has trivial power when the two random vectors are nonlinearly dependent but component-wise uncorrelated. As a remedy, Zhu et al.(2018) proposed a test by aggregating the pairwise squared sample distance covariances and studied its asymptotic behavior under the HDLSS setup.
This paper presents a new class of metrics to quantify the homogeneity of distributions and independence between two high-dimensional random vectors. The core of our methodology is a new way of defining the distance between sample points (interpoint distance) in the high-dimensional Euclidean spaces. In the first part of this work, we show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in the sense that it only detects the [*equality of means and the traces of covariance matrices*]{} in the high-dimensional setup. To overcome such a limitation, we propose a new class of metrics based on the new distance which inherits the nice properties of energy distance and maximum mean discrepancy in the low-dimensional setting and is capable of detecting the [*pairwise homogeneity of the low-dimensional marginal distributions*]{} in the HDLSS setup. We construct a high-dimensional two sample t-test based on the U-statistic type estimator of the proposed metric, which can be viewed as a generalization of the classical two-sample t-test with equal variances. We show under the HDLSS setting that the new two sample t-test converges to a central t-distribution under the null and it has nontrivial power for a broader class of alternatives compared to the energy distance. We further show that the two sample t-test converges to a standard normal limit under the null when the dimension and sample size both grow to infinity with the dimension growing more rapidly. It is worth mentioning that we develop an approach to unify the analysis for the usual energy distance and the proposed metrics. Compared to existing works, we make the following contribution.
- We derive the asymptotic variance of the generalized energy distance under the HDLSS setting and propose a computationally efficient variance estimator (whose computational cost is linear in the dimension). Our analysis is based on a pivotal t-statistic which does not require permutation or resampling-based inference and allows an asymptotic exact power analysis.
In the second part, we propose a new framework to construct dependence metrics to quantify the dependence between two high-dimensional random vectors $X$ and $Y$ of possibly different dimensions. The new metric, denoted by $\cal{D}^2(X, Y)$, generalizes both the distance covariance and HSIC. It completely characterizes independence between $X$ and $Y$ and inherits all other desirable properties of the distance covariance and HSIC for fixed dimensions. In the HDLSS setting, we show that the proposed population dependence metric behaves as an aggregation of group-wise (generalized) distance covariances. We construct an unbiased U-statistic type estimator of $\cal{D}^2(X, Y)$ and show that with growing dimensions, the unbiased estimator is asymptotically equivalent to the sum of group-wise squared sample (generalized) distance covariances. Thus it can quantify [*group-wise non-linear dependence*]{} between two high-dimensional random vectors, going beyond the scope of the distance covariance based on the usual Euclidean distance and HSIC which have been recently shown only to capture the componentwise linear dependence in high dimension, see Zhu et al. (2018). We further propose a t-test based on the new metrics to perform high-dimensional independence testing and study its asymptotic size and power behaviors under both the HDLSS and high dimension medium sample size (HDMSS) setups. In particular, under the HDLSS setting, we prove that the proposed t-test converges to a central t-distribution under the null and a noncentral t-distribution with a random noncentrality parameter under the alternative. Through extensive numerical studies, we demonstrate that the newly proposed t-test can capture group-wise nonlinear dependence which cannot be detected by the usual distance covariance and HSIC in the high dimensional regime. Compared to the marginal aggregation approach in Zhu et al. (2018), our new method enjoys two major advantages.
- Our approach provides a neater way of generalizing the notion of distance and kernel-based dependence metrics. The newly proposed metrics completely characterize dependence in the low-dimensional case and capture group-wise nonlinear dependence in the high-dimensional case. In this sense, our metric can detect a wider range of dependence compared to the marginal aggregation approach.
- The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data.
*Notation*. Let $X = (X_1, \dots X_p) \in \bb{R}^p$ and $Y = (Y_1, \dots, Y_q)$ $\in \bb{R}^q$ be two random vectors of dimensions $p$ and $q$ respectively. Denote by $\Vert\cdot\Vert_p$ the Euclidean norm of $\mathbb{R}^p$ (we shall use it interchangeably with $\Vert\cdot\Vert$ when there is no confusion). Let $0_p$ be the origin of $\bb{R}^p$. We use $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$ to denote that $X$ is independent of $Y$, and use $``X \overset{d}{=} Y"$ to indicate that $X$ and $Y$ are identically distributed. Let $(X',Y')$, $(X'', Y'')$ and $(X''', Y''')$ be independent copies of $(X,Y)$. We utilize the order in probability notations such as stochastic boundedness $O_p$ (big O in probability), convergence in probability $o_p$ (small o in probability) and equivalent order $\asymp$, which is defined as follows: for a sequence of random variables $\{Z_n\}_{n=1}^{\infty}$ and a sequence of real numbers $\{a_n\}_{n=1}^{\infty}$, $Z_n \asymp_p a_n$ if and only if $Z_n/a_n = O_p(1)$ and $a_n/Z_n = O_p(1)$ as $n \to \infty$. For a metric space $(\cal{X}, d_{\cal{X}})$, let $\cal{M}(\cal{X})$ and $\cal{M}_1(\cal{X})$ denote the set of all finite signed Borel measures on $\cal{X}$ and all probability measures on $\cal{X}$, respectively. Define $\cal{M}^1_{d_{\cal{X}}}(\cal{X}):= \{v \in \cal{M}(\cal{X}) \,:\, \exists\, x_0 \in \cal{X} \; \text{s.t.}\;\int_{\cal{X}} d_{\cal{X}}(x,x_0)\, d|v|(x) < \infty\}$. For $\theta > 0$, define $\cal{M}_{\mathcal{K}}^{\theta}(\cal{X}):= \{v \in \cal{M}(\cal{X}) \,:\, \int_{\cal{X}} \mathcal{K}^{\theta}(x,x)\, d|v|(x) < \infty\}$, where $\mathcal{K}: \cal{X} \times \cal{X} \to \bb{R}$ is a bivariate kernel function. Define $\cal{M}^1_{d_{\cal{Y}}}(\cal{Y})$ and $\cal{M}_{\mathcal{K}}^{\theta}(\cal{Y})$ in a similar way. For a matrix $A = (a_{kl})_{k,l=1}^n \in \bb{R}^{n \times n}$, define its $\cal{U}$-centered version $\tilde{A} = (\tilde{a}_{kl}) \in \bb{R}^{n \times n}$ as follows $$\begin{aligned}
\label{U centering def}
\tilde{a}_{kl} = \begin{cases}
a_{kl}\, -\,\mathlarger{ \frac{1}{n-2}} \dis\sum_{j=1}^n a_{kj} \,-\, \frac{1}{n-2} \dis\sum_{i=1}^n a_{il}\, +\, \frac{1}{(n-1)(n-2)} \dis\sum_{i,j=1}^n a_{ij}, \; & k \neq l,\\
0, & k=l, \end{cases}\end{aligned}$$ for $k, l = 1, \dots, \, n$. Define $$(\tilde{A} \cdot \tilde{B}):=\frac{1}{n(n-3)} \dis \sum_{k\neq l} \tilde{a}_{kl} \tilde{b}_{kl}$$ for $\tilde{A} = (\tilde{a}_{kl})$ and $\tilde{B} = (\tilde{b}_{kl})\in \bb{R}^{n \times n}$. Denote by $\textrm{tr}(A)$ the trace of a square matrix $A$. $A \otimes B$ denotes the kronecker product of two matrices $A$ and $B$. Let $\Phi(\cdot)$ be the cumulative distribution function of the standard normal distribution. Denote by $t_{a,b}$ the noncentral t-distribution with $a$ degrees of freedom and noncentrality parameter $b$. Write $t_a=t_{a,0}$. Denote by $q_{\alpha, a}$ and $Z_{\alpha}$ the upper $\alpha$ quantile of the distribution of $t_{a}$ and the standard normal distribution, respectively, for $\alpha \in (0,1)$. Also denote by $\chi^2_{a}$ the chi-square distribution with $a$ degrees of freedom. Denote $U\sim$ Rademacher$(0.5)$ if $P(U=1) = P(U=-1) = 0.5$. Let $\mathbbm{1}_A$ denote the indicator function associated with a set $A$. Finally, denote by $\lfloor a \rfloor$ the integer part of $a\in\mathbb{R}$.
An overview: distance and kernel-based metrics {#sec:overview}
==============================================
Energy distance and MMD {#ed sec}
-----------------------
Energy distance (see Székely et al.(2004, 2005), Baringhaus and Franz(2004)) or the Euclidean energy distance between two random vectors $X,Y\in \bb{R}^p$ and $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$ with ${{\mathbb{E}}}\Vert X \Vert_p < \infty$ and ${{\mathbb{E}}}\Vert Y \Vert_p < \infty$, is defined as $$\label{ed def}
ED(X,Y) \;=\; 2\,{{\mathbb{E}}}\Vert X-Y \Vert_p - {{\mathbb{E}}}\Vert X-X'\Vert_p - {{\mathbb{E}}}\Vert Y-Y'\Vert_p \; ,$$ where $(X',Y')$ is an independent copy of $(X,Y)$. Theorem 1 in Székely et al.(2005) shows that $ED(X,Y) \geq 0$ and the equality holds if and only if $X \overset{d}{=} Y$. In general, for an arbitrary metric space $(\cal{X}, d)$, the generalized energy distance between $X \sim P_X$ and $Y \sim P_Y$ where $P_X, P_Y \in \cal{M}_1(\cal{X}) \cap \cal{M}^1_{d}(\cal{X})$ is defined as $$\label{ed def general}
ED_d(X,Y) \;=\; 2\,{{\mathbb{E}}}\,d(X,Y) - {{\mathbb{E}}}\,d(X,X') - {{\mathbb{E}}}\,d(Y,Y') \;.$$
\[negative type\] A metric space $(\cal{X}, d)$ is said to have negative type if for all $n\geq 1$, $x_1, \dots, x_n \in \cal{X}$ and $\alpha_1, \dots, \alpha_n \in \bb{R}$ with $\sum_{i=1}^n \alpha_i = 0$, we have $$\begin{aligned}
\label{def:neg type}
\sum_{i, j =1}^n \alpha_i\, \alpha_j\, d(x_i, x_j) \leq 0\;.\end{aligned}$$ The metric space $(\cal{X}, d)$ is said to be of strong negative type if the equality in (\[def:neg type\]) holds only when $\alpha_i = 0$ for all $i \in \{1,\dots, n\}$.
If $(\cal{X}, d)$ has strong negative type, then $ED_d(X,Y)$ completely characterizes the homogeneity of the distributions of $X$ and $Y$ (see Lyons(2013) and Sejdinovic et al.(2013) for detailed discussions). This quantification of homogeneity of distributions lends itself for reasonable use in one-sample goodness-of-fit testing and two sample testing for equality of distributions.
On the machine learning side, Gretton et al.(2012) proposed a kernel-based metric, namely maximum mean discrepancy (MMD), to conduct two-sample testing for equality of distributions. We provide some background before introducing MMD.
(RKHS)\[RKHS\] Let $\cal{H}$ be a Hilbert space of real valued functions defined on some space $\cal{X}$. A bivariate function $\mathcal{K} : \cal{X} \times \cal{X} \to \bb{R}$ is called a reproducing kernel of $\cal{H}$ if :
1. $\forall x \in \cal{X}, \mathcal{K}(\cdot,x) \in \cal{H}$
2. $\forall x \in \cal{X}, \forall f \in \cal{H},\; \langle f,\mathcal{K}(\cdot,x)\rangle_{\cal{H}} = f(x)$
where $\langle \cdot,\cdot\rangle_{\cal{H}}$ is the inner product associated with $\cal{H}$. If $\cal{H}$ has a reproducing kernel, it is said to be a reproducing kernel Hilbert space (RKHS).
By Moore-Aronszajn theorem, for every positive definite function (also called a kernel) $\mathcal{K}: \cal{X} \times \cal{X} \to \bb{R}$, there is an associated RKHS $\cal{H}_\mathcal{K}$ with the reproducing kernel $\mathcal{K}$. The map $\Pi : \cal{M}_1(\cal{X}) \to \cal{H}_\mathcal{K}$, defined as $\Pi(P) = \int_{\cal{X}} \mathcal{K}(\cdot,x) \, dP(x)$ for $P \in \cal{M}_1(\cal{X})$ is called the mean embedding function associated with $\mathcal{K}$. A kernel $\mathcal{K}$ is said to be characteristic to $\cal{M}_1(\cal{X})$ if the map $\Pi$ associated with $\mathcal{K}$ is injective. Suppose $\mathcal{K}$ is a characteristic kernel on $\cal{X}$. Then the MMD between $X \sim P_X$ and $Y \sim P_Y$, where $P_X, P_Y \in \cal{M}_1(\cal{X}) \cap \cal{M}^{1/2}_\mathcal{K}(\cal{X})$ is defined as $$\begin{aligned}
\label{MMD}
MMD_\mathcal{K}(X,Y)\;&=\; \Vert \,\Pi(P_X) \,-\, \Pi(P_Y) \,\Vert_{\cal{H}_\mathcal{K}}\,.\end{aligned}$$ By virtue of $\mathcal{K}$ being a characteristic kernel, $MMD_\mathcal{K}(X,Y)=0$ if and only if $X \overset{d}{=} Y$. Lemma 6 in Gretton et al.(2012) shows that the squared MMD can be equivalently expressed as $$\label{MMD equiv expr}
MMD^2_\mathcal{K}(X,Y) \;=\; {{\mathbb{E}}}\,\mathcal{K}(X,X') \,+\, {{\mathbb{E}}}\,\mathcal{K}(Y,Y') \,-\, 2\,{{\mathbb{E}}}\,\mathcal{K}(X,Y) \; .$$
Theorem 22 in Sejdinovic et al.(2013) establishes the equivalence between (generalized) energy distance and MMD. Following is the definition of a kernel induced by a distance metric (refer to Section 4.1 in Sejdinovic et al.(2013) for more details).
(Distance-induced kernel and kernel-induced distance)\[dist-induced kernel\] Let $(\cal{X}, d)$ be a metric space of negative type and $x_0 \in \cal{X}$. Denote $\mathcal{K}: \cal{X} \times \cal{X} \to \bb{R}$ as $$\begin{aligned}
\label{def:dist-induced kernel}
\mathcal{K}(x,x')\; =\; \frac{1}{2}\, \left\{d(x,x_0) + d(x',x_0) - d(x,x')\right\} .\end{aligned}$$ The kernel $\mathcal{K}$ is positive definite if and only if $(\cal{X}, d)$ has negative type, and thus $\mathcal{K}$ is a valid kernel on $\cal{X}$ whenever $d$ is a metric of negative type. The kernel $\mathcal{K}$ defined in (\[def:dist-induced kernel\]) is said to be the distance-induced kernel induced by $d$ and centered at $x_0$. One the other hand, the distance $d $ can be generated by the kernel $\mathcal{K}$ through $$\begin{aligned}
\label{eq-dist}
d(x,x')=\mathcal{K}(x,x)+\mathcal{K}(x',x')-2\mathcal{K}(x,x').\end{aligned}$$
Proposition 29 in Sejdinovic et al.(2013) establishes that the distance-induced kernel $\mathcal{K}$ induced by $d$ is characteristic to $\cal{M}_1(\cal{X}) \cap \cal{M}^{1}_\mathcal{K}(\cal{X})$ if and only if $(\cal{X}, d)$ has strong negative type. Therefore, MMD can be viewed as a special case of the generalized energy distance in (\[ed def general\]) with $d$ being the metric induced by a characteristic kernel.
Suppose $\{X_i\}^{n}_{i=1}$ and $\{Y_i\}^{m}_{i=1}$ are i.i.d samples of $X$ and $Y$ respectively. A U-statistic type estimator of $E_d(X,Y)$ is defined as $$\begin{aligned}
\label{unif U est ED}
E_{n,m}(X,Y)=\frac{2}{n m}\sum_{k=1}^{n}\sum^{m}_{l=1}d(X_k, Y_l)-\frac{1}{n(n-1)}\sum_{k\neq l}^{n}d(X_k,X_l)-\frac{1}{m(m-1)}\sum_{k\neq l}^{m}d(Y_k,Y_l)\,.\end{aligned}$$
In Section \[sec:new-homo\], we shall propose a new class of metrics for quantifying the homogeneity of high-dimensional distributions. This new class can be viewed as a particular case of the general measures in (\[ed def general\]) with a suitably chosen distance $d$ to accommodate the high dimensionality. It thus inherits all the nice properties of $E_{d}(X, Y)$ in the low-dimensional context (see Proposition \[prop ED U stat\] and Theorem \[th ED U stat\] in the supplementary material). With the specific choice of distance, the new metrics can detect a broader range of inhomogeneity between high-dimensional distributions compared to Euclidean energy distance.
Distance covariance and HSIC
----------------------------
Distance covariance (dCov) was first introduced in the seminal paper by Székely et al. (2007) to quantify the dependence between two random vectors of arbitrary (fixed) dimensions. Consider two random vectors $X\in \bb{R}^p$ and $Y \in \bb{R}^q$ with ${{\mathbb{E}}}\Vert X \Vert_p < \infty$ and ${{\mathbb{E}}}\Vert Y \Vert_q < \infty$. The Euclidean dCov between $X$ and $Y$ is defined as the positive square root of $$\begin{aligned}
dCov^2(X,Y)=\frac{1}{c_{p}c_{q}}\int_{{\mathbb
R}^{p+q}}\frac{|f_{X,Y}(t,s)-f_X(t)f_Y(s)|^2}{\Vert t \Vert_p^{1+p}\,\Vert s \Vert_q^{1+q}}dtds,\end{aligned}$$ where $f_X$, $f_Y$ and $f_{X,Y}$ are the individual and joint characteristic functions of $X$ and $Y$ respectively, and, $c_{p}=\pi^{(1+p)/2}/\,\Gamma((1+p)/2)$ is a constant with $\Gamma(\cdot)$ being the complete gamma function.
The key feature of dCov is that it completely characterizes independence between two random vectors of arbitrary dimensions, or in other words $dCov(X,Y)=0$ if and only if $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$. According to Remark 3 in Székely et al.(2007), dCov can be equivalently expressed as $$\label{alt dcov}
dCov^2(X,Y) \;=\; {{\mathbb{E}}}\,\Vert X-X'\Vert_p \Vert Y-Y'\Vert_q \,+\, {{\mathbb{E}}}\,\Vert X-X'\Vert_p\,{{\mathbb{E}}}\,\Vert Y-Y'\Vert_q \,-\, 2 \,{{\mathbb{E}}}\,\Vert X-X'\Vert_p \Vert Y-Y''\Vert_q. \\$$ Lyons(2013) extends the notion of dCov from Euclidean spaces to general metric spaces. For arbitrary metric spaces $(\cal{X}, d_{\cal{X}})$ and $(\cal{Y}, d_{\cal{Y}})$, the generalized dCov between $X \sim P_X \in \cal{M}_1(\cal{X}) \cap \cal{M}^1_{d_{\cal{X}}}(\cal{X})$ and $Y \sim P_Y \in \cal{M}_1(\cal{Y}) \cap \cal{M}^1_{d_{\cal{Y}}}(\cal{Y})$ is defined as $$\begin{aligned}
\label{dCov Lyons}
D^2_{d_{\cal{X}}, d_{\cal{Y}}} (X,Y) \;=\; {{\mathbb{E}}}\,d_{\cal{X}}(X,X') d_{\cal{Y}}(Y,Y') \,+\, {{\mathbb{E}}}\,d_{\cal{X}}(X,X')\,{{\mathbb{E}}}\,d_{\cal{Y}}(Y,Y') \,-\, 2 \,{{\mathbb{E}}}\,d_{\cal{X}}(X,X') d_{\cal{Y}}(Y,Y'').\end{aligned}$$ Theorem 3.11 in Lyons(2013) shows that if $(\cal{X}, d_{\cal{X}})$ and $(\cal{Y}, d_{\cal{Y}})$ are both metric spaces of strong negative type, then $D_{d_{\cal{X}}, d_{\cal{Y}}} (X,Y)=0$ if and only if $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$. In other words, the complete characterization of independence by dCov holds true for any metric spaces of strong negative type. According to Theorem 3.16 in Lyons(2013), every separable Hilbert space is of strong negative type. As Euclidean spaces are separable Hilbert spaces, the characterization of independence by dCov between two random vectors in $(\bb{R}^p, \Vert \cdot \Vert_p)$ and $(\bb{R}^q, \Vert \cdot \Vert_q)$ is just a special case.
Hilbert-Schmidt Independence Criterion (HSIC) was introduced as a kernel-based independence measure by Gretton et al.(2005, 2007). Suppose $\cal{X}$ and $\cal{Y}$ are arbitrary topological spaces, $\mathcal{K}_{\cal{X}}$ and $\mathcal{K}_{\cal{Y}}$ are characteristic kernels on $\cal{X}$ and $\cal{Y}$ with the respective RKHSs $\cal{H}_{\mathcal{K}_{\cal{X}}}$ and $\cal{H}_{\mathcal{K}_{\cal{Y}}}$. Let $\mathcal{K} = \mathcal{K}_{\cal{X}} \otimes \mathcal{K}_{\cal{Y}}$ be the tensor product of the kernels $\mathcal{K}_{\cal{X}}$ and $\mathcal{K}_{\cal{Y}}$, and, $\cal{H}_\mathcal{K}$ be the tensor product of the RKHSs $\cal{H}_{\mathcal{K}_{\cal{X}}}$ and $\cal{H}_{\mathcal{K}_{\cal{Y}}}$. The HSIC between $X \sim P_X \in \cal{M}_1(\cal{X}) \cap \cal{M}^{1/2}_\mathcal{K}(\cal{X})$ and $Y \sim P_Y \in \cal{M}_1(\cal{Y}) \cap \cal{M}^{1/2}_\mathcal{K}(\cal{Y})$ is defined as $$\begin{aligned}
\label{HSIC}
HSIC_{\mathcal{K}_{\mathcal{X}},\mathcal{K}_\mathcal{Y}}(X,Y) \;&=\; \Vert \,\Pi(P_{XY}) \,-\, \Pi(P_X P_Y) \,\Vert_{\cal{H}_\mathcal{K}},\end{aligned}$$ where $P_{XY}$ denotes the joint probability distribution of $X$ and $Y$. The HSIC between $X$ and $Y$ is essentially the MMD between the joint distribution $P_{XY}$ and the product of the marginals $P_X$ and $P_Y$. Clearly, $HSIC_{\mathcal{K}_{\mathcal{X}},\mathcal{K}_\mathcal{Y}}(X,Y) = 0$ if and only if $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$. Gretton et al.(2005) shows that the squared HSIC can be equivalently expressed as $$\begin{aligned}
\label{HSIC equiv def}
HSIC^2_{\mathcal{K}_{\mathcal{X}},\mathcal{K}_\mathcal{Y}}(X,Y) \;&=\; {{\mathbb{E}}}\,\mathcal{K}_{\cal{X}}(X,X') \mathcal{K}_{\cal{Y}}(Y,Y') \,+\, {{\mathbb{E}}}\,\mathcal{K}_{\cal{X}}(X,X')\,{{\mathbb{E}}}\,\mathcal{K}_{\cal{Y}}(Y,Y') \,-\, 2 \,{{\mathbb{E}}}\,\mathcal{K}_{\cal{X}}(X,X') \mathcal{K}_{\cal{Y}}(Y,Y'').\end{aligned}$$ Theorem 24 in Sejdinovic et al.(2013) establishes the equivalence between the generalized dCov and HSIC.
For an observed random sample $(X_i,Y_i)^{n}_{i=1}$ from the joint distribution of $X$ and $Y$, a U-statistic type estimator of the generalized dCov in (\[dCov Lyons\]) can be defined as $$\label{ustat dcov}
\widetilde{D_n^2}_{\,;\,d_{\cal{X}},d_{\cal{Y}}}(X,Y)\; =\; (\tilde{A} \cdot \tilde{B})\;=\; \frac{1}{n(n-3)} \dis \sum_{k\neq l} \tilde{a}_{kl} \tilde{b}_{kl} \; ,$$ where $\tilde{A}, \tilde{B}$ are the $\cal{U}$-centered versions (see (\[U centering def\])) of $A=\big(d_{\cal{X}}(X_k, X_l)\big)_{k,l=1}^n$ and $B=\big(d_{\cal{Y}}(Y_k, Y_l)\big)_{k,l=1}^n$, respectively. We denote $\widetilde{D_n^2}_{\,;\,d_{\cal{X}},d_{\cal{Y}}}(X,Y)$ by $dCov^2_n(X,Y)$ when $d_{\cal{X}}$ and $d_{\cal{Y}}$ are Euclidean distances.
New distance for Euclidean space {#new distance}
================================
We introduce a family of distances for Euclidean space, which shall play a central role in the subsequent developments. For $x \in \mathbb{R}^{\tilde{p}}$, we partition $x$ into $p$ sub-vectors or groups, namely $x=(x_{(1)},\dots,x_{(p)})$, where $x_{(i)}\in \mathbb{R}^{d_i}$ with $\sum_{i=1}^{p}d_i=\tilde{p}$. Let $\rho_i$ be a metric or semimetric (see for example Definition 1 in Sejdinovic et al.(2013)) defined on $\mathbb{R}^{d_i}$ for $1\leq i\leq p$. We define a family of distances for $\mathbb{R}^{\tilde{p}}$ as $$\label{Kdef}
K_{\bf{d}}(x,x') \, := \, \sqrt{ \,\rho_1 (x_{(1)}, x_{(1)}') \, + \,\dots \,+ \,\rho_p (x_{(p)}, x_{(p)}') \,}\,,$$ where $x, x' \in \bb{R}^{\tilde{p}}$ with $x = (x_{(1)},\dots,x_{(p)})$ and $x' = (x'_{(1)},\dots,x'_{(p)})$, and $\textbf{d}=(d_1,d_2,\dots,d_p)$ with $d_i\in\mathbb{Z}_+$ and $\sum_{i=1}^{p}d_i=\tilde{p}$.
\[metric\] Suppose each $\rho_i$ is a metric of strong negative type on $\mathbb{R}^{d_i}$. Then $\left(\bb{R}^{\tilde{p}}, K_{\bf{d}}\right)$ satisfies the following two properties:
1. $K_{\bf{d}}:\bb{R}^{\tilde{p}} \times \bb{R}^{\tilde{p}} \rightarrow [0,\infty)$ is a valid metric on $\bb{R}^{\tilde{p}}$;
2. $\left(\bb{R}^{\tilde{p}},K_{\bf{d}}\right)$ has strong negative type.
In a special case, suppose $\rho_i$ is the Euclidean distance on $\bb{R}^{d_i}$. By Theorem 3.16 in Lyons(2013), $(\bb{R}^{d_i}, \rho_i)$ is a separable Hilbert space, and hence has strong negative type. Then the Euclidean space equipped with the metric $$\begin{aligned}
\label{sp case}
K_{\bf{d}}(x,x') \, = \, \sqrt{ \,\Vert x_{(1)} - x_{(1)}' \Vert \, + \,\dots \,+ \,\Vert x_{(p)} - x_{(p)}' \Vert \,}\,.\end{aligned}$$ is of strong negative type. Further, if all the components $x_{(i)}$ are unidimensional, i.e., $d_i = 1$ for $1 \leq i \leq p$, then the metric boils down to $$\label{Kdef_sp}
K_{\bf{d}}(x,x') \;= \; \Arrowvert x-x' \Arrowvert_1^{1/2} \;= \;\sqrt{\dis\sum_{j=1}^p |x_j - x'_j|} \; ,$$ where $\Arrowvert x \Arrowvert_{1} = \sum_{j=1}^p |x_j|$ is the $l_1$ or the absolute norm on $\bb{R}^p$. If $$\begin{aligned}
\label{rho for Eucl}
\rho_i(x_{(i)},x_{(i)}')=\Vert x_{(i)} - x_{(i)}' \Vert^2, \quad 1\leq i\leq p,\end{aligned}$$ then $K_{\bf{d}}$ reduces to the usual Euclidean distance. We shall unify the analysis of our new metrics with the classical metrics by considering $K_{\bf{d}}$ which is defined in (\[Kdef\]) with
1. each $\rho_i$ being a metric of strong negative type on $\mathbb{R}^{d_i}$;
2. each $\rho_i$ being a semimetric defined in (\[rho for Eucl\]).
The first case corresponds to the newly proposed metrics while the second case leads to the classical metrics based on the usual Euclidean distance. Remarks \[rem3.1\] and \[rem3.2\] provide two different ways of generalizing the class in (\[Kdef\]). To be focused, our analysis below shall only concern about the distances defined in (\[Kdef\]). In the numerical studies in Section \[sec:num\], we consider $\rho_i$ to be the Euclidean distance and the distances induced by the Laplace and Gaussian kernels (see Definition \[dist-induced kernel\]) which are of strong negative type on $\mathbb{R}^{d_i}$ for $1\leq i \leq p$.
\[rem3.1\] A more general family of distances can be defined as $$\begin{aligned}
K_{\mathbf{d},r}(x,x')=\Big(\rho_1(x_{(1)},x_{(1)}')+\cdots+\rho_p(x_{(p)},x_{(p)}')\Big)^{r}, \quad 0<r<1.\end{aligned}$$ According to Remark 3.19 of Lyons (2013), the space $(\mathbb{R}^{\tilde{p}},K_{\mathbf{d},r})$ is of strong negative type. The proposed distance is a special case with $r=1/2.$
\[rem3.2\] Based on the proposed distance, one can construct the generalized Gaussian and Laplacian kernels as $$f(K_{\mathbf{d}}(x,x')/\gamma)=\begin{cases}
\exp(-K_{\mathbf{d}}^2(x,x')/\gamma^2), \quad & f(x)=\exp(-x^2) \text{ for Gaussian kernel},\\
\exp(-K_{\mathbf{d}}(x,x')/\gamma), \quad & f(x)=\exp(-x) \text{ for Laplacian kernel}.
\end{cases}$$ If $K_{\bf{d}}$ is translation invariant, then by Theorem 9 in Sriperumbudur et al.(2010) it can be verified that $f(K_{\mathbf{d}}(x,x')/\gamma)$ is a characteristic kernel on $\mathbb{R}^{\tilde{p}}$. As a consequence, the Euclidean space equipped with the distance $$K_{\mathbf{d},f}(x,x')=f(K_{\mathbf{d}}(x,x)/\gamma)+f(K_{\mathbf{d}}(x',x')/\gamma)-2f(K_{\mathbf{d}}(x,x')/\gamma)$$ is of strong negative type.
\[rem3.3\] In Sections \[sec:new-homo\] and \[sec:ACdcov\] we develop new classes of homogeneity and dependence metrics to quantify the pairwise homogeneity of distributions or the pairwise non-linear dependence of the low-dimensional groups. A natural question to arise in this regard is how to partition the random vectors optimally in practice. We present some real data examples in Section \[sub:real\] of the main paper where all the group sizes have been considered to be one (as a special case of the general theory proposed in this paper), and an additional real data example in Section \[addl data ex\] of the supplement where the data admits some natural grouping. We believe this partitioning can be very much problem specific and may require subject knowledge. We leave it for future research to develop an algorithm to find the optimal groups using the data and perhaps some auxiliary information.
Homogeneity metrics {#sec:new-homo}
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Consider $X, Y \in \bb{R}^{\tilde{p}}$. Suppose $X$ and $Y$ can be partitioned into $p$ sub-vectors or groups, viz. $X = \left(X_{(1)}, X_{(2)}, \dots, X_{(p)} \right)$ and $Y = \left(Y_{(1)}, Y_{(2)}, \dots, Y_{(p)} \right)$, where the groups $X_{(i)}$ and $Y_{(i)}$ are $d_i$ dimensional, $1\leq i\leq p$, and $p$ might be fixed or growing. We assume that $X_{(i)}$ and $Y_{(i)}$’s are finite (low) dimensional vectors, i.e., $\{d_i\}_{i=1}^p$ is a bounded sequence. Clearly $\tilde{p} = \sum_{i=1}^p d_i = O(p)$. Denote the mean vectors and the covariance matrices of $X$ and $Y$ by $\mu_X$ and $\mu_Y$, and, $\Sigma_X$ and $\Sigma_Y$, respectively. We propose the following class of metrics $\cal{E}$ to quantify the homogeneity of the distributions of $X$ and $Y$: $$\begin{aligned}
\label{ED HD def}
\cal{E}(X,Y) \;=\; 2\,{{\mathbb{E}}}\,K_{\bf{d}}(X,Y)\, -\, {{\mathbb{E}}}\,K_{\bf{d}}(X,X') \,-\, {{\mathbb{E}}}\,K_{\bf{d}}(Y,Y') \; ,\end{aligned}$$ with $\textbf{d}=(d_1,\dots,d_p)$. We shall drop the subscript $\textbf{d}$ below for the ease of notation.
\[ass0\] Assume that $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i^{1/2}(X_{(i)},0_{d_i})< \infty$ and $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i^{1/2}(Y_{(i)},0_{d_i})< \infty$.
Under Assumption \[ass0\], $\cal{E}$ is finite. In Section \[ld E\] of the supplement we illustrate that in the low-dimensional setting, $\cal{E}(X,Y)$ completely characterizes the homogeneity of the distributions of $X$ and $Y$.
Consider i.i.d. samples $\{X_k\}^{n}_{k=1}$ and $\{Y_l\}^{m}_{l=1}$ from the respective distributions of $X$ and $Y \in \mathbb{R}^{\tilde{p}}$, where $X_k = (X_{k(1)}, \dots, X_{k(p)})$, $Y_l=(Y_{l(1)},\dots,Y_{l(p)})$ for $1 \leq k \leq n$, $1\leq l \leq m$ and $X_{k(i)}, Y_{l(i)}\in\mathbb{R}^{d_i}.$ We propose an unbiased U-statistic type estimator $\cal{E}_{n,m}(X,Y)$ of $\cal{E}(X,Y)$ as in equation (\[unif U est ED\]) with $d$ being the new metric $K$. We refer the reader to Section \[ld E\] of the supplement, where we show that $\cal{E}_{n,m}(X,Y)$ essentially inherits all the nice properties of the U-statistic type estimator of generalized energy distance and MMD.
We define the following quantities which will play an important role in our subsequent analysis: $$\begin{aligned}
\label{tau def original}
\tau_X^2={{\mathbb{E}}}\,K(X,X')^2,\quad \tau_Y^2={{\mathbb{E}}}\,K(Y,Y')^2,\quad \tau^2={{\mathbb{E}}}\,K(X,Y)^2.\end{aligned}$$ In Case S2 (i.e., when $K$ is the Euclidean distance), we have $$\begin{aligned}
\label{tau for Eucl}
\tau_X^2 = 2\textrm{tr}\Sigma_X,\quad \tau_Y^2 = 2\textrm{tr}\Sigma_Y,\quad \tau^2 = \textrm{tr}\Sigma_X + \textrm{tr} \Sigma_Y + \Vert \mu_X - \mu_Y \Vert^2.\end{aligned}$$ Under the null hypothesis $H_0 : X \overset{d}{=} Y$, it is clear that $\tau_X^2 = \tau_Y^2 = \tau^2$. In the subsequent discussion we study the asymptotic behavior of $\cal{E}$ in the high-dimensional framework, i.e., when $p$ grows to $\infty$ with fixed $n$ and $m$ (discussed in Subsection \[subsec:ED HDLSS\]) and when $n$ and $m$ grow to $\infty$ as well (discussed in Subsection \[subsec:ED HDMSS\] in the supplement). We point out some limitations of the test for homogeneity of distributions in the high-dimensional setup based on the usual Euclidean energy distance. Consequently we propose a test based on the proposed metric and justify its consistency for growing dimension.
High dimension low sample size (HDLSS) {#subsec:ED HDLSS}
--------------------------------------
In this subsection, we study the asymptotic behavior of the Euclidean energy distance and our proposed metric $\cal{E}$ when the dimension grows to infinity while the sample sizes $n$ and $m$ are held fixed. We make the following moment assumption.
\[ass2 : ED\] There exist constants $a, a', a'', A, A', A''$ such that uniformly over $p$, $$\begin{aligned}
&\; 0 < a \leq \dis \inf_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( X_{(i)}, X_{(i)}'\,) \leq \dis \sup_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( X_{(i)}, X_{(i)}'\,) \leq A < \infty,\\
&\; 0 < a' \leq \dis \inf_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( Y_{(i)}, Y_{(i)}'\,) \leq \dis \sup_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( Y_{(i)}, Y_{(i)}'\,) \leq A' < \infty,\\
&\; 0 < a'' \leq \dis \inf_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( X_{(i)}, Y_{(i)}\,) \leq \dis \sup_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i( X_{(i)}, Y_{(i)}\,) \leq A'' < \infty.\end{aligned}$$
Under Assumption \[ass2 : ED\], it is not hard to see that $\tau_X, \tau_Y, \tau \asymp p^{1/2}$. The proposition below provides an expansion for $K$ evaluated at random samples.
\[K taylor : ED\] Under Assumption \[ass2 : ED\], we have $$\begin{aligned}
&\frac{K(X,X')}{\tau_X} = 1 + \frac{1}{2} L_X(X,X') + R_X(X,X'),\\
&\frac{K(Y,Y')}{\tau_Y} = 1 + \frac{1}{2} L_Y(Y,Y') + R_Y(Y,Y'),\end{aligned}$$ and $$\frac{K(X,Y)}{\tau} = 1 + \frac{1}{2} L(X,Y) + R(X,Y),$$ where $$L_X(X,X') := \frac{K^2(X,X') - \tau_X^2}{\tau_X^2}, \;\; L_Y(Y,Y') := \frac{K^2(Y,Y') - \tau_Y^2}{\tau_Y^2},\;\;L(X,Y) := \frac{K^2(X,Y) - \tau^2}{\tau^2},$$ and $R_X(X,X'), R_Y(Y,Y'), R(X,Y)$ are the remainder terms. In addition, if $L_X(X,X'), L_Y(Y,Y')$ and $L(X,Y)$ are $o_p(1)$ random variables as $p \to \infty$, then $R_X(X,X') = O_p\left(L^2_X(X,X')\right)$, $R_Y(Y,Y') = O_p\left(L^2_Y(Y,Y')\right)$ and $R(X,Y) = O_p\left(L^2(X,Y)\right)$.
Henceforth we will drop the subscripts $X$ and $Y$ from $L_X, L_Y, R_X$ and $R_Y$ for notational convenience. Theorem \[th homo decomp\] and Lemma \[lemma 1\] below provide insights into the behavior of $\cal{E}(X,Y)$ in the high-dimensional framework.
\[ass0.6\] Assume that $L(X,Y) = O_p(a_{p})$, $L(X,X') = O_p(b_{p})$ and $L(Y,Y') = O_p(c_{p})$, where $a_{p}, b_{p}, c_{p}$ are positive real sequences satisfying $a_{p}=o(1)$, $b_{p}=o(1)$, $c_{p}=o(1)$ and $\tau a^{2}_{p} + \tau_X b^{2}_{p} + \tau_Y c^{2}_{p} = o(1)$.
\[assumption justification\] To illustrate Assumption \[ass0.6\], we observe that under assumption \[ass2 : ED\] we can write $$\begin{aligned}
\var \left(L(X,X')\right) \;&=\; O\Big(\frac{1}{p^2}\Big) \dis \sum_{i,j=1}^p \text{cov}\left( \rho_i( X_{(i)}, X_{(i)}')\,, \rho_j( X_{(j)}, X_{(j)}') \right) \;=\; O\Big(\frac{1}{p^2}\Big) \dis \sum_{i,j=1}^p \text{cov}\left( Z_i, Z_j\right)\,,\end{aligned}$$ where $Z_i := \rho_i( X_{(i)}, X_{(i)}')$ for $1\leq i \leq p$. Assume that $ \sup_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i^2( X_{(i)}, 0_{d_i}) < \infty$, which implies $ \sup_{1\leq i \leq p} {{\mathbb{E}}}\,Z_i^2 < \infty$. Under certain strong mixing conditions or in general certain weak dependence assumptions, it is not hard to see that $\sum_{i,j=1}^p \text{cov}\left( Z_i, Z_j\right) = O(p)$ as $p\to \infty$ (see for example Theorem 1.2 in Rio(1993) or Theorem 1 in Doukhan et al.(1999)). Therefore we have $\var \left(L(X,X')\right) = O(\frac{1}{p})$ and hence by Chebyshev’s inequality, we have $L(X,X') = O_p(\frac{1}{\sqrt{p}})$. We refer the reader to Remark 2.1.1 in Zhu et al.(2019) for illustrations when each $\rho_i$ is the squared Euclidean distance.
\[th homo decomp\] Suppose Assumptions \[ass2 : ED\] and \[ass0.6\] hold. Further assume that the following three sequences $$\left\{\frac{\sqrt{p}L^2(X,Y)}{1+L(X,Y)}\right\},~\left\{\frac{\sqrt{p} L^2(X,X')}{1+L(X,X')}\right\},~\left\{\frac{\sqrt{p} L^2(Y,Y')}{1+L(Y,Y')}\right\}$$ indexed by $p$ are all uniformly integrable. Then we have $$\label{th homo decomp eqn}
\cal{E}(X,Y) \;=\; 2\tau - \tau_X -\tau_Y \,+\,o(1).$$
\[refer to rem\_ui\] Remark \[rem\_ui\] in the supplementary materials provides some illustrations on certain sufficient conditions under which $\{\sqrt{p}L^2(X,Y)/(1+L(X,Y))\}$, $\{\sqrt{p}L^2(X,X')/(1+L(X,X'))\}$ and\
$\{\sqrt{p}L^2(Y,Y')/(1+L(Y,Y'))\}$ are uniformly integrable.
\[pop approx E\] To illustrate that the leading term in equation (\[th homo decomp eqn\]) indeed gives a close approximation of the population $\cal{E} (X,Y)$, we consider the special case when $K$ is the Euclidean distance. Suppose $X \sim N_p(0,I_p)$ and $Y=X+N$ where $N \sim N_p(0,I_p)$ with $N {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}X$. Clearly from (\[tau for Eucl\]) we have $\tau_X^2=2p$, $\tau_Y^2=4p$ and $\tau^2=3p$. We simulate large samples of sizes $m=n=5000$ from the distributions of $X$ and $ Y$ for $p=20, 40, 60, 80$ and $100$. The large sample sizes are to ensure that the U-statistic type estimator of $\cal{E} (X,Y)$ gives a very close approximation of the population $\cal{E} (X,Y)$. In Table \[table:pop approx E\] we list the ratio between $\cal{E} (X,Y)$ and the leading term in (\[th homo decomp eqn\]) for the different values of $p$, which turn out to be very close to $1$, demonstrating that the leading term in (\[th homo decomp eqn\]) indeed approximates $\cal{E} (X,Y)$ reasonably well.
$p=20$ $p=40$ $p=60$ $p=80$ $p=100$
-------- -------- -------- -------- ---------
0.995 0.987 0.992 0.997 0.983
: Ratio of $\cal{E} (X,Y)$ and the leading term in (\[th homo decomp eqn\]) for different values of $p$.[]{data-label="table:pop approx E"}
\[lemma 1\] Assume $\tau,\tau_X, \tau_Y < \infty$. We have
1. In Case S1, $2\tau -\tau_X - \tau_Y = 0$ if and only if $X_{(i)} \overset{d}{=} Y_{(i)}$ for $i \in \{1, \dots, p\}$;
2. In Case S2, $2\tau -\tau_X - \tau_Y = 0$ if and only if $\mu_X = \mu_Y$ and $\textrm{tr}\, \Sigma_X = \textrm{tr}\, \Sigma_Y$.
It is to be noted that assuming $\tau,\tau_X, \tau_Y < \infty$ does not contradict with the growth rate $\tau, \tau_X, \tau_Y=O(p^{1/2})$. Clearly under $H_0$, $2\tau -\tau_X - \tau_Y = 0$ irrespective of the choice of $K$. In view of Lemma \[lemma 1\] and Theorem \[th homo decomp\], in Case S2, the leading term of $\cal{E}(X,Y)$ becomes zero if and only if $\mu_X = \mu_Y$ and $\textrm{tr}\, \Sigma_X = \textrm{tr}\, \Sigma_Y$. In other words, when dimension grows high, the Euclidean energy distance can only capture the equality of the means and the first spectral means, whereas our proposed metric captures the pairwise homogeneity of the low dimensional marginal distributions of $X_{(i)}$ and $Y_{(i)}$. Clearly $X_{(i)} \overset{d}{=} Y_{(i)}$ for $1\leq i\leq p$ implies $\mu_X = \mu_Y$ and $\textrm{tr}\, \Sigma_X = \textrm{tr}\, \Sigma_Y$. Thus the proposed metric can capture a wider range of inhomogeneity of distributions than the Euclidean energy distance.
Define $$\begin{aligned}
d_{kl}(i):=\rho_i(X_{k(i)}, Y_{l(i)}) \,-\, {{\mathbb{E}}}\,\left[\rho_i(X_{k(i)}, Y_{l(i)})|X_{k(i)}\right] \,-\, {{\mathbb{E}}}\,\left[\rho_i(X_{k(i)}, Y_{l(i)})|Y_{l(i)}\right] \,+\, {{\mathbb{E}}}\,\left[\rho_i(X_{k(i)}, Y_{l(i)})\right],\end{aligned}$$ as the double-centered distance between $X_{k(i)}$ and $Y_{l(i)}$ for $1\leq i \leq p$, $1\leq k\leq n$ and $1\leq l\leq m$. Similarly define $d^X_{kl}(i)$ and $d^Y_{kl}(i)$ as the double-centered distances between $X_{k(i)}$ and $X_{l(i)}$ for $1\leq k \neq l\leq n$, and, $Y_{k(i)}$ and $Y_{l(i)}$ for $1\leq k \neq l\leq m$, respectively. Further define $H(X_k, Y_l) := \frac{1}{\tau} \sum_{i=1}^p d_{kl}(i)$ for $1\leq k\leq n\,,\, 1\leq l \leq m$, $H(X_k, X_l) := \frac{1}{\tau_X} \sum_{i=1}^p d^X_{kl}(i)$ for $1\leq k \neq l \leq n$ and $H(Y_k, Y_l)$ in a similar way.
We impose the following conditions to study the asymptotic behavior of the (unbiased) U-statistic type estimator of $\cal{E}(X,Y)$ in the HDLSS setup.
\[ass6\] For fixed $n$ and $m$, as $p \to \infty$, $$\begin{pmatrix}
H(X_k, Y_l) \\
H(X_s, X_t) \\
H(Y_u, Y_v)
\end{pmatrix}_{k,l,\, s<t,\, u<v} \overset{d}{\longrightarrow} \;\;\begin{pmatrix}
a_{kl} \\ b_{st} \\ c_{uv}
\end{pmatrix}_{k,l,\, s<t,\, u<v} \, ,$$ where $\{a_{kl}, b_{st}, c_{uv}\}_{k,l,\, s<t,\, u<v}$ are jointly Gaussian with zero mean. Further we assume that $$\begin{aligned}
var(a_{kl}) \; &:= \; \sigma^2 \; =\; \lim_{p \to \infty}\, {{\mathbb{E}}}\left[H^2(X_k, Y_l)\right],\\
var(b_{st}) \; &:= \; \sigma^2_X \; =\; \lim_{p \to \infty}\, {{\mathbb{E}}}\left[H^2(X_s, X_t)\right],\\
var(c_{uv}) \; &:= \; \sigma^2_Y \; =\; \lim_{p \to \infty}\, {{\mathbb{E}}}\left[H^2(Y_u, Y_v)\right].\end{aligned}$$ $\{a_{kl}, b_{st}, c_{uv}\}_{k,l,\, s<t,\, u<v}$ are all independent with each other.
Due to the double-centering property and the independence between the two samples, it is straightforward to verify that $\{H(X_k,Y_l), H(X_s,X_t), H(Y_u,Y_v)\}_{k,l,s<t,u<t}$ are uncorrelated with each other. So it is natural to expect that the limit $\{a_{kl}, b_{st}, c_{uv}\}_{k,l,\, s<t,\, u<v}$ are all independent with each other.
The above multi-dimensional central limit theorem is classic and can be derived under suitable moment and weak dependence assumptions on the components of $X$ and $Y$, such as mixing or near epoch dependent conditions. We refer the reader to Doukhan and Neumann(2008) for a review on central limit theorem results under weak dependence assumptions.
We describe a new two-sample t-test for testing the null hypothesis $H_0 : X \overset{d}{=} Y.$ The t statistic can be constructed based on either the Euclidean energy distance or the new homogeneity metrics. We show that the t-tests based on different metrics can have strikingly different power behaviors under the HDLSS setup. The major difficulty here is to introduce a consistent and computationally efficient variance estimator. Towards this end, we define a quantity called Cross Distance Covariance (cdCov) between $X$ and $Y$, which plays an important role in the construction of the t-test statistic: $$\begin{aligned}
cdCov^2_{n,m}(X,Y):=\frac{1}{(n-1)(m-1)}\sum^{n}_{k=1}\sum_{l=1}^{m}\widehat{K}(X_k,Y_l)^2,\end{aligned}$$ where $$\begin{aligned}
\widehat{K}(X_k, Y_l)\;=\;K(X_{k},Y_{l})-\frac{1}{n}\sum^{n}_{i=1}K(X_{i},Y_{l})-\frac{1}{m}\sum^{m}_{j=1}K(X_{k},Y_{j})+\frac{1}{nm}\sum_{i=1}^{n}\sum^{m}_{j=1}K(X_{i},Y_{j}).\end{aligned}$$ Let $v_s := s(s-3)/2$ for $s = m, n$. We introduce the following quantities $$\begin{aligned}
\label{quantities}
\begin{split}
m_0\; &:=\; \frac{\sigma^2\,(n-1)(m-1) + \sigma_X^2 \,v_n + \sigma_Y^2 \,v_m}{(n-1)(m-1)+v_n+v_m}\,,\\
\sigma_{nm} \;&:=\; \sqrt{\frac{\sigma^2}{nm} + \frac{\sigma^2_X}{2n(n-1)} + \frac{\sigma^2_Y}{2m(m-1)}}\, ,
\\ a_{nm} \;&:=\; \sqrt{\frac{1}{nm} + \frac{1}{2n(n-1)} + \frac{1}{2m(m-1)}} \,,\\
\Delta\;&:=\; \lim_{p \to \infty} 2\tau -\tau_X - \tau_Y,
\end{split}\end{aligned}$$ where $\sigma^2,\sigma_X^2$ and $\sigma_Y^2$ are defined in Assumption \[ass6\]. Under Assumption \[ass0.5\], further define $$\begin{aligned}
&m_0^*:= \dis \lim_{m,n \to \infty} m_0 \; = \; \frac{2\alpha_0 \, \sigma^2 + \sigma_X^2 + \sigma_Y^2 \, \alpha_0^2}{2\alpha_0 + 1 + \alpha_0^2},\\
&a^*_0 := \dis \lim_{m,n \to \infty}\, \frac{a_{nm}}{\sigma_{nm}} \;=\; \Big(\frac{2\alpha_0 + \alpha_0^2 + 1}{2\alpha_0 \, \sigma^2 + \alpha_0^2 \,\sigma_X^2 + \sigma_Y^2 }\Big)^{1/2} \,.\end{aligned}$$ We are now ready to introduce the two-sample t-test $$T_{n,m}\;:=\;\frac{\cal{E}_{n,m}(X,Y)}{a_{nm}\,\sqrt{S_{n,m}}},$$ where $$S_{n,m}\;:=\;\frac{4(n-1)(m-1)\,cdCov^2_{n,m}(X,Y)\,+\,4v_n\, \widetilde{\cal{D}_n^2}(X,X)\,+\,4v_m\, \widetilde{\cal{D}_n^2}(Y,Y)}{(n-1)(m-1)+v_n+v_m}$$ is the pool variance estimator with $\widetilde{\cal{D}^2_n}(X,X)$ and $\widetilde{\cal{D}^2_m}(Y,Y)$ being the unbiased estimators of the (squared) distance variances defined in equation (\[ustat dcov\]). It is interesting to note that the variability of the sample generalized energy distance depends on the distance variances as well as the cdCov. It is also worth mentioning that the computational complexity of the pool variance estimator and thus the t-statistic is linear in $p$.
To study the asymptotic behavior of the test, we consider the following class of distributions on $(X,Y)$: $$\begin{aligned}
\mathcal{P}=&\Big\{(P_X,P_Y):~X\sim P_X,~Y\sim P_Y,~E[\tau L(X,Y)-\tau_X L(X,X')|X]=o_p(1),
\\&E[\tau L(X,Y)-\tau_Y L(Y,Y')|Y]=o_p(1)\Big\}.\end{aligned}$$ If $P_X=P_Y$ (i.e., under the $H_0$), it is clear that $(P_X,P_Y)\in \mathcal{P}$ irrespective of the metrics in the definition of $L$. Suppose $\|X-\mu_X\|^2-\text{tr}(\Sigma_X)=O_p(\sqrt{p})$ and $\|Y-\mu_Y\|^2-\text{tr}(\Sigma_Y)=O_p(\sqrt{p})$, which hold under weak dependence assumptions on the components of $X$ and $Y$. Then in Case S2 (i.e., $K$ is the Euclidean distance), a set of sufficient conditions for $(P_X,P_Y)\in\mathcal{P}$ is given by $$\begin{aligned}
&(\mu_X-\mu_Y)^\top (\Sigma_X+\Sigma_Y)(\mu_X-\mu_Y)=o(p),\quad \tau-\tau_X=o(\sqrt{p}), \quad \tau-\tau_Y=o(\sqrt{p}),\end{aligned}$$ which suggests that the first two moments of $P_X$ and $P_Y$ are not too far away from each other. In this sense, $\mathcal{P}$ defines a class of local alternative distributions (with respect to the null $H_0: P_X=P_Y$).
We now state the main result of this subsection.
\[th KED HDLSS\] In both Cases S1 and S2, under Assumptions \[ass2 : ED\], \[ass0.6\] and \[ass6\] as $p \to \infty$ with $n$ and $m$ remaining fixed, and further assuming that $(P_X,P_Y)\in\mathcal{P}$, we have $$\begin{aligned}
&\frac{\cal{E}_{n,m}(X,Y) - (2\tau -\tau_X - \tau_Y)}{a_{nm}\,\sqrt{S_{n,m}}} \; \overset{d}{\longrightarrow} \; \frac{\sigma_{nm} \, Z}{a_{nm}\,\sqrt{M}}\,,\end{aligned}$$ where $$M \overset{d}{=} \frac{\sigma^2\,\chi^2_{(n-1)(m-1)} + \sigma_X^2 \chi^2_{v_n} + \sigma_Y^2 \chi^2_{v_m}}{(n-1)(m-1)+v_n+v_m}\,,$$ $\chi^2_{(n-1)(m-1)},\, \chi^2_{v_n},\, \chi^2_{v_m}$ are independent chi-squared random variables, and $Z \sim N(0,1)$. In other words, $$T_{n,m} \;\overset{d}{\longrightarrow} \; \frac{\sigma_{nm} \, N(\Delta/\sigma_{nm}, 1)}{a_{nm}\,\sqrt{M}}\,,$$ where $\sigma_{nm}$ and $a_{nm}$ are defined in equation (\[quantities\]). In particular, under $H_0$, we have $$\begin{aligned}
&T_{n,m}\overset{d}{\longrightarrow} \; t_{(n-1)(m-1)+v_n+v_m}.\end{aligned}$$
Based on the asymptotic behavior of $T_{n,m}$ for growing dimensions, we propose a test for $H_0$ as follows: at level $\alpha \in (0,1)$, reject $H_0$ if $T_{n,m} > q_{\alpha,(n-1)(m-1) + v_n + v_m}$ and fail to reject $H_0$ otherwise, where $P(t_{(n-1)(m-1) + v_n + v_m}> q_{\alpha,(n-1)(m-1) + v_n + v_m})=\alpha.$ For a fixed real number $t$, define $$\begin{aligned}
\label{eqn exact power}
\begin{split}
\phi_{n,m}(t) \;:=& \; \dis \lim_{p \to \infty} P(T_{n,m} \leq t) \;=\; {{\mathbb{E}}}\,\left[ P \left(\frac{\sigma_{nm} \, N(\Delta/\sigma_{nm}, 1)}{a_{nm}\,\sqrt{M}} \leq t \,\,\Big|\,\, M \right)\right]\\ =& \;{{\mathbb{E}}}\,\,\left[ \Phi\left(\frac{a_{nm}\, \sqrt{M}\, t-\Delta}{\sigma_{nm}} \right)\right]\,.
\end{split}\end{aligned}$$ The asymptotic power curve for testing $H_0$ based on $T_{n,m}$ is given by $1-\phi_{m,n}(t)$. The following proposition gives a large sample approximation of the power curve.
\[ass0.5\] As $m, n \to \infty$, $m/n \to \alpha_0$ where $\alpha_0 > 0$.
\[power\] Suppose $\Delta=\Delta_0/\sqrt{nm}$ where $\Delta_0$ is a constant with respect to $n,m$. Then for any bounded real number $t$ as $n, m \to \infty$ and under Assumption \[ass0.5\], we have $$\dis \lim_{m, n \to \infty} \phi_{n,m}(t) \;=\; \Phi \left(a^*_0 \sqrt{m_0^*}\,\, t\, -\, \Delta^*_0 \right) \;,$$ where $$\Delta^*_0 = \Delta_0 \lim_{m, n \to \infty}\frac{1}{\sigma_{nm} \sqrt{nm}}= \Delta_0 \, \Big(\frac{2\alpha_0}{2\sigma^2\, \alpha_0 + \sigma_X^2\, \alpha_0^2 + \sigma_Y^2}\Big)^{1/2}.$$
Under the alternative, if $\Delta_0 \to \infty$ as $n, m \to \infty$, we have $$\dis \lim_{m, n \to \infty} \left\{ 1 - \phi_{n,m} ( q_{\alpha,(n-1)(m-1) + v_n + v_m}) \right\}\; = \; 1,$$ thereby justifying the consistency of the test.
\[rm:power\] We first derive the power function $1 - \phi_{n,m}(t)$ under the assumption that $n$ and $m$ are fixed. The main idea behind Proposition \[power\] where we let $n,m\rightarrow\infty$ is to see whether we get a reasonably good approximation of power when $n,m$ are large. In a sense we are doing sequential asymptotics, first letting $p\rightarrow\infty$ and deriving the power function, and then deriving the leading term by letting $n,m\rightarrow\infty$. This is a quite common practice in Econometrics (see for example Phillips and Moon(1999)). The aim is to derive a leading term for the power when $n,m$ are fixed but large. Consider $\Delta = s/\sqrt{nm}$ (as in Proposition \[power\]) and set $\sigma^2=\sigma_X^2=\sigma_Y^2=1$. In Figure \[fig\_power\] below, we plot the exact power (computed from (\[eqn exact power\]) with $50,000$ Monte Carlo samples from the distribution of $M$) with $n=m=5$ and $10$, $t=q_{\alpha, (n-1)(m-1) + v_n + v_m}$ and $\alpha=0.05$, over different values of $s$. We overlay the large sample approximation of the power function (given in Proposition \[power\]) and observe that the approximation works reasonably well even for small sample sizes. Clearly larger $s$ results in better power and $s=0$ corresponds to trivial power.
We now discuss the power behavior of $T_{n,m}$ based on the Euclidean energy distance. In Case S2, it can be seen that $$\begin{aligned}
\label{sigma_x^2}
\sigma_X^2=\lim_{p \to \infty}\frac{1}{\tau^2_X}\dis\sum_{i,i'=1}^{p}4\, \textrm{tr}\,\Sigma_X^2(i,i'),\end{aligned}$$ where $\Sigma_X^2(i,i')$ is the covariance matrix between $X_{(i)}$ and $X_{(i')}$, and similar expressions for $\sigma_Y^2$. In case S2 (i.e., when $K$ is the Euclidean distance), if we further assume $\mu_X = \mu_Y$, it can be verified that $$\begin{aligned}
\label{sigma^2}
\sigma^2 \; = \; \dis\lim_{p \to \infty}\,\frac{1}{\tau^2}\dis\sum_{i,i'=1}^{p}4\,\textrm{tr} \big( \Sigma_X(i,i')\, \Sigma_Y(i,i')\big)\,.\end{aligned}$$
Hence in Case S2, under the assumptions that $\mu_X = \mu_Y$, $\textrm{tr}\, \Sigma_X = \textrm{tr}\, \Sigma_Y$ and $\textrm{tr}\, \Sigma_X^2 = \textrm{tr}\, \Sigma_Y^2 = \textrm{tr}\, \Sigma_X \Sigma_Y$, it can be easily seen from equations (\[tau for Eucl\]), (\[sigma\_x\^2\]) and (\[sigma\^2\]) that $$\begin{aligned}
\label{Eucl cases}
\tau_X^2 = \tau_Y^2 = \tau^2,\quad \sigma_X^2 = \sigma_Y^2 = \sigma^2,
$$ which implies that $\Delta^*_0=0$ in Proposition \[power\]. Consider the following class of alternative distributions $$H_A = \{(P_X,P_Y): P_X\neq P_Y,~\mu_X = \mu_Y,~\textrm{tr}\,\Sigma_X = \textrm{tr}\,\Sigma_Y,\, \textrm{tr}\, \Sigma_X^2 = \textrm{tr}\, \Sigma_Y^2 = \textrm{tr}\, \Sigma_X \Sigma_Y\}.$$ According to Theorem \[th KED HDLSS\], the t-test $T_{n,m}$ based on Euclidean energy distance has trivial power against $H_A.$ In contrast, the t-test based on the proposed metrics has non-trivial power against $H_A$ as long as $
\Delta^*_0>0.$\
To summarize our contributions :
- We show that the Euclidean energy distance can only detect the equality of means and the traces of covariance matrices in the high-dimensional setup. To the best of our knowledge, such a limitation of the Euclidean energy distance has not been pointed out in the literature before.
- We propose a new class of homogeneity metrics which completely characterizes homogeneity of two distributions in the low-dimensional setup and has nontrivial power against a broader range of alternatives, or in other words, can detect a wider range of inhomogeneity of two distributions in the high-dimensional setup.
- Grouping allows us to detect homogeneity beyond univariate marginal distributions, as the difference between two univariate marginal distributions is automatically captured by the difference between the marginal distributions of the groups that contain these two univariate components.
- Consequently we construct a high-dimensional two-sample t-test whose computational cost is linear in $p$. Owing to the pivotal nature of the limiting distribution of the test statistic, no resampling-based inference is needed.
\[discussion on power\] Although the test based on our proposed statistic is asymptotically powerful against the alternative $H_A$ unlike the Euclidean energy distance, it can be verified that it has trivial power against the alternative $H_{A'} = \{(X,Y) : X_{(i)} \overset{d}{=} Y_{(i)}, 1\leq i \leq p\}$. Thus although it can detect differences between two high-dimensional distributions beyond the first two moments (as a significant improvement to the Euclidean energy distance), it cannot capture differences beyond the equality of the low-dimensional marginal distributions. We conjecture that there might be some intrinsic difficulties for distance and kernel-based metrics to completely characterize the discrepancy between two high-dimensional distributions.
Dependence metrics {#sec:ACdcov}
==================
In this section, we focus on dependence testing of two random vectors $X \in \bb{R}^{\tilde{p}}$ and $Y \in \bb{R}^{\tilde{q}}$. Suppose $X$ and $Y$ can be partitioned into $p$ and $q$ groups, viz. $X = \left(X_{(1)}, X_{(2)}, \dots, X_{(p)} \right)$ and $Y = \left(Y_{(1)}, Y_{(2)}, \dots, Y_{(q)} \right)$, where the components $X_{(i)}$ and $Y_{(j)}$ are $d_i$ and $g_j$ dimensional, respectively, for $1\leq i\leq p, 1\leq j\leq q$. Here $p, q$ might be fixed or growing. We assume that $X_{(i)}$ and $Y_{(j)}$’s are finite (low) dimensional vectors, i.e., $\{d_i\}_{i=1}^p$ and $\{g_j\}_{j=1}^q$ are bounded sequences. Clearly, $\tilde{p} = \sum_{i=1}^p d_i = O(p)$ and $\tilde{q} = \sum_{j=1}^q g_j = O(q)$. We define a class of dependence metrics $\cal{D}$ between $X$ and $Y$ as the positive square root of $$\label{ACdcov def}
\cal{D}^2(X,Y) \;:=\; {{\mathbb{E}}}\,K_{\bf{d}}(X,X')\, K_{\bf{g}}(Y,Y') \, + \, {{\mathbb{E}}}\,K_{\bf{d}}(X,X') \, {{\mathbb{E}}}\,K_{\bf{g}}(Y,Y') \, - \, 2\, {{\mathbb{E}}}\,K_{\bf{d}}(X,X') \, K_{\bf{g}}(Y,Y'') \,,$$ where ${\bf{d}}=(d_1,\dots,d_p)$ and ${\bf{g}}=(g_1,\dots,g_q)$. We drop the subscripts ${\bf{d}}, {\bf{g}}$ of $K$ for notational convenience.
To ensure the existence of $\cal{D}$, we make the following assumption.
\[ass1\] Assume that $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i^{1/2}(X_{(i)},0_{d_i})< \infty$ and $\sup_{1\leq i\leq q}{{\mathbb{E}}}\rho_i^{1/2}(Y_{(i)},0_{g_i})< \infty$.
In Section \[ld D\] of the supplement we demonstrate that in the low-dimensional setting, $\cal{D}(X,Y)$ completely characterizes independence between $X$ and $Y$. For an observed random sample $(X_k,Y_k)^{n}_{k=1}$ from the joint distribution of $X$ and $Y$, define $D^X:=(d^X_{kl}) \in \bb{R}^{n\times n}$ with $d^X_{kl} := K(X_k,X_l)$ and $k,l \in \{1, \dots, n\}$. Define $d^Y_{kl}$ and $D^Y$ in a similar way. With some abuse of notation, we consider the U-statistic type estimator $\widetilde{\cal{D}^2_n}(X,Y)$ of $\cal{D}^2$ as defined in (\[ustat dcov\]) with $d_{\mathcal{X}}$ and $d_{\mathcal{Y}}$ being $K_\mathbf{d}$ and $K_\mathbf{g}$ respectively. In Section \[ld D\] of the supplement, we illustrate that $\widetilde{\cal{D}^2_n}(X,Y)$ essentially inherits all the nice properties of the U-statistic type estimator of generalized dCov and HSIC.
In the subsequent discussion we study the asymptotic behavior of $\cal{D}$ in the high-dimensional framework, i.e., when $p$ and $q$ grow to $\infty$ with fixed $n$ (discussed in Subsection \[sec:ACdcov-HDLSS\]) and when $n$ grows to $\infty$ as well (discussed in Subsection \[sec:D HDMSS\] in the supplement).
High dimension low sample size (HDLSS) {#sec:ACdcov-HDLSS}
--------------------------------------
In this subsection, our goal is to explore the behavior of $\cal{D}^2(X,Y)$ and its unbiased U-statistic type estimator in the HDLSS setting where $p$ and $q$ grow to $\infty$ while the sample size $n$ is held fixed. Denote $\tau^2_{XY} = \tau^2_X \tau^2_Y = {{\mathbb{E}}}\, K^2(X,X')\, {{\mathbb{E}}}\,K^2(Y,Y').$ We impose the following conditions.
\[ass D pop taylor\] ${{\mathbb{E}}}\, [L^2(X, X')] = O(a_p'^2)$ and ${{\mathbb{E}}}\, [L^2(Y, Y')] = O(b_q'^2)$, where $a_p'$ and $b_q'$ are positive real sequences satisfying $a_p' = o(1)$, $b_q' = o(1)$, $\tau_{XY}\, a_p'^2 b_q' = o(1)$ and $\tau_{XY}\, a_p' b_q'^2 = o(1)$. Further assume that ${{\mathbb{E}}}\,[R^2(X,X')] = O(a_p'^4)$ and ${{\mathbb{E}}}\,[R^2(Y,Y')] = O(b_q'^4)$.
We refer the reader to Remark \[assumption justification\] in Section \[sec:new-homo\] for illustrations about some sufficient conditions under which we have $\var \left(L(X, X')\right)\,=\,{{\mathbb{E}}}\,L^2(X, X')\,=\,O(\frac{1}{p})$, and similarly for $L(Y, Y')$. Remark \[rem\_ui\] in the supplement illustrates certain sufficient conditions under which ${{\mathbb{E}}}\,[R^2(X,X')] = O(\frac{1}{p^2})$, and similarly for $R(Y,Y')$.
\[D pop taylor\] Under Assumptions \[ass2 : ED\] and \[ass D pop taylor\], we have $$\label{eq D pop taylor}
\cal{D}^2 (X,Y) = \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q D^2_{\rho_i, \rho_j}(X_{(i)},Y_{(j)}) \, + \, \mathcal{R} \; ,$$ where $\mathcal{R}$ is the remainder term such that $\mathcal{R} = O(\tau_{XY}\, a_p'^2 b_q' + \tau_{XY}\, a_p' b_q'^2) = o(1)$.
Theorem \[D pop taylor\] shows that when dimensions grow high, the population $\cal{D}^2 (X,Y)$ behaves as an aggregation of group-wise generalized dCov and thus essentially captures group-wise non-linear dependencies between $X$ and $Y$.
\[rem to th5.3\] Consider a special case where $d_i =1$ and $g_j=1$, and $\rho_i$ and $\rho_j$ are Euclidean distances for all $1\leq i\leq p$ and $1\leq j\leq q$. Then Theorem \[D pop taylor\] essentially boils down to $$\label{eq D pop taylor sp case 1}
\cal{D}^2 (X,Y) = \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q dCov^2(X_{i},Y_{j}) \, + \, \mathcal{R} \; ,$$ where $\mathcal{R} = o(1)$. This shows that in a special case (when we have unit group sizes), $\cal{D}^2 (X,Y)$ essentially behaves as an aggregation of cross-component dCov between $X$ and $Y$. If $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances, or in other words if each $\rho_i$ and $\rho_j$ are squared Euclidean distances, then using equation (\[alt dcov\]) it is straightforward to verify that $D^2_{\rho_i, \rho_j}(X_{i},Y_{j}) = 4\, cov^2(X_i, Y_j)$ for all $1\leq i\leq p$ and $1\leq j\leq q$. Consequently we have $$\label{eq D pop taylor sp case 2}
\cal{D}^2 (X,Y) = dCov^2(X,Y) = \frac{1}{\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q cov^2(X_{i},Y_{j}) \, + \, \mathcal{R}_1 \; ,$$ where $\mathcal{R}_1 = o(1)$, which essentially presents a population version of Theorem 2.1.1 in Zhu et al.(2019) as a special case of Theorem \[D pop taylor\].
\[pop approx\] To illustrate that the leading term in equation (\[eq D pop taylor\]) indeed gives a close approximation of the population $\cal{D}^2 (X,Y)$, we consider the special case when $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances and $p=q$. Suppose $X \sim N_p(0,I_p)$ and $Y=X+N$ where $N \sim N_p(0,I_p)$ with $N {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}X$. Clearly we have $\tau_X^2=2p$, $\tau_Y^2=4p$, $D^2_{\rho_i, \rho_j}(X_{i},Y_{j}) = 4\, cov^2(X_i, Y_j) = 4$ for all $1\leq i=j\leq p$ and $D^2_{\rho_i, \rho_j}(X_{i},Y_{j}) = 0$ for all $1\leq i\neq j\leq p$. From Remark \[rem to th5.3\], it is clear that in this case we essentially have $\cal{D}^2 (X,Y) = dCov^2(X,Y)$. We simulate a large sample of size $n=5000$ from the distribution of $(X, Y)$ for $p=20, 40, 60, 80$ and $100$. The large sample size is to ensure that the U-statistic type estimator of $\cal{D}^2 (X,Y)$ (given in (\[ustat dcov\])) gives a very close approximation of the population $\cal{D}^2 (X,Y)$. We list the ratio between $\cal{D}^2 (X,Y)$ and the leading term in (\[eq D pop taylor\]) for the different values of $p$, which turn out to be very close to $1$, demonstrating that the leading term in (\[eq D pop taylor\]) indeed approximates $\cal{D}^2 (X,Y)$ reasonably well.
$p=20$ $p=40$ $p=60$ $p=80$ $p=100$
-------- -------- -------- -------- ---------
0.980 0.993 0.994 0.989 0.997
: Ratio of $\cal{D}^2 (X,Y)$ and the leading term in (\[eq D pop taylor\]) for different values of $p$.[]{data-label="table:pop approx D"}
The following theorem explores the behavior of the population $\cal{D}^2 (X,Y)$ when $p$ is fixed and $q$ grows to infinity, while the sample size is held fixed. As far as we know, this asymptotic regime has not been previously considered in the literature. In this case, the Euclidean distance covariance behaves as an aggregation of martingale difference divergences proposed in Shao and Zhang (2014) which measures conditional mean dependence. Figure \[fig1\] below summarizes the curse of dimensionality for the Euclidean distance covariance under different asymptotic regimes.
\[th MDD\] Under Assumption \[ass2 : ED\] and the assumption that ${{\mathbb{E}}}\,[R^2(Y,Y')] = O(b_q'^4)$ with $\tau_Y\,b_q'^2 = o(1)$, as $q \to \infty$ with $p$ and $n$ remaining fixed, we have $$\begin{aligned}
\cal{D}^2 (X,Y) \;&=\; \frac{1}{2\tau_{Y}} \dis \sum_{j=1}^q D^2_{K_{\textbf{d}}\,, \rho_j}(X,Y_{(j)}) \, + \, \mathcal{R},\end{aligned}$$ where $\mathcal{R}$ is the remainder term such that $\mathcal{R} = O(\tau_{Y}\, b_q'^2) = o(1)$.
\[rem for th MDD\] In particular, when both $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances, we have $$\begin{aligned}
\cal{D}^2 (X,Y) \;&=\; dCov^2(X,Y)\;=\; \frac{1}{\tau_{Y}} \dis \sum_{j=1}^{\tilde{q}} MDD^2(Y_{j}|X) \, + \, \mathcal{R},\end{aligned}$$ where $MDD^2(Y_{j}|X)=-{{\mathbb{E}}}[(Y_j-{{\mathbb{E}}}Y_j)(Y_j'-{{\mathbb{E}}}Y_j)\|X-X'\|]$ is the martingale difference divergence which completely characterizes the conditional mean dependence of $Y_j$ given $X$ in the sense that $E[Y_j|X]=E[Y_j]$ almost surely if and only if $MDD^2(Y_{j}|X)=0.$
![Curse of dimensionality for the Euclidean distance covariance under different asymptotic regimes[]{data-label="fig1"}](diag.pdf)
Next we study the asymptotic behavior of the sample version $\widetilde{\cal{D}^2_n} (X,Y)$.
\[ass2.1\] Assume that $L(X,X') = O_p(a_p)$ and $L(Y,Y') = O_p(b_q)$, where $a_p$ and $b_q$ are positive real sequences satisfying $a_p = o(1)$, $b_q = o(1)$, $\tau_{XY}\, a_p^2 b_q = o(1)$ and $\tau_{XY}\, a_p b_q^2 = o(1)$.
We refer the reader to Remark \[assumption justification\] in Section \[sec:new-homo\] for illustrations about Assumption \[ass2.1\].
\[ACdCov taylor thm\] Under Assumptions \[ass2 : ED\] and \[ass2.1\], it can be shown that $$\label{ACdCov taylor}
\widetilde{\cal{D}^2_n} (X,Y) = \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q \widetilde{D^2_n}_{\,;\,\rho_i, \rho_j}(X_{(i)},Y_{(j)}) \, + \, \mathcal{R}_n \; ,$$ where $X_{(i)}, Y_{(j)}$ are the $i^{th}$ and $j^{th}$ groups of $X$ and $Y$, respectively, $1 \leq i \leq p$, $1 \leq j \leq q$ , and $\mathcal{R}_n$ is the remainder term. Moreover $\mathcal{R}_n = O_p(\tau_{XY}\, a_p^2 b_q + \tau_{XY}\, a_p b_q^2) = o_p(1)$, i.e., $\mathcal{R}_n$ is of smaller order compared to the leading term and hence is asymptotically negligible.
The above theorem generalizes Theorem 2.1.1 in Zhu et al.(2019) by showing that the leading term of $\widetilde{\cal{D}^2_n} (X,Y)$ is the sum of all the group-wise (unbiased) squared sample generalized dCov scaled by $\tau_{XY}\,$. In other words, in the HDLSS setting, $\widetilde{\cal{D}^2_n} (X,Y)$ is asymptotically equivalent to the aggregation of group-wise squared sample generalized dCov. Thus $\widetilde{\cal{D}^2_n} (X,Y)$ can quantify group-wise non-linear dependencies between $X$ and $Y$, going beyond the scope of the usual Euclidean dCov.
\[rem to th5.5\] Consider a special case where $d_i =1$ and $g_j=1$, and $\rho_i$ and $\rho_j$ are Euclidean distances for all $1\leq i\leq p$ and $1\leq j\leq q$. Then Theorem \[ACdCov taylor thm\] essentially states that $$\label{ACdCov taylor sp case 1}
\widetilde{\cal{D}^2_n} (X,Y) = \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q dCov^2_n(X_{i},Y_{j}) \, + \, \mathcal{R}_n \; ,$$ where $\mathcal{R}_n = o_p(1)$. This demonstrates that in a special case (when we have unit group sizes), $\widetilde{\cal{D}^2_n} (X,Y)$ is asymptotically equivalent to the marginal aggregation of cross-component distance covariances proposed by Zhu et al.(2019) as dimensions grow high. If $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances, then Theorem \[ACdCov taylor thm\] essentially boils down to Theorem 2.1.1 in Zhu et al.(2019) as a special case.
\[approximation\] To illustrate the approximation of $\widetilde{\cal{D}^2_n} (X,Y)$ by the aggregation of group-wise squared sample generalized dCov given by Theorem \[ACdCov taylor thm\], we simulated the datasets in Examples \[eg1\].1, \[eg1\].2, \[eg2\].1 and \[eg2\].2 $100$ times each with $n=50$ and $p=q=50$. For each of the datasets, the difference between $\widetilde{\cal{D}^2_n} (X,Y)$ and the leading term in the RHS of equation (\[ACdCov taylor\]) is smaller than $0.01$ $100\%$ of the times, which illustrates that the approximation works reasonably well.
The following theorem illustrates the asymptotic behavior of $\widetilde{\cal{D}^2_n} (X,Y)$ when $p$ is fixed and $q$ grows to infinity while the sample size is held fixed. Under this setup, if both $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances, the leading term of $\widetilde{\cal{D}^2_n} (X,Y)$ is the sum of the group-wise unbiased U-statistic type estimators of $MDD^2(Y_j | X)$ for $1\leq j \leq q$, scaled by $\tau_{Y}\,$. In other words, the sample Euclidean distance covariance behaves as an aggregation of sample martingale difference divergences.
\[th MDD sample\] Under Assumption \[ass2 : ED\] and the assumption that $L(Y,Y') = O_p(b_q)$ with $b_q=o(1)$ and $\tau_Y\,b_q^2 = o(1)$, as $q \to \infty$ with $p$ and $n$ remaining fixed, we have $$\begin{aligned}
\widetilde{\cal{D}^2_n}(X,Y) \;&=\; \frac{1}{2\tau_{Y}} \dis \sum_{j=1}^q \widetilde{\cal{D}^2_n}_{\,;\,K_{\textbf{d}} \,, \rho_j}(X,Y_{(j)}) \, + \, \mathcal{R}_n,\end{aligned}$$ where $\mathcal{R}_n$ is the remainder term such that $\mathcal{R}_n = O_p(\tau_{Y}\, b_q^2) = o_p(1)$.
\[rem for th MDD sample\] In particular, when both $K_{\textbf{d}}$ and $K_{\textbf{g}}$ are Euclidean distances, we have $$\begin{aligned}
\widetilde{\cal{D}^2_n} (X,Y) \;&=\; dCov^2_n(X,Y) \;=\; \frac{1}{\tau_{Y}} \dis \sum_{j=1}^{\tilde{q}} MDD^2_n(Y_{j}|X) \, + \, \mathcal{R}_n,\end{aligned}$$ where $MDD^2_n(Y_{j}|X)$ is the unbiased U-statistic type estimator of $MDD^2(Y_{j}|X)$ defined as in (\[ustat dcov\]) with $d_{\cal{X}}(x,x')=\|x-x'\|$ for $x,x'\in\mathbb{R}^{\tilde{p}}$ and $d_{\cal{Y}}(y,y')=|y-y'|^2/2$ for $y,y'\in\mathbb{R}$, respectively.
Now denote $X_k = (X_{k(1)}, \dots, X_{k(p)})$ and $Y_k=(Y_{k(1)},\dots,Y_{k(q)})$ for $1\leq k\leq n$. Define the leading term of $\widetilde{\cal{D}^2_n} (X,Y)$ in equation (\[ACdCov taylor\]) as $$L := \,\frac{1}{4\tau_{XY}\,} \sum_{i=1}^p \sum_{j=1}^q \widetilde{D^2_n}_{\,;\,\rho_i, \rho_j}(X_{(i)},Y_{(j)})\,.$$ It can be verified that $$L\, = \,\frac{1}{4\tau_{XY}\,} \sum_{i=1}^p \sum_{j=1}^q \left(\tilde{D}^X(i) \cdot \tilde{D}^Y(j)\right) \,,$$ where $\tilde{D}^X(i), \tilde{D}^Y(j)$ are the $\cal{U}$-centered versions of $D^X(i) = \left(d^X_{kl}(i)\right)_{k,l=1}^n$ and $D^Y(j) = \left(d^Y_{kl}(j)\right)_{k,l=1}^n$, respectively.
As an advantage of using the double-centered distances, we have for all $1\leq i,i' \leq p$, $1\leq j, j' \leq q$ and $\{k,l\} \neq \{u, v\},$ $$\begin{aligned}
\label{double}
{{\mathbb{E}}}\left[d^X_{kl}(i)\, d^X_{uv}(i')\right] \; = \; {{\mathbb{E}}}\left[d^Y_{kl}(j)\,d^Y_{uv}(j')\right] \; = \;{{\mathbb{E}}}\left[d^X_{kl}(i)\, d^Y_{uv}(j)\right]\;= \;0.\end{aligned}$$ See for example the proof of Proposition 2.2.1 in Zhu et al.(2019) for a detailed explanation.
\[ass3\] For fixed $n$, as $p, q \to \infty$, $$\begin{pmatrix}
\frac{1}{2\,\tau_X} \dis \sum_{i=1}^p d^X_{kl}(i) \\
\frac{1}{2\,\tau_Y} \dis \sum_{j=1}^q d^Y_{uv}(j)
\end{pmatrix}_{k<l,\, u<v} \overset{d}{\longrightarrow} \;\;\begin{pmatrix}
d_{kl}^1 \\ \\ d_{uv}^2
\end{pmatrix}_{k<l,\, u<v} \, ,$$ where $\{d_{kl}^1,\, d_{uv}^2\}_{k<l,\, u<v}$ are jointly Gaussian. Further we assume that $$\begin{aligned}
&var(d_{kl}^1):= \sigma_X^2= \lim_{p \to \infty}\, \frac{1}{4\tau_X^2} \dis \sum_{i,i'=1}^p D^2_{\rho_i, \rho_{i'}} \left(X_{(i)}, X_{(i')} \right),\\
&var(d_{kl}^2):= \sigma_Y^2 = \lim_{q \to \infty}\, \frac{1}{4\tau_Y^2} \dis \sum_{j,j'=1}^q D^2_{\rho_j, \rho_{j'}} \left(Y_{(j)}, Y_{(j')} \right),\\
&\cov\,(d_{kl}^1, d_{kl}^2) := \sigma_{XY}^2 = \lim_{p, q \to \infty}\, \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q D^2_{\rho_i, \rho_j} \left(X_{(i)}, Y_{(j)} \right).\end{aligned}$$
In view of (\[double\]), we have $\cov\,(d_{kl}^1, d_{uv}^1) = \cov\,(d_{kl}^2, d_{uv}^2) = \cov\,(d_{kl}^1, d_{uv}^2) = 0$ for $\{k,l\} \neq \{u, v\}$. Theorem \[ACdCov taylor thm\] states that for growing $p$ and $q$ and fixed $n$, $\widetilde{\cal{D}^2_n} (X,Y)$ and $L$ are asymptotically equivalent. By studying the leading term, we obtain the limiting distribution of $\widetilde{\cal{D}^2_n} (X,Y)$ as follows.
\[ACdcov:dist\_conv\] Under Assumptions \[ass2 : ED\], \[ass2.1\] and \[ass3\], for fixed $n$ and $p, q \to \infty$, $$\begin{aligned}
&\widetilde{\cal{D}^2_n} (X,Y)\; \overset{d}{\longrightarrow} \; \frac{1}{\nu} d^{1\top} M d^2 \,,\\
&\widetilde{\cal{D}^2_n} (X,X)\; \overset{d}{\longrightarrow} \; \frac{1}{\nu} d^{1\top} M d^1\; \overset{d}{=} \; \frac{\sigma_X^2}{\nu} \chi^2_{\nu}\,,\\
&\widetilde{\cal{D}^2_n} (Y,Y)\; \overset{d}{\longrightarrow} \; \frac{1}{\nu} d^{2\top} M d^2\; \overset{d}{=} \; \frac{\sigma_Y^2}{\nu} \chi^2_{\nu}\,,\end{aligned}$$ where $M$ is a projection matrix of rank $\nu=\frac{n(n-3)}{2}$, and $$\begin{pmatrix}
d^1 \\ d^2
\end{pmatrix} \; \sim \; N \left( 0\,, \begin{pmatrix}
\sigma_X^2 \,I_{\frac{n(n-1)}{2}} \;\; \sigma_{XY}^2 \,I_{\frac{n(n-1)}{2}} \\ \\
\sigma_{XY}^2 \,I_{\frac{n(n-1)}{2}} \;\; \sigma_Y^2 \,I_{\frac{n(n-1)}{2}}
\end{pmatrix} \right) \,.$$
To perform independence testing, in the spirit of Székely and Rizzo(2014), we define the studentized test statistic $$\label{student t}
\cal{T}_n \; := \; \sqrt{\nu-1}\, \frac{\widetilde{\cal{DC}^2_n} (X,Y)}{\sqrt{1 - \left(\widetilde{\cal{DC}^2_n} (X,Y)\right)^2}}\; ,$$ where $$\widetilde{\cal{DC}^2_n} (X,Y) \; = \; \frac{\widetilde{\cal{D}^2_n} (X,Y)}{\sqrt{\widetilde{\cal{D}^2_n} (X,X) \, \widetilde{\cal{D}^2_n} (Y,Y)}}\,.$$
Define $\psi = \sigma_{XY}^2 / \sqrt{\sigma_X^2 \sigma_Y^2}$. The following theorem states the asymptotic distributions of the test statistic $\cal{T}_n$ under the null hypothesis $\tilde{H}_0: X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$ and the alternative hypothesis $\tilde{H}_A: X {\centernot{{\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}}}Y$.
\[HDLSS dist conv\] Under Assumptions \[ass2 : ED\], \[ass2.1\] and \[ass3\], for fixed $n$ and $p, q \to \infty$, $$\begin{aligned}
& P_{\tilde{H}_0} \left(\cal{T}_n \leq t \right) \; \longrightarrow \; P\left(t_{\nu -1} \leq t \right),\\
& P_{\tilde{H}_A} \left(\cal{T}_n \leq t \right) \; \longrightarrow \; {{\mathbb{E}}}\left[P\left(t_{\nu -1, W} \leq t |W\right) \right],\end{aligned}$$ where $t$ is any fixed real number and $W \sim \sqrt{\frac{\psi^2}{1-\psi^2}\, \chi^2_{\nu}}$.
For an explicit form of ${{\mathbb{E}}}\left[P\left(t_{\nu -1, W} \leq t |W \right)\right]$, we refer the reader to Lemma 3 in the appendix of Zhu et al.(2019). Now consider the local alternative hypothesis $\tilde{H}_{A}^*$: $X {\centernot{{\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}}}Y$ with $\psi=\psi_0/\sqrt{\nu}$, where $\psi_0$ is a constant with respect to $n$. The following proposition gives an approximation of ${{\mathbb{E}}}\left[P\left(t_{\nu -1, W} \leq t |W \right)\right]$ under the local alternative hypothesis $\tilde{H}_{A}^*$ when $n$ is allowed to grow.
\[local alt power\] Under $\tilde{H}_{A}^*$, as $n \to \infty$ and $t = O(1)$, $${{\mathbb{E}}}\left[P\left(t_{\nu -1, W} \leq t |W\right)\right] \; = \; P\left(t_{\nu -1, \,\psi_0} \leq t \right) \; + \; O\Big(\frac{1}{\nu}\Big)\,.$$
The following summarizes our key findings in this section.
- **Advantages of our proposed metrics over the Euclidean dCov and HSIC :**
i) Our proposed dependence metrics completely characterize independence between $X$ and $Y$ in the low-dimensional setup, and can detect group-wise non-linear dependencies between $X$ and $Y$ in the high-dimensional setup as opposed to merely detecting component-wise linear dependencies by the Euclidean dCov and HSIC (in light of Theorem 2.1.1 in Zhu et al.(2019)).
ii) We also showed that with $p$ remaining fixed and $q$ growing high, the Euclidean dCov can only quantify conditional mean independence of the components of $Y$ given $X$ (which is weaker than independence). To the best of our knowledge, this has not been pointed out in the literature before.
- **Advantages over the marginal aggregation approach by Zhu et al.(2019) :**
i) In the low-dimensional setup, our proposed dependence metrics can completely characterize independence between $X$ and $Y$, whereas the metric proposed by Zhu et al.(2019) can only capture pairwise dependencies between the components of $X$ and $Y$.
ii) We provide a neater way of generalizing dCov and HSIC between $X$ and $Y$ which is shown to be asymptotically equivalent to the marginal aggregation of cross-component distance covariances proposed by Zhu et al.(2019) as dimensions grow high. Also grouping or partitioning the two high-dimensional random vectors (which again may be problem specific) allows us to detect a wider range of alternatives compared to only detecting component-wise non-linear dependencies, as independence of two univariate marginals is implied from independence of two higher dimensional marginals containing the two univariate marginals.
iii) The computational complexity of the (unbiased) squared sample $\cal{D}(X,Y)$ is $O(n^2(p+q))$. Thus the computational cost of our proposed two-sample t-test only grows linearly with the dimension and therefore is scalable to very high-dimensional data. Although a naive aggregation of marginal distance covariances has a computational complexity of $O(n^2 pq)$, the approach of Zhu et al.(2019) essentially corresponds to the use of an additive kernel and the computational cost of their proposed estimator can also be made linear in the dimensions if properly implemented.
[|L|L|L|]{} Choice of $\rho_i(x,x')$ & Asymptotic behavior of the proposed homogeneity metric & Asymptotic behavior of the proposed dependence metric\
the semi-metric $\Vert x-x' \Vert^2$ & Behaves as a sum of squared Euclidean distances & Behaves as a sum of squared Pearson correlations\
metric of strong negative type on $\mathbb{R}^{d_i}$ & Behaves as a sum of groupwise energy distances with the metric $\rho_i$ & Behaves as a sum of groupwise dCov with the metric $\rho_i$\
$k_i(x,x) + k_i(x',x') - 2k_i(x,x')$, where $k_i$ is a characteristic kernel on $\mathbb{R}^{d_i}\times \mathbb{R}^{d_i}$ & Behaves as a sum of groupwise MMD with the kernel $k_i$ & Behaves as a sum of groupwise HSIC with the kernel $k_i$\
Numerical studies {#sec:num}
=================
Testing for homogeneity of distributions {#subsec homo num}
----------------------------------------
We investigate the empirical size and power of the tests for homogeneity of two high dimensional distributions. For comparison, we consider the t-tests based on the following metrics:
I. $\cal{E}$ with $\rho_i$ as the Euclidean distance for $1\leq i\leq p$;
II. $\cal{E}$ with $\rho_i$ as the distance induced by the Laplace kernel for $1\leq i\leq p$;
III. $\cal{E}$ with $\rho_i$ as the distance induced by the Gaussian kernel for $1\leq i\leq p$;
IV. the usual Euclidean energy distance;
V. MMD with the Laplace kernel;
VI. MMD with the Gaussian kernel.
We set $d_i =1$ in Examples \[eg1:ed\] and \[eg2:ed\], and $d_i =2$ in Example \[eg3:ed\] for $1\leq i\leq p$.
\[eg1:ed\] Consider $X_k = (X_{k1}, \dots, X_{kp})$ and $Y_l = (Y_{l1}, \dots, Y_{lp})$ with $k=1,\dots, n$ and $l=1,\dots, m$. We generate i.i.d. samples from the following models:
1. $X_k \sim N(0, I_p)$ and $Y_l \sim N(0, I_p)$.
2. $X_k \sim N(0, \Sigma)$ and $Y_l \sim N(0, \Sigma)$, where $\Sigma = (\sigma_{ij})_{i,j=1}^p$ with $\sigma_{ii}=1$ for $i=1, \dots, p$, $\sigma_{ij} = 0.25$ if $1 \leq |i-j| \leq 2$ and $\sigma_{ij} = 0$ otherwise.
3. $X_k \sim N(0, \Sigma)$ and $Y_l \sim N(0, \Sigma)$, where $\Sigma = (\sigma_{ij})_{i,j=1}^p$ with $\sigma_{ij} = 0.7^{|i-j|}$.
\[eg2:ed\] Consider $X_k = (X_{k1}, \dots, X_{kp})$ and $Y_l = (Y_{l1}, \dots, Y_{lp})$ with $k=1,\dots, n$ and $l=1,\dots, m$. We generate i.i.d. samples from the following models:
1. $X_k \sim N(\mu, I_p)$ with $\mu = (1, \dots, 1)\in\mathbb{R}^p$ and $Y_{li} \overset{ind}{\sim}$ Poisson$(1)$ for $i=1,\dots,p$.
2. $X_k \sim N(\mu, I_p)$ with $\mu = (1, \dots, 1)\in \mathbb{R}^p$ and $Y_{li} \overset{ind}{\sim}$ Exponential$(1)$ for $i=1,\dots,p$.
3. $X_k \sim N(0, I_p)$ and $Y_l = (Y_{l1}, \dots, Y_{l\lfloor\beta p\rfloor}, Y_{l(\lfloor\beta p\rfloor +1)}, \dots , Y_{lp})$, where $Y_{l1}, \dots, Y_{l \lfloor\beta p\rfloor} \overset{i.i.d.}{\sim}$ Rademacher$(0.5)$ and $Y_{l(\lfloor\beta p\rfloor +1)}, \dots , Y_{lp} \overset{i.i.d.}{\sim} N(0,1)$.
4. $X_k \sim N(0, I_p)$ and $Y_l = (Y_{l1}, \dots, Y_{l \lfloor\beta p\rfloor}, Y_{l(\lfloor\beta p\rfloor +1)}, \dots , Y_{lp})$, where $Y_{l1}, \dots, Y_{l\lfloor \beta p\rfloor} \overset{i.i.d.}{\sim}$ Uniform$(-\sqrt{3}, \sqrt{3})$ and $Y_{l(\lfloor\beta p\rfloor +1)}, \dots , Y_{lp} \overset{i.i.d.}{\sim} N(0,1)$.
5. $X_k = R^{1/2} Z_{1k}$ and $Y_l = R^{1/2} Z_{2l}$, where $R = (r_{ij})_{i,j=1}^p$ with $r_{ii}=1$ for $i=1, \dots, p$, $r_{ij} = 0.25$ if $1 \leq |i-j| \leq 2$ and $r_{ij} = 0$ otherwise, $Z_{1k} \sim N(0, I_p)$ and $Z_{2l} = \underbrace{(Z_{2l1}, \dots, Z_{2lp})}_{ \overset{i.i.d.}{\sim} Exponential(1)} -\, 1.
$
\[eg3:ed\] Consider $X_k = (X_{k(1)}, \dots, X_{k(p)})$ and $Y_l = (Y_{l(1)}, \dots, Y_{l(p)})$ with $k=1,\dots, n$ and $l=1,\dots, m$ and $d_i =2$ for $1\leq i \leq p$. We generate i.i.d. samples from the following models:
1. $X_{k(i)} \sim N(\mu, \Sigma_1)$ and $Y_{l(i)} \sim N(\mu, \Sigma_2)$ with $X_{k(i)} {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}X_{k(j)}$ and $Y_{l(i)} {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y_{l(j)}$ for $1\leq i \neq j \leq p$, where $\mu=(1,1)^\top $, $\Sigma_1 = \begin{pmatrix}
1 & 0.9\\
0.9 & 1
\end{pmatrix}$and $\Sigma_2 = \begin{pmatrix}
1 & 0.1\\
0.1 & 1
\end{pmatrix}$.
2. $X_{k(i)} \sim N(\mu, \Sigma)$ with $X_{k(i)} {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}X_{k(j)}$ for $1\leq i \neq j \leq p$, where $\mu=(1,1)^\top $, $\Sigma = \begin{pmatrix}
1 & 0.7\\
0.7 & 1
\end{pmatrix}$. The components of $Y_l$ are i.i.d. Exponential$(1)$.
Note that for Examples \[eg1:ed\] and \[eg2:ed\], the metric defined in equation (\[Kdef\]) essentially boils down to the special case in equation (\[Kdef\_sp\]). We try small sample sizes $n=m=50$, dimensions $p=q=50, 100$ and $200$, and $\beta = 1/2$. Table \[table1:ed\] reports the proportion of rejections out of $1000$ simulation runs for the different tests. For the tests V and VI, we chose the bandwidth parameter heuristically as the median distance between the aggregated sample observations. For tests II and III, the bandwidth parameters are chosen using the median heuristic separately for each group.
In Example \[eg1:ed\], the data generating scheme suggests that the variables $X$ and $Y$ are identically distributed. The results in Table \[table1:ed\] show that the tests based on both the proposed homogeneity metrics and the usual Euclidean energy distance and MMD perform more or less equally good, and the rejection probabilities are quite close to the $10\%$ or $5\%$ nominal level. In Example \[eg2:ed\], clearly $X$ and $Y$ have different distributions but $\mu_X = \mu_Y$ and $\Sigma_X = \Sigma_Y$. The results in Table \[table1:ed\] indicate that the tests based on the proposed homogeneity metrics are able to detect the differences between the two high-dimensional distributions beyond the first two moments unlike the tests based on the usual Euclidean energy distance and MMD, and thereby outperform the latter in terms of empirical power. In Example \[eg3:ed\], clearly $\mu_X = \mu_Y$ and $\textrm{tr}\, \Sigma_X = \textrm{tr}\, \Sigma_Y$ and the results show that the tests based on the proposed homogeneity metrics are able to detect the in-homogeneity of the low-dimensional marginal distributions unlike the tests based on the usual Euclidean energy distance and MMD.
---------------------------------------------------------------------- ----- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
(r)[6-7]{}(r)[8-9]{}(r)[10-11]{}(r)[12-13]{}(r)[14-15]{}(r)[16-17]{} $n$ $m$ $p$ 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
(1) 50 50 50 0.109 0.062 0.109 0.058 0.106 0.063 0.109 0.068 0.110 0.069 0.109 0.070
(1) 50 50 100 0.124 0.073 0.119 0.053 0.121 0.063 0.116 0.067 0.114 0.068 0.117 0.068
(1) 50 50 200 0.086 0.043 0.099 0.048 0.088 0.035 0.090 0.045 0.086 0.043 0.090 0.045
(2) 50 50 50 0.114 0.069 0.108 0.054 0.118 0.068 0.116 0.077 0.115 0.073 0.116 0.078
(2) 50 50 100 0.130 0.069 0.133 0.073 0.124 0.070 0.126 0.067 0.123 0.068 0.124 0.067
(2) 50 50 200 0.099 0.048 0.103 0.041 0.092 0.047 0.097 0.040 0.095 0.039 0.097 0.040
(3) 50 50 50 0.100 0.064 0.107 0.057 0.099 0.060 0.112 0.072 0.105 0.067 0.110 0.073
(3) 50 50 100 0.103 0.062 0.113 0.061 0.113 0.063 0.097 0.060 0.100 0.057 0.098 0.059
(3) 50 50 200 0.108 0.062 0.115 0.062 0.117 0.064 0.091 0.055 0.093 0.056 0.090 0.055
(1) 50 50 50 1 1 1 1 0.995 0.994 0.102 0.067 0.111 0.069 0.103 0.066
(1) 50 50 100 1 1 1 1 1 1 0.120 0.066 0.120 0.071 0.119 0.066
(1) 50 50 200 1 1 1 1 1 1 0.111 0.057 0.111 0.057 0.111 0.057
(2) 50 50 50 1 1 1 1 1 1 0.126 0.085 0.154 0.105 0.119 0.073
(2) 50 50 100 1 1 1 1 1 1 0.098 0.058 0.108 0.066 0.094 0.055
(2) 50 50 200 1 1 1 1 1 1 0.111 0.055 0.114 0.056 0.108 0.054
(3) 50 50 50 1 1 1 1 1 0.999 0.118 0.069 0.117 0.072 0.120 0.070
(3) 50 50 100 1 1 1 1 1 1 0.102 0.067 0.106 0.065 0.103 0.067
(3) 50 50 200 1 1 1 1 1 1 0.103 0.046 0.103 0.049 0.102 0.046
(4) 50 50 50 0.452 0.328 0.863 0.771 0.552 0.421 0.114 0.061 0.111 0.061 0.114 0.061
(4) 50 50 100 0.640 0.491 0.990 0.967 0.761 0.637 0.098 0.063 0.104 0.063 0.098 0.062
(4) 50 50 200 0.840 0.733 1 0.999 0.933 0.876 0.105 0.042 0.108 0.042 0.105 0.043
(5) 50 50 50 1 1 1 1 1 1 0.128 0.078 0.163 0.098 0.115 0.077
(5) 50 50 100 1 1 1 1 1 1 0.098 0.053 0.115 0.063 0.091 0.051
(5) 50 50 200 1 1 1 1 1 1 0.100 0.050 0.103 0.054 0.098 0.050
(1) 50 50 50 1 1 1 1 1 1 0.157 0.098 0.223 0.137 0.156 0.098
(1) 50 50 100 1 1 1 1 1 1 0.158 0.089 0.188 0.124 0.157 0.090
(1) 50 50 200 1 1 1 1 1 1 0.122 0.074 0.161 0.091 0.121 0.074
(2) 50 50 50 1 1 1 1 1 1 0.140 0.078 0.190 0.118 0.137 0.075
(2) 50 50 100 1 1 1 1 1 1 0.139 0.080 0.171 0.105 0.136 0.080
(2) 50 50 200 1 1 1 1 1 1 0.109 0.053 0.127 0.069 0.108 0.053
---------------------------------------------------------------------- ----- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
: Empirical size and power for the different tests of homogeneity of distributions.[]{data-label="table1:ed"}
\
\[rem on choice of d\_i\] In Example \[eg3:ed\].1, marginally the $p$-many two-dimensional groups of $X$ and $Y$ are not identically distributed, but each of the $2p$ unidimensional components of $X$ and $Y$ have identical distributions. Consequently, choosing $d_i = 1$ for $1\leq i \leq p$ leads to trivial power of even our proposed tests, as is evident from Table \[table1:ed rem\] below. This demonstrates that grouping allows us to detect a wider range of alternatives.
---------------------------------------------------------------------- ----- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
(r)[6-7]{}(r)[8-9]{}(r)[10-11]{}(r)[12-13]{}(r)[14-15]{}(r)[16-17]{} $n$ $m$ $p$ 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
(1) 50 50 50 0.144 0.087 0.133 0.076 0.143 0.086 0.174 0.107 0.266 0.170 0.175 0.105
(1) 50 50 100 0.145 0.085 0.134 0.070 0.142 0.085 0.157 0.098 0.223 0.137 0.156 0.098
(1) 50 50 200 0.126 0.063 0.101 0.058 0.111 0.065 0.158 0.089 0.188 0.124 0.157 0.090
---------------------------------------------------------------------- ----- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
: Empirical power in Example \[eg3:ed\].1 if we choose $d_i = 1$ for $1\leq i \leq p$.[]{data-label="table1:ed rem"}
\
Testing for independence
------------------------
We study the empirical size and power of tests for independence between two high dimensional random vectors. We consider the t-tests based on the following metrics:
I. $\cal{D}$ with $d_i=1$ and $\rho_i$ be the Euclidean distance for $1\leq i\leq p$;
II. $\cal{D}$ with $d_i=1$ and $\rho_i$ be the distance induced by the Laplace kernel for $1\leq i\leq p$;
III. $\cal{D}$ with $d_i=1$ and $\rho_i$ be the distance induced by the Gaussian kernel for $1\leq i\leq p$;
IV. the usual Euclidean distance covariance;
V. HSIC with the Laplace kernel;
VI. HSIC with the Gaussian kernel.
The numerical examples we consider are motivated from Zhu et al. (2019).
\[eg1\] Consider $X_k = (X_{k1}, \dots, X_{kp})$ and $Y_k = (Y_{k1}, \dots, Y_{kp})$ for $k=1,\dots, n$. We generate i.i.d. samples from the following models :
1. $X_k \sim N(0, I_p)$ and $Y_k \sim N(0, I_p)$.
2. $X_k \sim AR(1), \phi = 0.5$, $Y_k \sim AR(1), \phi = -0.5$, where $AR(1)$ denotes the autoregressive model of order $1$ with parameter $\phi$.
3. $X_k \sim N(0, \Sigma)$ and $Y_k \sim N(0, \Sigma)$, where $\Sigma = (\sigma_{ij})_{i,j=1}^p$ with $\sigma_{ij} = 0.7^{|i-j|}$.
\[eg2\] Consider $X_k = (X_{k1}, \dots, X_{kp})$ and $Y_k = (Y_{k1}, \dots, Y_{kp})$, $k=1,\dots, n$. We generate i.i.d. samples from the following models :
1. $X_k \sim N(0, I_p)$ and $Y_{kj} = X_{kj}^2$ for $j=1,\dots, p$.
2. $X_k \sim N(0, I_p)$ and $Y_{kj} = \log|X_{kj}|$ for $j=1,\dots, p$.
3. $X_k \sim N(0, \Sigma)$ and $Y_{kj} = X_{kj}^2$ for $j=1,\dots, p$, where $\Sigma = (\sigma_{ij})_{i,j=1}^p$ with $\sigma_{ij} = 0.7^{|i-j|}$.
\[eg3\] Consider $X_k = (X_{k1}, \dots, X_{kp})$ and $Y_k = (Y_{k1}, \dots, Y_{kp})$, $k=1,\dots, n$. Let $\circ$ denote the Hadamard product of matrices. We generate i.i.d. samples from the following models:
1. $X_{kj} \sim U(-1,1)$ for $j=1,\dots, p$, and $Y_k = X_k \circ X_k$.
2. $X_{kj} \sim U(0,1)$ for $j=1,\dots, p$, and $Y_{k} = 4 X_k \circ X_k - 4X_k +2$.
3. $X_{kj} = \sin(Z_{kj})$ and $Y_{kj} = \cos(Z_{kj})$ with $Z_{kj} \sim U(0,2\pi)$ and $j=1,\dots,p$.
For each example, we draw $1000$ simulated datasets and perform tests for independence between the two variables based on the proposed dependence metrics, and the usual Euclidean dCov and HSIC. We try a small sample size $n=50$ and dimensions $p=50, 100$ and $200$. For the tests II, III, V and VI, we chose the bandwidth parameter heuristically as the median distance between the sample observations. Table \[table1\] reports the proportion of rejections out of the $1000$ simulation runs for the different tests.
In Example \[eg1\], the data generating scheme suggests that the variables $X$ and $Y$ are independent. The results in Table \[table1\] show that the tests based on the proposed dependence metrics perform almost equally good as the other competitors, and the rejection probabilities are quite close to the $10\%$ or $5\%$ nominal level. In Examples \[eg2\] and \[eg3\], the variables are clearly (componentwise non-linearly) dependent by virtue of the data generating scheme. The results indicate that the tests based on the proposed dependence metrics are able to detect the componentwise non-linear dependence between the two high-dimensional random vectors unlike the tests based on the usual Euclidean dCov and HSIC, and thereby outperform the latter in terms of empirical power.
--------------------------------------------------------------------- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
(r)[5-6]{}(r)[7-8]{}(r)[9-10]{}(r)[11-12]{}(r)[13-14]{}(r)[15-16]{} $n$ $p$ 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5%
(1) 50 50 0.115 0.053 0.109 0.055 0.106 0.053 0.112 0.060 0.112 0.053 0.111 0.061
(1) 50 100 0.106 0.057 0.090 0.046 0.095 0.048 0.111 0.060 0.112 0.059 0.113 0.060
(1) 50 200 0.076 0.031 0.084 0.046 0.084 0.042 0.096 0.035 0.090 0.038 0.095 0.035
(2) 50 50 0.101 0.052 0.096 0.061 0.094 0.053 0.096 0.050 0.103 0.054 0.096 0.052
(2) 50 100 0.080 0.036 0.083 0.035 0.086 0.042 0.081 0.041 0.088 0.044 0.083 0.041
(2) 50 200 0.117 0.051 0.098 0.056 0.103 0.052 0.104 0.048 0.103 0.052 0.106 0.048
(3) 50 50 0.093 0.056 0.098 0.052 0.097 0.056 0.091 0.052 0.080 0.050 0.087 0.052
(3) 50 100 0.104 0.052 0.085 0.046 0.091 0.054 0.104 0.048 0.105 0.051 0.102 0.048
(3) 50 200 0.105 0.059 0.110 0.057 0.103 0.051 0.106 0.055 0.099 0.052 0.105 0.056
(1) 50 50 1 1 1 1 1 1 0.267 0.172 0.534 0.398 0.277 0.182
(1) 50 100 1 1 1 1 1 1 0.171 0.102 0.284 0.180 0.167 0.102
(1) 50 200 1 1 1 1 1 1 0.130 0.075 0.194 0.108 0.128 0.073
(2) 50 50 1 1 1 1 1 1 0.154 0.092 0.199 0.130 0.154 0.091
(2) 50 100 1 1 1 1 1 1 0.109 0.050 0.128 0.064 0.108 0.049
(2) 50 200 1 1 1 1 1 1 0.099 0.057 0.107 0.060 0.097 0.057
(3) 50 50 1 1 1 1 1 1 0.654 0.546 0.981 0.959 0.708 0.631
(3) 50 100 1 1 1 1 1 1 0.418 0.309 0.790 0.700 0.455 0.343
(3) 50 200 1 1 1 1 1 1 0.277 0.188 0.504 0.391 0.284 0.193
(1) 50 50 1 1 1 1 1 1 0.129 0.072 0.193 0.105 0.130 0.071
(1) 50 100 1 1 1 1 1 1 0.145 0.069 0.158 0.091 0.145 0.069
(1) 50 200 1 1 1 1 1 1 0.113 0.065 0.123 0.067 0.113 0.065
(2) 50 50 1 1 1 1 1 1 0.129 0.072 0.193 0.105 0.130 0.071
(2) 50 100 1 1 1 1 1 1 0.145 0.069 0.158 0.091 0.145 0.069
(2) 50 200 1 1 1 1 1 1 0.113 0.065 0.123 0.067 0.113 0.065
(3) 50 50 0.540 0.388 1 1 0.859 0.760 0.110 0.057 0.108 0.063 0.111 0.056
(3) 50 100 0.550 0.416 1 1 0.857 0.761 0.108 0.063 0.112 0.063 0.108 0.062
(3) 50 200 0.542 0.388 1 1 0.872 0.765 0.106 0.049 0.111 0.051 0.106 0.050
--------------------------------------------------------------------- ----- ----- ----- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
: Empirical size and power for the different tests of independence.[]{data-label="table1"}
\
Real data analysis {#sub:real}
------------------
### Testing for homogeneity of distributions {#testing-for-homogeneity-of-distributions}
We consider the two sample testing problem of homogeneity of two high-dimensional distributions on Earthquakes data. The dataset has been downloaded from UCR Time Series Classification Archive (<https://www.cs.ucr.edu/~eamonn/time_series_data_2018/>). The data are taken from Northern California Earthquake Data Center. There are 368 negative and 93 positive earthquake events and each data point is of length 512.
Table \[table:real\] shows the p-values corresponding to the different tests for the homogeneity of distributions between the two classes. Here we set $d_i=1$ for tests I-III. Clearly the tests based on the proposed homogeneity metrics reject the null hypothesis of equality of distributions at $5\%$ level. However the tests based on the usual Euclidean energy distance and MMD fail to reject the null at $5\%$ level, thereby indicating no significant difference between the distributions of the two classes.
I II III IV V VI
----------------------- ----------------------- ------------------------ --------- --------- ---------
$2.27\times 10^{-93}$ $3.19\times 10^{-86}$ $9.74\times 10^{-110}$ $0.070$ $0.068$ $0.070$
: p-values corresponding to the different tests for homogeneity of distributions for Earthquakes data.[]{data-label="table:real"}
### Testing for independence
We consider the daily closed stock prices of $p=127$ companies under the finance sector and $q=125$ companies under the healthcare sector on the first dates of each month during the time period between January 1, 2017 and December 31, 2018. The data has been downloaded from Yahoo Finance via the R package ‘quantmod’. At each time $t$, denote the closed stock prices of these companies from the two different sectors by $X_t = (X_{1 t}, \dots, X_{p t})$ and $Y_t = (Y_{1 t}, \dots, Y_{q t})$ for $1\leq t \leq 24$. We consider the stock returns $S^X_t = (S^X_{1 t}, \dots, S^X_{p t})$ and $S^Y_t = (S^Y_{1 t}, \dots, S^Y_{q t})$ for $1\leq t \leq 23$, where $S^X_{i t} = \log \frac{X_{i, t+1}}{X_{i t}}$ and $S^Y_{j t} = \log \frac{Y_{j, t+1}}{Y_{j t}}$ for $1\leq i \leq p$ and $1\leq j \leq q$. It seems intuitive that the stock returns for the companies under two different sectors are not totally independent, especially when a large number of companies are being considered. Table \[table:real independence\] shows the p-values corresponding to the different tests for independence between $\{S^X_t\}_{t=1}^{23}$ and $\{S^Y_t\}_{t=1}^{23}$, where we set $d_i=g_i=1$ for the proposed tests. The tests based on the proposed dependence metrics deliver much smaller p-values compared to the tests based on traditional metrics. We note that the tests based on the usual dCov and HSIC with the Laplace kernel fail to reject the null at $5\%$ level, thereby indicating cross-sector independence of stock return values. These results are consistent with the fact that the dependence among financial asset returns is usually nonlinear and thus cannot be fully characterized by traditional metrics in the high dimensional setup.
I II III IV V VI
----------------------- ----------------------- ----------------------- --------- --------- ---------
$5.70\times 10^{-13}$ $2.36\times 10^{-10}$ $7.99\times 10^{-11}$ $0.120$ $0.093$ $0.040$
: p-values corresponding to the different tests for cross-sector independence of stock returns data.[]{data-label="table:real independence"}
We present an additional real data example on testing for independence in high dimensions in Section \[addl data ex\] of the supplement. There the data admits a natural grouping, and our results indicate that our proposed tests for independence exhibit better power when we consider the natural grouping than when we consider unit group sizes. It is to be noted that considering unit group sizes makes our proposed statistics essentially equivalent to the marginal aggregation approach proposed by Zhu et al.(2019). This indicates that grouping or clustering might improve the power of testing as they are capable of detecting a wider range of dependencies.
Discussions
===========
In this paper, we introduce a family of distances for high dimensional Euclidean spaces. Built on the new distances, we propose a class of distance and kernel-based metrics for high-dimensional two-sample and independence testing. The proposed metrics overcome certain limitations of the traditional metrics constructed based on the Euclidean distance. The new distance we introduce corresponds to a semi-norm given by $$B(x)=\sqrt{\rho_1(x_{(1)})+\dots,\rho_p(x_{(p)})},$$ where $\rho_i(x_{(i)})=\rho_i(x_{(i)},0_{d_i})$ and $x = (x_{(1)},\dots,x_{(p)})\in\mathbb{R}^{\tilde{p}}$ with $x_{(i)}=(x_{i,1},\dots,x_{i,d_i}).$ Such a semi-norm has an interpretation based on a tree as illustrated by Figure \[fig\].
![An interpretation of the semi-norm $B(\cdot)$ based on a tree[]{data-label="fig"}](tree.pdf){height="6cm" width="10cm"}
Tree structure provides useful information for doing grouping at different levels/depths. Theoretically, grouping allows us to detect a wider range of alternatives. For example, in two-sample testing, the difference between two one-dimensional marginals is always captured by the difference between two higher dimensional marginals that contain the two one-dimensional marginals. The same thing is true for dependence testing. Generally, one would like to find blocks which are nearly independent, but the variables inside a block have significant dependence among themselves. It is interesting to develop an algorithm for finding the optimal groups using the data and perhaps some auxiliary information. Another interesting direction is to study the semi-norm and distance constructed based on a more sophisticated tree structure. For example, in microbiome-wide association studies, phylogenetic tree or evolutionary tree which is a branching diagram or “tree” showing the evolutionary relationships among various biological species. Distance and kernel-based metrics constructed based on the distance utilizing the phylogenetic tree information is expected to be more powerful in signal detection. We leave these topics for future investigation.
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[**Supplement to “A New Framework for Distance and Kernel-based Metrics in High Dimensions"**]{}\
The supplement is organized as follows. In Section \[ld app\] we explore our proposed homogeneity and dependence metrics in the low-dimensional setup. In Section \[HDMSS app\] we study the asymptotic behavior of our proposed homogeneity and dependence metrics in the high dimension medium sample size (HDMSS) framework where both the dimension(s) and the sample size(s) grow. Section \[addl data ex\] illustrates an additional real data example for testing for independence in the high-dimensional framework. Finally, Section \[technical\] contains additional proofs of the main results in the paper and Sections \[ld app\] and \[HDMSS app\] in the supplement.
Low-dimensional setup {#ld app}
=====================
In this section we illustrate that the new class of homogeneity metrics proposed in this paper inherits all the nice properties of generalized energy distance and MMD in the low-dimensional setting. Likewise, the proposed dependence metrics inherit all the desirable properties of generalized dCov and HSIC in the low-dimensional framework.
Homogeneity metrics {#ld E}
-------------------
Note that in either Case S1 or S2, the Euclidean space equipped with distance $K$ is of strong negative type. As a consequence, we have the following result.
\[ED HD homo\] $\cal{E}(X,Y) = 0$ if and only if $X \overset{d}{=} Y$, in other words $\cal{E}(X,Y)$ completely characterizes the homogeneity of the distributions of $X$ and $Y$.
The following proposition shows that $\cal{E}_{n,m}(X,Y)$ is a two-sample U-statistic and an unbiased estimator of $\cal{E} (X,Y)$.
\[prop ED U stat\] The U-statistic type estimator enjoys the following properties:
1. $\cal{E}_{n,m}$ is an unbiased estimator of the population $\cal{E}$.
2. $\cal{E}_{n,m}$ admits the following form : $$\cal{E}_{n,m}(X,Y) \;=\; \frac{1}{\binom{n}{2} \,\binom{m}{2}} \dis \sum_{1 \leq i < j \leq n}\, \sum_{1 \leq k < l \leq m} h(X_i, X_j ; Y_k, Y_l)\,,$$ where $$h(X_i, X_j ; Y_k, Y_l) \;=\; \frac{1}{2}\Big( K(X_i, Y_k) \,+\,K(X_i, Y_l)\,+\,K(X_j, Y_k)\,+\,K(X_j, Y_l)\Big) \,-\, K(X_i, X_j) \,-\, K(Y_k, Y_l)\,.$$
The following theorem shows the asymptotic behavior of the U-statistic type estimator of $\cal{E}$ for fixed $p$ and growing $n$.
\[th ED U stat\] Under Assumption \[ass0.5\] and the assumption that $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i(X_{(i)},0_{d_i})< \infty$ and $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i(Y_{(i)},0_{d_i})< \infty$, as $m, n \to \infty$ with $p$ remaining fixed, we have the following:
1. $\cal{E}_{n,m} (X,Y) \,\overset{a.s.}{\longrightarrow}\, \cal{E} (X,Y)$.
2. When $X \overset{d}{=} Y$, $\cal{E}_{n,m}$ has degeneracy of order $(1,1)$, and $$\frac{(m-1)(n-1)}{n+m} \, \cal{E}_{n,m} (X,Y) \, \overset{d}{\longrightarrow} \, \sum_{k=1}^{\infty} \lambda_k^2 \left(Z_k^2 \,-\, 1 \right)\,,$$ where $\{Z_k\}$ is a sequence of independent $N(0,1)$ random variables and $\lambda_k$’s depend on the distribution of $(X, Y)$.
Proposition \[prop ED U stat\], Theorem \[ED HD homo\] and Theorem \[th ED U stat\] demonstrate that $\cal{E}$ inherits all the nice properties of generalized energy distance and MMD in the low-dimensional setting.
Dependence metrics {#ld D}
------------------
Note that Proposition \[metric\] in Section \[new distance\] and Proposition 3.7 in Lyons(2013) ensure that $\cal{D}(X,Y)$ completely characterizes independence between $X$ and $Y$, which leads to the following result.
\[characterization\] Under Assumption \[ass1\], $\cal{D}(X,Y) = 0$ if and only if $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$.
The following proposition shows that $\widetilde{\cal{D}^2_n}(X,Y)$ is an unbiased estimator of $\cal{D}^2 (X,Y)$ and is a U-statistic of order four.
\[unbiased and ustat form\] The U-statistic type estimator $\widetilde{\cal{D}^2_n}$ (defined in (\[ustat dcov\]) in the main paper) has the following properties:
1. $\widetilde{\cal{D}^2_n}$ is an unbiased estimator of the squared population $\cal{D}^2$.
2. $\widetilde{\cal{D}^2_n}$ is a fourth-order U-statistic which admits the following form: $$\widetilde{\cal{D}^2_n} \;=\; \frac{1}{\binom{n}{4}} \dis \sum_{i<j<k<l} h_{i,j,k,l}\,,$$ where $$\begin{aligned}
h_{i,j,k,l} \,&= \, \frac{1}{4!} \dis \sum_{(s,t,u,v)}^{(i,j,k,l)} (d^X_{st} d^Y_{st} + d^X_{st} d^Y_{uv} - 2 d^X_{st} d^Y_{su}) \\ &= \, \frac{1}{6} \dis \sum_{s<t, u<v}^{(i,j,k,l)} (d^X_{st} d^Y_{st} + d^X_{st} d^Y_{uv}) - \frac{1}{12} \dis \sum_{(s,t,u)}^{(i,j,k,l)} d^X_{st} d^Y_{su}\,,\end{aligned}$$ the summation is over all possible permutations of the $4$-tuple of indices $(i,j,k,l)$. For example, when $(i,j,k,l)=(1,2,3,4)$, there exist 24 permutations, including $(1,2,3,4),\dots,(4,3,2,1)$. Furthermore, $\widetilde{\cal{D}^2_n}$ has degeneracy of order 1 when $X$ and $Y$ are independent.
The following theorem shows the asymptotic behavior of the U-statistic type estimator of $\cal{D}^2$ for fixed $p, q$ and growing $n$.
\[ACdCov U-stat prop\] Under Assumption \[ass1\], with fixed $p, q$ and $n \to \infty$, we have the following as $n \to \infty$:
1. $\widetilde{\cal{D}^2_n}(X,Y) \overset{a.s.}{\longrightarrow} \cal{D}^2(X,Y)$;
2. When $\cal{D}^2(X,Y)=0$ (i.e., $X {\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}Y$), $n\,\widetilde{\cal{D}^2_n}(X,Y)\, \overset{d}{\longrightarrow} \dis\sum_{i=1}^{\infty} \tilde{\lambda}^2_i (Z_i^2 -1)$, where $Z_i's$ are i.i.d. standard normal random variables and $ \tilde{\lambda}_i$’s depend on the distribution of $(X, Y)$;
3. When $\cal{D}^2(X,Y)>0$, $n\,\widetilde{\cal{D}^2_n}(X,Y) \, \overset{a.s.}{\longrightarrow}\infty$.
Proposition \[unbiased and ustat form\], Theorem \[characterization\] and Theorem \[ACdCov U-stat prop\] demonstrate that in the low-dimensional setting, $\cal{D}$ inherits all the nice properties of generalized dCov and HSIC.
High dimension medium sample size (HDMSS) {#HDMSS app}
=========================================
Homogeneity metrics {#subsec:ED HDMSS}
-------------------
In this subsection, we consider the HDMSS setting where $p \to \infty$ and $n, m \to \infty$ at a slower rate than $p$. Under $H_0$, we impose the following conditions to obtain the asymptotic null distribution of the statistic $T_{n,m}$ under the HDMSS setup.
\[assED\_HDMSS\] As $n, m$ and $p \to \infty$, $$\begin{aligned}
& \frac{1}{n^2}\, \frac{{{\mathbb{E}}}\,\left[H^4(X,X')\right]}{\left({{\mathbb{E}}}\,\left[H^2(X, X')\right] \right)^2} \;=\; o(1),\quad \frac{1}{n}\, \frac{{{\mathbb{E}}}\,\left[H^2(X,X'')\, H^2(X', X'')\right]}{\left({{\mathbb{E}}}\,\left[H^2(X, X')\right] \right)^2} \;=\; o(1),\\ & \frac{{{\mathbb{E}}}\,\left[H(X,X'')\, H(X', X'')\,H(X, X''')\,H(X', X''')\right]}{\left({{\mathbb{E}}}\,\left[H^2(X, X')\right] \right)^2} \;=\; o(1).\end{aligned}$$
\[rem\_ass4.6\] We refer the reader to Section 2.2 in Zhang et al.(2018) and Remark A.2.2 in Zhu et al.(2019) for illustrations of Assumption \[assED\_HDMSS\] where $\rho_i$ has been considered to be the Euclidean distance or the squared Euclidean distance, respectively, for $1\leq i\leq p$.
\[assED\_HDMSS4\] Suppose ${{\mathbb{E}}}\, [L^2(X, X')] = O(\alpha_p^2)$ where $\alpha_p$ is a positive real sequence such that $\tau_X \alpha_p^2 = o(1)$ as $p \to \infty$. Further assume that as $n, p \to \infty$, $$\frac{n^4 \,\tau_X^4\, {{\mathbb{E}}}\,\left[ R^4(X,X')\right]}{\left( {{\mathbb{E}}}\,\left[ H^2(X,X')\right] \right)^2} = o(1)\,.$$
\[expl for assED\_HDMSS4\] We refer the reader to Remark \[assumption justification\] in the main paper which illustrates some sufficient conditions under which $\alpha_p = O(\frac{1}{\sqrt{p}})$ and consequently $\tau_X \alpha_p^2 = o(1)$ holds, as $\tau_X \asymp p^{1/2}$. In similar lines of Remark \[rem\_ui\] in Section \[technical\] of the supplementary material, it can be argued that ${{\mathbb{E}}}\,\left[ R^4(X,X')\right] = O\left(\frac{1}{p^4}\right)$. If we further assume that Assumption \[ass6\] holds, then we have ${{\mathbb{E}}}\,\left[ H^2(X,X')\right] \asymp 1$. Combining all the above, it is easy to verify that $\frac{n^4 \,\tau_X^4\, {{\mathbb{E}}}\,\left[ R^4(X,X')\right]}{\left( {{\mathbb{E}}}\,\left[ H^2(X,X')\right] \right)^2} = o(1)$ holds provided $n = o(p^{1/2})$.
The following theorem illustrates the limiting null distribution of $T_{n,m}$ under the HDMSS setup. We refer the reader to Section \[technical\] of the supplement for a detailed proof.
\[th\_ED\_HDMSS\] Under $H_0$ and Assumptions \[ass0.5\], \[assED\_HDMSS\] and \[assED\_HDMSS4\], as $n, m$ and $p \to \infty$, we have
$$T_{n,m} \; \overset{d}{\longrightarrow} \; N(0,1).$$
Dependence metrics {#sec:D HDMSS}
------------------
In this subsection, we consider the HDMSS setting where $p, q \to \infty$ and $n \to \infty$ at a slower rate than $p, q$. The following theorem shows that similar to the HDLSS setting, under the HDMSS setup, $\widetilde{\cal{D}^2_n}$ is asymptotically equivalent to the aggregation of group-wise generalized dCov. In other words $\widetilde{\cal{D}^2_n} (X,Y)$ can quantify group-wise nonlinear dependence between $X$ and $Y$ in the HDMSS setup as well.
\[ass4\] ${{\mathbb{E}}}[L_X(X,X')^2] = \alpha_p^2$, ${{\mathbb{E}}}[L_X(X,X')^4] = \gamma_p^2$, ${{\mathbb{E}}}[L_Y(Y,Y')^2] = \beta_q^2$ and ${{\mathbb{E}}}[L_Y(Y,Y')^4] = \lambda_q^2$, where $\alpha_p, \gamma_p, \beta_q, \lambda_q$ are positive real sequences satisfying $n \alpha_p = o(1)$, $n \beta_q = o(1)$, $\tau_X^2 (\alpha_p \gamma_p + \gamma_p^2) = o(1)$, $\tau_Y^2 (\beta_q \lambda_q + \lambda_q^2) = o(1)$, and $\tau_{XY}\, (\alpha_p \lambda_q + \gamma_p \beta_q + \gamma_p \lambda_q ) = o(1)$.
\[assumption justification for rem5.5\] Following Remark \[assumption justification\] in the main paper, we can write $L(X,X') = O(\frac{1}{p}) \sum_{i=1}^p \left(Z_i - {{\mathbb{E}}}\, Z_i \right)$, where $Z_i = \rho_i( X_{(i)}, X_{(i)}')$ for $1\leq i \leq p$. Assume that $ \sup_{1\leq i \leq p} {{\mathbb{E}}}\,\rho_i^4( X_{(i)}, 0_{d_i}) < \infty$, which implies $ \sup_{1\leq i \leq p} {{\mathbb{E}}}\,Z_i^4 < \infty$. Under certain weak dependence assumptions, it can be shown that ${{\mathbb{E}}}\,\big( \sum_{i=1}^p ( Z_i - {{\mathbb{E}}}\, Z_i)\big)^4 = O(p^2)$ as $p\to \infty$ (see for example Theorem 1 in Doukhan et al.(1999)). Therefore we have ${{\mathbb{E}}}[L(X,X')^4] = O(\frac{1}{p^2})$. It follows from H[ö]{}lder’s inequality that ${{\mathbb{E}}}[L(X,X')^2 ] = O(\frac{1}{p})$. Similar arguments can be made about ${{\mathbb{E}}}[L(Y,Y')^4]$ and ${{\mathbb{E}}}[L(Y,Y')^2]$ as well.
\[ACdCov hd\] Under Assumptions \[ass2 : ED\] and \[ass4\], we can show that $$\label{ACdCov taylor HDMSS}
\widetilde{\cal{D}^2_n} (X,Y) = \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q \widetilde{D^2_n}_{\,;\,\rho_i, \rho_j}(X_{(i)},Y_{(j)}) \, + \, \mathcal{R}_n \; ,$$ where $\mathcal{R}_n$ is the remainder term satisfying that $\mathcal{R}_n = O_p(\tau_{XY}\, (\alpha_p \lambda_q + \gamma_p \beta_q + \gamma_p \lambda_q )) = o_p(1)$, i.e., $\mathcal{R}_n$ is of smaller order compared to the leading term and hence is asymptotically negligible.
The following theorem states the asymptotic null distribution of the studentized test statistic $\mathcal{T}_n$ (given in equation (\[student t\]) in the main paper) under the HDMSS setup. Define $$U(X_k, X_l) := \frac{1}{\tau_X} \dis \sum_{i=1}^p d^X_{kl}(i), \quad \text{and} \quad V(Y_k, Y_l) := \frac{1}{\tau_Y} \dis \sum_{i=1}^q d^Y_{kl}(i).$$
\[ass5\] Assume that $$\begin{aligned}
&\frac{{{\mathbb{E}}}\left[U(X, X')\right]^4}{\sqrt{n}\left({{\mathbb{E}}}[U(X, X')]^2 \right)^2} \,=\, o(1),
\\ &\frac{{{\mathbb{E}}}\left[ U(X, X')\, U(X', X'')\, U(X'', X''')\,U(X''', X) \right]}{\left({{\mathbb{E}}}[U(X, X')]^2 \right)^2}\, =\, o(1),\end{aligned}$$ and the same conditions hold for $Y$ in terms of $V(Y,Y')$.
\[rem HDdCov HDMSS\] We refer the reader to Section 2.2 in Zhang et al.(2018) and Remark A.2.2 in Zhu et al.(2019) for illustrations of Assumption \[assED\_HDMSS\] where $\rho_i$ has been considered to be the Euclidean distance or the squared Euclidean distance, respectively.
We can show that under $H_0$, the studentized test $\mathcal{T}_n$ converge to the standard normal distribution under the HDMSS setup.
\[HDMSS dist conv\] Under $H_0$ and Assumptions \[ass4\]-\[ass5\], as $n, p, q \to \infty$, we have $\mathcal{T}_n \, \overset{d}{\longrightarrow} \, N(0,1)\,.$
Additional real data example {#addl data ex}
============================
We consider the monthly closed stock prices of $\tilde{p}=33$ companies under the oil and gas sector and $\tilde{q}=34$ companies under the transport sector between January 1, 2017 and December 31, 2018. The companies under both the sectors are clustered or grouped according to their countries. The data has been downloaded from Yahoo Finance via the R package ‘quantmod’. Under the oil and gas sector, we have $p=18$ countries or groups, viz. USA, Australia, UK, Canada, China, Singapore, Hong Kong, Netherlands, Colombia, Italy, Norway, Bermuda, Switzerland, Brazil, South Africa, France, Turkey and Argentina, with $d = (5,1,2,5,4,1,1,2,1,1,1,1,2,1,1,2,1,1)$. And under the transport sector, we have $q=14$ countries or groups, viz. USA, Brazil, Canada, Greece, China, Panama, Belgium, Bermuda, UK, Mexico, Chile, Monaco, Ireland and Hong Kong, with $g = (5,1,2,6,4,1,1,3,1,3,1,4,1,1)$. At each time $t$, denote the closed stock prices of these companies from the two different sectors by $X_t = (X_{1 t}, \dots, X_{p t})$ and $Y_t = (Y_{1 t}, \dots, Y_{q t})$ for $1\leq t \leq 24$. We consider the stock returns $S^X_t = (S^X_{1 t}, \dots, S^X_{p t})$ and $S^Y_t = (S^Y_{1 t}, \dots, S^Y_{q t})$ for $1\leq t \leq 23$, where $S^X_{i t l} = \log \frac{X_{i, t+1, l}}{X_{i t l}}$ and $S^Y_{j t l'} = \log \frac{Y_{j, t+1, l'}}{Y_{j t l'}}$ for $1\leq l\leq d_i$, $1\leq i \leq p$, $1\leq l'\leq g_j$ and $1\leq j \leq q$.
The intuitive idea is, stock returns of oil and gas companies should affect the stock returns of companies under the transport sector, and here both the random vectors admit a natural grouping based on the countries. Table \[table:real independence 2\] shows the p-values corresponding to the different tests for independence between $\{S^X_t\}_{t=1}^{23}$ and $\{S^Y_t\}_{t=1}^{23}$. The tests based on the proposed dependence metrics considering the natural grouping deliver much smaller p-values compared to the tests based on the usual dCov and HSIC which fail to reject the null hypothesis of independence between $\{S^X_t\}_{t=1}^{23}$ and $\{S^Y_t\}_{t=1}^{23}$. This makes intuitive sense as the dependence among financial asset returns is usually nonlinear in nature and thus cannot be fully characterized by the usual dCov and HSIC in the high dimensional setup.
I II III IV V VI
--------- --------- --------- --------- --------- ---------
$0.036$ $0.048$ $0.087$ $0.247$ $0.136$ $0.281$
: p-values corresponding to the different tests for cross-sector independence of stock returns data considering the natural grouping based on countries.[]{data-label="table:real independence 2"}
Table \[table:real independence 2.5\] shows the p-values corresponding to the different tests for independence when we disregard the natural grouping and consider $d_i =1$ and $g_j =1$ for all $1\leq i \leq p$ and $1\leq j \leq q$. Considering unit group sizes makes our proposed statistics essentially equivalent to the marginal aggregation approach proposed by Zhu et al.(2019). In this case the proposed tests have higher p-values than when we consider the natural grouping, indicating that grouping or clustering might improve the power of testing as they are capable of detecting a wider range of dependencies.
I II III IV V VI
--------- --------- --------- --------- --------- ---------
$0.092$ $0.209$ $0.226$ $0.247$ $0.136$ $0.281$
: p-values corresponding to the different tests for cross-sector independence of stock returns data considering unit group sizes.[]{data-label="table:real independence 2.5"}
Technical Appendix {#technical}
==================
To prove (1), note that if $d$ is a metric on a space $\cal{X}$, then so is $d^{1/2}$. It is easy to see that $K^2$ is a metric on $\bb{R}^{\tilde{p}}$. To prove (2), note that $(\bb{R}^{d_i}, \rho_i)$ has strong negative type for $1 \leq i \leq p$. The rest follows from Corollary 3.20 in Lyons(2013).
It is easy to verify that $\cal{E}_{n,m}$ is an unbiased estimator of $\cal{E}$ and is a two-sample U-statistic with the kernel $h$.
The first part of the proof follows from Theorem 1 in Sen(1977) and the observation that ${{\mathbb{E}}}\,\left[|h| \log^+ |h|\right] \leq {{\mathbb{E}}}[h^2]$. The power mean inequality says that for $a_i \in \mathbb{R},\, 1\leq i \leq n, \, n\geq 2$ and $r>1$, $$\begin{aligned}
\label{power mean ineq}
\left|\dis \sum_{i=1}^n a_i \right|^r \; \leq \; n^{r-1} \, \dis \sum_{i=1}^n |a_i|^r \,.\end{aligned}$$ Using the power mean inequality, it is easy to see that the assumptions $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i(X_{(i)},0_{d_i})< \infty$ and $\sup_{1\leq i\leq p}{{\mathbb{E}}}\rho_i(Y_{(i)},0_{d_i})< \infty$ ensure that ${{\mathbb{E}}}[h^2] < \infty$. For proving the second part, define $h_{1,0}(X)=\; {{\mathbb{E}}}\, \left[ h(X, X'; Y, Y') | X \right]$ and $h_{0,1}(Y)=\; {{\mathbb{E}}}\, \left[ h(X, X'; Y, Y') | Y \right]$ Clearly, when $X\overset{d}{=}Y$, $h_{1,0}(X)$ and $h_{0,1}(Y)$ are degenerate at $0$ almost surely. Following Theorem 1.1 in Neuhaus(1977), we have $$\frac{(m-1)(n-1)}{n+m} \, \cal{E}_{n m} (X,Y) \, \overset{d}{\longrightarrow} \, \sum_{k=1}^{\infty} \sigma_k^2 \left[(a_k U_k + b_k V_k )^2 \,-\, (a_k^2 + b_k^2) \right]\,,$$ where $\{U_k\}, \{V_k\}$ are two sequences of independent $N(0,1)$ variables, independent of each other, and $(\sigma_k, a_k, b_k)$’s depend on the distribution of $(X,Y)$. The proof can be completed by some simple rearrangement of terms.
The proof is essentially similar to the proof of Proposition 2.1.1 in Zhu et al. (2019), replacing the Euclidean distance between, for example, $X$ and $X'$, viz. $\Vert X-X' \Vert_{\tilde{p}}$, by the new distance metric $K(X,X')$. To show that $R(X,X')=O_p(L^2(X,X'))$ if $L(X,X')=o_p(1)$, we define $f(x) = \sqrt{1+x}$. By the definition of the Lagrange’s form of the remainder term from Taylor’s expansion, we have $$R(X,X') = \int_0^{L(X,X')} f''(t) \left(\,L(X,X') - t\,\right)\, dt\,.$$ Using $R$ and $L$ interchangeably with $R(X,X')$ and $L(X,X')$ respectively, we can write $$\label{eq_taylor}
\begin{split}
|R| \; & \leq \; |L| \, \left[ \int_0^L f''(t) \, \mathbbm{1}_{L>0} \, dt \;+\; \int_L^0 f''(t) \, \mathbbm{1}_{L<0} \, dt \,\right]\\
&= \;\frac{|L|}{2}\, \big|1 - \frac{1}{\sqrt{1+L}}\big|\\
&= \; \frac{|L|}{2}\, \frac{|L|}{1 + L + \sqrt{1+L}}\\
&\leq \; \frac{L^2}{2(1+L)}\,.
\end{split}$$ It is clear that $R(X,X')=O_p(L^2(X,X'))$ provided that $L(X,X')=o_p(1)$.
Observe that ${{\mathbb{E}}}\,L(X,Y) = {{\mathbb{E}}}\,L(X,X') = {{\mathbb{E}}}\,L(Y,Y') = 0$. By Proposition \[K taylor : ED\], $$\begin{aligned}
\cal{E}(X,Y) \;&=\; 2\,{{\mathbb{E}}}\,\left[\tau + \tau\,R(X,Y)\right] \,-\, {{\mathbb{E}}}\,\left[\tau_X + \tau_X\,R(X,X')\right] \,-\, {{\mathbb{E}}}\,\left[\tau_Y + \tau_Y\,R(Y,Y')\right] \\
&=\; 2\tau -\tau_X - \tau_Y \,+\, \cal{R}_{\cal{E}}\,.\end{aligned}$$ Clearly $\vert \cal{R}_{\cal{E}} \vert\; \leq \; 2\,\tau\,{{\mathbb{E}}}\,\left[\,\vert R(X,Y)\vert \,\right] \,+\, \tau_X\,{{\mathbb{E}}}\,\left[\,\vert R(X,X')\vert \,\right] \,+\, \tau_Y\,{{\mathbb{E}}}\,\left[\,\vert R(Y,Y')\vert \,\right].$ By (\[eq\_taylor\]) and Assumption \[ass0.6\], we have $$\tau |R(X,Y)|\leq \frac{\tau L^2(X,Y)}{2(1+L(X,Y))}=O(\tau a^{2}_p)=o_p(1).$$ As $\{\sqrt{p} L^2(X,Y)/(1+L(X,Y))\}$ is uniformly integrable and $\tau\asymp \sqrt{p}$, we must have $\tau {{\mathbb{E}}}[|R(X,Y)|]=o(1)$. The other terms can be handled in a similar fashion.
\[rem\_ui\] Write $L(X,Y) = \frac{1}{\tau^2} (A_p - {{\mathbb{E}}}\,A_p) = \frac{1}{\tau^2} \sum_{i=1}^p (Z_i - {{\mathbb{E}}}Z_i)$, where $A_p := \sum_{i=1}^p Z_i$ and $Z_i := \rho_i (X_i, Y_i)$ for $1\leq i \leq p$. Assume $\sup_i {{\mathbb{E}}}\rho_i^8 (X_i, 0_{d_i}) < \infty$ and $\sup_i {{\mathbb{E}}}\rho_i^8 (X_i, 0_{d_i}) < \infty$, which imply $\sup_i {{\mathbb{E}}}Z_i^8 < \infty$. Denote $L(X,Y)$ by $L$ and $R(X,Y)$ by $R$ for notational simplicities. Further assume that $E \exp(t A_p)=O((1-\theta_1 t)^{-\theta_2 p})$ for $\theta_1, \theta_2 >0$ and $\theta_2 \,p > 4$ uniformly over $t<0$ (which is clearly satisfied when $Z_i$’s are independent and ${{\mathbb{E}}}\exp(t Z_i)\leq a_1(1-a_2 t)^{-a_3}$ uniformly over $t<0$ and $1\leq i\leq p$ for some $a_1,a_2,a_3>0$ with $a_3\, p > 4$). Under certain weak dependence assumptions, it can be shown that:
1. $\{\sqrt{p} L^2/(1+L)\}$ is uniformly integrable;
2. ${{\mathbb{E}}}\, R^2 = O(\frac{1}{p^2})$.
Similar arguments hold for $L(X,X')$ and $R(X,X')$, and, $L(Y,Y')$ and $R(Y,Y')$ as well.
To prove the first part, define $L_p := \sqrt{p} L^2/(1+L)$. Following Chapter 6 of Resnick(1999), it suffices to show that $\sup_p {{\mathbb{E}}}\,L_p^2 < \infty$. Towards that end, using H[ö]{}lder’s inequality we observe $$\begin{aligned}
\label{eqn rem_ui}
{{\mathbb{E}}}\,L_p^2 \;& \leq \left({{\mathbb{E}}}(p^2 L^8) \right)^{1/2} \, \left({{\mathbb{E}}}\Big[\frac{1}{(1+L)^4}\Big]\right)^{1/2}\,.\end{aligned}$$
With $\sup_i {{\mathbb{E}}}Z_i^8 < \infty$ and under certain weak dependence assumptions, it can be shown that ${{\mathbb{E}}}(A_p - {{\mathbb{E}}}A_p)^8 = O(p^4)$ (see for example Theorem 1 in Doukhan et al.(1999)). Consequently we have ${{\mathbb{E}}}\,L^8 = O(\frac{1}{p^4})$ , as $\tau\asymp \sqrt{p}$. Clearly this yields ${{\mathbb{E}}}\,(p^2 L^8) = O(\frac{1}{p^2})$.
Now note that $$\begin{aligned}
\label{eqn rem_ui 2}
{{\mathbb{E}}}\Big[\frac{1}{(1+L)^4}\Big] \;&=\; \tau^8\, {{\mathbb{E}}}\left(\frac{1}{A_p^4}\right)\,.\end{aligned}$$ Equation (3) in Cressie et al.(1981) states that for a non-negative random variable $U$ with moment-generating function $M_U(t) = {{\mathbb{E}}}\exp(tU)$, one can write $$\begin{aligned}
\label{eqn rem_ui 2.5}
{{\mathbb{E}}}(U^{-k}) &= (\Gamma(k))^{-1} \dis \int_0^{\infty} t^{k-1} M_U(-t) \, dt\,,\end{aligned}$$ for any positive integer $k$, provided both the integrals exist. Using equation (\[eqn rem\_ui 2.5\]), the assumptions stated in Remark \[rem\_ui\] and basic properties of beta integrals, some straightforward calculations yield $$\begin{aligned}
\label{eqn 3 calc}
{{\mathbb{E}}}\left(\frac{1}{A_p^4}\right) \;&\leq\; C_1 \, \int_0^{\infty} \frac{t^{4-1}}{(1+\theta_1 t)^{\theta_2 p}} \, dt \;=\; C_2 \, \frac{\Gamma(\theta_2 p -4)}{\Gamma(\theta_2 p)}\,,\end{aligned}$$ where $C_1, C_2$ are positive constants, which clearly implies that ${{\mathbb{E}}}\left(\frac{1}{A_p^4}\right) = O(\frac{1}{p^4})$. This together with equation (\[eqn rem\_ui 2\]) implies that ${{\mathbb{E}}}\Big[\frac{1}{(1+L)^4}\Big] = O(1)$, as $\tau\asymp \sqrt{p}$.
Combining all the above, we get from (\[eqn rem\_ui\]) that ${{\mathbb{E}}}\,L_p^2 = O(\frac{1}{p})$ and therefore $\sup_p {{\mathbb{E}}}\,L_p^2 < \infty$, which completes the proof of the first part.
To prove the second part, note that following the proof of Proposition \[K taylor : ED\] and H[ö]{}lder’s inequality we can write $$\begin{aligned}
\label{R_expr}
{{\mathbb{E}}}\,R^2 = O\left( {{\mathbb{E}}}\,\left[\frac{L^4}{(1+L)^2} \right]\right) = O\left( \left({{\mathbb{E}}}(L^8) \right)^{1/2} \, \left({{\mathbb{E}}}\Big[\frac{1}{(1+L)^4}\Big] \right)^{1/2} \right)\,.\end{aligned}$$ Following the arguments as in the proof of the first part, clearly we have ${{\mathbb{E}}}\,L^8 = O(\frac{1}{p^4})$ and ${{\mathbb{E}}}\Big[\frac{1}{(1+L)^4}\Big] = O(1)$. From this and equation (\[R\_expr\]), it is straightforward to verify that ${{\mathbb{E}}}\, R^2 = O(\frac{1}{p^2})$, which completes the proof of the second part.
To see (2), first observe that the sufficient part is straightforward from equation (\[tau for Eucl\]) in the main paper. For the necessary part, denote $a = \textrm{tr}\, \Sigma_X$, $b = \textrm{tr}\, \Sigma_Y$ and $c = \Vert \mu_X - \mu_Y \Vert^2$. Then we have $2\,\sqrt{a + b + c} = \sqrt{2 a} + \sqrt{2 b}$. Some straightforward calculations yield $(\sqrt{2 a} - \sqrt{2 b})^2 + 4\,c = 0$ which implies the rest.
To see (1), again the sufficient part is straightforward from equation (\[tau def original\]) in the paper and the form of $K$ given in equation (\[Kdef\]) in the paper. For the necessary part, first note that as $(\bb{R}^{d_i}, \rho_i)$ is a metric space of strong negative type for $1 \leq i \leq p$, there exists a Hilbert space $\cal{H}_i$ and an injective map $\phi_i : \bb{R}^{d_i} \to \cal{H}_i$ such that $\rho_i(z,z') = \Vert \phi_i(z) - \phi_i(z') \Vert^2_{\cal{H}_i}$, where $\langle\cdot,\cdot\rangle_{\cal{H}_i}$ is the inner product defined on $\cal{H}_i$ and $\Vert \cdot \Vert_{\cal{H}_i}$ is the norm induced by the inner product (see Proposition 3 in Sejdinovic et al.(2013) for detailed discussions). Further, if $k_i$ is a distance-induced kernel induced by the metric $\rho_i$, then by Proposition 14 in Sejdinovic et al.(2013), $\cal{H}_i$ is the RKHS with the reproducing kernel $k_i$ and $\phi_i(z)$ is essentially the canonical feature map for $\cal{H}_i$, viz. $\phi_i(z) : z \mapsto k_i(\cdot,z)$. It is easy to see that $$\begin{aligned}
\tau_X^2\,=&\,{{\mathbb{E}}}\,\sum^{p}_{i=1}\|\phi_i(X_{(i)})-\phi_i(X_{(i)}')\|^2_{\cal{H}_i}\,=\,2\,{{\mathbb{E}}}\sum^{p}_{i=1}\|\phi_i(X_{(i)})-{{\mathbb{E}}}\,\phi_i(X_{(i)})\|^2_{\cal{H}_i},\\
\tau_Y^2\,=&\,{{\mathbb{E}}}\,\sum^{p}_{i=1}\|\phi_i(Y_{(i)})-\phi_i(Y_{(i)}')\|^2_{\cal{H}_i}\,=\,2\,{{\mathbb{E}}}\sum^{p}_{i=1}\|\phi_i(Y_{(i)})-{{\mathbb{E}}}\,\phi_i(Y_{(i)})\|^2_{\cal{H}_i},\\
\tau^2\,=&\,{{\mathbb{E}}}\,\sum^{p}_{i=1}\|\phi_i(X_{(i)})-\phi_i(Y_{(i)})\|^2_{\cal{H}_i}\,=\,\tau_X^2/2+\tau_Y^2/2+\zeta^2,\end{aligned}$$ where $\zeta^2=\sum^{p}_{i=1}\|{{\mathbb{E}}}\,\phi(X_{(i)})-{{\mathbb{E}}}\,\phi(Y_{(i)})\|^2_{\cal{H}_i}$. Thus $2\tau - \tau_X - \tau_Y = 0$ is equivalent to $$\begin{aligned}
4(\tau_X^2/2+\tau_Y^2/2+\zeta^2)=(\tau_X+\tau_Y)^2=\tau_X^2+\tau_Y^2+2\tau_X\tau_Y.\end{aligned}$$ which implies that $$4\zeta^2+(\tau_X-\tau_Y)^2=0.$$ Therefore, $2\tau - \tau_X - \tau_Y = 0$ holds if and only if (1) $\zeta=0$, i.e., ${{\mathbb{E}}}\,\phi_i(X_{(i)})={{\mathbb{E}}}\,\phi_i(Y_{(i)})$ for all $1\leq i\leq p$, and, (2) $\tau_X=\tau_Y$, i.e., $${{\mathbb{E}}}\sum^{p}_{i=1}\|\phi_i(X_{(i)})-{{\mathbb{E}}}\,\phi_i(X_{(i)})\|_{\cal{H}_i}^2\,=\,{{\mathbb{E}}}\sum^{p}_{i=1}\|\phi_i(Y_{(i)})-{{\mathbb{E}}}\,\phi_i(Y_{(i)})\|_{\cal{H}_i}^2.$$ Now if $X \sim P$ and $Y \sim Q$, then note that $${{\mathbb{E}}}\,\phi_i(X_{(i)}) \,=\, \dis\int_{\bb{R}^{d_i}} k_i(\cdot, z)\, dP_i (z) \,=\, \Pi_i(P_i)\;\;\; \textrm{and} \;\;\; {{\mathbb{E}}}\,\phi_i(Y_{(i)}) \,=\, \dis\int_{\bb{R}^{d_i}} k_i(\cdot, z)\, dQ_i (z) \,=\, \Pi_i(Q_i)\,,$$ where $\Pi_i$ is the mean embedding function (associated with the distance induced kernel $k_i$) defined in Section \[ed sec\], $P_i$ and $Q_i$ are the distributions of $X_{(i)}$ and $Y_{(i)}$, respectively. As $\rho_i$ is a metric of strong negative type on $\bb{R}^{d_i}$, the induced kernel $k_i$ is characteristic to $\cal{M}_1(\bb{R}^{d_i})$ and hence the mean embedding function $\Pi_i$ is injective. Therefore condition (1) above implies $X_{(i)} \overset{d}{=} Y_{(i)}$.
Now we introduce some notation before presenting the proof of Theorem \[th KED HDLSS\]. The key of our analysis is to study the variance of the leading term of $\cal{E}_{n,m}(X,Y)$ in the HDLSS setup, propose the variance estimator and study the asymptotic behavior of the variance estimator. It will be shown later (in the proof of Theorem \[th KED HDLSS\]) that the leading term in the Taylor’s expansion of $\cal{E}_{n,m}(X,Y) - (2\tau - \tau_X - \tau_Y)$ can be written as $L_1 + L_2$, where $$\begin{aligned}
\label{L decomp.1}
\begin{split}
L_1\;&:=\;\frac{1}{nm\tau}\sum_{k=1}^{n}\sum_{l=1}^{m}\sum_{i=1}^{p}d_{kl}(i)-\frac{1}{n(n-1)\tau_X}\sum_{k< l}\sum_{i=1}^{p}d^X_{kl}(i)
-\frac{1}{m(m-1)\tau_Y}\sum_{k< l}\sum_{i=1}^{p}d^Y_{kl}(i)\\
\;&:=\,L_1^1-L_1^2-L_1^3 \;,
\end{split}\end{aligned}$$ where $L_1^i$’s are defined accordingly and $$\begin{aligned}
\label{L decomp.2}
\begin{split}
L_2 \;:=&\; \frac{1}{nm\tau}\sum_{k=1}^{n}\sum_{l=1}^{m}\sum_{i=1}^{p}\Big({{\mathbb{E}}}\,[\rho_i(X_{k(i)},Y_{l(i)})|X_{k(i)}] + [\rho_i(X_{k(i)},Y_{l(i)})|Y_{l(i)}] - 2\,{{\mathbb{E}}}\,\rho_i(X_{k(i)},Y_{l(i)})\Big) \\ & \;-\frac{1}{n(n-1)\tau_X}\sum_{k< l}\sum_{i=1}^{p}\Big({{\mathbb{E}}}\,[\rho_i(X_{k(i)},X_{l(i)})|X_{k(i)}] + [\rho_i(X_{k(i)},X_{l(i)})|X_{l(i)}] - 2\,{{\mathbb{E}}}\,\rho_i(X_{k(i)},X_{l(i)})\Big)
\\&-\frac{1}{m(m-1)\tau_Y}\sum_{k< l}\sum_{i=1}^{p}\Big({{\mathbb{E}}}\,[\rho_i(Y_{k(i)},Y_{l(i)})|Y_{k(i)}] + [\rho_i(Y_{k(i)},Y_{l(i)})|Y_{l(i)}] - 2\,{{\mathbb{E}}}\,\rho_i(Y_{k(i)},Y_{l(i)})\Big)\,.
\end{split}\end{aligned}$$ By the double-centering properties, it is easy to see that $L_1^i$ for $1\leq i\leq 3$ are uncorrelated. Define $$\begin{aligned}
\label{V def supp}
\begin{split}
V :=& \frac{1}{n m \tau^2}\sum_{i,i'=1}^{p}{{\mathbb{E}}}\,[d_{kl}(i)\,d_{kl}(i')] \;+\;\frac{1}{2n(n-1)\tau^2_X}\sum_{i,i'=1}^{p}{{\mathbb{E}}}\,[d_{kl}^X(i)\,d_{kl}^X(i')]\\ & +\;\frac{1}{2m(m-1)\tau^2_Y}\sum_{i,i'=1}^{p}{{\mathbb{E}}}\,[d_{kl}^Y(i)\,d_{kl}^Y(i')]\\ :=& \;V_1\; +\; V_2 \;+\; V_3,
\end{split}\end{aligned}$$ where $V_i$’s are defined accordingly. Further let $$\begin{aligned}
\label{V tilde def supp 1}
\widetilde{V_1} \;:=\; nm V_1\;,\; \widetilde{V_2} \;:=\; 2n(n-1) V_2 \;,\; \widetilde{V_3} \;:=\; 2m(m-1) V_3\,.\end{aligned}$$ It can be verified that $$\begin{aligned}
{{\mathbb{E}}}\,[d^X_{kl}(i)\,d^X_{kl}(i')] \;=\; D^2_{\rho_i, \rho_{i'}} (X_{(i)}, X_{(i')})\,.\end{aligned}$$ Thus we have $$\begin{aligned}
\label{V tilde def supp 2}
\widetilde{V_2}\;=\; \frac{1}{\tau^2_X} \dis \sum_{i,i'=1}^p D^2_{\rho_i, \rho_{i'}} (X_{(i)}, X_{(i')}) \;\;\;\; \textrm{and} \;\;\;\; \widetilde{V_3}\;=\; \frac{1}{\tau^2_Y} \dis \sum_{i,i'=1}^p D^2_{\rho_i, \rho_{i'}} (Y_{(i)}, Y_{(i')})\,.\end{aligned}$$
We study the variances of $L_1^i$ for $1\leq i\leq 3$ and propose some suitable estimators. The variance for $L_1^2$ is given by $$\begin{aligned}
var(L_1^2) \;=&\;\frac{1}{n^2(n-1)^2\tau_X^2}\sum_{i,i'=1}^{p}\sum_{k<l}{{\mathbb{E}}}\,[d^X_{kl}(i)\,d^X_{kl}(i')]
\;=\;V_2\,.\end{aligned}$$ Clearly $$\frac{n(n-1)V_2}{2}\;=\;\frac{1}{4\tau^2_X}\dis\sum_{i,i'=1}^{p}D^2_{\rho_i, \rho_j}(X_{(i)},X_{(i')})\,.$$ From Theorem \[ACdCov taylor thm\] in Section \[sec:ACdcov-HDLSS\], we know that for fixed $n$ and growing $p$, $\widetilde{\cal{D}_n^2}(X,X)$ is asymptotically equivalent to $\frac{1}{4\tau^2_X} \sum_{i,i'=1}^{p}\widetilde{D^2_n}_{\,;\,\rho_i, \rho_j}(X_{(i)},X_{(i')})$. Therefore an estimator of $\widetilde{V_2}$ is given by $4\,\widetilde{\cal{D}_n^2}(X,X)$. Note that the computational cost of $\widetilde{\cal{D}_n^2}(X,X)$ is linear in $p$ while direct calculation of its leading term $\frac{1}{4\tau^2_X} \sum_{i,i'=1}^{p}\widetilde{D^2_n}_{\,;\,\rho_i, \rho_j}(X_{(i)},X_{(i')})$ requires computation in the quadratic order of $p$. Similarly it can be shown that the variance of $L_1^3$ is $V_3$ and $\widetilde{V_3}$ can be estimated by $4\,\widetilde{\cal{D}_m^2}(Y,Y)$. Likewise some easy calculations show that the variance of $L_1^1$ is $V_1$. Define $$\label{eq rho hat}
\begin{split}
\hat{\rho_i}(X_{k(i)},Y_{l(i)})\;:=&\;\rho_i(X_{k(i)},Y_{l(i)})\,-\,\frac{1}{n}\sum^{n}_{a=1}\rho_i(X_{a(i)},Y_{l(i)})\,-\,\frac{1}{m}\sum^{m}_{b=1}\rho_i(X_{k(i)},Y_{b(i)})
\\&+\,\frac{1}{nm}\sum_{a=1}^{n}\sum^{m}_{b=1}\rho_i(X_{a(i)},Y_{b(i)})\;,
\end{split}$$ and $$\begin{aligned}
\label{expr for R hat}
\hat{R}(X_k, Y_l)\;:=\;R(X_{k},Y_{l})-\frac{1}{n}\sum^{n}_{a=1}R(X_{a},Y_{l})-\frac{1}{m}\sum^{m}_{b=1}R(X_{k},Y_{b})+\frac{1}{nm}\sum_{a=1}^{n}\sum^{m}_{b=1}R(X_{a},Y_{b})\,.\end{aligned}$$ It can be verified that $$\hat{\rho_i}(X_{k(i)},Y_{l(i)})\; =\;d_{kl}(i)\,-\,\frac{1}{n}\sum^{n}_{a=1}d_{al}(i)\,-\,\frac{1}{m}\sum^{m}_{b=1}d_{kb}(i)\,+\,\frac{1}{nm}\sum_{a=1}^{n}\sum^{m}_{b=1}d_{ab}(i).$$ Observe that $$\begin{aligned}
\label{eq 0.8}
{{\mathbb{E}}}\,[\hat{\rho_i}(X_{k(i)},Y_{l(i)})\rho_{i'}(X_{k(i')},Y_{l(i')})]\;=\;(1-1/n)(1-1/m)\,{{\mathbb{E}}}\,[d_{kl}(i)\,d_{kl}(i')]\,.\end{aligned}$$ Let $\hat{\bf A}_i=(\hat{\rho_i}(X_{k(i)},Y_{l(i)}))_{k,l},\;{\bf A}_i=(\rho_i(X_{k(i)},Y_{l(i)}))_{k,l}\,\in \mathbb{R}^{n\times m}$. Note that $$\begin{aligned}
\label{eq 0.9}
\begin{split}
&\;\frac{1}{(n-1)(m-1)}\,{{\mathbb{E}}}\,\sum_{k=1}^{n}\sum_{l=1}^{m}\hat{\rho_i}(X_{k(i)},Y_{l(i)})\hat{\rho_i}(X_{k(i')},Y_{l(i')})\\
=&\;\frac{1}{(n-1)(m-1)}\,{{\mathbb{E}}}\,\text{tr}(\hat{{\bf A}}_i\hat{{\bf A}}_{i'}^\top)
\\=&\;\frac{1}{(n-1)(m-1)}\,{{\mathbb{E}}}\,\text{tr}(\hat{{\bf A}}_i {\bf A}_{i'}^\top)
\\=&\;\frac{1}{(n-1)(m-1)}\,{{\mathbb{E}}}\,\sum_{k=1}^{n}\sum_{l=1}^{m}\,\rho_i(X_{k(i')},Y_{l(i')})\,\hat{\rho_i}(X_{k(i)},Y_{l(i)})
\\=&\;{{\mathbb{E}}}\,[d_{kl}(i)\,d_{kl}(i')],
\end{split}\end{aligned}$$ which suggests that $$\begin{aligned}
\breve{V}_1=\frac{1}{nm\tau^2}\sum_{i,i'=1}^{p}\frac{1}{(n-1)(m-1)}\sum_{k=1}^{n}\sum_{l=1}^{m}\hat{\rho_i}(X_{k(i)},Y_{l(i)})\,\hat{\rho_i}(X_{k(i')},Y_{l(i')})\end{aligned}$$ is an unbiased estimator for $V_1.$ However, the computational cost for $\breve{V}_1$ is linear in $p^2$ which is prohibitive for large $p.$ We aim to find a joint metric whose computational cost is linear in $p$ whose leading term is proportional to $\breve{V}_1.$ It can be verified that $cdCov^2_{n,m}(X,Y)$ is asymptotically equivalent to $$\begin{aligned}
\frac{1}{4\tau^2}\sum_{i,i'=1}^{p}\frac{1}{(n-1)(m-1)}\sum_{k=1}^{n}\sum_{l=1}^{m}\hat{\rho_i}(X_{k(i)},Y_{l(i)})\hat{\rho_i}(X_{k(i')},Y_{l(i')})\;.\end{aligned}$$ This can be seen from the observation that $$\begin{aligned}
\begin{split}
4\,cdCov^2_{n,m}(X,Y) \;&=\; \frac{1}{\tau^2} \dis \sum_{i,i'=1}^p \frac{1}{(n-1)(m-1)} \sum_{k=1}^n \sum_{l=1}^m \hat{\rho}_i (X_{k(i)}, Y_{l(i)})\,\hat{\rho}_{i'} (X_{k(i')}, Y_{l(i')}) \\ & \qquad +\; \frac{\tau^2}{(n-1)(m-1)} \dis \sum_{k=1}^n \sum_{l=1}^m \hat{R}^2(X_k, Y_l) \\
&\qquad + \; \frac{1}{(n-1)(m-1)} \dis \sum_{k=1}^n \sum_{l=1}^m \frac{1}{\tau} \dis \sum_{i=1}^p \hat{\rho}_i(X_{k(i)}, Y_{(li)}) \,\, \tau \hat{R}(X_k, Y_l).
\end{split}\end{aligned}$$ Using the H[ö]{}lder’s inequality as well as the fact that $\tau^2 \,\hat{R}^2(X_k, Y_l)$ is $O_p(\tau^2 a^4_p) = o_p(1)$ under Assumption \[ass0.6\]. Therefore, we can estimate $\widetilde{V}_1$ by $4cdCov^2_{n,m}(X,Y)$. Thus the variance of $L_1$ is $V$ which can be estimated by $$\begin{aligned}
\label{V hat def}
\begin{split}
\hat{V} \;&:= \;\frac{1}{n m}\, 4\, cdCov_{n,m}^2(X,Y) \;+\;\frac{1}{2n(n-1)}\, 4\,\widetilde{\cal{D}_n^2}(X,X) \;+\; \frac{1}{2m(m-1)}\,4\, \widetilde{\cal{D}_m^2}(Y,Y) \,\\
&:= \;\hat{V}_1\; +\; \hat{V}_2 \;+\; \hat{V}_3\,.
\end{split}\end{aligned}$$
Using Proposition \[K taylor : ED\], some algebraic calculations yield $$\begin{aligned}
&\cal{E}_{nm}(X,Y)-(2\tau-\tau_X-\tau_Y)
\\=&\;\frac{\tau}{nm}\sum_{k=1}^{n}\sum_{l=1}^{m}L(X_k,Y_l)-\frac{\tau_X}{2n(n-1)}\sum_{k\neq l}^n L(X_k,X_l)-\frac{\tau_Y}{2m(m-1)}\sum_{k\neq l}^m L(Y_k,Y_l)\;+\;R_{n,m}
\\=& \;\frac{1}{nm\tau}\sum_{k=1}^{n}\sum_{l=1}^{m}\sum_{i=1}^{p}\big(\rho_i(X_{k(i)},Y_{l(i)})-{{\mathbb{E}}}\,\rho_i(X_{k(i)},Y_{l(i)})\big)
\\&-\frac{1}{2n(n-1)\tau_X}\sum_{k\neq l}^n \sum_{i=1}^{p}\big(\rho_i(X_{k(i)},X_{l(i)})-{{\mathbb{E}}}\,\rho_i(X_{k(i)},X_{l(i)})\big)
\\&-\frac{1}{2m(m-1)\tau_Y}\sum_{k\neq l}^m \sum_{i=1}^{p} \big(\rho_i(Y_{k(i)},Y_{l(i)})-{{\mathbb{E}}}\, \rho_i(Y_{k(i)},Y_{l(i)})\big)\;+ \;R_{n,m},\end{aligned}$$ where $$\begin{aligned}
\label{remainder negl}
R_{n,m} \;=\; \frac{2\tau}{nm}\sum_{k=1}^{n}\sum_{l=1}^{m}R(X_k,Y_l)-\frac{\tau_X}{n(n-1)}\sum_{k\neq l}^n R(X_k,X_l)-\frac{\tau_Y}{m(m-1)}\sum_{k\neq l}^m R(Y_k,Y_l)\; .\end{aligned}$$ By Assumption \[ass0.6\], $R_{n,m} = O_p(\tau a^{2}_{p} + \tau_X b^{2}_{p} + \tau_Y c^{2}_{p}) = o_p(1)$ as $p \to \infty$. Denote the leading term above by $L$. We can rewrite $L$ as $L_1+L_2$, where $L_1$ and $L_2$ are defined in equations (\[L decomp.1\]) and (\[L decomp.2\]), respectively. Some calculations yield that $$\begin{aligned}
\label{L2 expression}
\begin{split}
L_2 \; =& \; \frac{1}{n} \dis \sum_{k=1}^n \left[ \frac{1}{\tau} \sum_{i=1}^p {{\mathbb{E}}}\,[\rho_i(X_{k(i)},Y_{(i)})|X_{k(i)}] \; -\; \frac{1}{\tau_X} \sum_{i=1}^p {{\mathbb{E}}}\,[\rho_i(X_{k(i)},X_{(i)}')|X_{k(i)}]\,\right]\;-\; (\tau-\tau_X) \\
& \; + \frac{1}{m} \dis \sum_{l=1}^m \left[ \frac{1}{\tau} \sum_{i=1}^p {{\mathbb{E}}}\,[\rho_i(X_{(i)},Y_{l(i)})|Y_{l(i)}] \; -\; \frac{1}{\tau_Y} \sum_{i=1}^p {{\mathbb{E}}}\,[\rho_i(Y_{l(i)},Y_{(i)}')|Y_{l(i)}]\,\right] \; - \; (\tau - \tau_Y)\\
=& \; \frac{1}{n} \dis \sum_{k=1}^n {{\mathbb{E}}}\,\left[\tau L(X_k, Y) - \tau_X L(X_k, X') \,|\, X_k \right] \;+\; \frac{1}{m} \dis \sum_{l=1}^m {{\mathbb{E}}}\,\left[\tau L(X, Y_l) - \tau_X L(Y_l, Y') \,|\, Y_l \right]\,.
\end{split}\end{aligned}$$ For $(P_X,P_Y)\in\mathcal{P}$, we have $L_2=o_p(1)$.
Under Assumption \[ass6\], the asymptotic distribution of $L_1$ as $p \to \infty$ is given by $$\begin{aligned}
L_1 \overset{d}{\longrightarrow} N \Big(0\,,\,\frac{\sigma^2}{nm} + \frac{\sigma_X^2}{2n(n-1)} + \frac{\sigma_Y^2}{2m(m-1)}\Big).\end{aligned}$$ Define the vector $d_{\textrm{vec}} := \left(\frac{1}{\tau} \sum_{i=1}^p d_{kl}(i) \right)_{1\leq k \leq n, \, 1\leq l \leq m}$. It can be verified that $$\begin{aligned}
\label{dvec expr}
4(n-1)(m-1)\,cdCov^2_{n,m}(X,Y) \;&= \; d_{\textrm{vec}}^{\top} \,A\, d_{\textrm{vec}}\end{aligned}$$ where $A=A_1 + A_2 + A_3 + A_4$ with $A_1 = I_n \otimes I_m$, $A_2 = - I_n \otimes \frac{1}{m}1_m 1_m^{\top}$, $A_3 = -\frac{1}{n}1_n 1_n^{\top} \otimes I_m$ and $A_4 = \frac{1}{nm}1_{nm} 1_{nm}^{\top}$. Here $\otimes$ denotes the Kronecker product. It is not hard to see that $A^2 = A$ and $\textrm{rank}(A) = (n-1)(m-1)$. Therefore by Assumption \[ass6\], we have as $p \to \infty$, $$\begin{aligned}
4(n-1)(m-1)\,cdCov^2_{n,m}(X,Y)\;\overset{d}{\rightarrow }\;\sigma^2\chi^2_{(n-1)(m-1)}.\end{aligned}$$ By Theorem \[ACdcov:dist\_conv\], we have as $p \to \infty$, $$\begin{aligned}
& 4\,\widetilde{\cal{D}_n^2}(X,X) \;\overset{d}{\rightarrow} \; \frac{\sigma_X^2}{v_n} \chi^2_{v_n}\;,\;\; \textrm{i.e.}, \;\; 4\,v_n\,\widetilde{\cal{D}_n^2}(X,X) \;\overset{d}{\rightarrow} \; \sigma_X^2 \, \chi^2_{v_n}\,,\end{aligned}$$ and similarly $$\begin{aligned}
4\,v_m\,\widetilde{\cal{D}_m^2}(Y,Y) \;\overset{d}{\rightarrow} \; \sigma_Y^2 \, \chi^2_{v_m}\,.\end{aligned}$$ By Assumption \[ass6\], $\chi^2_{(n-1)(m-1)},\chi^2_{v_n}$ and $\chi^2_{v_m}$ are mutually independent. The proof can be completed by combining all the arguments above and using the continuous mapping theorem.
Note that as $n, m \to \infty$, $$\begin{aligned}
{{\mathbb{E}}}\,[(M-m_0)^2] \; &= \; \frac{2(n-1)(m-1)\sigma^4 \, + 2v_n\sigma_X^4 \, + 2v_m\sigma_Y^4}{\left\{\,(n-1)(m-1) \,+ \,v_n \,+\, v_m \,\right\}^2}\; = \; o(1),\end{aligned}$$ where $m_0={{\mathbb{E}}}[M]$. Therefore by Chebyshev’s inequality, $M - m_0 = o_p(1)$ as $n, m \to \infty$. As a consequence, we have $M \overset{p}{\longrightarrow} m_0^*$ as $n, m \to \infty$. Observing that $\Phi$ is a bounded function, the rest follows from Lebesgue’s Dominated Convergence Theorem.
Under $H_0$, without any loss of generality define $U_1=X_1,\dots,U_n=X_n,U_{n +1}:= Y_1,\dots,
U_{n + m}:=Y_{m}$. Further define $$\begin{aligned}
\label{phi def supp}
\phi_{i_1 i_2}:=\phi(U_{i_1},U_{i_2})=\begin{cases} -\frac{1}{n(n -1)}\;H(U_{i_1}, U_{i_2}) \;\; & \; \textrm{if}\;\, i_1, i_2 \in \{1, \dots \,,n\} \,,\\ \qquad \frac{1}{n m}\;H(U_{i_1}, U_{i_2}) \;\; & \; \textrm{if}\;\, i_1 \in \{1, \dots \,,n\},i_2 \in \{n +1, \dots \,,n+m\}\,, \\ -\frac{1}{m(m -1)}\,H(U_{i_1}, U_{i_2}) \;\; & \; \textrm{if}\;\, i_1, i_2 \in \{n +1, \dots \,,n+m\}\,. \end{cases}\end{aligned}$$ It can be verified that $\cov(\phi_{i_1 i_2},\, \phi_{i_1' i_2'})=0$ if the cardinality of the set $\{i_1, i_2\} \cap \{i_1', i_2'\}$ is less than $2$. Define $$\breve{T}_{n,m} \;=\;\frac{\cal{E}_{n,m}(X,Y)}{\sqrt{V}}.$$
\[lemma\_supp\_1\] Under $H_0$ and Assumptions \[ass0.5\], \[assED\_HDMSS\] and \[assED\_HDMSS4\], as $n, m$ and $p \to \infty$, we have $$\breve{T}_{n, m} \; \overset{d}{\longrightarrow} \; N(0,1)\,.$$
Set $N=n+m.$ Define $V_{Nj} := \sum_{i=1}^{j-1} \phi_{ij}$ for $2\leq j\leq N$, $S_{N r} := \sum_{j=2}^r V_{N j} = \sum_{j=2}^r \sum_{i=1}^{j-1} \phi_{ij}$ for $2\leq r \leq N$, and $\mathcal{F}_{N,r} := \sigma(X_1, \dots\,, X_r)$. Then the leading term of $\cal{E}_{n m}(X,Y)$, viz., $L_1$ (see equation (\[L decomp.1\])) can be expressed as $$L_1 \;=\; S_{NN} = \dis \sum_{j=2}^N V_{Nj} \;=\; \sum_{j=2}^N \sum_{i=1}^{j-1} \phi_{ij}\;=\;\dis \sum_{1\leq i_1<i_2\leq n} \phi_{i_1 i_2} \;+ \; \sum_{i_1=1}^{n} \sum_{i_2=n+1}^{N} \phi_{i_1 i_2} \;+\; \sum_{n+1\leq i_1<i_2\leq N} \phi_{i_1 i_2}\,.$$ By Corollary 3.1 of Hall and Heyde(1980), it suffices to show the following :
1. For each $N$, $\{S_{Nr}, \mathcal{F}_{N,r}\}_{r=1}^N$ is a sequence of zero mean and square integrable martingales,
2. $\frac{1}{V} \dis \sum_{j=2}^N {{\mathbb{E}}}\,\left[V_{Nj}^2 \,|\, \mathcal{F}_{N,j-1}\right] \; \overset{P}{\longrightarrow} \; 1\,$,
3. $\frac{1}{V} \dis \sum_{j=2}^N {{\mathbb{E}}}\,\left[V_{Nj}^2 \,\mathbbm{1}(|V_{Nj}| > \epsilon \sqrt{V}) \,|\, \mathcal{F}_{N,j-1}\,\right] \; \overset{P}{\longrightarrow} \; 0\,,\;\;\; \forall \; \epsilon > 0$.
To show (1), it is easy to see that $S_{Nr}$ is square integrable, ${{\mathbb{E}}}(S_{Nr}) = \dis \sum_{j=2}^r \sum_{i=1}^{j-1} {{\mathbb{E}}}(\phi_{ij}) = 0$, and, $\mathcal{F}_{N,1} \subseteq \mathcal{F}_{N,2} \subseteq\, \dots \, \subseteq\mathcal{F}_{N,N}$. We only need to show ${{\mathbb{E}}}(S_{Nq}\,|\,\mathcal{F}_{N,r}) = S_{Nr}$ for $q>r$. Now ${{\mathbb{E}}}(S_{Nq}\,|\,\mathcal{F}_{N,r}) = \dis \sum_{j=2}^q \sum_{i=1}^{j-1} {{\mathbb{E}}}(\phi_{ij}\,|\,\mathcal{F}_{N,r})$. If $j \leq r <q$ and $i<j$, then ${{\mathbb{E}}}(\phi_{ij}\,|\,\mathcal{F}_{N,r}) = \phi_{ij}$. If $r<j\leq q$, then :
1. if $r<i<j\leq q$, then ${{\mathbb{E}}}(\phi_{ij}\,|\,\mathcal{F}_{N,r}) = {{\mathbb{E}}}(\phi_{ij}) = 0$,
2. if $i\leq r<j\leq q$, then ${{\mathbb{E}}}(\phi_{ij}\,|\,\mathcal{F}_{N,r}) = 0$ (due to $\mathcal{U}$-centering).
Therefore ${{\mathbb{E}}}(S_{Nq}\,|\,\mathcal{F}_{N,r}) = S_{Nr}$ for $q>r$. This completes the proof of (1).\
To show (2), define $L_j(i,k) := {{\mathbb{E}}}\,[\phi_{i j}\, \phi_{k j}\,|\,\mathcal{F}_{N,j-1}]$ for $i, k < j \leq N$, and $$\eta_N := \dis \sum_{j=2}^N {{\mathbb{E}}}\,\left[V_{Nj}^2\,|\,\mathcal{F}_{N,j-1}\,\right] = \sum_{j=2}^N \sum_{i,k=1}^{j-1} {{\mathbb{E}}}[\phi_{i j}\, \phi_{k j}\,|\,\mathcal{F}_{N,j-1}] = \sum_{j=2}^N \sum_{i,k=1}^{j-1} L_j(i,k)\,.$$ Note that ${{\mathbb{E}}}\, [L_j(i,k)] = 0$ for $i \neq k$. Clearly $$\begin{aligned}
\label{eq 1}
{{\mathbb{E}}}[\eta_N] \;&=\; \dis\sum_{j=2}^N {{\mathbb{E}}}[V_{Nj}^2] \;= \;\dis\sum_{j=2}^N \sum_{i,k=1}^{j-1} {{\mathbb{E}}}[\phi_{i j}\, \phi_{k j}] \;=\; \sum_{j=2}^N \sum_{i=1}^{j-1} {{\mathbb{E}}}[\phi^2_{i j}] = V\,.\end{aligned}$$ By virtue of Chebyshev’s inequality, it will suffice to show $\text{var}(\frac{\eta_N}{V}) = o(1)$. Note that $$\begin{aligned}
\label{exp L}
\begin{split}
& \qquad {{\mathbb{E}}}\, [L_j(i,k) \, L_{j'}(i',k')] \\&= \begin{cases}
{{\mathbb{E}}}\,\left[\phi^2(U_i,U_j)\phi^2(U_i,U_{j'}')\right] & \; \; i=k=i'=k'\,,\\
{{\mathbb{E}}}\,\left[\phi(U_i,U_j) \phi(U_k,U_j) \phi(U_i,U_{j'}') \phi(U_{k},U'_{j'})\right] & \; \; i=i'\neq k=k' \;\; \textrm{or} \;\; i=k'\neq k=i'\,,\\
{{\mathbb{E}}}\,\left[\phi^2(U_i,U_j)\right] {{\mathbb{E}}}\,\left[U^2(U_{i'},U_{j'})\right] & \; \; i=k \neq i'=k'\,.
\end{cases}
\end{split}\end{aligned}$$ In view of equation (\[phi def supp\]), it can be verified that the above expression for ${{\mathbb{E}}}\, L_j(i,k) \, L_{j'}(i',k')$ holds true for $j=j'$ as well. Therefore $$\begin{aligned}
\text{var}\,(\eta_N^2) \;&=\; \dis \sum_{j,j'=2}^N \sum_{i,k=1}^{j-1} \sum_{i',k'=1}^{j'-1} \text{cov}\,(L_j(i,k) \,,L_{j'}(i',k'))\\
&=\; \dis \sum_{j=j'} \Bigg\{\sum_{i=1}^{j-1} \text{cov}\,\left(\phi^2(U_i,U_j),\phi^2(U_i,U_{j}')\right) \, +\, 2\sum_{i \neq k}^{j-1}{{\mathbb{E}}}\,\left[\phi(U_i,U_j) \phi(U_k,U_j) \phi(U_i,U_{j}') \phi(U_{k},U'_{j})\right]\Bigg\}\\
& \; + \; 2\sum_{2 \leq j < j' \leq N} \Bigg\{\sum_{i=1}^{j-1} \text{cov}\,\left(\phi^2(U_i,U_j),\phi^2(U_i,U_{j'}')\right) \\&\,+\, 2\sum_{i \neq k}^{j-1}{{\mathbb{E}}}\,\left[\phi(U_i,U_j) \phi(U_k,U_j) \phi(U_i,U_{j'}') \phi(U_{k},U'_{j'})\right]\Bigg\}\,.\end{aligned}$$ Under Assumption \[ass0.5\] and $H_0$, it can be verified that $$\begin{aligned}
\label{var L 2nd}
\begin{split}
\text{var}(\eta_N)=O\Big(\frac{1}{N^5}\, {{\mathbb{E}}}\,\left[H^2(X, X'') H^2(X', X'') \right] +\frac{1}{N^4}{{\mathbb{E}}}\left[ H(X, X'') H(X', X'') H(X, X''') H(X', X''') \right] \Big),
\end{split}\end{aligned}$$ and $$\begin{aligned}
\label{eqn V order}
V^2\;& \asymp \; \,\frac{1}{N^4}\, \left({{\mathbb{E}}}\, \left[H^2(X,X')\right]\right)^2 \,.\end{aligned}$$ Therefore under Assumption \[assED\_HDMSS\] and $H_0$, we have $$\text{var}\left(\frac{\eta_N}{V}\right)=o(1),$$ which completes the proof of (2). To show (3), note that it suffices to show $$\frac{1}{V^2} \dis \sum_{j=2}^N {{\mathbb{E}}}\left[\,V^4_{Nj} \, | \, \mathcal{F}_{N,j-1}\,\right] \; \overset{P}{\longrightarrow} \; 0\;.$$ Observe that $$\begin{aligned}
\dis \sum_{j=2}^N {{\mathbb{E}}}\left[V^4_{Nj} \,\right] \; &= \; \sum_{j=2}^N {{\mathbb{E}}}\left(\, \sum_{i=1}^{j-1} \phi_{ij}\,\right)^4 \;
\\&= \; \sum_{j=2}^N \sum_{i=1}^{j-1} {{\mathbb{E}}}[\phi^4(U_i,U_j)] \; + \; 3 \,\sum_{j=2}^{N} \sum_{i_1 \neq i_2}^{j-1} {{\mathbb{E}}}[\phi^2(U_{i_1},U_j) \,\phi^2(U_{j_2},U_j)] \,.\end{aligned}$$ Under Assumption \[ass0.5\], we have $$\begin{aligned}
\dis \sum_{j=2}^N {{\mathbb{E}}}\left[\,V^4_{Nj} \,\right] \; &= \; O\,\Big(\frac{1}{N^6}\, {{\mathbb{E}}}\, \left[H^4(X,X') \right] \;+\; \frac{1}{N^5}\, {{\mathbb{E}}}\,\left[H^2(X, X'') H^2(X', X'') \right] \Big) \,.\end{aligned}$$ This along with the observation from equation (\[var L 2nd\]) and Assumption \[assED\_HDMSS\] complete the proof of (3).
Finally to see that $\frac{R_{n,m}}{\sqrt{V}} = o_p(1)$, note that from equation (\[remainder negl\]) we can derive using power mean inequality that ${{\mathbb{E}}}\,R_{n,m}^2 \leq C\, \tau^2 \,{{\mathbb{E}}}\,\left[ R^2(X,X')\right]$ for some positive constant $C$. Using this, equation (\[eqn V order\]), Chebyshev’s inequality and H[ö]{}lder’s inequality, we have for any $\epsilon >0$ $$\begin{aligned}
\label{rem/v neg}
\begin{split}
P\left(\Big\vert\frac{R_{n,m}}{\sqrt{V}}\Big\vert > \epsilon \right)\; & \leq \; \frac{{{\mathbb{E}}}\,R_{n,m}^2}{\epsilon^2 \; V} \;\leq \; C' \, \frac{N^2 \,\tau^2\, {{\mathbb{E}}}\,\left[ R^2(X,X')\right]}{\epsilon^2 \; {{\mathbb{E}}}\,\left[ H^2(X,X')\right]} \;\leq \; \frac{C'}{\epsilon^2} \, \left(\frac{N^4 \,\tau^4\, {{\mathbb{E}}}\,\left[ R^4(X,X')\right]}{\left( {{\mathbb{E}}}\,\left[ H^2(X,X')\right] \right)^2} \right)^{1/2}\,,
\end{split}\end{aligned}$$ for some positive constant $C'$. From this and Assumptions \[ass0.5\] and \[assED\_HDMSS4\], we get $\frac{R_{n,m}}{\sqrt{V}} = o_p(1)$, as $N \asymp n$. This completes the proof of the lemma.
\[lemma\_supp\_2\] Under $H_0$ and Assumptions \[ass0.5\] and \[assED\_HDMSS4\], as $n, m$ and $p \to \infty$, we have $$\frac{\left|{{\mathbb{E}}}\,[\hat{V}_i] - V_i\right|}{V_i} = o(1)\;\; , \;\; 1 \leq i \leq 3\,,$$ where $V_i$ and $\hat{V}_i$, $1\leq i \leq 3$ are defined in equations (\[V def supp\]) and (\[V hat def\]), respectively in the supplementary material.
We first deal with $\hat{V}_2$. Note that $$\widetilde{\cal{D}^2_n} (X,X) \,=\, \frac{1}{n(n-3)} \dis \sum_{k\neq l} \left(\widetilde{D}^X_{kl}\right)^2\,,$$ where $$\begin{aligned}
\label{expr to be used later}
\begin{split}
\widetilde{D}^X_{kl} \;=&\; K(X_k, X_l) \;-\; \frac{1}{n-2}\dis \sum_{b=1}^n K(X_k, X_{b}) \;-\; \frac{1}{n-2}\dis \sum_{a =1}^n K( X_{a}, X_l)
\\& \;+\; \frac{1}{(n-1)(n-2)}\dis \sum_{a,b =1}^n K(X_{a}, X_{b})\\ \; \;&= \;\; \frac{1}{2\tau} \dis \sum_{i=1}^p {{\widetilde \rho}}_i(X_{k(i)}, X_{l(i)}) \;+\; \tau \widetilde{R}(X_k, X_l)\,,
\end{split}\end{aligned}$$ using Proposition \[K taylor : ED\]. As a consequence, we can write $$\begin{aligned}
\label{expr for D_n(X,X)}
\begin{split}
\widetilde{\cal{D}^2_n} (X,X) \;&=\; \frac{1}{4\tau^2} \dis \sum_{i,i'=1}^p \widetilde{D^2_n}_{\,;\,\rho_i, \rho_{i'}} (X_{(i)}, X_{(i')}) \;+\; \frac{ \tau^2}{n(n-3)} \dis \sum_{k\neq l} \widetilde{R}^2(X_k, X_l) \\
&\qquad + \; \frac{1}{n(n-3)} \dis \sum_{k\neq l} \frac{1}{\tau} \dis \sum_{i=1}^p {{\widetilde \rho}}_i(X_{k(i)}, X_{(li)}) \,\, \tau \widetilde{R}(X_k, X_l) \,.
\end{split}\end{aligned}$$ Note that following Step 3 in Section 1.6 in the supplementary material of Zhang et al.(2018), we can write $$\begin{aligned}
\widetilde{R}(X_k, X_l) \;&=\; \frac{n-3}{n-1} \bar{R}(X_k, X_l) \,-\, \frac{n-3}{(n-1)(n-2)} \dis \sum_{b\notin \{k,l\}} \bar{R}(X_k, X_{b}) \,-\, \frac{n-3}{(n-1)(n-2)} \dis \sum_{a \notin \{k,l\}} \bar{R}(X_{a}, X_l) \,\\ & \;\;+\, \frac{1}{(n-1)(n-2)}\dis \sum_{a,b \notin \{k,l\}} \bar{R}(X_{a}, X_{b})\,,\end{aligned}$$ where $\bar{R}(X,X')=R(X,X')-E[R(X,X')|X]-E[R(X,X')|X']+E[R(X,X')]$. Using the power mean inequality, it can be verified that ${{\mathbb{E}}}\,[\widetilde{R}^2(X_k, X_l)]\leq C\, {{\mathbb{E}}}\,[\bar{R}^2(X_k, X_l)]$ for some positive constant $C$. Using this and the H[ö]{}lder’s inequality, the expectation of the third term in the summation in equation (\[expr for D\_n(X,X)\]) can be bounded as follows $$\begin{aligned}
&\left|{{\mathbb{E}}}\, \left[ \frac{1}{n(n-3)} \dis \sum_{k\neq l} \frac{1}{\tau} \dis \sum_{i=1}^p {{\widetilde \rho}}_i(X_{k(i)}, X_{l(i)}) \,\, \tau \widetilde{R}(X_k, X_l) \right]\right| \\
\leq \; &\frac{1}{n(n-3)} \dis \sum_{k\neq l} \left( {{\mathbb{E}}}\, \left[\left( \frac{1}{\tau} \dis \sum_{i=1}^p {{\widetilde \rho}}_i(X_{k(i)}, X_{l(i)}) \right)^2\right] \, \tau^2\, {{\mathbb{E}}}\, \left[\bar{R}^2(X_k, X_l)\right]\,\right)^{1/2}\\
\leq \; & C' \, \left( \left(\frac{1}{\tau^2} \dis \sum_{i,i'=1}^p D^2_{\rho_i, \rho_{i'}} (X_{(i)}, X_{(i')}) \right) \,\tau^2\, {{\mathbb{E}}}\, \left[\bar{R}^2(X, X')\right]\, \right)^{1/2}\,\end{aligned}$$ for some positive constant $C'$. Combining all the above, we get $$\begin{aligned}
\vert {{\mathbb{E}}}\,(\hat{V}_2) - V_2 \vert \leq& \frac{C_1}{n(n-1)}\, \tau^2 \,{{\mathbb{E}}}\, \bar{R}^2(X, X')
\\&\,+\, \frac{C_2}{n(n-1)}\, \left( \left(\frac{1}{\tau^2} \dis \sum_{i,i'=1}^p D^2_{\rho_i, \rho_{i'}} (X_{(i)}, X_{(i')}) \right) \tau^2\, {{\mathbb{E}}}\left[ \bar{R}^2(X, X')\right] \right)^{1/2},\end{aligned}$$ for some positive constants $C_1$ and $C_2$. As $V_2=\frac{1}{2n(n-1)}E[H^2(X,X')]$, $$\frac{\left|{{\mathbb{E}}}[\hat{V}_2] - V_2\right|}{V_2} = o(1) \quad \text{is satisfied if}\quad
\frac{\tau^2\, {{\mathbb{E}}}\left[\bar{R}^2(X,X')\right]}{{{\mathbb{E}}}[H^2(X,X')]} \;=\; o(1)\,.$$ Using power mean inequality and Jensen’s inequality, it is not hard to verify that ${{\mathbb{E}}}\left[\bar{R}^4(X, X')\right] = O\left({{\mathbb{E}}}\, \left[R^4(X, X')\right]\right)$. Using this and H[ö]{}lder’s inequality, we have $$\frac{\tau^2\, {{\mathbb{E}}}\left[\bar{R}^2(X,X')\right]}{{{\mathbb{E}}}[H^2(X,X')]} \;= \; O\left(\left(\frac{\tau^4\, {{\mathbb{E}}}\, [R^4(X,X')]}{\left( {{\mathbb{E}}}\,[ H^2(X,X')] \right)^2} \right)^{1/2}\right)\,.$$ Clearly Assumption \[assED\_HDMSS4\] implies $\frac{\tau^4\, {{\mathbb{E}}}\, [R^4(X,X')]}{\left( {{\mathbb{E}}}\,[ H^2(X,X')] \right)^2} = o(1)$, which in turn implies $$\frac{\tau^2\, {{\mathbb{E}}}\left[\bar{R}^2(X,X')\right]}{{{\mathbb{E}}}[H^2(X,X')]} \;=\; o(1)\,.$$ Similar expressions can be derived for $\hat{V}_3$ as well. For the term involving $\hat{V}_1$, in the similar fashion, we can write $$\begin{aligned}
\label{expr for first term}
\begin{split}
{{\mathbb{E}}}\, \left[4\, cdCov^2_{n,m}(X,Y) \right] \;&=\; \frac{1}{\tau^2} \dis \sum_{i,i'=1}^p \frac{1}{(n-1)(m-1)} \sum_{k=1}^n \sum_{l=1}^m {{\mathbb{E}}}\left[ \hat{\rho}_i (X_{k(i)}, Y_{l(i)})\,\hat{\rho}_{i'} (X_{k(i')}, Y_{l(i')}) \right] \\ & \qquad +\; \tau^2 \frac{1}{(n-1)(m-1)} \dis \sum_{k=1}^n \sum_{l=1}^m {{\mathbb{E}}}\left[\hat{R}^2(X_k, Y_l)\right] \\
&\qquad + \; \frac{1}{(n-1)(m-1)} \dis \sum_{k=1}^n \sum_{l=1}^m \frac{1}{\tau} \dis \sum_{i=1}^p {{\mathbb{E}}}\left[\hat{\rho}_i(X_{k(i)}, Y_{(li)}) \,\, \tau \hat{R}(X_k, Y_l)\right] \,,
\end{split}\end{aligned}$$ where the expression for $\hat{R}(X_k, Y_l)$ is given in equation (\[expr for R hat\]). Following equation (\[eq 0.9\]) we can write $$\frac{1}{\tau^2} \dis \sum_{i,i'=1}^p \frac{1}{(n-1)(m-1)} \sum_{k=1}^n \sum_{l=1}^m {{\mathbb{E}}}\left[ \hat{\rho}_i (X_{k(i)}, Y_{l(i)})\,\hat{\rho}_{i'} (X_{k(i')}, Y_{l(i')}) \right] \;=\; {{\mathbb{E}}}\left[H^2(X,Y)\right] \,.$$ Therefore in view of equations (\[V def supp\]), (\[eq 0.8\]) and (\[V hat def\]), using the power mean inequality we can write $$\begin{aligned}
\vert {{\mathbb{E}}}\,(\hat{V}_1) - V_1 \vert &\leq \frac{C_1'}{nm}\, \tau^2 \,{{\mathbb{E}}}\, \bar{R}^2(X, Y) \,+\, \frac{C_2'}{nm}\, \left( \left(\frac{1}{\tau^2} \dis \sum_{i,i'=1}^p {{\mathbb{E}}}\left[d_{kl}(i) d_{kl}(i') \right] \right) \tau^2\, {{\mathbb{E}}}\left[ \bar{R}^2(X, Y)\right] \right)^{1/2},\end{aligned}$$ for some positive constants $C_1'$ and $C_2'$. Then under $H_0$ and Assumptions \[ass0.5\] and \[assED\_HDMSS4\], we have $$\begin{aligned}
\frac{\left|{{\mathbb{E}}}\,(\hat{V}_1) - V_1\right|}{V_1} = o(1)\,.\end{aligned}$$
\[lemma\_supp\_3\] Under $H_0$ and Assumptions \[ass0.5\], \[assED\_HDMSS\] and \[assED\_HDMSS4\], as $n, m$ and $p \to \infty$, we have $$\frac{\text{var}(\hat{V}_i)}{V_i^2} = o(1) , \;\; 1 \leq i \leq 3\,.$$
Again we deal with $\hat{V}_2$ first. To simplify the notations, denote $A_{ij} = K(X_i, X_j)$ and $\widetilde{A}_{ij} = \widetilde{D}^X_{ij}$ for $1 \leq i \neq j \leq n$. Observe that $$\begin{aligned}
\label{var decomp}
\begin{split}
\text{var}\left( \widetilde{\cal{D}^2_n}(X,X) \right) \;&=\; \text{var} \left( \frac{1}{n(n-3)} \dis \sum_{i \neq j} \widetilde{A}^2_{ij} \right)\\
& \asymp \; \frac{1}{n^4} \, \left[\dis \sum_{i<j} \text{var}(\widetilde{A}^2_{ij}) \;+\; \dis \sum_{i<j<j'} \text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{jj'}) \;+\; \dis \sum_{\substack{i<j, i'<j'\\ \{i,j\} \cap \{i',j'\}= \phi}} \text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{i'j'}) \right]\,.
\end{split}\end{aligned}$$ As in the proof of Lemma \[lemma\_supp\_2\], we can write $$\label{zhang 2018}
\begin{split}
\widetilde{A}_{ij} \;=&\; \frac{n-3}{n-1} \bar{A}_{ij} \,-\, \frac{n-3}{(n-1)(n-2)} \dis \sum_{l \notin \{i,j\}} \bar{A}_{il} \,-\, \frac{n-3}{(n-1)(n-2)} \dis \sum_{k \notin \{i,j\}} \bar{A}_{kj}
\\&+\, \frac{1}{(n-1)(n-2)} \dis \sum_{k,l \notin \{i,j\}} \bar{A}_{kl}\,,
\end{split}$$ where the four summands are uncorrelated with each other. Using the power mean inequality, it can be shown that $$\begin{aligned}
{{\mathbb{E}}}\,(\widetilde{A}^4_{ij}) \;\leq \; C\, {{\mathbb{E}}}\,(\bar{A}^4_{ij}) \;=\; C\, {{\mathbb{E}}}\,\left[ \bar{K}^4(X,X') \right],\end{aligned}$$ for some positive constant $C$, where $\bar{K}(X,X')=K(X,X') - E[K(X,X')|X] - E[K(X,X')|X'] + E[K(X,X')]$ (similarly define $\bar{L}(X,X')$). Therefore the first summand in equation (\[var decomp\]) scaled by $\widetilde{V_2}^2$ is $o(1)$ as $n, p \to \infty$, provided $$\frac{1}{n^2} \, \frac{{{\mathbb{E}}}\,\left[ \bar{K}^4(X,X') \right]}{\widetilde{V_2}^2} \;=\; o(1)\,,$$ where $\widetilde{V_2}$ is defined in equations (\[V tilde def supp 1\]) and (\[V tilde def supp 2\]). Note that $$\bar{K}(X,X') \;=\; \frac{\tau_X}{2}\, \bar{L}(X,X') \;+\; \tau_X\,\bar{R}(X,X')\,.$$ Using the power mean inequality we can write $$\frac{1}{n^2} \, \frac{{{\mathbb{E}}}\,\left[ \bar{K}^4(X,X') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} \,\leq \, C_0\, \frac{1}{n^2} \,\frac{\tau^4_X \,{{\mathbb{E}}}\,\left[ \bar{L}^4(X,X') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} \,+\, C_0'\,\frac{1}{n^2} \,\frac{\tau^4_X\,{{\mathbb{E}}}\,\left[ \bar{R}^4(X,X') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2}\,$$ for some positive constants $C_0$ and $C_0'$. It is easy to see that $$\begin{aligned}
\label{eqn used later}
\bar{L}(X_k,X_l) \,&=\, \frac{1}{\tau^2_X}\,\bar{K}^2(X_k,X_l) \,=\, \frac{1}{\tau^2_X}\, \dis \sum_{i=1}^p d^X_{kl}(i) \,=\, \frac{1}{\tau_X}\, H(X_k, X_l)\,.\end{aligned}$$ From equation (\[eqn used later\]) it is easy to see that the condition $$\frac{1}{n^2} \,\frac{\tau^4_X \,{{\mathbb{E}}}\,\left[ \bar{L}^4(X,X') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} = o(1)\quad \text{is equivalent to} \quad \frac{1}{n^2} \,\frac{{{\mathbb{E}}}\,\left[ H^4(X,X') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} = o(1).$$ For the third summand in equation (\[var decomp\]), observe that $$\begin{aligned}
\label{eq 0.7}
\begin{split}
\widetilde{A}^2_{ij}=& O(1)\bar{A}^2_{ij} + O\left(\frac{1}{n^2}\right) \dis \sum_{l,l' \notin \{i,j\}} \bar{A}_{il} \bar{A}_{il'} + O\left(\frac{1}{n^2}\right) \dis \sum_{k,k' \notin \{i,j\}} \bar{A}_{kj} \bar{A}_{k'j} + O\left(\frac{1}{n^4}\right) \dis \sum_{k,k',l,l' \notin \{i,j\}} \bar{A}_{kl} \bar{A}_{k'l'} \\
& + O\left(\frac{1}{n}\right) \,\bar{A}_{ij} \dis \sum_{l \notin \{i,j\}} \bar{A}_{il} \,+\, O\left(\frac{1}{n}\right) \,\bar{A}_{ij} \dis \sum_{k \notin \{i,j\}} \bar{A}_{kj} \,+\, O\left(\frac{1}{n^2}\right) \,\bar{A}_{ij} \dis \sum_{k,l \notin \{i,j\}} \bar{A}_{kl}
\\&+O\left(\frac{1}{n^2}\right) \, \dis \sum_{k,l \notin \{i,j\}} \bar{A}_{il} \bar{A}_{kj} + O\left(\frac{1}{n^3}\right) \, \dis \sum_{k,l,l' \notin \{i,j\}} \bar{A}_{il} \bar{A}_{kl'} \,+\, O\left(\frac{1}{n^3}\right) \, \dis \sum_{k,k',l \notin \{i,j\}} \bar{A}_{kl} \bar{A}_{k'j}\,.
\end{split}\end{aligned}$$ Likewise $\widetilde{A}^2_{i'j'}$ admits a similar expression as in equation (\[eq 0.7\]). We claim that when $\{i,j\} \cap \{i',j'\}= \phi $, the leading term of $\text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{i'j'})$ is $O \left(\frac{1}{n^2}\, {{\mathbb{E}}}\,(\bar{A}^4_{ij})\right)$. To see this first note that $\bar{A}_{ij}$ is independent of $\bar{A}_{i'j'}$ when $\{i,j\} \cap \{i',j'\}= \phi $. Using the double-centering properties, it can be verified that $$\text{cov}\left(\bar{A}^2_{i'j'}, \bar{A}_{ij} \dis \sum_{l \notin \{i,j\}} \bar{A}_{il} \right)\; = \; \text{cov} \left(\bar{A}^2_{i'j'}, \bar{A}_{ij} \dis \sum_{k \notin \{i,j\}} \bar{A}_{kj} \right) \;=\; \text{cov} \left(\bar{A}^2_{i'j'}, \bar{A}_{ij} \dis \sum_{k,l \notin \{i,j\}} \bar{A}_{kl} \right) = 0.$$ To compute the quantity $\text{cov}\left( \bar{A}^2_{i'j'} \;,\, O\left(\frac{1}{n^2}\right)\dis \sum_{l,l' \notin \{i,j\}} \bar{A}_{il} \bar{A}_{il'} \right)$, consider the following cases:
1. When $l=l'=i'$ or $l=l'=j'$ or $l=i', l'=j'$, $\text{cov} \left(\bar{A}^2_{i'j'}, \bar{A}_{il} \bar{A}_{il'}\right)$ boils down to $\text{cov}(\bar{A}^2_{i'j'}, \bar{A}^2_{ii'})$ or $\text{cov}(\bar{A}^2_{i'j'}, \bar{A}^2_{ij'})$ or $\text{cov}(\bar{A}^2_{i'j'}, \bar{A}_{ii'} \bar{A}_{ij'})$.
2. When $l=i, l'\notin \{i,j,i',j'\}$ or $l=j', l'\notin \{i,j,i',j'\}$, $\text{cov} \left(\bar{A}^2_{i'j'}, \bar{A}_{il} \bar{A}_{il'}\right)$ boils down to $\text{cov}(\bar{A}^2_{i'j'}, \bar{A}_{ii'} \bar{A}_{il'})$ or $\text{cov}(\bar{A}^2_{i'j'}, \bar{A}_{ij'} \bar{A}_{il'})$, which can be easily verified to be zero.
3. When $\{l, l'\} \cap \{i', j'\} = \phi$, $\text{cov} \left(\bar{A}^2_{i'j'}, \bar{A}_{il} \bar{A}_{il'}\right)$ is again zero.
Similar arguments can be made about $$\text{cov}\left( \bar{A}^2_{i'j'} \;,\, O\left(\frac{1}{n^2}\right)\dis \sum_{k,k' \notin \{i,j\}} \bar{A}_{kj} \bar{A}_{k'j} \right)~~\text{ and }~~\text{cov}\left( \bar{A}^2_{i'j'} \;,\, O\left(\frac{1}{n^2}\right)\dis \sum_{k,l \notin \{i,j\}} \bar{A}_{il} \bar{A}_{kj}\right).$$ With this and using H[ö]{}lder’s inequality, it can be verified that when $\{i,j\} \cap \{i',j'\}= \phi $, the leading term of $\text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{i'j'})$ is $O \left(\frac{1}{n^2}\, {{\mathbb{E}}}\,(\bar{A}^4_{ij})\right)$. Therefore the third summand in equation (\[var decomp\]) scaled by $\widetilde{V_2}^2$ can be argued to be $o(1)$ in similar lines of the argument for the first summand in equation (\[var decomp\]).
For the second summand in equation (\[var decomp\]), in the similar line we can argue that the leading term of $\text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{jj'})$ is $$O \left(\frac{1}{n}\right) {{\mathbb{E}}}\left[\bar{A}_{ij}^4 \right] \;+\; O(1) \, {{\mathbb{E}}}\left[ \bar{A}_{ij}^2 \bar{A}_{jj'}^2 \right]\,.$$ Therefore the leading term of $\frac{1}{n^4}\dis \sum_{i<j<j'}\text{cov}(\widetilde{A}^2_{ij}, \widetilde{A}^2_{jj'})$ is $$O \left(\frac{1}{n^2}\right) {{\mathbb{E}}}\left[\bar{A}_{ij}^4 \right] \;+\; O\left(\frac{1}{n}\right) \, {{\mathbb{E}}}\left[ \bar{A}_{ij}^2 \bar{A}_{jj'}^2 \right]\,.$$ For the second term above, using the power mean inequality we can write $$\begin{aligned}
\frac{1}{n} \, \frac{{{\mathbb{E}}}\,\left[ \bar{A}_{ij}^2 \,\bar{A}_{jj'}^2 \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} \;&\leq \; C_3 \, \frac{1}{n} \,\frac{\tau^4 \,{{\mathbb{E}}}\,\left[ \bar{L}^2(X,X')\,\bar{L}^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} \,+\, C_3'\frac{1}{n} \,\frac{\tau^4\,{{\mathbb{E}}}\,\left[\bar{L}^2(X,X')\, \bar{R}^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2}\,\\
& \qquad + \, C_3''\,\frac{1}{n} \,\frac{\tau^4\,{{\mathbb{E}}}\,\left[\bar{R}^2(X,X')\, \bar{R}^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2}\\
&= \, C_3 \, \frac{1}{n} \,\frac{ \,{{\mathbb{E}}}\,\left[ H^2(X,X')\,H^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2} \,+\, C_3'\frac{1}{n} \,\frac{\tau^2\,{{\mathbb{E}}}\,\left[H^2(X,X')\, \bar{R}^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2}\,\\
& \qquad + \, C_3''\,\frac{1}{n} \,\frac{\tau^4\,{{\mathbb{E}}}\,\left[\bar{R}^2(X,X')\, \bar{R}^2(X',X'') \right]}{\left({{\mathbb{E}}}\,\left[H^2(X,X')\right]\right)^2}\end{aligned}$$ for some positive constants $C_3, C_3'$ and $C_3''$. Using H[ö]{}lder’s inequality it can be seen that the second summand in equation (\[var decomp\]) scaled by $\widetilde{V_2}^2$ is $o(1)$ as $n, p \to \infty$ under Assumptions \[assED\_HDMSS\] and \[assED\_HDMSS4\]. This completes the proof that $$\frac{\text{var}(\hat{V}_2)}{V_2^2} = o(1)\,.$$ A similar line of argument and the simple observation that $$\begin{aligned}
\hat{K}(X_k, Y_l)\;&=\;K(X_{k},Y_{l})-\frac{1}{n}\sum^{n}_{a=1}K(X_{a},Y_{l})-\frac{1}{m}\sum^{m}_{b=1}K(X_{k},Y_{b})+\frac{1}{nm}\sum_{a=1}^{n}\sum^{m}_{b=1}K(X_{a},Y_{b})\,\\
&= \;\bar{K}(X_{k},Y_{l})-\frac{1}{n}\sum^{n}_{a=1} \bar{K}(X_{a},Y_{l})-\frac{1}{m}\sum^{m}_{b=1} \bar{K}(X_{k},Y_{b})+\frac{1}{nm}\sum_{a=1}^{n}\sum^{m}_{b=1}\bar{K}(X_{a},Y_{b})\,\end{aligned}$$ will show that under Assumptions \[ass0.5\], \[assED\_HDMSS\] and \[assED\_HDMSS4\], $$\frac{\text{var}(\hat{V}_1)}{V_1^2} = o(1)\qquad \textrm{and} \qquad \frac{\text{var}(\hat{V}_3)}{V_3^2} = o(1)\,.$$
\[lemma\_supp\_4\] Under $H_0$ and Assumptions \[ass0.5\], \[assED\_HDMSS\] and \[assED\_HDMSS4\], as $n, m$ and $p \to \infty$, we have $\hat{V}/V \overset{P}{\to} 1 \,.$
It is enough to show that $${{\mathbb{E}}}\,\left[\left(\frac{\hat{V}}{V} - 1\right)^2\right] = o(1)\;, \;\; \textrm{i.e.}\,, \;\; \frac{\text{var}(\hat{V}) + \left({{\mathbb{E}}}\,[\hat{V}] - V\right)^2}{V^2} = o(1)\,.$$It suffices to show the following $$\begin{aligned}
\frac{\text{var}(\hat{V}_i)}{V_i^2} = o(1) \;\;\;\; \textrm{and} \;\;\;\; \frac{\left({{\mathbb{E}}}\,[\hat{V}_i] - V_i\right)^2}{V_i^2} = o(1), \quad 1 \leq i \leq 3.\end{aligned}$$ The proof can be completed using Lemmas \[lemma\_supp\_2\] and \[lemma\_supp\_3\].
The proof essentially follows from Lemma \[lemma\_supp\_1\] and \[lemma\_supp\_4\].
The proof of the first part follows similar lines of the proof of Proposition 1 in Székely et al.(2014), replacing the Euclidean distance between $X$ and $X'$, viz. $\Vert X-X' \Vert_{\tilde{p}}$, by $K(X,X')$. The second part of the proposition has a proof similar to Lemma 2.1 in Yao et al.(2018) and Section 1.1 in the Supplement of Yao et al.(2018).
The first two parts of the theorem immediately follow from Proposition 2.6 and Theorem 2.7 in Lyons(2013), respectively and the parallel U-statistics theory (see for example Serfling(1980)). The third part follows from the first part and the fact that $\cal{D}$ is non-zero for two dependent random vectors.
Following the definition of $\cal{D}(X,Y)$ and applying Proposition \[K taylor : ED\], we can write $$\begin{aligned}
\frac{1}{\tau_{XY}}\,\cal{D}^2 (X,Y) \; &= \; {{\mathbb{E}}}\,\frac{K(X,X')}{\tau_X}\, \frac{K(Y,Y')}{\tau_Y} \, + \, {{\mathbb{E}}}\,\frac{K(X,X')}{\tau_X} \, {{\mathbb{E}}}\,\frac{K(Y,Y')}{\tau_Y} \, - \, 2\, {{\mathbb{E}}}\,\frac{K(X,X')}{\tau_X}\, \frac{K(Y,Y'')}{\tau_Y}\\
&= \;\; {{\mathbb{E}}}\, \left(1 + \frac{1}{2} L(X, X') + R(X, X') \right) \, \left(1 + \frac{1}{2} L(Y, Y') + R(Y, Y') \right)\\
& \qquad + \; {{\mathbb{E}}}\, \left(1 + \frac{1}{2} L(X, X') + R(X, X') \right) \, {{\mathbb{E}}}\,\left(1 + \frac{1}{2} L(Y, Y') + R(Y, Y') \right)\\
& \qquad - \; 2\, {{\mathbb{E}}}\, \left(1 + \frac{1}{2} L(X, X') + R(X, X') \right) \, \left(1 + \frac{1}{2} L(Y, Y'') + R(Y, Y'') \right)\,\\
&= \;\; L \; + \; R,\end{aligned}$$ where $$\begin{aligned}
L \; &= \; \frac{1}{4} \, \left[ \,{{\mathbb{E}}}\, L(X,X') L(Y,Y') \,\, + \,\,{{\mathbb{E}}}\,L(X,X') \, {{\mathbb{E}}}\,L(Y,Y') \,\, - \,\, 2\, {{\mathbb{E}}}\, L(X,X') L(Y,Y'') \, \right],\end{aligned}$$ and $$\begin{aligned}
R \; =& \; {{\mathbb{E}}}\, \left[ \, \frac{1}{2} L(X,X') R(Y,Y') \,+\, \frac{1}{2} R(X,X') L(Y,Y') \, +\, R(X,X') R(Y,Y') \,\right]\\
& -\,2\,{{\mathbb{E}}}\, \left[ \, \frac{1}{2} L(X,X') R(Y,Y'') \,+\, \frac{1}{2} R(X,X') L(Y,Y'') \, +\, R(X,X') R(Y,Y'') \,\right]
\\&+ {{\mathbb{E}}}\, R(X,X') \, {{\mathbb{E}}}\,R(Y,Y').\end{aligned}$$ Some simple calculations yield $$\begin{aligned}
L \; &=\; \frac{1}{4\tau^2_{XY}} \, \left\{ \,{{\mathbb{E}}}\, [K^2(X,X') K^2(Y,Y')] \,\, + \,\,{{\mathbb{E}}}\,[K^2(X,X')] \, {{\mathbb{E}}}\,[K^2(Y,Y')] \,\, - \,\, 2\, {{\mathbb{E}}}\, [K^2(X,X') K^2(Y,Y'')] \, \right\}\\
&= \; \frac{1}{4\tau^2_{XY}} \, \dis\sum_{i=1}^p \sum_{j=1}^q \Big\{\,{{\mathbb{E}}}\, [\rho_i(X_{(i)}, X_{(i)}') \, \rho_j(Y_{(j)}, Y_{(j)}')] \, + \, {{\mathbb{E}}}\, [\rho_i(X_{(i)}, X_{(i)}')] \, {{\mathbb{E}}}\,[\rho_j(Y_{(j)}, Y_{(j)}')] \\ &\qquad \qquad \qquad \qquad -\,2 \,{{\mathbb{E}}}\, [\rho_i(X_{(i)}, X_{(i)}') \, \rho_j(Y_{(j)}, Y_{(j)}'')] \Big\}\\
&= \; \frac{1}{4\tau^2_{XY}} \, \dis\sum_{i=1}^p \sum_{j=1}^q D^2_{\rho_i,\,\rho_j}(X_{(i)}, Y_{(j)})\,.\end{aligned}$$ To observe that the remainder term is negligible, note that under Assumption \[ass D pop taylor\], $$\begin{aligned}
{{\mathbb{E}}}\,[ L(X, X') R(Y,Y')] \; & \leq \; \left(\, {{\mathbb{E}}}\, [L(X, X')^2] \, {{\mathbb{E}}}\, [R(Y, Y')^2] \,\right)^{1/2} \; = \; O(a_p' b_q'^2)\;,\\
{{\mathbb{E}}}\,[ R(X, X') L(Y,Y')] \; & \leq \; \left(\, {{\mathbb{E}}}\, [R(X, X')^2 ]\, {{\mathbb{E}}}\, [L(Y, Y')^2] \,\right)^{1/2} \; = \; O(a_p'^2 b_q')\;,\\
{{\mathbb{E}}}\,[ R(X, X') R(Y,Y')] \; & \leq \; \left(\, {{\mathbb{E}}}\, [R(X, X')^2] \, {{\mathbb{E}}}\, [R(Y, Y')^2] \,\right)^{1/2} \; = \; O(a_p'^2 b_q'^2)\;,\end{aligned}$$ Clearly, $\mathcal{R} = \tau_{XY} R = O(\tau_{XY}\, a_p'^2 b_q' + \tau_{XY}\, a_p' b_q'^2)$.
The proof is essentially similar to the proof of Theorem \[D pop taylor\]. Note that using Proposition \[K taylor : ED\], we can write $$\begin{aligned}
\frac{1}{\tau_Y}\,\cal{D}^2 (X,Y) \; &= \; {{\mathbb{E}}}\,K(X,X')\, \frac{K(Y,Y')}{\tau_Y} \, + \, {{\mathbb{E}}}\,K(X,X') \, {{\mathbb{E}}}\,\frac{K(Y,Y')}{\tau_Y} \, - \, 2\, {{\mathbb{E}}}\,K(X,X')\, \frac{K(Y,Y'')}{\tau_Y}\\
&= \;\; {{\mathbb{E}}}\, K(X,X') \, \left(1 + \frac{1}{2} L(Y, Y') + R(Y, Y') \right)\\
& \qquad + \; {{\mathbb{E}}}\, K(X,X') \, {{\mathbb{E}}}\,\left(1 + \frac{1}{2} L(Y, Y') + R(Y, Y') \right)\\
& \qquad - \; 2\, {{\mathbb{E}}}\, K(X,X') \, \left(1 + \frac{1}{2} L(Y, Y'') + R(Y, Y'') \right)\,\\
&= \;\; L \; + \; R,\end{aligned}$$ where $$\begin{aligned}
L \; &=\; \frac{1}{2\tau^2_{Y}} \, \dis \sum_{j=1}^q \Big\{\,{{\mathbb{E}}}\, [K(X,X') \, \rho_j(Y_{(j)}, Y_{(j)}')] \, + \, {{\mathbb{E}}}\, [K(X,X') \, {{\mathbb{E}}}\,[\rho_j(Y_{(j)}, Y_{(j)}')] \; -\,2 \,{{\mathbb{E}}}\, [K(X,X') \, \rho_j(Y_{(j)}, Y_{(j)}'')] \Big\}\\
&= \; \frac{1}{2\tau^2_{Y}} \, \sum_{j=1}^q D^2_{K,\,\rho_j}(X, Y_{(j)})\,,\end{aligned}$$ and $$\begin{aligned}
R \; =& \; {{\mathbb{E}}}\, \left[ \, K(X,X') R(Y,Y') \,\right]\;+\; {{\mathbb{E}}}\,\left[ K(X,X')\right] \, {{\mathbb{E}}}\left[ R(Y,Y')\right]\; -\; 2\,{{\mathbb{E}}}\, \left[ \, K(X,X') R(Y,Y'') \,\right]\,.\end{aligned}$$ Under the assumption that ${{\mathbb{E}}}\,[R^2(Y,Y')] = O(b_q'^4)$, using H[ö]{}lder’s inequality it is easy to see that $\tau_Y R = O(\tau_{Y}\, b_q'^2) = o(1)$.
Following equation (\[expr to be used later\]), we have for $1\leq k\neq l \leq n$ $$\begin{aligned}
\widetilde{D}^X_{kl} \;&=\; \frac{\tau_X}{2} \widetilde{L}(X_k, X_l) \;+\; \tau_X \widetilde{R}(X_k, X_l) \;=\; \frac{1}{2\tau_X} \dis \sum_{i=1}^p {{\widetilde \rho}}_i(X_{k(i)}, X_{l(i)}) \;+\; \tau_X \widetilde{R}(X_k, X_l)\,,\\
\widetilde{D}^Y_{kl} \;&=\; \frac{\tau_Y}{2} \widetilde{L}(Y_k, Y_l) \;+\; \tau_Y \widetilde{R}(Y_k, Y_l) \;=\; \frac{1}{2\tau_Y} \dis \sum_{j=1}^q {{\widetilde \rho}}_i(Y_{k(j)}, Y_{l(j)}) \;+\; \tau_Y \widetilde{R}(Y_k, Y_l)\,.\end{aligned}$$ From equation (\[ustat dcov\]) in the main paper it is easy to check that $$\begin{aligned}
\widetilde{\cal{D}^2_n} (X,Y) \;&=\; \frac{1}{4\tau_{XY}} \dis \sum_{i=1}^p \sum_{j=1}^q \widetilde{D^2_n}_{\,;\,\rho_i, \rho_{j}} (X_{(i)}, Y_{(j)}) \;+\; \frac{\tau_{XY}}{2n(n-3)} \dis \sum_{k\neq l} \widetilde{L}(X_k, X_l) \widetilde{R}(Y_k, Y_l) \\
&\qquad + \; \frac{\tau_{XY}}{2n(n-3)} \dis \sum_{k\neq l} \widetilde{L}(Y_k, Y_l) \widetilde{R}(X_k, X_l) \;+\; \frac{\tau_{XY}}{n(n-3)} \dis \sum_{k\neq l} \widetilde{R}(X_k, X_l)\, \widetilde{R}(Y_k, Y_l) \,.\end{aligned}$$ Under Assumption \[ass2.1\], using H[ö]{}lder’s inequality and power mean inequality, it can be verified that $$\begin{aligned}
\dis \sum_{k\neq l} \widetilde{L}(X_k, X_l) \widetilde{R}(Y_k, Y_l) \; & \leq \; \left(\, \dis \sum_{k\neq l} \widetilde{L}(X_k, X_l)^2 \, \sum_{k\neq l} \widetilde{R}(Y_k, Y_l)^2 \,\right)^{1/2} \; = \; O_p(a_p b^2_q)\;,\\
\dis \sum_{k\neq l} \widetilde{L}(Y_k, Y_l) \widetilde{R}(X_k, X_l) \; & \leq \; \left(\, \dis \sum_{k\neq l} \widetilde{L}(Y_k, Y_l)^2 \, \sum_{k\neq l} \widetilde{R}(X_k, X_l)^2 \,\right)^{1/2} \; = \; O_p(a^2_p b_q)\;,\\
\dis \sum_{k\neq l} \widetilde{R}(X_k, X_l) \widetilde{R}(Y_k, Y_l) \; & \leq \; \left(\, \dis \sum_{k\neq l} \widetilde{R}(X_k, X_l)^2 \, \sum_{k\neq l} \widetilde{R}(Y_k, Y_l)^2 \,\right)^{1/2} \; = \; O_p(a_p^2 b^2_q)\;.\end{aligned}$$ This completes the proof of the theorem.
Following equation (\[expr to be used later\]), we have for $1\leq k\neq l \leq n$ $$\begin{aligned}
\widetilde{D}^Y_{kl} \;&=\; \frac{1}{2\tau_Y} \dis \sum_{j=1}^q {{\widetilde \rho}}_j(Y_{k(j)}, Y_{l(j)}) \;+\; \tau_Y \widetilde{R}(Y_k, Y_l)\,,\end{aligned}$$ and therefore $$\begin{aligned}
\widetilde{\cal{D}^2_n} (X,Y) \;&=\; \frac{1}{2\tau_{Y}} \dis \sum_{j=1}^q \widetilde{D^2_n}_{\,;\,K, \rho_{j}} (X, Y_{(j)}) \;+\; \frac{\tau_{Y}}{n(n-3)} \dis \sum_{k\neq l} \widetilde{K}(X_k, X_l) \widetilde{R}(Y_k, Y_l) \,.\end{aligned}$$ Using power mean inequality, it can be verified that $ \sum_{k\neq l} \widetilde{K}(X_k, X_l) \widetilde{R}(Y_k, Y_l)\,=\, O_p(b^2_q)$. This completes the proof of the theorem.
The proof follows similar lines of the proof Theorem 2.2.1 in Zhu et al.(2019), with the distance metric being the one from the class of metrics we proposed in equation (\[Kdef\]).
The proof of the theorem follows similar lines of the proof of Proposition 2.2.2 in Zhu et al.(2019).
The decomposition into the leading term follows the similar lines of the proof of Theorem \[ACdCov taylor thm\]. The negligibility of the remainder term can be shown by mimicking the proof of Theorem 3.1.1 in Zhu et al.(2019).
It essentially follows similar lines of Proposition 3.2.1 in Zhu et al.(2019).
[9]{}
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Q:
Show that $\int_{|z|=3} \frac{1}{z^2-1} dz = 0$
Here's a homework problem I'm having some trouble with:
Show that $$ \int_{|z|=3} \frac{1}{z^2-1} dz = 0$$
So far, I've shown using Cauchy's Integral Formula that $$ \int_{|z-1|=1} \frac{1}{z^2-1} dz = \pi i$$ and $$\int_{|z+1|=1} \frac{1}{z^2-1} dz = - \pi i$$ where $|z-1|=1, |z+1|=1$ are circles of radius $1$ centered at $1$ and $-1$, respectively.
Now, I'm trying to use these results to solve the problem. It seems as if I will somehow add the integrals up to obtain the result $0$, but I'm having trouble extending the two circles to the circle of radius $3$ about the origin. My thought was to split the circle $|z|=3$ into two semicircles, separated on the imaginary axis, but I'm not quite sure how to do it. Any help would be greatly appreciated.
A:
Yes, split the path $p$ of $|z| = 3$ into two semicircles as you said to create left semicircle $p_1$ and right semicircle $p_2$. Since
$$\int_{-3i}^{3i} f(z)~dz + \int_{3i}^{-3i} f(z) ~ dz = 0$$
you will get that (in the case that $p_1$ and $p_2$ are both defined to be counter clockwise paths):
$$\begin{align}
\int_p f(z)~dz &= \int_{\text{left arc}} f(z)~dz + \int_{\text{right arc}} f(z)~dz \\
&= \int_{\text{left arc}} f(z)~dz + \int_{-3i}^{3i} f(z)~dz + \int_{3i}^{-3i} f(z)~dz + \int_{\text{right arc}} f(z)~dz\\
&= \int_{p_1} f(z)~dz + \int_{p_2} f(z)~dz
\end{align}$$
From Cauchy's Integral Formula we know that:
The analytic integral around closed path $q$ only depends on the poles inside of $q$
$p_1$ and $|z+1|=1$ contain the same poles, because the second is inside the first
$p_2$ and $|z-1|=1$ contain the same poles, because the second is inside the first
So
$$\begin{align}\int_p f(z)~dz &=
\int_{p_1} f(z)~dz + \int_{p_2} f(z)~dz \\
&= \int_{|z+1|=1} f(z)~dz + \int_{|z-1|=2} f(z)~dz \\
&= -\pi i + \pi i \\
&= 0\end{align}$$
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Do I need a carbon monoxide detector in my home?
Carbon monoxide is a colorless, odorless and tasteless gas. Carbon monoxide is produced due to incomplete combustion of burned fuels (i.e. oil, natural gas, wood, etc.). Many household appliances can produce carbon monoxide including:
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Message From the Chief
Welcome from Fire Chief Phil Hyman
Welcome to the City of Chattanooga's Fire Department Webpage. Whether you are a citizen of Chattanooga or a visitor to our great city, rest assured an elite group of men and women stand ready to protect you and your family 24/7!
Since 1871, we have had a singular mission: reduce the loss of life and minimize property damage for the people in our growing city. We've worked wisely to that end, investing in technology and recruiting a highly trained and dedicated force of firefighters. We have challenged ourselves to continue to innovate and evolve in order to perform our duties in the most effective and efficient ways possible. Today, more than 400 firefighters wear a patch on their sleeves that reminds them of the faith and trust that has been placed in their hands. It's a symbol of our commitment to excellence and an honor we take very seriously, from our fire stations to our administrative offices and everywhere in between.
By budgeting for outcomes in five key areas, our department has served an important role in Chattanooga becoming America's best mid-sized city.
- Safer Streets: The department has 26 companies ready to respond to incidents involving fire, hazardous materials, confined space, high angle rescue, building collapse, medical emergencies, and mass casualty incidents.
- Stronger Neighborhoods: Our department has 20 stations rooted in our communities, involved in day to day activities outside of emergency response, and being positive influences on neighborhood children and adults.
- Smarter Students: Fire prevention and community fire stations team up to bring fire safety and educational programs to all ages and demographics in our city.
- Growing Economy: The CFD is a ISO Class 1 department that provides fire protection for large company investments and the city's workforce, as well as all residential communities within Chattanooga.
- High-Performing Government: The CFD serves alongside a Mayor and City Council who are progressive in funding public safety while prioritizing budget expenditures throughout the city.
Please view other areas of our webpage for employment opportunities and the history of our department. We have a dynamic department that strives to provide the best possible services to the citizens of Chattanooga, Tennessee! Please also let us know if you have any suggestions for improving the website or our department.
Thanks for visiting!
Community Involvement
The Chattanooga Fire Department (CFD) places a high priority on connecting with the citizens it serves. Our firefighters are involved with community groups and events throughout the year, on and off the job.
Here is a list of many other ways Chattanooga firefighters connect with the citizens they serve:
Did You Know?
Did you know that the CFD Honor Guard can provide ceremonial assistance for your event? Contact Captain David Thompson, Jr., for more information or to request the CFD Honor Guard ([email protected]).
Did you know that the CFD teams up with the American Red Cross twice a year to distribute free smoke alarms in at-risk neighborhoods?
Did you know that our firefighters hand out stuffed animals to children involved in fires and car wrecks? It's a small gesture, but the stuffed animals help the children cope with one of the most traumatic experiences of their young lives.
Did you know that the Fire Prevention Bureau conducts public education classes for children on a regular basis? Using the Fire Safety House and Sparky, the Bureau teaches children how to protect themselves in the event of a fire. If you're interested in having the Fire Safety House visit your school or venue, please download the Fire Safety House Teacher's Packet. It has all the info you need to set up and run a successful fire safety presentation, then fill in a request form and mail it to us or fax it to (423) 643-5611.
Did you know that Chattanooga firefighters raise money for the Muscular Dystrophy Association (MDA) through the annual Fill-the-Boot Campaign?
Did you know that our firefighters provide 'After the Fire' brochures to fire victims? These brochures provide useful information on what to do after the fire, such as what information to gather for insurance purposes, how to clean up belongings that are salvageable, and how to replace lost documents.
Did you know that Station 1 on E. Main Street has a Community Room that can be used by citizens for community meetings and other events? All you have to do is call Station 1 at 267-1463 to find out how to reserve the room. More community rooms will be added at future fire stations.
Did you know that our firefighters offer free blood pressure checks at all of our fire stations? Just stop by your neighborhood Chattanooga fire station during the hours of 8:00 a.m. and 7:00 p.m. any day of the week and the firefighters will be glad to check your blood pressure for free.
Did you know that our firefighters conduct an annual food drive in August to replenish supplies at the Chattanooga Area Food Bank?
Did you know that Chattanooga firefighters have constructed two houses for Habitat for Humanity? The firefighters used their considerable construction skills to build the houses, on their own time, for two deserving families.
Did you know that our fire stations keep their bay doors open whenever possible to present a friendly face to the community? We want you to know that we're there when you need us, and that we appreciate your support! | http://propertytax.chattanooga.gov/fire-department/news-releases/24-fire-department?start=65 |
Are there more chickens or cows or fish in existence?
chickens
there's more chickens
What are all the animals people eat?
What has more legs 8 cows or 10 chickens?
Animals that have been selectively bred?
What is 3a plus 5a?
This should not be a problem, when you consider the following situations: ==> You have 3 cows. You get 5 more cows. Now you have 8 cows. ==> You have 3 tomatoes. You get 5 more tomatoes. Now you have 8 tomatoes. ==> You have 3 books. You get 5 more books. Now you have 8 books. Am I going too fast for you ? ==> You have 3 chickens. You get 5 more chickens…
What types of agriculture is available in Spain?
How many types of animals are there on Farmville Facebook?
Cows, Alien Cows, Pink Cows, Brown Cows, Geese, Ducks, Sheep, Black Sheep, Pigs, Horses, Goats, Reindeer, Clumsy Reindeer, White Chickens, Brown Chickens, Black Chickens, Golden Chickens, Elephants, Baby Elephant, Orange Tabby, Grey Tabby, White Kitty, Black Cat, Bull, Penguin, Rabbit, Turkey, Lamb, Wild Turkey, Lop-eared Bunny, Donkeys, Turtle, Jackalope, Peacock, Buffalo, Ox, Deer, Clydesdale, Calf, Brown Calf, Pink Calf, Alien Calf, Swan, Brown Goose... maybe more
What animals lived in ancient India?
What kind of Animals live in Samoa?
Which of the animals have bones on the inside?
Can you give me a long list of the farm animals?
How do animals provide for humans and animals?
What animals were in the Indus valley civilization?
What kind of pets are there in Bangladesh?
What are the native animals in El Salvador?
What are the plants and animals in El Salvador?
What animals on farm?
Which animals are producers?
How common are chickens?
What is the general outline of the structural and functional organisation of the nervous system of farm animals?
Why are fish not animals?
Because there are so many fish in the sea. Ask your dad. He knows best. There are so many fish and they are underwater that they are not animals. If you are vegetarian you can eat a fish because there are so many that eating one won't make a difference. If you eat a cow there are less cows. But there will ALWAYS be more fish. Learn more at Fish.com
What animals were raised on mission San Jose?
What is a group of dairy cattle?
Are there more people or chickens?
Do all women smell like fish?
Since every organism has unique biochemistry, and the more biochemical systems are involved in the existence of an organism, the more potential for variety, no, it is highly unlikely that at any point in human existence all females will have pheromonic emissions that are similar to the olfactory stimulation caused by most fish. If you experience a sensation of smelling fish when in contact with all female pheromones, it is possible that you have either…
How do you prevent the chickens from plucking the feathers out on other chickens?
You raise chickens there are more than 50 in your barn?
Can you have more than ducks chickens one horse cows goats and sheep in harvest moon a wonderful life for the ps2?
What are the animals in zanzibar island?
What do people in rural communities eat?
Which one is correct chicken or chickens?
Where do most cows come from?
What wild life live in lowa?
What country has more cows than people?
Can all ducks live with chickens?
Ducks are gerally more aggresive and tend to peck chickens, and cause neither the ducks or the chickens to lay. I wouldn't recomend keeping them together. Though several species of ducks can settle down with chickens, especially if you have more chickens then ducks, and perhaps you have had the chickens longer, I see this behavior in my animals at home. | https://www.answers.com/Q/Are_there_more_chickens_or_cows_or_fish_in_existence |
1 Preheat oven to 180C. Put 20g butter in a microwave-safe jug and microwave on high for 20 seconds or until melted. Brush 4 soufflé dishes (1 cup capacity each) with butter. Put 1 tsp sugar in each soufflé dish and shake so the sides and base are coated. Shake off excess sugar.
2 Melt remaining butter in a saucepan over medium heat. Add flour and stir until well combined and mixture is bubbling. Remove saucepan from heat and slowly add ½ cup lime juice and milk, stirring constantly.
Return saucepan to heat and cook, stirring constantly, for 2–3 minutes or until mixture comes to the boil and thickens. Reduce heat and simmer for 1 minute. Remove from heat. Cool for 10 minutes, then stir through 1/3 cup sugar, egg yolks and lime rind. Set aside to cool.
3 Using an electric mixer, beat egg whites and salt in a bowl until stiff peaks form. Add remaining sugar, 1 tablespoon at a time, beating well after each addition until thick and glossy.
4 Add a third of the egg white mixture to the lime mixture and gently fold together with a large metal spoon. Add remaining egg white mixture and gently fold (without overworking the mixture). Spoon mixture into prepared soufflé dishes.
5 Bake for 15–20 minutes or until well risen and golden. When cooked, sprinkle with icing sugar and serve immediately. | https://www.mindfood.com/recipe/lime-souffle-recipe-dessert-sweet-winter/ |
CROSS-REFERENCE TO RELATED APPLICATION(S)
0001 This application claims the benefit of U.S. Provisional Patent Application No. 60/262,029 filed Jan. 15, 2001 which is hereby incorporated by reference as if set forth in full herein.
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
0002 Many laboratories use automated equipment for handling and transporting test tubes for analytical procedures. Such equipment requires precise location of multiple test tubes in a rack used with the automated equipment. Conventional racks for accurately positioning and securing test tubes typically contain multiple plates or vertical channels to align and secure the test tubes. This invention provides an improved rack and clamps for holding test tubes in precise alignment with respect to each other and the rack in which the tubes are mounted. The rack includes a top plate and a bottom plate and insertable test tube clamps for positioning and securing the test tubes into position.
0003 The present invention provides an apparatus including a rack having a top plate and a bottom plate with openings in the top plate vertically aligned with corresponding cavities formed in the bottom plate. The plates are spaced apart and secured together by legs. A test tube clamp which fits securely in the opening formed in the top plate of the rack is also provided. The clamp includes a ring section and a shoulder as well as a plurality of downwardly extending fingers to secure the test tube securely in position and aligned with the vertically aligned cavity in the bottom of the rack. The fingers include at least one set of fingers extending to a first depth and a second set of fingers extending to a second depth so the tube is contacted and held in two planes. The clamp can secure test tubes having various diameters centrally within the opening.
BRIEF DESCRIPTION OF THE DRAWING
0004FIG. 1 is a perspective view of an exemplary rack;
0005FIG. 1A is a fragmentary elevation, partly in section of a leg which secures the rack plates together;
0006FIG. 2 is a perspective view of an exemplary clamp;
0007FIG. 3 is a fragmentary view of an exemplary clamp mounted in an opening of the top plate of a rack;
0008FIG. 4 is a top view of an exemplary clamp;
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0009FIG. 5 is a cross-sectional view of the exemplary clamp taken along line - of FIG. 4;
0010FIG. 6 is a perspective view of another exemplary clamp; and
0011FIG. 7 is a side view including fragmentary cross sections of the exemplary clamp shown in FIG. 6.
DETAILED DESCRIPTION
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0012 Referring to the drawings, FIG. 1 is a perspective view of a rack , which includes a horizontal rectangular top plate or shelf with a plurality of circular openings . Top plate is molded integrally with four downwardly extending legs at each corner of the top plate. If necessary, say to resist deformation when subjected to high temperatures of about 130 C. and low temperatures of about 70 C., one or more additional downwardly extending legs or pillars (not shown) are molded integrally with the bottom surface of the top plate, and are spaced from the edges of the top plate by distance at least equal to about the elements of the holes. The rack is preferably formed of a plastic material such as polysulfone but other materials may be used alternatively. Materials of formation are chosen for durability and high and low temperature tolerance.
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0013 The lower end of each leg is secured to a respective corner of a rectangular bottom plate of the same size as the top plate. Leg may optionally be internally reinforced with steel or other rigid metals.
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0014 Bottom plate is preferably molded of the same material as top plate , and includes a plurality of upwardly opening conical cavities molded in the top surface of bottom plate . Each cavity is a frustro-conical shape to receive the lower end of a test tube, and has a respective opening extending through the bottom plate to facilitate cleaning. Each cavity is vertically aligned with a corresponding circular opening formed in top plate . As shown in FIG. 1, the top and bottom plates each define a respective substantially flat plane, and are substantially parallel.
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0015 As shown in FIG. 1A, each leg of the top plate includes a downwardly extending dowel pin which fits into a matching bore A at each corner of the bottom plate so that the top plate and bottom plate are precisely aligned to ensure that each circular opening formed in top plate directly overlies a corresponding cavity in bottom plate . The dowel may be formed of steel or other rigid materials. If additional legs are used at intermediate locations on the top plate (as referred to above), each such leg carries a downwardly extending dowel pin which makes a snug fit in a matching base in the bottom plate.
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0016 A downwardly extending broad flange is molded integrally with one of the long edges of top plate and with two adjacent legs. The vertical dimension of the flange is about one half the distance between the top and bottom plates but other dimensions may be used alternatively. Rectangular recess is an alignment marker formed in the outer face of flange and is shaped to receive a label (not shown) for identifying the rack. The opposite edge of the top plate does not have a broad flange. This insures that the label is always placed on the correct side of the rack for proper orientation in automated handling. This feature is especially useful in automated pick and place equipment for positioning the rack and for removing and inserting the test tubes into and from the rack. According to another exemplary embodiment, the flange with the formed recess may extend from leg to leg along a side of the rack and be located superjacent an edge of bottom plate and extend upward when the rack is assembled.
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0017 Another aspect of the present invention is the clamp for positioning and securing a test tube centrally within openings such as circular openings of top plate . FIGS. - show an exemplary clamp which is constructed to be inserted to make a pressed fit in circular opening of top plate . FIG. 2 is a perspective view of clamp according to an exemplary embodiment of the invention. The clamp may be formed of a copolymer material chosen to have the following characteristics: ease of manufacture, strength, durability, flexibility, high and low temperature tolerance and low surface friction. In an exemplary embodiment, a copolymer such as DuPont Celcom 90 may be used. Clamp is constructed such that a single clamp may secure test tubes of various diameters within an opening such as circular opening such that, regardless of test tube diameter, the longitudinal axis of the grasped tube is coaxial with the axis passing through the center of the hole in which the clamp is inserted. With respect to exemplary rack , clamps retain test tubes, regardless of diameter, such that the longitudinal axis of the gripped test tube is coaxial with the axis passing through the center of circular opening and the respective underlying cavity to which circular opening is aligned. This ensures accurate alignment and permits reliable handling of test tubes of different sizes with automated equipment. The openings may preferably be formed in compliance with NCCLS (National Committee for Clinical Laboratory Standards) and may include a pitch on the order of 22 mm, but other configurations may be used alternatively. Clamp fits securely in circular opening when inserted therein but may be removed after use.
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0018 Clamp includes a plurality of downwardly and inwardly extending fingers to align and secure the test tube securely into position. The fingers and ring of the clamp are integrally molded of any suitable plastic which provides rigidity around the ring portion and flexibility and resiliency for the fingers as well as the other characteristics described above. The fingers include at least one set of fingers extending to a first depth and a second set of fingers extending to a second depth. Each of the fingers preferentially extend centrally toward the center axis of the ring from which they extend. The fingers are free at their lower ends so that the fingers in one set can grasp the test tube in a first horizontal plane and the fingers in the other set can grasp the test tube in a second horizontal plane spaced vertically from the first horizontal plane.
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0019FIG. 3 is a fragmentary view showing exemplary clamp mounted in circular opening of top plate . FIG. 4 is a top view of the clamp, and FIG. 5 is a view taken on line - of FIG. 4. Each circular opening of top plate includes an annular inwardly extending and upwardly facing shoulder on which rests a complementary annular outwardly extending and downwardly facing shoulder of an annular ring of clamp . Annular ring is sized to be tightly secured when pressed into in circular opening of top plate .
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0020 As shown in FIG. 3, ring defines a substantially flat plane A, which is parallel to the flat planes defined by the top and bottom plates of the rack. Clamp is adapted receive a test tube therein. Annular ring forms the top portion of clamp and includes inner face of annular ring which tapers upwardly and outwardly at the upper end of the ring to facilitate insertion of a test tube (not shown) into clamp . Clamp includes a first set of three downwardly extending long fingers spaced at equal intervals around the clamp ring, and a second set of three downwardly extending short fingers spaced at equal intervals around the ring and spaced equidistant between adjacent long fingers. According to other exemplary embodiments, other configurations and numbers of fingers may be used. Long fingers comprise one set of fingers which extend to a first depth and short fingers comprise a second set of fingers which extend to a second depth less than the first depth.
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0021 As shown clearly in FIGS. 2, 3 and , short fingers extend downwardly and inwardly toward each other, and are about one half the length of long fingers , which also extend downwardly toward each other. The distal or free ends of the long fingers in the first are more closely spaced to each other than are the short finger in the second set. Short fingers may have a flat or inwardly facing convex surface according to various exemplary embodiments. Long fingers extend downwardly and inwardly and each presents an inwardly facing convex surface. Each of the set of short fingers and the set of long figures include portions internal to the periphery of circular opening . In the preferred embodiment, each of the fingers extend downwardly inward and toward the center axis of circular opening and annular ring . The fingers of each set of fingers may take on various shapes as they extend downwardly inward. The fingers of each of the respective sets of fingers preferably have the same shape.
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0022 When each circular opening of top plate is fitted with a clamp as shown in FIG. 3, test tubes can easily be inserted in, and removed from, rack with great precision, even though the test tubes may not be of the same size, and may not be located with the bottom of the test tubes resting in corresponding cavity underlying a respective circular opening . Test tubes of various sizes (for example ranging from 8 to 16 mm in diameter) are firmly and precisely grasped so that the longitudinal axis of each test tube is coaxial with the axis passing through the center of circular opening and respective underlying cavity because the grasped test tube is secured at contact points (the points at which the test tube contacts each of the respective sets of fingers) in two different horizontal planes. The particular contact point for each of the sets of fingers depends upon the curvature, degree of inward extension and length of the fingers. The contact point for the fingers of each set of fingers lie essentially in a horizontal plane. The horizontal plane defined by contact points of one set of fingers is vertically spaced from the horizontal plane defined by the other set of fingers.
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0023FIGS. 6 and 7 show another exemplary embodiment of the clamp of the present invention. A clamp is constructed to make a pressed fit into an opening formed in a top shelf of an exemplary rack and includes features as previously described in conjunction with exemplary clamp . Clamp includes annular ring , downwardly facing shoulder , inner face , long fingers and short fingers . Clamp additionally includes a third set of fingersperipheral fingers . Peripheral fingers extend essentially straight down and are essentially the same length as long fingers in the illustrated exemplary embodiment although they may have different lengths in alternative embodiments. As described in conjunction with short fingers and long fingers of clamp , long fingers and short fingers of clamp each extend downwardly inward. In the exemplary embodiment, long fingers and short fingers each include an arcuate shape although other shapes may be used in other embodiments. Also in the exemplary embodiment, long fingers and short fingers extend centrally inward toward the center axis of annular ring from which they extend. Clamp includes peripheral fingers peripherally spaced about annular ring and having a short finger or a long finger alternately interposed between each neighboring set of peripheral fingers . Each of the three long fingers and the three short fingers are spaced at equal intervals around the ring and interposed equidistant between adjacent peripheral fingers . Different numbers of fingers and various other arrangements may be used in other exemplary embodiments. | |
What Sensors Do Medical Robots Have?
- November 18, 2021
- 0
There are three sensors on the medical robot: a light sensor, a speed sensor, and a camera. In addition to the light sensor, the speed sensor, and the camera are used to see the patient.
Table of contents
What Type Of Sensors Are Used In Robots?
What Sensors Does A Medical Robot Have And How Does The Medical Robot Use These Sensors?
There are three types of sensors on the medical robot: proximity sensors, range sensors, and timbre sensors. By using proximity sensors, the robot can detect a human presence in the work volume, which can be used to reduce accidents. Distance between a sensor and a work part is calculated using range sensors.
What Sensors Does The Da Vinci Surgical Robot Have?
Surgical tools with greater range of motion than the human hand are available. A surgeon console with high-definition 3D vision is also available. A variety of optical encoders, Hall sensors, magnetic encoders, and infrared sensors can be used.
What Do Sensors Allow Robots?
In order to interact with a robot, sensors allow it to overcome obstacles and collect information from the surrounding environment. To mimic the abilities of living beings, humanoid robots need a large number of sensors.
What Could Be A Sensor In A Robot?
In order for robots to see what is around them, they need to use sensors. LIDAR (Light Detection And Ranging) is an example of a sensor that is used in some robots. An object is illuminated by a laser and reflected back by a laser. In order to map its environment, the robot analyzes these reflections.
What Sensors Do Industrial Robots Have?
Industrial robots are commonly equipped with two-dimensional visual sensors, three-dimensional visual sensors, force/torque sensors, and collision detection sensors.
Why Sensors Are Used In Robotics?
The sensors in robots allow them to react in a flexible manner to their environment. A robot can see and feel with the help of sensors, and this would enable it to perform complex tasks more effectively. A robot must be able to recognize its position and its movement in order to control its own actions.
Is The Type Of Robot Internal Sensor?
It is possible to equip a robot with two types of sensors: internal sensors, which establish its configuration in its own coordinate axes, and external sensors, which allow it to position itself in a specific way.
What Are Sensors Used For In Robotics?
An estimated robot’s condition and environment can be determined by robotic sensors. In order to enable appropriate behavior, these signals are passed to the controller. Human sensory organs are used to develop sensors in robots. In order for robots to function effectively, they must have extensive information about their environment.
What Are Five Uses For Sensors On A Robot?
Assembly, welding, monitoring marine life, rescue and recovery, and even vacuuming the floor are some of these tasks. In order for a robot to meet its specific needs, it will need a variety of sensors.
What Sensors Are Used In Autonomous Robots?
In real-world situations, GPS devices, gyroscopes, and magnetometers are commonly used to provide this data. In devices and robotics systems that must navigate longer distances or traverse unfamiliar terrain, these sensors are among the most common and useful.
How Does The Da Vinci Surgical Robot Use Its Sensors?
Furthermore, the sensors can prevent surgeons from accidentally applying too much force, which could result in injury.
What Are The Components Of The Davinci Surgical Robot?
What Tools Does The Da Vinci Robot Use?
In order to operate on an area, a small surgical incision is made and long, delicate instruments are inserted. In addition to an endoscopic camera with magnification, lighting, and temperature control, the surgeon can also use a control arm to view inside the body using the tools. | https://www.wovo.org/what-sensors-do-medical-robots-have/ |
Why is a sunflower yellow?
It reflects yellow light
Which forms of light are lower in energy and frequency than the light that our eyes can see?
Infrared and Radio
What is the fundamental difference between 2 chemical elements?
They have different numbers of protons in their nucleus
Suppose you look at a spectrum of visible light by looking through a prism or diffraction grating. How can you decide whether it is an emission line spectrum or an absorption line spectrum?
An emission line spectrum consists of bright lines on a dark background, while an absorption line spectrum consists of dark lines on a rainbow background
According to the laws of thermal radiation, hotter objects emit photons with _________.
A shorter average wavelength
The spectra of most galaxies show redshifts. This means that their spectral lines _________.
Have wavelengths that are longer than normal
Compared to the Sun, a star whose spectrum peaks in the infrared is.......?
Cooler
Everything looks red through a red filter because...?
The filter transmits red light and absorbs other colors
From lowest energy to highest energy, What is the correct order of the different categories of electromagnetic radiation?
Radio, infrared, visible light, ultraviolet, X rays, gamma rays
Each of the following describes an "Atom 1" and an "Atom 2." In which case are the two atoms isotopes of each other?
A) Atom 1: nucleus with 1 proton and 0 neutrons, surrounded by 1 electron
Atom 2: nucleus with 2 protons and 2 neutrons, surrounded by 2 electrons
B) Atom 1: nucleus with 92 protons and 143 neutrons, surrounded by 92 electrons
Atom 2: nucleus with 92 protons and 146 neutrons, surrounded by 92 electrons
C) Atom 1: nucleus with 4 protons and 5 neutrons, surrounded by 4 electrons
Atom 2: nucleus with 5 protons and 5 neutrons, surrounded by 4 electrons
D) Atom 1: nucleus with 8 protons and 8 neutrons, surrounded by 8 electrons
Atom 2: nucleus with 8 protons and 8 neutrons, surrounded by 7 electrons
E) Atom 1: nucleus with 6 protons and 8 neutrons, surrounded by 6 electrons
Atom 2: nucleus with 7 protons and 8 neutrons, surrounded by 7 electrons
B) Atom 1: nucleus with 92 protons and 143 neutrons, surrounded by 92 electrons
Atom 2: nucleus with 92 protons and 146 neutrons, surrounded by 92 electrons
What happens when an atom loses an electron?
It becomes ionized
What is the study of energy levels in atoms called?
Quantum mechanics
What is the wavelength of a wave?
The distance between two adjacent peaks of the wave
How much electrical charge does an atom with 6 protons, 6 neutrons, and 5 electrons have?
A positive charge of +1
How do you do describe the phase of matter at extremely high temperatures (Millions of degrees)?
Plasma consisting of positively charged ions and free electrons
What is the process of Dissociation?
When the bonds between atoms in a molecule are broken
If you heat a gas so that collisions are continually bumping electrons to higher energy levels, when the electrons fall back to lower energy levels the gas produces....?
An emission line spectrum
When white light passes through a cool cloud of gas, we see...?
An absorption line spectrum
What is Thermal radiation?
Radiation that depends only on the emitting object's temperature
True or False:
Any object moving relative to Earth will have a Doppler shift
False
Rank the forms of light from left to right in order of increasing wavelength
Gamma ray, X-Ray, Ultraviolet, visible light, Infrared, Radio waves
Rank the forms of light from left to right in order of increasing frequency
Radio waves, Infrared, Visible light, Ultraviolet, X-ray, Gamma rays
Rank the forms of light from left to right in order of increasing energy
Radio waves, Infrared, Visible light, ultraviolet, X-rays, Gamma rays
Rank the forms of light from left to right in order of increasing speed
They are all the same. The speed of light is constant.
What procedures would allow you to make a spectrum of the Sun?
Pass a narrow beam of sunlight through a prism.
Why does the Sun's spectrum contain black lines over an underlying rainbow?
The Sun's hot interior produces a continuous rainbow of color, but cooler gas at the surface absorbs light at particular wavelengths
What does the yellow-green region (brightest/intense portion) of the Sun's spectrum tell us?
The approximate temperature of the Sun's surface
What should we do if we want to find out what the Sun is made of?
Compare the wavelengths of lines in the Sun's spectrum to the wavelengths of lines produced by chemical elements in the laboratory
What kind of graph would represent a portion of the Sun's visible light spectrum?
A curve with both smooth parts and dips
Why does the emission line spectrum of a neon "OPEN" sign appear reddish-orange?
Neon atoms emit many more yellow and red photons than blue and violet photons
The absorption line spectrum shows what we see when we look at a hot light source (such as a star or light bulb) directly behind a cooler cloud of gas. Suppose instead that we are looking at the gas cloud but the light source is off to the side instead of directly behind it. In that case, the spectrum would __________.
Be an emission line spectrum
What type of visible light spectrum does the Sun produce?
An absorption line spectrum
Power
The rate of energy flow (measured in units called watts)
Watts
- Unit of measure for Power
- 1 Watt = 1 joule/per second
Spectrum
(Prism) Split light into a rainbow
When do we see the color white?
When the colors of the rainbow are mixed in equal proportions
White light
- Light from the Sun or a light bulb
- Contains all the colors of the rainbow
When do we see the color black?
When there is no light, and hence, no color
Diffraction Grating
Producing a spectrum by using a piece of glass/plastic etched with many closely spaced lines
Emission
- A light bulb emits visible light
- The energy of the lights comes from electrical potential energy supplied to the light bulb
Absorption
When you place your hand near an IL bulb, you hand absorbs some of the light
- This absorbed energy warms your hand
Transmission
- Some forms of matter, such as glass or air, transmit light
- ^^Which means allowing it to pass through
Reflection/Scattering
- Light can bounce off matter
- Reflection is when all the bouncing light is going in the same general direction
- Scattering is when the bouncing light is more random
What do we call materials that transmit light?
Transparent
What do we call materials that absorb light?
Opaque
True or False:
Red glass transmits red light, but absorbs other colors
True
Pebble and floating leaf example:
You through a pebble into a pond
- The creates waves
- As the waves reach the leaf, it will rise and fall with the peaks and troughs of the wave
- But the leaf doesn't actually travel with the wave
- The wave moves outward
- But the molecules of the water move up and down
- The waves carry energy outward from where the pebble landed, but don't carry matter along with them
- A particle is a thing
- A wave is a pattern revealed by its interaction with particles
Wavelength
- The distance from 1 peak to the next
- Or 1 trough to the next
Frequency
The number of peaks passing by any point each second
Hertz
Unit for frequency
What does the speed of waves tell us?
- How fast their peaks travel across the pond
- Also tells us how fast the energy carried travels from one place to the next
Wavelength x Frequency = Speed
Wavelength x Frequency = Speed
Field
Describes the strength of force that a particle would experience at any point in space
EX) Earths gravitational field describes the strength of gravity at any distance from earth
Electromagnetic Wave
- What light is
- Light waves are traveling vibrations of both electric and magnetic fields
- Hence, ELECTRO-MAGNETIC fields
- Vibrations of an electric field in an electromagnetic wave will cause any charged particles (such as an electron) to bob up and down
The longer the wavelength, the lower the frequency (and vice versa)
The longer the wavelength, the lower the frequency (and vice versa)
Photons
- Light particles
- Have properties of both particles and waves
Electromagnetic spectrum
- Spectrum of light
- Represents all the forms of light
- All consists of photons that travel through space at the speed of light
Electromagnetic Radiation
Light itself
Where is visible light found on the Electromagnetic spectrum?
- Near the middle
- Wavelengths range from 400 nanometers at the blue or violet end of the rainbow to about 700 nanometers at the red end
Infrared
- Light with wavelength longer than red light
- Lies beyond the red end of the rainbow
Radio Waves
- Longest wavelength of light
Microwaves
Between the border of infrared light and radio waves
Ultraviolet
- Light with wavelengths shorter than blue light
- Lies beyond the blue (or violet) end of the rainbow
- Carries enough energy to damage skin cell and cause cancer or sunburns
X-Rays
- Light with shorter wavelengths than UVs
- Have enough energy to penetrate skin and muslces, but not bone
- Which is why its used in medicine
Gamma Rays
THE shortest-wavelength light
What kind of light do molecules moving in a warm object emit?
Infrared light
EX) Heat sensors
Nucleus contains most of the atoms mass
Nucleus contains most of the atoms mass
Atomic number
Number of protons in an atoms nucleus
Atomic Mass number
Combined number of protons and neutrons in an atom
Isotopes
- Elements with different number of neutrons
- The number of neutrons isn't the same as the number of protons
What causes the different phases changes of matter?
- Differing strengths of the bonds between neighboring atoms and molecules
- Phase changes occur when one type of bond is broken and replaced by another
- This can be caused by changes in pressure or temperature
EX) increased temp increases the kinetic energy of the particles, which enables the particles to break the bonds holding them to their neighbors
- Ice has a much stronger bond
- Water has a looser/weaker bond
Sublimation
Vaporization from a solid
Molecular Dissociation
- When high enough temps cause molecule collisions to become violent enough that they can break the bonds holding individual water molecules together
- The molecules then split into pieces
Ionization
Stripping electrons from atoms
Plasma
Hot gas in which atoms have been ionized
Energy Levels (of an atom)
- Electrons can have only particular amounts of energy, and not other energies in between
- EX) Step ladder
- The final level is "Ionization level"
- This is where the electron has gained enough energy to escape the atom
- This makes the atom ionized
Ground State
- Lowest possible energy level
- AKA Level 1
Spectrum of a traditional incandescent light bulb?
- Continuous Spectrum
- Its a rainbow that spans a broad range of wavelengths without interruption
- Graph is smooth and continuous
Spectrum of a thin/low density cloud of gas?
- Emission lines (against a black background)
- Called an emission line spectrum
- The cloud only emits light at specific wavelengths
- Graph is spiked from the bottom (spikes are where the wavelengths we see on the spectrum are)
Spectrum of a cloud of gas between us and a light bulb?
- Dark Absorption lines (over the background rainbow)
- Called an absorption line spectrum
- Black lines over the rainbow where the wavelengths are
- Graph is arched up and has dips (where the wavelengths of the light is)
Where does the energy an electron loses go when it falls down an energy level?
- Energy goes toward emitting a photon of light
- Has the exact amount of energy that the electron lost
- So it has a specific wavelength and frequency
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Related questions
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How do you think nature formed the salt deposits in Michigan, Ohio, New York and Ontario? | https://quizlet.com/161218164/ch-5-light-and-matter-flash-cards/ |
Endangered turtles bred at Brookfield Zoo released into DuPage wild
The tiny turtle squirming in Dan Thompson's hands looks like a content creature.
The shape of its mouth gives the appearance of a smile and the impression that the young reptile is blissfully unaware it takes a village to reach this rite of passage.
Every fall, Thompson wades into knee-deep, murky water to release Blanding's turtles into the marshes of the Forest Preserve District of DuPage County. After a year in captivity, the turtles have grown from quarter-sized hatchlings to plucky juveniles, eager to experience the natural world.
Each youngster carries a spotted shell and the hopes of conservationists who have spent 25 years working to offset population losses.
"They're all special, really," said Thompson, a district ecologist.
But the batch released Wednesday are unique in their own way. They are the first Blanding's turtles bred and hatched at Brookfield Zoo's captive breeding pond, a genetic reservoir built with females from the forest preserves.
"We're just very happy to see them move on and contribute to a population," said Andy Snider, the zoo's curator of herpetology and aquatics.
Brookfield contributed 10 turtles to Wednesday's release, while another dozen from the Peggy Notebaert Nature Museum in Chicago ventured into grassy marshes.
The species is endangered in Illinois, under threat from habitat loss and fragmentation, predators, roadside mortalities as females cross traffic to get to nesting sites, and an illegal black market. The forest preserve district doesn't disclose the exact location of their release to protect the turtles from collectors.
District ecologists launched the species recovery program in the mid-1990s after observing a troubling population decline. Older generations outnumbered more vulnerable, younger turtles as raccoons and other midlevel predators flourished with suburban sprawl.
"We were dealing with a really small population of wild adult turtles, and that's all we had," Thompson said. "We did not have any kind of age structure."
The district tracks turtle births with radio transmitters attached to females. When they're ready to nest around early June, ecologists bring the turtles to Willowbrook Wildlife Center in Glen Ellyn so the reptiles can lay their eggs away from the threat of raccoons.
After an incubation period, the hatchlings spend about a year at zoos, museums and other partner institutions until the turtles are large enough to have a better chance at survival in the wild.
Then the process starts all over again. A slowing-maturing species, Blanding's turtles don't start reproducing until their midteens to their 20s.
"We've been fortunate that we've hit that threshold, and we've actually had some of our head starts in the wild survive to maturity and are reproducing and providing our third generation of turtles now," Thompson said.
In 2011, Brookfield Zoo began to develop a breeding pond to produce captive-bred youngsters to send into the wild. Last year, some of the nearly 25 females in the pond were old enough to successfully breed and lay fertile eggs in a nesting beach.
In the spring and summer months, the zoo also uses outdoor enclosures in Dragonfly Marsh to "head start" the district's hatchlings.
Nearly 3,500 turtles have hatched out since the inception of the forest preserve district's program.
Ecologists hope that eventually, the region will see signs of a self-sustaining population, but conservation efforts take time. They keep at it knowing that their work preserving the Great Lakes native supports biodiversity.
"We're really certainly at a time right now that we really need to be much more vigilant about how we care for the planet and how we live to make sure that we have less impact," Thompson said. | https://www.dailyherald.com/article/20200902/news/200909803/ |
How many pushups should a 13 year old do?
If you can do pushups in a set, but no pull-ups, or sit-ups, then clearly there’s an issue. Having said that, at 13 you’d hitting a pretty optimal strength-weight ratio, so there’s not a whole lot of excuses for not being able to do at least 20–30 in a set. How many push-ups should a 14 year old do?
Is it OK for a 12 year old to do push-ups?
Routines including push-ups, sit-ups and light calisthenics are completely safe for children not yet of middle school age. … By utilizing light resistance with bands and bodyweight exercises, children will be able to increase strength without the dangers of pumping iron.
How many pushups should I do by age?
Depending on your age, you also should be able to do a specific number of push-ups and sit-ups in one minute: Men between 50 and 59 years old should be able to do 15 to 19 push-ups and 20 to 24 sit-ups. Women of the same age should be able to do seven to 10 push-ups and 15 to 19 sit-ups.
How many push-ups can a 14 year old do?
For a 14-year-old to be in the 50th percentile, a boy had to perform 24 push-ups and a girl, 10. A score of just 3 for a girl or 11 for a boy was considered poor and put them in the 10th percentile. Percentiles aren’t considered at all at the Cooper Institute’s FitnessGram.
How many pushups can a girl do?
Table: push-up test norms for WOMEN
|Age||17-19||20-29|
|Excellent||> 30||> 32|
|Good||22-30||24-32|
|Above Average||11-21||14-23|
|Average||7-10||9-13|
Do push-ups stunt your growth?
Push-ups for Grown-ups
It almost goes without saying that there’s no evidence out that to support push-ups stunting growth in adults. … You don’t have to worry about stunting your growth, but do pay attention to proper form to maximize your results and minimize the risk of injury.
How many push-ups a day is good?
There is no limit to how many push-ups one can do in a day. Many people do more than 300 push-ups a day. But for an average person, even 50 to 100 push-ups should be enough to maintain a good upper body, provided it is done properly. You can start with 20 push-ups, but do not stick to this number.
Can kids do sit-ups?
Kids can start with body weight exercises (such as sit-ups and push-ups) and work on technique without using weights. When proper technique is mastered, a relatively light weight can be used with a high number of repetitions (8–15). Increase the weight, number of sets, or types of exercises as strength improves.
Is 13 push-ups good?
Looking at the “good” category, the average number of push-ups for each age group is: … 30 to 39 years old: 17 to 21 push-ups for men, 13 to 19 push-ups for women. 40 to 49 years old: 13 to 16 push-ups for men, 11 to 14 push-ups for women. 50 to 59 years old: 10 to 12 push-ups for men, seven to 10 push-ups for women.
How many push-ups should 11 year old do?
Push-ups (Boys)
|Age|
|Rating||6||11|
|90||11||30|
|70||7||23|
|50||7||15|
Is push-ups good for kids?
Resistance tubing and body-weight exercises, such as pushups, are other effective options. Emphasize proper technique. Form and technique are more important than the amount of weight your child lifts. Your child can gradually increase the resistance or number of repetitions as he or she gets older.
Is 30 push ups in a row good?
A good standard to aim for when it comes to muscular strength for most men is the ability to perform 30 push-ups in a row. If you find that once you get to 20 you start to falter in form, you need to work on improving your overall strength capacity. | https://fitforyouonline.com/gymnastics/how-much-push-ups-should-a-12-year-old-do.html |
Bačka Palanka have over 1.5 goals in their last 5 games.
Bačka Palanka have conceded over 0.5 goals in their last 7 games.
Bačka Palanka have over 0.5 first half goals in their last 7 games.
Bačka Palanka have over 0.5 second half goals in their last 5 games.
Mačva Šabac have under 3.5 goals in their last 29 games.
Bačka Palanka have over 0.5 goals in 100% of their games in the last 2 months (total games 9).
Radnički Niš have won over 2.5 corners in their last 19 games.
Partizan have scored over 0.5 goals in their last 4 games.
Partizan have scored in the 0.5 half in their last 4 games.
Partizan have over 0.5 first half goals in their last 7 games.
Partizan have won over 2.5 corners in their last 16 games.
Partizan have over 0.5 goals in 100% of their games in the last 2 months (total games 11).
Partizan have over 0.5 first half goals in 91% of their games in the last 2 months (total games 11).
Crvena Zvezda have won their last 9 games.
Crvena Zvezda have over 2.5 goals in their last 9 games.
Crvena Zvezda have over 1.5 goals in their last 10 games.
Crvena Zvezda have scored over 2.5 goals in their last 4 games.
Crvena Zvezda have scored over 1.5 goals in their last 9 games.
Crvena Zvezda have scored over 0.5 goals in their last 31 games.
Crvena Zvezda have scored in the 0.5 half in their last 9 games.
Crvena Zvezda have over 0.5 first half goals in their last 28 games.
Crvena Zvezda have over 0.5 second half goals in their last 10 games.
Crvena Zvezda have won over 2.5 corners in their last 4 games.
Crvena Zvezda have over 0.5 goals in 100% of their games in the last 2 months (total games 11).
Crvena Zvezda have over 1.5 goals in 91% of their games in the last 2 months (total games 11).
Crvena Zvezda have over 2.5 goals in 82% of their games in the last 2 months (total games 11).
Crvena Zvezda have over 0.5 first half goals in 100% of their games in the last 2 months (total games 11).
Rad Beograd have lost their last 5 games.
Rad Beograd have under 2.5 goals in their last 4 games.
Rad Beograd have under 3.5 goals in their last 9 games.
Rad Beograd have conceded over 0.5 goals in their last 5 games.
Rad Beograd have over 0.5 second half goals in their last 5 games.
Radnik Surdulica have under 3.5 goals in their last 6 games.
Radnik Surdulica have over 0.5 first half goals in their last 4 games.
Radnik Surdulica have over 0.5 goals in 100% of their games in the last 2 months (total games 10).
Radnik Surdulica have over 0.5 first half goals in 90% of their games in the last 2 months (total games 10).
Voždovac have scored over 0.5 goals in their last 8 games.
Voždovac have over 0.5 goals in 100% of their games in the last 2 months (total games 9).
Mladost Lučani have over 0.5 first half goals in 91% of their games in the last 2 months (total games 11).
Zemun have conceded over 0.5 goals in their last 7 games.
Zemun have won over 3.5 corners in their last 4 games.
Spartak Subotica have won their last 4 games.
Spartak Subotica have under 3.5 goals in their last 5 games.
Spartak Subotica have under 2.5 goals in their last 4 games.
Spartak Subotica have scored over 0.5 goals in their last 4 games.
Napredak have under 3.5 goals in their last 5 games.
Napredak have over 0.5 first half goals in their last 4 games. | https://afootballreport.com/predictions/serbia/super-liga/MTEyMw== |
Ramon van Handel is an associate professor, and a member of the executive committee of the Program in Applied and Computational Mathematics (PACM), at Princeton University. He received his PhD in 2007 from Caltech. His honors include the NSF CAREER award, the Presidential Early Career Award for Scientists and Engineers (PECASE), the Erlang Prize, the Princeton University Graduate Mentoring Award, and several teaching awards. He has served on the editorial boards of the Annals of Applied Probability and Probability Theory and Related Fields, and is the probability editor for the IMS Textbooks and Monographs series. Ramon’s research interests lie broadly in probability theory and its interactions with other areas of mathematics. He is particularly fascinated by the development of probabilistic principles and methods that explain the common structure in a variety of pure and applied mathematical problems. His recent interests are focused on high-dimensional phenomena in probability, analysis, and geometry. He has also worked on conditional phenomena in probability and ergodic theory, and on applications of noncommutative probability. This Medallion lecture was given at the Seminar on Stochastic Processes in March 2022.
Nonasymptotic random matrix theory
Classical random matrix theory is largely concerned with the spectral properties of special models of random matrices, such as matrices with i.i.d. entries or invariant ensembles, whose asymptotic behavior as the dimension increases has been understood in striking detail. On the other hand, suppose we are given a random matrix with an essentially arbitrary pattern of entry means and variances, dependencies, and distributions. What can we say about its spectrum? Beside lacking most of the special features that facilitate the analysis of classical random matrix models, such questions are inherently nonasymptotic in nature: when we are asked to study the spectral properties of a given, arbitrarily structured random matrix, there is no associated sequence of models of increasing dimension that enables us to formulate asymptotic questions.
It may appear hopeless that anything useful can be proved at this level of generality. Nonetheless, a set of tools known as “matrix concentration inequalities” makes it possible at least to crudely estimate the range of the spectrum of very general random matrices up to logarithmic factors in the dimension. Due to their versatility and ease of use, these inequalities have had a considerable impact on a wide variety of applications in pure mathematics, applied mathematics, and statistics. On the other hand, it is well known that these inequalities fail to capture the correct order of magnitude of the spectrum even in the simplest examples of random matrices. Until very recently, results of this kind were essentially the only available tool for the study of generally structured random matrices.
In my lecture, I describe a new approach to such questions (developed in joint work with Bandeira, Boedihardjo, and Brailovskaya) that has opened the door to a drastically improved nonasymptotic understanding of the spectral properties of generally structured random matrices. In this new theory, we introduce certain deterministic infinite-dimensional operators, constructed using methods of free probablity, that may be viewed a the “Platonic ideals” associated random matrices. Our theory shows, in a precise nonasymptotic sense, that the spectrum of an arbitrarily structured random matrix is accurately captured by that of the associated Platonic ideal under remarkably mild conditions. The resulting sharp inequalities are easily applicable in concrete situations, and capture the correct behavior of many examples for which no other approach is known. | https://imstat.org/2022/07/18/ims-medallion-lecture-ramon-van-handel/ |
An updated driving theory test is being introduced in the Isle of Man.
It has been developed by officers from the Department of Infrastructure in conjunction with their counterparts in Jersey and Guernsey.
The government says collaborative working has enabled the theory test questions to ‘reflect the unique aspects of driving in an island, as well as highlighting the differences that will be encountered if driving in the UK’.
It has also helped to cut the cost of the project.
The revised driving theory test is scheduled to come into effect for learner drivers and motorcycle riders from Thursday, September 4, replacing the current version which has been in place since 2004.
The computer-based test, which features multiple choice questions and answers and a section on hazard perception, must be passed before candidates can apply to take their practical driving examination.
Infrastructure Minister Phil Gawne MHK said: ‘We are continually striving to improve driving standards in the Isle of Man to make our roads as safe as possible. The updated test should ensure that new drivers and riders have greater theory-based knowledge at the start of their motoring experience. It is important for learners to have a good understanding of the rules of the road before they get behind the wheel to take their practical driving test.’
He added: ‘I am delighted that we have been able to work in partnership with our colleagues in Jersey and Guernsey to develop the new theory test. Sharing expertise and costs has been beneficial for all parties.’
From September 4, learner drivers and motorcyclists taking the theory test will face questions designed to place greater emphasis on the demands of modern motoring, including safety and environmental issues, vehicle knowledge and being more considerate to other road users.
A new feature of the test is a voice-over option, which allows candidates to listen to the questions and answers being read out in English if required. This is designed to assist those who experience difficulties with reading.
All new provisional licence applicants receive a training package, including a CD featuring example questions and hazard perception clips, so they can practise before taking the theory test. This CD includes additional information and explanations, so anyone clicking a wrong answer can discover why it is incorrect and be guided towards the right answer.
The new CD asks people which island they live in – and selecting the ‘Isle of Man’ will ensure that no questions specific to Jersey or Guernsey will appear.
Candidates applying to take the theory test after September 4 should practise using the updated training package. Anyone who has the previous version can exchange it free of charge at the Isle of Man College of Further and Higher Education or at the Vehicle Test Centre on Ballafletcher Road, Tromode. | http://www.iomtoday.co.im/news/isle-of-man-news/new-driving-theory-test-to-be-introduced-next-month-1-6798206 |
A water-source heat pump system includes water source intermediate heat exchanger (25), water-source water system which is connected with water source intermediate heat exchanger (25) and is equipped with water pump (23) and expansion water tank (24), the first water-cooling cold water units (1), second water-cooling cold water units (2), auxiliary pumps (26) and the connecting pipes, among them the connection between the first water-cooling cold water units (1) and the second water-cooling cold water units (2) has two method of series and parallel connection: the parallel method is the condenser (3) and the evaporator (4) of the first water-cooling cold water units (1) are parallel with the condenser (5) and the evaporator (6) of the second water-cooling cold water units (2) separately; the series connection is the water outlet pipe and inlet pipe of the condenser (5) of the second water-cooling cold water units (2) are series connected with the water outlet pipe and inlet pipe of the evaporator (4) of the first water-cooling cold water units (1). The water resource water utilizing temperature difference of the water-source heat pump in the invention is large, the water-taking recharge flow is small and the hot water outlet temperature can reach 45-98deg.C when heat-production operates. | |
This is a very special Hand Embellished Multiple of 'Elements' by Agnes Cecile.
These beautiful 17.25 x 24” inch pieces were carefully hand finished by the Artist during her visit to our Studio in November 2018 to create a piece as unique and special as the original itself.
17.25x24 inches on Fine Art Paper (28x32 inches framed).
"Elements" belongs to a series about the styles I went through the years. This one painting encloses two aspects: my interest in muted and soft shades of colors, and the use of natural elements beside the human figure.
Butterflies give me the freedom to play with delicate shapes and unlimited palette of colors.
In "Elements" I painted them in shy and muted tones, while in this springtime’s HPM I give them new life and energy.
An awakening in colors blooming from a placid white surrounding. | https://www.eyesonwalls.com/collections/agnes-cecile/products/elements |
In this example we explore the approximation properties of Chebyshev interpolation for entire functions; that is, functions that are analytic everywhere in the complex plane.
1. Analytic functions
In the following discussion, it will be helpful to utilise the notion of an r-ellipse, which we define as the image of a circle of radius r > 1 in the complex x-plane under the mapping x = (z + 1/z) / 2.
Here are some such ellipses, which we denote by Er:
rr = 1 + (1:10)/10; circ = exp(1i*chebfun('t',[0 2*pi])); clf, hold on for k = 1:numel(rr) rho = rr(k); plot((rho*circ + (rho*circ).^(-1))/2,LW,lw) end hold off, axis equal, box on
Suppose we have a function f that is analytic on [-1,1] and that can be analytically continued into an r-ellipse for some r > 1. Then [1, Chap. 8], the infinity-norm error arising from interpolating f with a polynomial in n+1 Chebyshev points is
max| f - p_n | <= 4 M / ( r^n (r-1) ) ,
where M is the maximum absolute value taken by f on the ellipse Er. This is a geometric rate of convergence. If we require an accuracy of 0 < e < 1 for our approximations, then it will suffice to obtain the smallest n satisfying
4 M / ( r^n (r-1) ) <= e .
Some trivial rearrangement of this expression gives
[log(4/e) - log(r-1) + log(M)]/log(r) <= n .
Choosing an n larger than this will ensure that the interpolant is of accuracy e.
2. Oscillatory entire functions
When the function f is entire, then one may expect the convergence to be even better than geometric, and this is indeed the case. Consider for example, for some positive integer N, the entire function
f(x) = sin(pi N x) .
Because f is analytic in the entire complex plane, the convergence result above must hold for any value of r > 1. An estimate for the parameter M may be obtained by observing that on a given ellipse, a complex exponential is maximised where the ellipse intersects the (negative) imaginary axis, i.e.,
M <= 1/2 exp[ pi N(r - 1/r)/2 ] .
Since this relationship is true for every r > 1, we must find the minimum value of the following expression over all r > 1,
[log(2/e) - log(r-1) + pi N(r - 1/r)/2] / log(r) .
For a given oscillation parameter N and precision e, this may be accomplished using Chebfun. This provides an interesting way to validate the performance of the Chebfun constructor. The plot below shows the function on the LHS of the equation above plotted for different values of N. The mimimum of each function -- and the estimate for the minimum Chebfun degree required for accuracy e = eps -- is plotted in each case as a red dot.
ee = eps; NN = 10:100:1010; clf, hold on, estimates = zeros(numel(NN),1); chebdegrees = estimates; for k = 1:numel(NN) N = NN(k); P = @(p) (log(2/ee) - log(p-1) + N*pi/2*(p-1./p))./log(p); PP = chebfun(P,[1.01 10]); [mn,pos]= min(PP); estimates(k) = mn; ff = chebfun(@(x) sin(pi*N*x),'eps',ee); chebdegrees(k) = length(ff)-1; plot(PP,LW,lw) plot(pos,mn,'.r',MS,ms) end text(8.02,200, sprintf('N = %3i',NN(1))) text(8.02,800, sprintf('N = %3i',NN(2))) text(8.02,1450, sprintf('N = %3i',NN(3))) text(8.02,2100, sprintf('N = %3i',NN(4))) text(8.02,2700, sprintf('N = %3i',NN(5))) text(8.02,3350, sprintf('N = %3i',NN(6))) hold off, xlabel('\rho') shg, grid on, ylim([0 3.5e3]), box on
How do these estimates for the length of the polynomial interpolant compare to Chebfun lengths resulting from Chebfun's adaptive construction process?
est = ceil(estimates); fprintf(' function estimate chebfun length \n') for k = 1:numel(NN) fprintf(' sin( %4i pi x) %4i %4i \n',... NN(k),est(k),chebdegrees(k)) end fprintf('\n')
function estimate chebfun length sin( 10 pi x) 69 65 sin( 110 pi x) 427 415 sin( 210 pi x) 761 745 sin( 310 pi x) 1090 1071 sin( 410 pi x) 1415 1393 sin( 510 pi x) 1739 1715 sin( 610 pi x) 2062 2037 sin( 710 pi x) 2384 2359 sin( 810 pi x) 2705 2677 sin( 910 pi x) 3025 2993 sin( 1010 pi x) 3346 3313
We see that the estimates from the analysis are very slightly too generous. This is due to our estimate for M not being the tightest upper bound possible. The reason that we used the bound for M given above is that the exponent is easily extracted by the log(M) term; using the tighter bound for M prohibits this.
References: | https://la.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/23972/versions/22/previews/chebfun/examples/approx/html/Entire.html |
Big bees do a better job: intraspecific size variation influences pollination effectiveness.
Abstract
Bumblebees (Bombus spp.) are efficient pollinators of many flowering plants, yet the pollen deposition performance of individual bees has not been investigated. Worker bumblebees exhibit large intraspecific and intra-nest size variation, in contrast with other eusocial bees; and their size influences collection and deposition of pollen grains.
Laboratory studies with B. terrestris workers and Vinca minor flowers showed that pollen grains deposited on stigmas in single visits (SVD) were significantly positively related to bee size; larger bees deposited more grains, while the smallest individuals, with proportionally shorter tongues, were unable to collect or deposit pollen in these flowers. Individuals did not increase their pollen deposition over time, so handling experience does not influence SVD in Vinca minor.
Field studies using Geranium sanguineum and Echium vulgare, and multiple visiting species, confirmed that individual size affects SVD. All bumblebee species showed size effects, though even the smallest individuals did deposit pollen, whereas there was no detectable effect with Apis with its limited size variation. Two abundant hoverfly species also showed size effects, particularly when feeding for nectar.
Mean size of foragers also varied diurnally, with larger individuals active earlier and later, so that pollination effectiveness varies through a day; flowers routinely pollinated by bees may best be served by early morning dehiscence and visits from larger individuals.
Thus, while there are well-documented species-level variations in pollination effectiveness, the fine-scale individual differences between foragers should also be taken into account when assessing the reproductive outputs of biotically-pollinated plants.
Full Text:PDF
DOI: http://dx.doi.org/10.26786/1920-7603%282014%2922
This work is licensed under a Creative Commons Attribution 3.0 License. | http://pollinationecology.org/index.php?journal=jpe&page=article&op=view&path%5B%5D=301 |
Q:
Trouble with two equations with 4 unknowns
I was wondering if I could receive assistance for the following system:
$$\begin{cases}(x/a)^{3.2}+(y/b)^{3.2}=1\\ a/b = 174.1/86\end{cases}$$
I'm looking for integer solutions or how to find them (if possible).
Edit: I have currently gotten this far:
$$\begin{cases} ((86x)^{3.2}+y^{3.2})/(174.1b)^{3.2}=1 \end{cases}$$
Thanks in advance!
Ken
A:
You don't need the equation for $a/b$. There is no solution of the first equation in integers $a,b,x,y$ with $x, y \ne 0$.
You want $r = x/a$ and $s = y/b$ to be nonzero rational numbers, with $r^{16/5} + s^{16/5} = 1$. Let $r^{1/5} = t$ and $s^{1/5} = u$, so that $t^{16} + s^{16} = 1$. By a case of Fermat's Last Theorem, it is known that
this has no rational solutions with $s,t\ne 0$. So at least one of $t$ and $u$ is irrational. WLOG assume $t$ is irrational, i.e. $r$ is not the $5$'th power of a rational number. Now the same is true of $r^{16}$. It can be shown for any rational $y$ that is not the $5$'th power of a rational, the polynomial $X^5 - y$ is irreducible. So $u = t^{16}$, a root of the polynomial $X^5 - r^{16}$, has the minimal polynomial $X^5 - r^{16}$.
Now $s^{16} = (1 - r^{16/5})^5 = (1 - t^{16})^5 = (1-u)^5$, and thus
$(X-1)^5 + s^{16}$ is a different monic polynomial of degree $5$ with $u$ as a root. That contradicts the assertion that $X^5 - r^{16}$ is the minimal polynomial of $u$.
We conclude that no such solution is possible.
| |
Florida is surrounded by water, so projections that sea levels will rise up to one foot — and even more in some areas, including Tampa Bay — by 2050 are bad news. This estimate comes from a new report issued by the National Oceanic and Atmospheric Administration, which gives a “what’s coming” look at tide, wind and storm-driven extreme water levels that will likely drive future coastal flood risk. Moderate flooding is expected to occur more than 10 times as often as it does now, according to NOAA.
U.S. Secretary of Commerce Gina M. Raimondo said the intent of the report is to help businesses and communities know what to expect and plan for the future. Much of the concern over sea level rise focuses on the threat to waterfront homes, coastal businesses and recreation spots. A less obvious area for concern is the impact to agricultural production.
Florida currently produces more than 300 different commodities, from tomatoes to strawberries, which generated roughly $7.4 billion in cash receipts in 2020. Agriculture in Florida, and particularly in South Florida, is a significant contributor to the state’s economic vitality.
Safeguarding and advancing agricultural production in south Florida means we must act now to reduce saltwater intrusion, which affects soil, surface and groundwater quality, and plant viability, given that many farmlands in Florida are near the coastline. For example, many areas of the South Dade Agricultural area, which as of 2017 had nearly 71,000 acres being farmed, have an elevation of less than 3 feet above sea level.
Even a slight increase in saltwater intrusion into Florida’s aquifers could have major repercussions in availability of fresh water for drinking and agricultural purposes. Consider that in South Florida, 90 percent of drinking water comes from underground aquifers. At the same time, agriculture depends heavily on groundwater.
Sea level rise is a complex and serious problem, with multiple factors requiring many approaches and dedicated effort over time.
While we may not be able to stop sea level rise quickly, we can take action to mitigate some of the impact of saltwater intrusion on Florida’s agricultural productivity and farmers’ livelihoods. Florida’s rising stature as a food producer necessitates taking this issue seriously and adopting both short-term mitigation measures and long-term adaptations, such as investing in and breeding salt-tolerant crops.
Short-term management practices could be implemented at farm and field levels and show effects quickly. One potential approach is to flush salt-affected soils with freshwater, a quick way to keep soil salinity low; however, this method is limited by availability of freshwater and potential for contamination of groundwater.
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Improving irrigation efficiency also is critical to reduce salt build up in the soil and the need to pump groundwater, which will, in turn, reduce flow of saltwater further inland.
The use of solid oxygen fertilizers has been demonstrated to provide plant roots with needed oxygen in soil depleted by salinity and also improve their resistance to salt and flooding. Gypsum, biochar and compost are soil amendments that could help counter negative effects of salt-soaked soil, as can the use of cover crops which reduce salt buildup. Farmers also could use raised beds to grow plants, elevating their roots.
Long-term management practices often take a more protracted process for development and adoption and will depend on growers’ willingness to adopt new practices and, potentially, invest capital in new or modified farm equipment.
Many major crops grown in South Florida are sensitive to high salinity and flooding, such as snap beans, strawberries, avocados and papaya. There are a few plants with limited tolerance to salinity and flooding, such as coconut, guava, jujube, dragon fruit and mango for tropical fruit growing area. Lastly, integrated pest management will be key to mitigate disease and pest impacts that arise with saltwater intrusion.
We are in a race against time to better understand the mechanisms and negative impacts of sea level rise and saltwater intrusion on freshwater resources, soil health, crops and our ability to play a role as a major food producer. Coming up with the best management practices to mitigate these impacts should be a priority to safeguard food and water security.
Haimanote Bayabil is an assistant professor of water resources in the Agricultural and Biological Engineering Department, Tropical Research and Education Center at the University of Florida.
Yuncong Li is a professor of soil quality in the Soil and Water Sciences Department, Tropical Research and Education Center at the University of Florida. | https://www.tampabay.com/opinion/2022/02/21/what-rising-sea-levels-means-for-agriculture-in-south-florida-column/ |
---
abstract: 'Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some point $p\in M$, we prove that the set of metrics in the conformal class of $g$ with prescribed constant positive $Q$-curvature is compact in $C^{4, \alpha}$ for any $0 <\alpha < 1$. We also give some estimates for dimension $6$ and $7$.'
address: 'Gang Li, Beijing International Center for Mathematical Research, Peking University, Beijing, China'
author:
- 'Gang Li$^\dag$'
title: 'A compactness theorem on Branson’s $Q$-curvature equation'
---
[^1]
Introduction
============
On a manifold $(M^n, g)$ of dimension $n\geq 5$, the $Q$-curvature of Branson [@Branson] is defined by $$\begin{aligned}
Q_g=-\frac{2}{(n-2)^2}|Ric_g|^2+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}R_g^2-\,\frac{1}{2(n-1)}\Delta_gR_g\end{aligned}$$ where $Ric_g$ is the Ricci curvature of $g$, $R_g$ is the scalar curvature of $g$ and $\Delta_g$ is the Laplacian operator with negative eigenvalues. The Paneitz operator [@Paneitz], which is the linear operator in the conformal transformation formula of the $Q$-curvature, is defined as $$\begin{aligned}
\label{operatorP}
P_g=\Delta_g^2-\text{div}_g(a_nR_g g -b_nRic_g)\nabla_g + \frac{n-4}{2}Q_g,\end{aligned}$$ with $a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}$ and $b_n=\frac{4}{n-2}$. In fact, under the conformal change $\tilde{g}=u^{\frac{4}{n-4}}g$ the transformation formula of the $Q$-curvature is given by $$\begin{aligned}
P_gu\,=\,\frac{n-4}{2}Q_{\tilde{g}}u^{\frac{n+4}{n-4}}.\end{aligned}$$ In comparison, the change of scalar curvature under the conformal change $\tilde{g}=u^{\frac{4}{n-2}}g$ satisfies $$\begin{aligned}
R_{\tilde{g}}=u^{-\frac{n+2}{n-2}}(-\frac{4(n-1)}{(n-2)})\Delta_g u\,+\,R_g u).\end{aligned}$$ 0.2cm
Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n\geq 5$. Assume that $R_g\geq 0$ and $Q_g\geq 0$ on $M$ with $Q_g$ not identically zero. For existence of solutions $u$ to the prescribed constant positive $Q$-curvature equation $$\begin{aligned}
\label{equation1}
P_gu\,=\,\frac{n-4}{2}\bar{Q}u^{\frac{n+4}{n-4}},\end{aligned}$$ with $\bar{Q}=\frac{1}{8}n(n^2-4)$, one may refer to Qing-Raske [@Qing-Raske1], Hebey-Robert [@Hebey-Robert], Gursky-Malchiodi [@Gursky-Malchiodi], Hang-Yang [@Hang-Yang], Gursky-Hang-Lin [@Gursky-Hang-Lin]. Recently, based on a nice maximum principle, Gursky and Malchiodi proved that
\[thma\] (Gursky-Malchiodi [@Gursky-Malchiodi]) For a closed Riemannian manifold $(M^n, g)$ of dimension $n\geq 5$, if $R_g\geq 0$ and $Q_g\geq 0$ on $M$ with $Q_g$ not identically zero, then there is a conformal metric $h=u^{\frac{4}{n-4}}g$ with positive scalar curvature and constant $Q$-curvature $Q_h=\bar{Q}$.
Moreover, they showed positivity of the Green’s function of the Paneitz operator. Also, for $n=5,\,6,\,7$, they proved a version of positive mass theorem( see Theorem \[thm1\]), which implies possibility to show compactness of the set of positive solutions to the prescribed constant $Q$-curvature problem in $C^{4, \alpha}(M)$ with $0< \alpha <1$.
For compactness results of solutions to the prescribed constant $Q$-curvature equation with different conditions, one would like to see Djadli-Hebey-Ledoux [@Djadli-Hebey-Ledoux], Hebey-Robert [@Hebey-Robert], Humbert-Raulot [@Humbert-Raulot], Qing-Raske [@Qing-Raske]. In Djadli-Hebey-Ledoux [@Djadli-Hebey-Ledoux], the authors studied the optimal Sobolev constant in the embedding $W^{2,2}\hookrightarrow L^{\frac{2n}{n-4}}$ where $P_g$ has constant coefficients. With some additional assumptions, they studied compactness of solutions to the related equations under $W^{2,2}$ bound and obtained existence of positive solutions for the corresponding equations. Under the assumption that the Paneitz operator and positive Green’s function, Hebey-Robert [@Hebey-Robert] considers compactness of positive solutions with $W^{2,2}$ bound in locally conformally flat manifolds with positive scalar curvature. They showed when the Green’s function satisfies a positive mass theorem, the conclusion holds. Later, Humbert-Raulot [@Humbert-Raulot] showed that the positive mass theorem holds automatically under the assumption in Hebey-Robert [@Hebey-Robert]. In Qing-Raske [@Qing-Raske], with the use of the developing map and moving plane method, they showed $L^{\infty}$ bound of solutions to the prescribed constant $Q$-curvature equation, for locally conformally flat manifolds with positive scalar curvature without additional assumptions. Combining Qing-Raske’s result with positivity of Green’s function, one can also easily get the full compactness result, see Theorem \[thmconformally\_flat\].
In this notes we want to study compactness of solutions to the prescribed constant $Q$-curvature equation in Theorem \[thma\], following Schoen’s outline of proof of compactness of solutions to the prescribed scalar curvature problem. For compactness results of solutions to the prescribed scalar curvature problem, following Schoen’s original outline, one can see Schoen ([@Schoen], [@Schoen1], [@Schoen-Zhang]), Li-Zhu [@Li-Zhu], Druet [@Druet], Chen-Lin [@Chen-Lin], Li-Zhang ([@Li-Zhang], [@Li-Zhang1]), Marques [@Marques], Khuri-Marques-Schoen [@Khuri-Marques-Schoen]. For non-compactness results, see Brendle [@Brendle], Brendle-Marques [@Brendle-Marques] and Wei-Zhao [@Wei-Zhao]. For compactness argument for the Nirenberg problem for a more general type conformal equation on the round sphere, see Jin-Li-Xiong [@Jin-Li-Xiong]. More precisely, we will follow the approach in Li-Zhu [@Li-Zhu] for solutions to constant $Q$-curvature problem in dimension $n=5$ under Gursky-Malchiodi’s setting. For $n=6, \,7$, we give some estimates along this direction.
Our main theorem is
\[thm27\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n=5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p)>0$ for some point $p\in M$. Assume that $(M, g)$ is not conformal equivalent to the round sphere. Then there exists $C>0$ depending on $M$ and $g$ such that for any positive solution to $(\ref{equation1})$, $$\begin{aligned}
C^{-1}\leq u \leq C,\end{aligned}$$ and for any $0<\alpha <1$, there exists $C'>0$ depending on $M$, $g$, $\alpha$ and $n$ such that $$\begin{aligned}
\|u\|_{C^{4, \alpha}}\leq C'.\end{aligned}$$
We will perform a contradiction argument between local information from a Pohozaev type identity relating to the constant $Q$-curvature equation and a global discussion provided by the positive mass theorem in Gursky-Malchiodi [@Gursky-Malchiodi]( see Theorem \[thm1\]). In comparison, for compactness of Yamabe problem, the application of positive mass theorem by Schoen and Yau [@Schoen-Yau] is crucial.
We give a direct modification of the maximum principle in Gursky-Malchiodi [@Gursky-Malchiodi] for manifolds with boundary, see Lemma \[lemmaximumprinciple\]. It turns out to be very useful in the proof of lower bound of the solutions away from the isolated blowup points( see Theorem \[thmlowerbound\]) and it plays a role of local maximum principle in estimating upper bounds of solutions near blowup points( see Lemma \[lemupperboundestimates1\]). To show upper bound of the solutions, we give upper bound estimates of a sequence of blowup solutions near isolated simple blowup points as in Li-Zhu [@Li-Zhu], see in Section \[section6\]. We are able to prove a Harnack type inequality near the isolated blowup points for $5\leq n \leq 9$, see Lemma \[lemH\]. Besides the prescribed $Q$-curvature equation, nonnegativity of scalar curvature is also important in the analysis of the limit space of blowing-up argument. With the aid of the Pohozaev type identity, we then show that in dimension $n=5$, each isolated blowup point is in fact an isolated simple blowup point. After that, proof of Theorem \[thm27\] is standard, except that more is involved for the blowing up limit in ruling out the bubble accumulations, see Proposition \[propdistanceofsingularities\].
In Marques [@Marques] and Li-Zhang [@Li-Zhang], by using a classification theorem by Chen-Lin [@Chen-Lin] of solutions to the linearized equation of the constant scalar curvature equation on $\mathbb{R}^n$ vanishing at infinity, better estimates are obtained for error terms in the Pohozaev type identity. If such a classification theorem still holds for linearized equation of the constant $Q$-curvature equation on $\mathbb{R}^n$, then argument in [@Marques] still works and Proposition \[propinequality\] still holds for $n=6$ and $n=7$, see in Remark \[remark6\]. We should remark that in these two dimensions, estimates on the Weyl tensor at the blowup points are not necessary. Once Proposition \[propinequality\] holds, Proposition \[propisolatedsingularpoints\] and Proposition \[propdistanceofsingularities\] hold for $n=6$ and $n=7$ automatically. That leads to the compactness result Theorem \[thm27\] for $n=6$ and $n=7$. But we are not able to show such a classification so far.
To end the introduction, we introduce definition of isolated blowup points and isolated simple blowup points.
Let $g_j$ be a sequence of Riemannian metric on a domain $\Omega \subseteq M$. Let $\{u_j\}_j$ be a sequence of positive solutions to $(\ref{equation1})$ under the background metric $g_j$ in $\Omega$. We call a point $\bar{x}\in \Omega$ an [*isolated blowup point*]{} of $\{u_j\}$ if there exist $\bar{C}>0$, $0< \delta < dist_{g_j}(\bar{x}, \partial \Omega)$ and $x_j \to \bar{x}$ as a local maximum of $u_j$ with $u_j(x_j)\to \infty$ satisfying $$\begin{aligned}
&B^{g_j}_{\delta}(\bar{x}) \subseteq \Omega,\\
&\label{boundground}u_j(x)\leq \bar{C} d_{g_j}(x, x_j)^{\frac{4-n}{2}},\,\,\,\,\text{for}\,\,d_{g_j}(x, x_j)\leq \delta,\end{aligned}$$ where $B^{g_j}_{\delta}$ is the $\delta$-geodesic ball with respect to the metric $g_j$, and $d_{g_j}(x, x_j)$ is the geodesic distance between $x$ and $x_j$ with respect to the metric $g_j$.
For an isolated blowup point $x_j\to \bar{x}$ of $u_j$, we define $$\begin{aligned}
\bar{u}_j(r)=\frac{1}{|\partial B^{g_j}_r(x_j)|}\int_{\partial B^{g_j}_r(x_j)}u_j ds_{g_j},\,\,0<r<\delta,\end{aligned}$$ and $$\begin{aligned}
\hat{u}_j(r)=r^{\frac{n-4}{2}}\bar{u}_j(r),\,\,0<r<\delta,\end{aligned}$$ with $B^{g_j}_r(x_j)$ that $r$-geodesic ball centered at $x_j$, $ds_{g_j}$ the area element and $|\partial B^{g_j}_r(x_j)|$ volume of $B^{g_j}_r(x_j)$.
\[def02\] We call $x_j\to \bar{x}$ an isolated simple blowup point if it is an isolated blowup point and there exists $0<\delta_1<\delta$ independent of $j$ such that $\hat{u}_j$ has precisely one critical point in $(0, \delta_1)$, for $j$ large.
[**Acknowledgements.**]{} The author would like to thank Doctor Jingang Xiong and Professor Lei Zhang for helpful discussion when he read [@Li-Zhang] for other purpose. The author is grateful to Professor Chiun-Chuan Chen for helpful discussion for understanding [@Chen-Lin].
The Green’s representation
==========================
In this section, we assume that $(M^n, g)$ is a closed Riemannian manifold of dimension $n\geq 5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p)>0$ for some point $p\in M$.
\[thm1\] (Gursky-Malchiodi, [@Gursky-Malchiodi]) For a closed Riemannian manifold $(M^n, g)$ of dimension $n\geq 5$, if $R_g\geq 0$, $Q_g\geq 0$ on $M$ and also $Q_g(p)>0$ for some point $p\in M$, then the following holds:
- The scalar curvature $R_g>0$ in $M$;
- the Paneitz operator $P_g$ is in fact positive and the Green’s function $G$ of $P_g$ is positive where $G: M\times M - \{(q, q), q\in M\}\,\to\,\mathbb{R}$. Also, if $u\in C^4(M)$ and $P_g u\geq 0$ on $M$, then either $u\equiv 0$ or $u>0$ on $M$;
- for any metric $g_1$ in the conformal class of $g$, if $Q_{g_1}\geq 0$, then $R_{g_1}>0$;
- for any distinct points $q_1, q_2 \in M$, $$\begin{aligned}
\label{expansion1}
G(q_1, q_2)=G(q_1,q_2)=c_nd_{g}(q_1,q_2)^{4-n}(1+f(q_1, q_2)),\end{aligned}$$ with $c_n=\frac{1}{(n-2)(n-4)\omega_{n-1}}$, $\omega_{n-1}=|S^{n-1}|$, and $d_g(q_1, q_2)$ distance between $q_1$ and $q_2$. Here $f$ is bounded and $f\to 0$ as $d_g(q_1, q_2)\to 0$ and $$\begin{aligned}
\label{ineqGrp}
|\nabla^jf|\leq C_jd_g(q_1, q_2)^{1-j}\end{aligned}$$ for $1 \leq j \leq 4$,
- (positive mass theorem) when the dimension $n= 5,\,6, $ or $7$, for any point $q_1\in M$, let $x=(x^1, ..., x^n)$ be the conformal normal coordinates ( see [@Lee-Parker]) centered at $q_1$ and $h$ be the corresponding conformal metric. For $q_2$ close to $q_1$ the Green’s function $G_h(q_2, q_1)$ of the Paneitz operator $P_h$ has the expansion $$\begin{aligned}
G_h(q_2, q_1)=c_nd_{h}(q_2,q_1)^{4-n}+\alpha + f(q_2)
\end{aligned}$$ with a constant $\alpha \geq 0$ and $f$ satisfying $(\ref{ineqGrp})$ and $f(q_2)\to 0$ as $q_2\to q_1$; moreover, $\alpha=0$ if and only if $(M^n, g)$ is conformally equivalent to the round sphere.
Let $u\in C^{4, \alpha}(M)$ be a solution to the equation $$\begin{aligned}
P_gu=f\geq 0.\end{aligned}$$ Then we have the Green’s representation $$\begin{aligned}
u(x)=\int_M G(x, y)f(y)dV_g(y),\end{aligned}$$ for $x\in M$.
Now let $u>0$ be a solution to the constant $Q$-curvature equation $(\ref{equation1})$. Using the Green’s representation, $$\begin{aligned}
u(x)=\frac{n-4}{2}\bar{Q} \int_M G(x, y)\,u^{\frac{n+4}{n-4}}(y)\,d V_g(y),\end{aligned}$$ we first show some basic estimates of the solution $u$.
\[lembounda\] For a closed Riemannian manifold $(M^n, g)$ of dimension $n\geq 5$ with $R_g>0$, $Q_g\geq 0$ on $M$ and $Q_g(p)>0$ for some point $p\in M$. Then there exists $C_1,\,C_2>0$ depending on $(M, g)$, so that for any solution $u$ to $(\ref{equation1})$, we have that $$\begin{aligned}
\inf_M u \leq C_1,\,\,\sup_M u \geq C_2.\end{aligned}$$
Let $u(q)=\inf_M u$. Then by Green’s representation, $$\begin{aligned}
u(q)&=\frac{(n-4)}{2}\bar{Q}\int_M\,G(q,y)\,u(y)^{\frac{n+4}{n-4}}\,dV_g(y)\\
&\geq u(q)^{\frac{n+4}{n-4}}*\frac{(n-4)}{2}\bar{Q}\int_M\,G(q,y)\,dV_g(y)\\
&\geq C_1^{-\frac{8}{n-4}} u(q)^{\frac{n+4}{n-4}}\end{aligned}$$ with $C_1$ independent of the solution $u$ and $q$, and the last inequality follows from $(\ref{expansion1})$. Therefore, the upper bound of $\inf_M u$ is established. Similar argument leads to lower bound of $\sup_M u$.
Next we give an integral type inequality, which shows that if $u$ is bounded from above, then we get lower bound of $u$.
\[lemlowerbound1\] For a closed Riemannian manifold $(M^n, g)$ with dimension $n\geq 5$, $R_g>0$, and also $Q_g\geq 0$ with $Q_g(p)>0$ for some point $p\in M$. Then we have the inequality $$\begin{aligned}
\inf_M u \geq C (\int_M G(z, y)^p\,u(y)^{\frac{8}{n-4}\alpha p}\, d V_g(y))^{-\frac{q}{p}}\end{aligned}$$ where $p=\frac{n+4}{n-4} - a$, $\frac{1}{p}+\frac{1}{q}=1$, and $\alpha=\frac{(n-4)a}{8p}$, for any fixed number $\frac{4}{n-4}< a <\frac{8}{n-4}$, and $z$ is the maximum point of $u$. $C=C(a, g)>0$ is a constant. In particular, uniform upper bound of $u$ implies uniform lower bound of $u$.
Let $u(x)=\inf_M u$ and $u(z)=\sup_M u$.
By the expansion formula $(\ref{expansion1})$, there exist two constants $C_3, C_4>0$ so that $$\begin{aligned}
\label{polebound}
0< C_3 < \frac{1}{C_4}d_g(z_1,z_2)^{4-n} \leq G(z_1, z_2) \leq C_4 d_g(z_1, z_2)^{4-n},\end{aligned}$$ for any two distinct points $z_1, z_2 \in M$.
By Green’s representation at the maximum point $z$, $$\begin{aligned}
u(z)&= \frac{(n-4)}{2}\bar{Q}\int_M G(z, y)\,u(y)^{\frac{n+4}{n-4}}\, d V_g(y)\\
&\leq \frac{(n-4)}{2}\bar{Q}u(z)\int_M G(z, y)\,u(y)^{\frac{8}{n-4}}\, d V_g(y)\end{aligned}$$ so that $$\begin{aligned}
1&\leq \frac{(n-4)}{2}\bar{Q}\int_M G(z, y)\,u(y)^{\frac{8}{n-4}(\alpha+(1-\alpha))}\, d V_g(y)\\
&\leq \frac{(n-4)}{2}\bar{Q}(\int_M G(z, y)^p\,u(y)^{\frac{8}{n-4}\alpha p}\, d V_g(y))^{\frac{1}{p}}\,(\int_Mu(y)^{\frac{8}{(n-4)}(1-\alpha)q}\,dv_g(y))^{\frac{1}{q}}\\
&= \frac{(n-4)}{2}\bar{Q}(\int_M G(z, y)^p\,u(y)^{\frac{8}{n-4}\alpha p}\, d V_g(y))^{\frac{1}{p}}\,(\int_Mu(y)^{\frac{n+4}{n-4}}\,dv_g(y))^{\frac{1}{q}},\end{aligned}$$ with $\alpha$, $p$, $q$ chosen as in the lemma. Here the second inequality is by H$\ddot{\text{o}}$lder’s inequality. The range of $a$ in the lemma keeps $0<\alpha<1$, $p>1$ and $q>1$, and also $p(4-n)>-n$ so that $G^p$ is integrable.
Therefore, combining with $(\ref{polebound})$ we have $$\begin{aligned}
\inf_M u &= u(x)=\frac{n-4}{2}\bar{Q}\int_M G(x, y) u(y)^{\frac{n+4}{n-4}}\,d V_g(y)\\
&\geq C' \int_M u(y)^{\frac{n+4}{n-4}}\,d V_g(y)\\
&\geq C (\int_M G(z, y)^p\,u(y)^{\frac{8}{n-4}\alpha p}\, d V_g(y))^{-\frac{q}{p}},\end{aligned}$$ where $C', C>0$ are uniform constants independent of $u$, $z$ and $x$.
Locally conformally flat manifolds
==================================
In Qing-Raske [@Qing-Raske], for locally conformally flat manifolds, upper bound for positive solutions to $(\ref{equation1})$ is given:
\[thmqr\] (Theorem 1.3 in [@Qing-Raske]) Let $(M^n g)$ be a closed locally conformally flat manifold of dimension $n \geq 5$ with positive Yamabe constant. Assume $(M, g)$ is not conformally equivalent to the round sphere. Then there exists $C>0$ so that for any positive function $u$ if the metric $g_1=u^{\frac{4}{n-4}}g$ is of positive scalar curvature and constant $Q$-curvature $1$, then $u\leq C$.
For estimate of lower bound of $u$, they need assumption on the so called Poincar$\acute{\text{e}}$ exponent. Now for our problem, since $R_g>0$, the Yamabe constant is positive. The above theorem applies. Combining with Lemma \[lemlowerbound1\], we obtain that
\[thmconformally\_flat\] Let $(M^n, g)$ be a closed locally conformally flat Riemannian manifold of dimension $n\geq 5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p)>0$ for some point $p\in M$, then there exists $C>0$ and $C'=C'(\alpha)$ for any $0<\alpha<1$, so that for any solution $u$ of $(\ref{equation1})$, $$\begin{aligned}
&\label{boundflatness} \frac{1}{C}< u < C,\\
&\label{boundflatness1} |u|_{C^{4, \alpha}}\leq C'.\end{aligned}$$
For $Q_{u^{\frac{4}{n-4}}g}=\bar{Q}$, by Theorem \[thm1\] the scalar curvature $R_{u^{\frac{4}{n-4}}g}>0$. The estimate $(\ref{boundflatness})$ follows from Theorem \[thmqr\] and Lemma \[lemlowerbound1\]. To establish $(\ref{boundflatness1})$, one can either use ellipticity of the equation $(\ref{equation1})$ with a bootstrapping argument or take derivatives on the Green’s representation and use $(\ref{expansion1})$. This completes the proof of Theorem \[thmconformally\_flat\].
A maximum principle
===================
In this section we give a maximum principle for smooth domains with boundary in the manifold $(M, g)$ defined in Lemma \[lembounda\], which is a modification of the maximum principle given by Gursky and Malchiodi, see Lemma \[lemmaximumprinciple\]. As an application, we give a lower bound estimate of the blowing up sequence.
\[lemboundScurvature\] Let $(\bar{\Omega}, g)$ be a compact Riemannian manifold with boundary $\partial \Omega$ of dimension $n\geq 5$. Let $\Omega$ be the interior of $\bar{\Omega}$. Assume the scalar curvature $R_g\geq 0$ in $\bar{\Omega}$ and $R_g>0$ at points on the boundary, and also $Q_g\geq 0$ in $\bar{\Omega}$. Then $R_g>0$ in $\bar{\Omega}$.
The proof is the same as that for closed manifolds. The $Q$- curvature is expressed as $$\begin{aligned}
Q_g=-\frac{1}{2(n-1)}\Delta_gR_g+c_1(n)R_g^2-c_2(n)|Ric|_g^2\end{aligned}$$ with $c_1(n), c_2(n)$ positive. By the non-negativity of $Q_g$, $$\begin{aligned}
\frac{1}{2(n-1)}\Delta_gR_g\leq c_1(n)R_g^2.\end{aligned}$$ By strong maximum principle and the boundary condition, $R_g>0$ in $\bar{\Omega}$.
\[lemmaximumprinciple\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n\geq 5$ with $R_g\geq 0$, and $Q_g\geq 0$. Let $\Omega \subseteq M$ be an open domain with smooth boundary $\partial \Omega$ so that $\bar{\Omega}=\Omega \bigcup \partial \Omega$. Assume that $u\in C^4(\bar{\Omega})$ satisfies that $$\begin{aligned}
P_gu\geq 0\,\,\text{in}\,\Omega,\end{aligned}$$ with $u>0$ on $\partial \Omega$. Let $\tilde{g} =u^{\frac{4}{n-4}}g$ be the conformal metric in a neighborhood $\mathcal {U}$ of $\partial \Omega$ where $u>0$. If the scalar curvature of $(\mathcal {U}, \tilde{g})$ satisfies $R_{\tilde{g}}(p)>0$ for all points $p \in \partial\Omega$, then $u>0$ in $\Omega$.
Our conditions on the boundary guarantee that all the argument is focused on the interior and then the argument is the same as in the proof of the maximum principle by Gursky and Malchiodi. For completeness, we present the proof.
We define the function $$\begin{aligned}
u_{\lambda}=(1-\lambda)+\lambda u\end{aligned}$$ for $\lambda\in [0, 1]$, so that $u_0=1$ and $u_1=u$. We assume that $$\begin{aligned}
\min_{\overline{\Omega}} u\leq 0.\end{aligned}$$ Then there exists $\lambda_0\in (0, 1]$ so that $$\begin{aligned}
\lambda_0=\inf\{\lambda \in(0, 1],\,\inf_{\overline{\Omega}} u_{\lambda}=0\}.\end{aligned}$$ By definition, for $0<\lambda<\lambda_0$, $u_{\lambda}>0$. For the metric $$\begin{aligned}
g_{\lambda}=u_{\lambda}^{\frac{4}{n-4}}g,\end{aligned}$$ the $Q$-curvature satisfies $$\begin{aligned}
Q_{g_{\lambda}}\geq 0\,\,\text{in}\,\Omega,\end{aligned}$$ for $0<\lambda<\lambda_0$. That follows from the conformal transformation formula $$\begin{aligned}
Q_{g_{\lambda}}&=\frac{2}{n-4}u_{\lambda}^{-\frac{(n+4)}{(n-4)}}P_gu_{\lambda}\\
&=\frac{2}{n-4}u_{\lambda}^{-\frac{(n+4)}{(n-4)}}((1-\lambda)P_g(1)+\lambda P_g u)\\
&=\frac{2}{n-4}u_{\lambda}^{-\frac{(n+4)}{(n-4)}}((1-\lambda)\frac{(n-4)}{2}Q_g+\lambda P_g u)\\
&\geq (1-\lambda)Q_gu_{\lambda}^{-\frac{n+4}{n-4}}\geq 0.\end{aligned}$$
Under the conformal transformation, the scalar curvature of $g_{\lambda}$ satisfies $$\begin{aligned}
R_{g_{\lambda}}&=u_{\lambda}^{-\frac{n}{n-4}}\big(-\frac{4(n-1)}{(n-4)}\Delta_g u_{\lambda}- \frac{8(n-1)}{(n-4)^2}\frac{|\nabla_gu_{\lambda}|^2}{u_{\lambda}}+R_gu_{\lambda}\big)\\
&=u_{\lambda}^{-\frac{n}{n-4}}\big(-\frac{4(n-1)}{(n-4)}\lambda\Delta_g u- \frac{8(n-1)}{(n-4)^2}\frac{\lambda^2|\nabla_gu|^2}{(1-\lambda)+\lambda u}+R_gu_{\lambda}\big)\\
&\geq u_{\lambda}^{-\frac{n}{n-4}}\big(-\frac{4(n-1)}{(n-4)}\lambda\Delta_g u- \frac{8(n-1)}{(n-4)^2}\frac{\lambda|\nabla_gu|^2}{ u}+ \lambda R_gu\big)\\
&=\lambda\big(\frac{u}{u_{\lambda}}\big)^{\frac{n}{n-4}}R_{\tilde{g}}>0\end{aligned}$$ on $\partial \Omega$ for $0<\lambda< \lambda_0$. Then by Lemma \[lemboundScurvature\], $$\begin{aligned}
R_{g_{\lambda}}>0 \,\,\text{in}\,\Omega\end{aligned}$$ for $0< \lambda < \lambda_0$. Again by the conformal transformation formula of scalar curvature, $$\begin{aligned}
\Delta_g u_{\lambda}\leq \frac{(n-4)}{4(n-1)}R_gu_{\lambda}\,\,\,\text{in}\,\,\Omega.\end{aligned}$$ By taking limit $\lambda\nearrow \lambda_0$, this also holds at $\lambda=\lambda_0$. But $$\begin{aligned}
u_{\lambda}=(1-\lambda)+\lambda u>0\end{aligned}$$ on $\partial \Omega$ for $0\leq\lambda\leq 1$. By strong maximum principle, $u_{\lambda_0}>0$ in $\bar{\Omega}$, contradicting with choice of $\lambda_0$. Therefore, for all $0\leq \lambda \leq 1$, $$\begin{aligned}
u_{\lambda}>0\,\,\, \text{in}\,\, \Omega.\end{aligned}$$ In particular, $u>0$ in $\Omega$.
0.3cm
\[thmlowerbound\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n\geq 5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. There exists $C>0$ so that if there exists a sequence of positive solutions $\{u_j\}_{j=1}^{\infty}$ of $(\ref{equation1})$ so that $$\begin{aligned}
M_j=u_j(x_j)=\sup_{M}u_j \to \infty\end{aligned}$$ as $j\to \infty$, then $$\begin{aligned}
\label{ineqnblowuplowerbound}
u_j(p)\geq C M_j^{-1}d_g^{4-n}(p, x_j)\end{aligned}$$ for any $p\in M$ such that $d_g(p, x_j)\geq M_j^{-\frac{2}{n-4}}$.
To prove the theorem, we only need to show that there exists $C>0$ so that for any blowing up sequence, there exists a subsequence so that $(\ref{ineqnblowuplowerbound})$ holds.
Let $x=(x^1,...,x^n)$ be normal coordinates in a small geodesic ball centered at $x_j$ with radius $\delta>0$ and $x_j$ the origin. Let $y=M_j^{\frac{2}{n-4}}x$ and the metric $h_j$ be given by $(h_j)_{pq}(y)=g_{pq}(M_j^{-\frac{2}{n-4}} y)$. Let $$\begin{aligned}
v_j(y)=M_j^{-1}u_j(\exp_{x_j}(M_j^{-\frac{2}{n-4}}y))\,\,\,\text{for}\,\,|y|\leq \delta M_j^{\frac{2}{n-4}}.\end{aligned}$$ Then $$\begin{aligned}
&0<\, v_j(y)\,\leq v_j(0)=1,\\
&P_{h_j}v_j(y)=\frac{(n-4)}{2}\bar{Q}v_j(y)^{\frac{n+4}{n-4}}\,\,\,\,\text{for}\,\,|y|\leq\delta M_j^{\frac{2}{n-4}}.\end{aligned}$$ Here $h_j$ converges to Euclidean metric on $\mathbb{R}^n$ in $C^k$ norm for any $k\geq 0$. By ellipticity, we have, after passing to a subsequence( still denoted as $\{v_j\}$), $v_j\to v$ in $C_{loc}^4(\mathbb{R}^n)$ and $v$ satisfies $$\begin{aligned}
&0\leq v(y) \leq v(0)=1\,\,\,\,\text{in}\,\,\mathbb{R}^n,\\
&\Delta^2v(y)=\frac{(n-4)}{2}\bar{Q}v(y)^{\frac{n+4}{n-4}}\,\,\,\,\text{in}\,\,\mathbb{R}^n.\end{aligned}$$ Also, since $R_{h_j}>0$ and $R_{u_j^{\frac{4}{n-4}}g}>0$ on $M$, by conformal transformation formula of scalar curvature, $$\begin{aligned}
\Delta_{h_j}v_j\leq \frac{(n-4)}{4(n-1)}R_{h_j}v_j.\end{aligned}$$ Passing to the limit we have $$\begin{aligned}
\Delta v(y)\leq 0\,\,\,\,\text{in}\,\,\mathbb{R}^n.\end{aligned}$$ By strong maximum principle, since $v(0)=1$, we have that $v(y)>0$ in $\mathbb{R}^n$. Then by the classification theorem of C.S. Lin([@Lin]), we have that $$\begin{aligned}
v(y)= \big(\frac{1}{1+4^{-1}|y|^2}\big)^{\frac{n-4}{2}}\,\,\,\,\text{in}\,\,\mathbb{R}^n.\end{aligned}$$ We will abuse the notation $v(|y|)=v(y)$. Therefore, for fixed $R>0$, for $j$ large, $$\begin{aligned}
\frac{1}{2}\big(\frac{1}{1+4^{-1}R^2}\big)^{\frac{n-4}{2}}M_j\leq u_j(\exp_{x_j}(x))\leq M_j\,\,\,\,\text{for}\,\,|x|\leq R M_j^{-\frac{2}{n-4}}.\end{aligned}$$ For any $\epsilon>0$, there exists $j_0>0$ so that for $j>j_0$, $$\begin{aligned}
\|v_j-v\|_{C^4}\leq \epsilon\,\,\,\,\text{for}\,\,|y|\leq 2.\end{aligned}$$ We define $\phi_j: M-\{x_j\}\to \mathbb{R}$ as $$\begin{aligned}
\phi_j(p)=u_j(p)-\tau M_j^{-1}G_{x_j}(p)\end{aligned}$$ with $G_{x_j}(p)=G(x_j, p)$ Green’s function of Paneitz operator and $\tau>0$ a small constant to be chosen. We will use maximum principle to show that for $\epsilon, \tau>0$ small, $$\begin{aligned}
\phi_j>0\,\,\,\,\text{in} \,\,M-B_{M_j^{-\frac{2}{n-4}}}(x_j),\,\,\,\text{for}\,\,j>j_0.\end{aligned}$$ Here $B_{M_j^{-\frac{2}{n-4}}}(x_j)$ denote the geodesic $M_j^{-\frac{2}{n-4}}$-ball centered at $x_j$ in $(M, g)$. If this holds, we will choose $\{u_j\}_{j>j_0}$ as the subsequence and the theorem is proved.
It is clear that $$\begin{aligned}
P_g\phi_j=P_gu_j=\frac{n-4}{2}\bar{Q}u_j^{\frac{n+4}{n-4}}>0\,\,\,\text{in} \,\,M-B_{M_j^{-\frac{2}{n-4}}}(x_j).\end{aligned}$$ To apply the maximum principle, we only need to verify sign of $\phi_j$ and related scalar curvature on $\partial B_{M_j^{-\frac{2}{n-4}}}(x_j)$.
First, for $|x|= M_j^{-\frac{2}{n-4}}$, we choose $\epsilon$ small so that for $j>j_0$ $$\begin{aligned}
u_j(\exp_{x_j}(x))=M_jv_j(M_j^{\frac{2}{n-4}}x)\geq \frac{1}{2}v(1)M_j;\end{aligned}$$ while by $(\ref{polebound})$, $$\begin{aligned}
M_j^{-1}G_{x_j}(\exp_{x_j}(x))\leq C_4 M_j.\end{aligned}$$ We take $\tau< \frac{v(1)}{4C_4}$. Then $$\begin{aligned}
\phi_j>0\,\,\,\text{on} \,\,\partial B_{M_j^{-\frac{2}{n-4}}}(x_j),\,\,\,\text{for}\,\,j>j_0.\end{aligned}$$ Now let $\tilde{g}_j=\phi_j^{\frac{4}{n-4}}g_j$ in small neighborhood of $\partial B_{M_j^{-\frac{2}{n-4}}}(x_j)$ where $\phi_j>0$. By conformal transformation, $$\begin{aligned}
R_{\tilde{g}_j}&=\phi_j^{-\frac{n}{n-4}}\big(-\frac{4(n-1)}{(n-4)}\Delta_g \phi_j- \frac{8(n-1)}{(n-4)^2}\frac{|\nabla_g\phi_j|^2}{\phi_j}+R_g\phi_j\big).\end{aligned}$$ Note that $R_g\phi_j>0$ on $\partial B_{M_j^{-\frac{2}{n-4}}}(x_j)$. We only need to show that $$\begin{aligned}
\label{positiveScurvature}
-\frac{4(n-1)}{(n-4)}(\Delta_g \phi_j+ \frac{2}{(n-4)}\frac{|\nabla_g\phi_j|^2}{\phi_j})>0 \,\,\,\text{on} \,\,\partial B_{M_j^{-\frac{2}{n-4}}}(x_j),\,\,\,\text{for}\,\,j>j_0.\end{aligned}$$ Recall that $$\begin{aligned}
(\Delta_g u_j+ \frac{2}{(n-4)}\frac{|\nabla_gu_j|^2}{u_j})=M_j^{1+\frac{4}{n-4}}(\Delta_{h_j}v_j+ \frac{2}{(n-4)}\frac{|\nabla_{h_j}v_j|^2}{v_j}).\end{aligned}$$ Also, $$\begin{aligned}
(\Delta_{h_j}v_j+ \frac{2}{(n-4)}\frac{|\nabla_{h_j}v_j|^2}{v_j})&\to (\Delta v+ \frac{2}{(n-4)}\frac{|\nabla v |^2}{v})\\
&=2(4-n)(|y|^2+4)^{-\frac{n}{2}}(|y|^2+2n)\,\,+\frac{2}{n-4}\frac{(4-n)^2(|y|^2+4)^{2-n}|y|^2}{(|y|^2+4)^{\frac{4-n}{2}}}\\
&=2(4-n)(|y|^2+4)^{-\frac{n}{2}}(|y|^2+2n)\,\,+\,2(n-4)(|y|^2+4)^{-\frac{n}{2}}|y|^2\\
&=4n(4-n)(|y|^2+4)^{-\frac{n}{2}}<0\,\,\,\,\text{at}\,\,|y|=1.\end{aligned}$$ Then we can choose $\epsilon< \frac{1}{100^n}|v|_{C^4(B_1(0))}$. Combining with the fact that $$\begin{aligned}
|D_g^kG_{p}(q)|\leq C_k d_g^{4-n-k}(p, q)\,\,\,\,\text{for}\,\,0\leq k\leq 4,\end{aligned}$$ for any distinct points $p, q \in M$ with constant $C_k>0$ independent of $p, q$, there exists $\tau>0$ only depending on $C_k$ and $\epsilon$ so that $$\begin{aligned}
&\tau M_j^{-1}|\Delta_g G_{x_j}(\exp_{x_j}(M_j^{-\frac{2}{n-4}}y))|\,<\,- M_j^{1+\frac{4}{n-4}}\frac{\Delta v}{4(2n+1)},\,\,\,\,\text{and}\\
&\frac{|\nabla_g\phi_j|^2}{\phi_j}\leq \frac{5}{4}M_j^{1+\frac{4}{n-4}}\frac{|\nabla v |^2}{v}\,\,\,\text{at}\,\,|y|=1,\,\,\,\,\text{for}\,\,j>j_0.\end{aligned}$$ Therefore, for $j>j_0$, $(\ref{positiveScurvature})$ holds, which implies that $$\begin{aligned}
R_{\tilde{g}_j}>0\,\,\text{on}\,\,\partial B_{M_j^{-\frac{2}{n-4}}}(x_j).
\end{aligned}$$ By Lemma \[lemmaximumprinciple\], $\phi_j> 0$ in $M - B_{M_j^{-\frac{2}{n-4}}}(x_j)$. Recall that $\epsilon$ and $\tau$ are chosen independent of choice of the sequence. This completes the proof of the theorem.
A Pohozaev type identity
========================
In this section we introduce a Pohozaev type identity related to the constant $Q$-curvature equation. It will provide local information of the solutions in later use.
Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n\geq 5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $u$ be a positive solutions to $(\ref{equation1})$. For any geodesic ball $\Omega=B_{\delta}(q)$ in $M$ with $2\delta$ less than injectivity radius of $(M, g)$, we let $x=(x^1,...,x^n)$ be geodesic normal coordinates centered at $q$ so that $g_{ij}(0)=\delta_{ij}$ and the Christoffel symbols $\Gamma_{ij}^k(0)=0$. In this section, the gradient $\nabla$, Laplacian $\Delta$, divergent $\text{div}$, volume element $dx$, area element $ds$, $\sigma$-ball $B_{\sigma}$ and $|x|^2=(x^1)^2+..+(x^n)^2$ are all with respect to the Euclidean metric. Define $$\begin{aligned}
&\mathcal {P}(u)=\int_{\Omega}(x\cdot \nabla u+ \frac{n-4}{2}u)\Delta^2u dx\\
&=\int_{\Omega} [\frac{n-4}{2}\text{div}(u\nabla(\Delta u)-\Delta u\nabla u)+\text{div}((x\cdot\nabla u)\nabla(\Delta u)-\nabla (x\cdot\nabla u)\Delta u + \frac{1}{2} (\Delta u)^2x)] dx\\
&=\int_{\partial \Omega} \,\frac{n-4}{2}(u\frac{\partial}{\partial \nu}(\Delta u)-\Delta u \frac{\partial}{\partial \nu}u)+ ((x\cdot\nabla u)\frac{\partial}{\partial \nu}(\Delta u)- \frac{\partial}{\partial \nu} (x\cdot\nabla u)\Delta u + \frac{1}{2} (\Delta u)^2x\cdot \nu) ds,\end{aligned}$$ where $\nu$ is the outer pointing normal vector of $\partial \Omega$ in Euclidean metric. Then using $(\ref{equation1})$ we have $$\begin{aligned}
\mathcal {P}(u)&=\int_{\Omega}(x\cdot \nabla u+ \frac{n-4}{2}u)(\Delta^2-P_g)u\,+\,(x\cdot \nabla u+ \frac{n-4}{2}u)\,P_gu\,dx\\
&=\int_{\Omega}(x\cdot \nabla u+ \frac{n-4}{2}u)(\Delta^2-P_g)u\,+\,\frac{n-4}{2}\bar{Q}\,(x\cdot \nabla u+ \frac{n-4}{2}u) u^{\frac{n+4}{n-4}} dx\\
&=\int_{\Omega}(x\cdot \nabla u+ \frac{n-4}{2}u)(\Delta^2-P_g)u\,+\,\frac{(n-4)^2}{4n}\bar{Q}\,\text{div}(u^{\frac{2n}{n-4}}x) dx\\
&=\int_{\Omega}(x\cdot \nabla u+ \frac{n-4}{2}u)(\Delta^2-P_g)u\,dx\,+\,\frac{(n-4)^2}{4n}\bar{Q}\int_{\partial \Omega}(x\cdot \nu)u^{\frac{2n}{n-4}}\, dx.\end{aligned}$$ Using the expression $(\ref{operatorP})$, we have $$\begin{aligned}
(\Delta^2-P_g)u=(\Delta^2-\Delta_g^2)u+\text{div}_g(a_nR_g g -b_nRic_g)\nabla_gu - \frac{n-4}{2}Q_gu.\end{aligned}$$ Since $\Gamma_{ij}^k(0)=0$ and $g_{ij}(0)=\delta_{ij}$, $$\begin{aligned}
(\Delta^2-\Delta_g^2)u&=(\delta^{pq}\delta^{ij}\nabla_p\nabla_q\nabla_i\nabla_j-g^{pq}g^{ij}\nabla^g_p\nabla^g_q\nabla^g_i\nabla^g_j)u\\
&=(\delta^{pq}\delta^{ij}- g^{pq}g^{ij})\nabla_p\nabla_q\nabla_i\nabla_ju+ O(|x|)|D^3u|+O(1)|D^2u|+O(1)|D u|\\
&=O(|x|^2)|D^4u|+O(|x|)|D^3u|+O(1)|D^2u|+O(1)|D u|.\end{aligned}$$ It follows that there exists $C>0$ depending on $|Rm_g|_{L^{\infty}(\Omega)}$, $|Q_g|_{C(\Omega)}$ and $|Ric_g|_{C^1(\Omega)}$ such that $$\begin{aligned}
\label{ineqbounderrorterms}
|(\Delta^2-P_g)u|\leq C(|x|^2|D^4u|+\,|x|\,|D^3u|+\,|D^2u|+\,|D u|+\,u).\end{aligned}$$
Upper bound estimates near isolated simple blowup points {#section6}
========================================================
In this section we perform a parallel approach of [@Li-Zhu] to show upper bound estimates of the solutions to $(\ref{equation1})$ near an isolated simple blowup point, see Proposition \[propupperbound\]. We start with a Hanark type inequality near an isolated blowup point.
\[lemH\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $5\leq n \leq 9$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $\{u_j\}$ be a sequence of positive solutions to $(\ref{equation1})$ and $x_j\to \bar{x}$ be an isolated blowup point. Then there exists a constant $C>0$ such that for any $0< r< \frac{\delta}{3}$, we have $$\begin{aligned}
\label{ineqH}
\max_{q\in B_{2r}(x_j)- B_{\frac{r}{2}}(x_j)}u_j(q) \leq C \min_{q\in B_{2r}(x_j)- B_{\frac{r}{2}}(x_j)}u_j(q).\end{aligned}$$
Let $x=(x^1,...,x^n)$ be geodesic normal coordinates centered at $x_j$. Here $\delta>0$ can be chosen small so that the coordinates exist. Let $y=r^{-1}x$. Define$$\begin{aligned}
v_j(y)=r^{\frac{n-4}{2}}u_j(\exp_{x_j}(ry))\,\,\,\text{for}\,\,|y| < 3.\end{aligned}$$ Then $$\begin{aligned}
&v_j(y)\leq \bar{C}|y|^{-\frac{n-4}{2}}\,\,\,\text{for}\,\,|y|<3,\\
&v_j(y)\leq 3^{\frac{n-4}{2}}\bar{C}\,\,\,\,\,\,\,\,\text{for}\,\,\frac{1}{3}<|y|<3.\end{aligned}$$ We denote $$\begin{aligned}
\Omega_r=B_{3r}(x_j)- B_{\frac{r}{3}}(x_j).\end{aligned}$$ By Green’s representation, $$\begin{aligned}
v_j(y)&=r^{\frac{n-4}{2}} u_j(exp_{x_j}(ry))=r^{\frac{n-4}{2}}\int_M G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)\\
&=r^{\frac{n-4}{2}}\int_{\Omega_r}G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)+\,r^{\frac{n-4}{2}}\,\int_{M-\Omega_r} G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)\end{aligned}$$ We [**claim**]{} that for $\frac{5}{12}\leq |y|\leq \frac{12}{5}$, if $$\begin{aligned}
\label{halfbounds}
v_j(y)\geq 2 r^{\frac{n-4}{2}}\int_{\Omega_r}G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q),\end{aligned}$$ then there exists $C>0$ independent of $j$, $x_j$, $r$ and $y$, such that for any $\frac{5}{12}\leq |z|\leq \frac{12}{5}$, $$\begin{aligned}
\label{inqH1}
v_j(z)\geq C v_j(y).\end{aligned}$$ In fact, by $(\ref{polebound})$, there exists $C>0$, such that $$\begin{aligned}
G(\exp_{x_j}(ry), q)\leq C G(\exp_{x_j}(rz), q)\end{aligned}$$ for $q\in M - \Omega_r$. Therefore, $$\begin{aligned}
\frac{1}{2}v_j(y)&\leq r^{\frac{n-4}{2}}\int_{M-\Omega_r}G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)\\
&\leq C r^{\frac{n-4}{2}} \int_{M-\Omega_r}G(exp_{x_j}(rz), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)\\
&\leq C v_j(z).\end{aligned}$$ This proves the [**claim**]{}.0.2cm
We denote $$\begin{aligned}
\mathcal {C}=\{y\in \mathbb{R}^n,\,\frac{5}{12}\leq |y|\leq \frac{12}{5},\,\,\text{so that}\,\,(\ref{halfbounds})\,\,\text{fails for}\,\,y\}.\end{aligned}$$ We choose $\frac{5}{12}\leq |y|\leq \frac{12}{5}$ with $$\begin{aligned}
v_j(y)\geq\frac{1}{2} \sup_{\frac{5}{12}\leq|z|\leq \frac{12}{5}}v_j(z).
\end{aligned}$$ If $y\notin \mathcal {C}$, then using the claim, we are done. If $y\in \mathcal {C}$, we will prove that the Harnack inequality $(\ref{ineqH})$ still holds.
By H$\ddot{\text{o}}$lder’s inequality, $$\begin{aligned}
u_j(\exp_{x_j}(ry))&\leq 2 \int_{\Omega_r}G(exp_{x_j}(ry), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q)\\
&\leq 2(\int_{\Omega_r}G(exp_{x_j}(ry),q)^{\alpha}d V_g(q))^{\frac{1}{\alpha}}(\int_{\Omega_r}u_j(q)^{\frac{n+4}{n-4}\beta}d V_g(q))^{\frac{1}{\beta}}\\
&\leq C(\alpha) r^{4-n+\frac{n}{\alpha}}(\int_{\Omega_r}u_j(q)^{\frac{n+4}{n-4}\beta}d V_g(q))^{\frac{1}{\beta}}\\
&\leq C(\alpha) r^{4-n+\frac{n}{\alpha}}\,(\bar{C}3^{\frac{n-4}{2}}r^{\frac{4-n}{2}})^{\frac{n+4}{n-4}(1-\frac{1}{\beta})}(\int_{\Omega_r}u_j(q)^{\frac{n+4}{n-4}}d V_g(q))^{\frac{1}{\beta}}\\
&\leq C(\alpha) r^{4-n+\frac{n}{\alpha}}\,(\bar{C}3^{\frac{n-4}{2}}r^{\frac{4-n}{2}})^{\frac{n+4}{n-4}(1-\frac{1}{\beta})}(\int_{\Omega_r}C_4(4r)^{n-4}G(\exp_{x_j}(rz), q)u_j(q)^{\frac{n+4}{n-4}}d V_g(q))^{\frac{1}{\beta}}\\
&\leq C(\alpha)r^{4-n+\frac{n}{\alpha}} (\bar{C}3^{\frac{n-4}{2}}r^{\frac{4-n}{2}})^{\frac{n+4}{n-4}(1-\frac{1}{\beta})}r^{\frac{n-4}{\beta}}u_j(\exp_{x_j}(rz))^{\frac{1}{\beta}}\\
&=C(\alpha, \bar{C},n)r^{(2-\frac{n}{2})(1-\frac{1}{\beta})}u_j(\exp_{x_j}(rz))^{\frac{1}{\beta}}.\end{aligned}$$ for any $\frac{1}{3}\leq |z|\leq 3$, where $1 < \alpha < \frac{n}{n-4}$, $\frac{1}{\alpha}+\frac{1}{\beta}=1$ so that $\beta> \frac{n}{4}$. Here we have used $(\ref{boundground})$ and $(\ref{polebound})$.
Since $$\begin{aligned}
\frac{n+4}{n-4}\,>\,\frac{n}{4}\end{aligned}$$ for $5\leq n\leq 9$, we set $\beta=\frac{n+4}{n-4}$ and obtain $$\begin{aligned}
\label{ineqHinside}
u_j(\exp_{x_j}(rz))&\geq C(\bar{C},n)\,r^4\,u_j(\exp_{x_j}(ry))^{\frac{n+4}{n-4}}\\
&\geq C(\bar{C},n)\,r^4\,(2^{-1}u_j(q))^{\frac{n+4}{n-4}},\end{aligned}$$ for all $q\in \,B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)$ and $\frac{1}{2}\leq |z|\leq 2$, where $5\leq n\leq 9$.
For any $\frac{1}{2}\leq |z|\leq 2$, $$\begin{aligned}
\label{ineqgradients}
|\nabla_g u_j|(\exp_{x_j}(rz))&\leq \frac{n-4}{2}\bar{Q}\int_{ B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)}|\nabla_gG(\exp_{x_j}(rz), q)|\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q)\\
&+\,\frac{n-4}{2}\bar{Q}\int_{M - \big( B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)\big)}|\nabla_gG(\exp_{x_j}(rz), q)|\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q).\end{aligned}$$ Note that for $\frac{1}{2}\leq |z|\leq 2$, $$\begin{aligned}
\label{ineqHoutside}
u_j(\exp_{x_j}(rz))&\geq \,\frac{n-4}{2}\bar{Q}\int_{M - \big( B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)\big)}G(\exp_{x_j}(rz), q)\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q)\\
&\geq C r \int_{M - \big( B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)\big)}|\nabla_g \,G(\exp_{x_j}(rz), q)|\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q),\end{aligned}$$ for a uniform constant $C$ independent of $j$ and the choice of points, where for the last inequality we have used $(\ref{expansion1})$.
Combining $(\ref{ineqHinside})$, $(\ref{ineqHoutside})$ and $(\ref{ineqgradients})$, for $\frac{1}{2}\leq |z|\leq 2$ we have the gradient estimate $$\begin{aligned}
|\nabla_g \log(u_j(\exp_{x_j}(rz)))|&=\frac{|\nabla_g u_j(\exp_{x_j}(rz))|}{u_j(\exp_{x_j}(rz))}\\
&\leq\,\frac{1}{u_j(\exp_{x_j}(rz))}\,\frac{n-4}{2}\bar{Q}\int_{ B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)}|\nabla_gG(\exp_{x_j}(rz), q)|\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q)\\
&+\,\frac{1}{u_j(\exp_{x_j}(rz))}\,\frac{n-4}{2}\bar{Q}\int_{M - \big( B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)\big)}|\nabla_gG(\exp_{x_j}(rz), q)|\,u_j(q)^{\frac{n+4}{n-4}}dV_g(q)\\
&\leq\,\frac{n-4}{2}\bar{Q}\int_{ B_{\frac{12r}{5}}(x_j)- B_{\frac{5r}{12}}(x_j)}|\nabla_gG(\exp_{x_j}(rz), q)|\,C(\bar{C}, n)^{-1}r^{-4}2^{-\frac{n+4}{n-4}}dV_g(q)\\
&+\,C^{-1}r^{-1}\\
&\leq C(\bar{C}, n)(r^{3}r^{-4}+r^{-1})\\
&=\,C(\bar{C}, n)r^{-1},\end{aligned}$$ where $C(\bar{C}, n)$ is some uniform constant depending on $\bar{C}$, the manifold and $n$. For any two points $p,\,q\in B_{2r}(x_j)-B_{\frac{r}{2}}(x_j)$, by the gradient estimate, $$\begin{aligned}
\frac{u_j(p)}{u_j(q)}\leq e^{C(\bar{C}, n)r^{-1}\,d_g(p,q)}\leq e^{4nC(\bar{C}, n)}.\end{aligned}$$ This completes the proof of Harnack inequality.
Next we show that near an isolated blowup point, after rescaling the functions $u_j$ converge to the standard solution in $\mathbb{R}^n$.
\[lemlimitmodel\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $5\leq n \leq 9$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $\{u_j\}$ be a sequence of positive solutions to $(\ref{equation1})$ and $x_j\to \bar{x}$ be an isolated blowup point. Let $M_j=u_j(x_j)$. For any given $R_j\to +\infty$ and positive numbers $\epsilon_j\to 0$, after possibly passing to a subsequence $u_{k_j}$ and $x_{k_j}$( still denoted as $u_j$ and $x_j$), it holds that $$\begin{aligned}
\label{ineqmeasurelimit}
&\|M_j^{-1}u_j(\exp_{x_j}(M_j^{-\frac{2}{n-4}}y))\,-\,\big(1+4^{-1}|y|^2\big)^{-\frac{n-4}{2}}\|_{C^4(B_{2R_j})}\\
&+\|M_j^{-1}u_j(\exp_{x_j}(M_j^{-\frac{2}{n-4}}y))\,-\,\big(1+4^{-1}|y|^2\big)^{-\frac{n-4}{2}}\|_{H^4(B_{2R_j})}\,\leq \epsilon_j,\end{aligned}$$ and $$\begin{aligned}
\label{ineqrulerlimit}
\frac{R_j}{\log(M_j)}\to 0,\,\,\text{as}\,\,j\to\,\infty.\end{aligned}$$
The proof is almost the same as in [@Li-Zhu]. Let $x=(x^1,...,x^n)$ be geodesic normal coordinates centered at $x_j$, $y=r^{-1}x$ and the metric $h=r^{-2}g$ be the rescaled metric so that $(h_j)_{pq}(y)=(g_j)_{pq}(ry)$ in normal coordinates. Define $$\begin{aligned}
v_j(y)=M_j^{-1}u_j(\exp_{x_j}(M_j^{-\frac{2}{n-4}}y))\,\,\,\text{for}\,\,|y| < \delta\,M_j^{\frac{2}{n-4}}.\end{aligned}$$ Then $v_j$ satisfies $$\begin{aligned}
&P_{h_j}v_j(y)=\frac{n-4}{2}\bar{Q}v_j(y)^{\frac{n+4}{n-4}},\,\,\text{for}\,\,|y|\leq\,\delta M_j^{\frac{2}{n-4}},\\
&\label{maximumpoint}v_j(0)=1,\,\,\nabla_{h_j}v_j(0)=0,\\
&\label{ineqboundoutside}0<v_j(y)\leq \bar{C} |y|^{-\frac{n-4}{2}},\,\,\text{for}\,\,|y|\leq\,\delta M_j^{\frac{2}{n-4}}.\end{aligned}$$ We next show that $v_j$ is uniformly bounded. Since $R_{h_j}>0$ and $R_{u_j^{\frac{4}{n-4}}g}>0$ on $M$, by conformal transformation formula of scalar curvature, $$\begin{aligned}
\label{ineqmaxiprinc}
\Delta_{h_j}v_j\leq \frac{(n-4)}{4(n-1)}R_{h_j}v_j,\end{aligned}$$ where $R_{h_j}\to 0$ uniformly in $|y|\leq 2$ as $j\to \infty$. Then the function $\eta_j(y)= (1+|y|^2)^{-1}v_j(y)$ satisfies $$\begin{aligned}
\Delta_{h_j}\eta_j+ \sum_{k=1}^nb_k(y)\partial_k \eta_j(y)\leq 0,\end{aligned}$$ in $|y|\leq 2$ with some function $b_k(y)$. By maximum principle, $$\begin{aligned}
\label{ineqboundS}
\eta_j(0)\geq \inf_{|y|=r}\eta_j(y)\,\,\,\text{for}\,\,0<r\leq 1.\end{aligned}$$ By the Harnack inequality $(\ref{ineqH})$ in Lemma \[lemH\], $$\begin{aligned}
\label{ineqHv}
\max_{|y|=r}v_j(y) \leq C \min_{|y|=r} v_j(y)\,\,\,\text{for}\,\,0<r\leq 1,\end{aligned}$$ where $C$ is independent of $r$ and $j$. The inequalities $(\ref{ineqboundS})$ and $(\ref{ineqHv})$ immediately derive $$\begin{aligned}
\max_{|y|=r}v_j(y) \leq C \min_{|y|=r} v_j(y)\leq C v_j(0)=C\,\,\,\text{for}\,\,0<r \leq 1.\end{aligned}$$ Combining this with $(\ref{ineqboundoutside})$, we have for $|y|\leq\,\delta M_j^{\frac{2}{n-4}}$, $$\begin{aligned}
v_j(y)\leq C,\end{aligned}$$ with $C$ independent of $j$, $y$ and $r$.
Standard elliptic estimates of $v_j$ imply that, after possibly passing to a subsequence, $v_j\to v$ in $C^4_{loc}$ in $\mathbb{R}^n$ where by $(\ref{maximumpoint})$ and $(\ref{ineqmaxiprinc})$, $v$ satisfies $$\begin{aligned}
&\Delta^2v(y)=\frac{n-4}{2}\bar{Q}v^{\frac{n+4}{n-4}},\,\,y\in \mathbb{R}^n,\\
&v(0)=1,\,\,\nabla v(0)=0,\\
&\Delta v(y)\leq 0,\,\,y\in \mathbb{R}^n,\\
&v(y)\geq 0,\,\,y\in \mathbb{R}^n.\end{aligned}$$ By strong maximum principle, $v(y)>0$ in $\mathbb{R}^n$. Then the classification theorem in [@Lin] gives that $$\begin{aligned}
v(y)=(1+4^{-1}|y|^2)^{-\frac{n-4}{2}}.\end{aligned}$$ Then the lemma follows.
\[remark01\] From Lemma \[lemlimitmodel\], we can see that the proof of Theorem \[thmlowerbound\] still works at the isolated blowup point $x_j\to \bar{x}$. Therefore, there exists $C>0$ independent of $j>0$ so that for any isolated blowup point $x_j\to \bar{x}$, $$\begin{aligned}
u_j(q)\geq C u_j(x_j)^{-1}d_g^{4-n}(q, x_j)\end{aligned}$$ any $q\in M$ such that $d_g(q, x_j)\geq u_j(x_j)^{-\frac{2}{n-4}}$.
We now state the upper bound estimate of $u_j$ near the isolated simple blowup points.
\[propupperbound\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $5\leq n \leq 9$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $\{u_j\}$ be a sequence of positive solutions to $(\ref{equation1})$ and $x_j\to \bar{x}$ be an isolated simple blowup point. Let $\delta_1$ and $\bar{C}$ be the constants defined in Definition \[def02\] and $(\ref{boundground})$. Then there exists a constant $C$ depending only on $\delta_1$, $\bar{C}$, $\|R_g\|_{C^1(B_{\delta_1}(\bar{x}))}$ and $\|Q_g\|_{C^1(B_{\delta_1}(\bar{x}))}$ such that $$\begin{aligned}
\label{inequpperboundlevel}
u_j(p)\leq C u_j(x_j)^{-1}d_g(p, x_j)^{4-n},\,\,\text{for}\,\,d_g(p, x_j)\leq \frac{\delta_1}{2},\end{aligned}$$ for $\delta_1>0$ small. Moreover, up to a subsequence, $$\begin{aligned}
\label{inequpperboundbehavior}
u_j(x_j)u_j(p)\to a G(\bar{x}, p)+b(p)\,\,\text{in}\,\,C_{loc}^4(B_{\delta_1}(\bar{x})-\{\bar{x}\}),\end{aligned}$$ where $G$ is Green’s function of the Paneitz operator $P_g$, $a>0$ is a constant and $b(p)\in C^4(B_{\frac{\delta_1}{2}}(\bar{x}))$ satisfies $P_gb=0$ in $B_{\frac{\delta_1}{2}}(\bar{x})$.
The proof of the proposition follows after a series of lemmas.
We first give a rough estimate of upper bound of $u_j$ near the isolated simple blowup points.
\[lemupperboundestimates1\] Under the condition in Proposition \[propupperbound\], assume $R_j\to \infty$ and $0< \epsilon_j<e^{-R_j}$ satisfy $(\ref{ineqmeasurelimit})$ and $(\ref{ineqrulerlimit})$. Denote $M_j=u_j(x_j)$. Then for any small number $0<\sigma<\frac{1}{100}$, there exists $0<\delta_2<\delta_1$ and $C>0$ independent of $j$ such that $$\begin{aligned}
&\label{ineqlembound1}M_j^{\lambda}u_j(p)\leq C d_g(p, x_j)^{4-n+\sigma},\\
&M_j^{\lambda}|\nabla_g^k u_j(p)|\leq C d_g(p, x_j)^{4-n-k+\sigma},\end{aligned}$$ for any $p$ in $R_jM_j^{-\frac{2}{n-4}}\leq d_g(p, x_j)\leq \delta_2$ and $1\leq k\leq 4$, where $\lambda=1-\frac{2}{n-4}\sigma$.
The outline of the proof is from [@Li-Zhu], while the use of our maximum principle here is more subtle. Let $x=(x^1,...,x^n)$ be geodesic normal coordinates centered at $x_j$ for $d_g(p, x_j)\leq \delta$. Let $r=|x|$. For any $0<\delta_2<\delta_1$ to be chosen, let $$\begin{aligned}
\Omega_j=\{p\in M,\,\,R_jM_j^{-\frac{2}{n-4}}\leq d_g(p, x_j)\leq \delta_2\}.\end{aligned}$$ We want to use maximum principle to get the upper bound of $u_j$. Before construction of the barrier function on $\Omega_j$, we first go through some properties of $u_j$.
From Lemma \[lemlimitmodel\], we know that $$\begin{aligned}
\label{ineqoutsidelim}
u_j(p)\leq C R_j^{4-n}M_j,\,\,\text{for}\,\,d_g(p, x_j)=R_jM_j^{-\frac{2}{n-4}},\end{aligned}$$ and there exists a critical point $r_0$ of $\hat{u}_j(r)$ in $0< r < R_jM_j^{-\frac{2}{n-4}}$; moreover, for $r>r_0$, $\hat{u}_j(r)$ is decreasing. By the assumption that $\bar{x}$ is an isolated simple blowup point, $\hat{u}_j$ is strictly decreasing for $R_jM_j^{-\frac{2}{n-4}}< r < \delta_1$. Therefore, combining with the Harnack inequality $(\ref{ineqH})$, for $p \in \Omega_j$ we have $$\begin{aligned}
d_g(p, x_j)^{\frac{n-4}{2}}u_j(p)&\leq C \bar{u}_j(d_g(p, x_j))\\
&\leq C R_j^{\frac{n-4}{2}}M_j^{-1}\bar{u}_j(R_jM_j^{-\frac{2}{n-4}})\\
&\leq C R_j^{\frac{n-4}{2}}M_j^{-1} R_j^{4-n}M_j\\
&=C R_j^{-\frac{n-4}{2}}.\end{aligned}$$ This leads to $$\begin{aligned}
\label{ineqdecreasing}
u_j(p)^{\frac{8}{n-4}}\leq C R_j^{-4} d_g(p, x_j)^{-4},\,\,\text{for}\,\,R_jM_j^{-\frac{2}{n-4}}< r < \delta_1.\end{aligned}$$ We now define a linear elliptic operator on $\Omega_j$ $$\begin{aligned}
L_j\phi=P_g\phi - \frac{n-4}{2}\bar{Q}u_j^{\frac{8}{n-4}}\phi,\,\,\text{for}\,\,\phi\in C^4(\Omega_j).\end{aligned}$$ Therefore $$\begin{aligned}
L_ju_j=0,\,\,\text{in}\,\,\Omega_j.\end{aligned}$$ Set $$\begin{aligned}
\varphi(p)=B\bar{M}_j\delta_2^{\sigma}d_g(p, x_j)^{-\sigma}+A M_j^{-1+\frac{2}{n-4}\sigma}d_g(p, x_j)^{-n+4+\sigma},\,\,p\in \Omega_j,\end{aligned}$$ where $A, B>0$ are constant to be determined, $0< \sigma <\frac{1}{100}$ and $$\begin{aligned}
\bar{M}_j=\sup_{d_g(p, x_j)=\delta_2}u_j\,\leq \bar{C} \delta_2^{-\frac{n-4}{2}}.\end{aligned}$$ There exists $C>0$, for $m>0$, $1\leq k\leq 4$ and any $p\in M$ fixed and $q \in M$ so that $d_g(p, q)<\delta_2$ with $\delta_2$ less than the injectivity radius, $$\begin{aligned}
\label{ineqdistance}
|D_g^k d_g(p, q)^{-m}|\leq C m^k d_g(p, q)^{-m-k}.\end{aligned}$$ It is easy to check that there exists $\delta_2>0$ independent of $j$ so that in $\Omega_j$ $$\begin{aligned}
&|(P_g-\Delta_0^2)|x|^{-\sigma}|\leq 100^{-1}|P_g (|x|^{-\sigma})|,\\
&|(P_g-\Delta_0^2)|x|^{-n+4+\sigma}|\leq 100^{-1}|P_g(|x|^{-n+4+\sigma})|,\end{aligned}$$ where $|x|=d_g(p, x_j)$ and $\Delta_0$ is the Euclidean Laplacian in the normal coordinates. It is easy to check that for $0<m< n-4$ and $0< r < \delta_2$, $$\begin{aligned}
\label{ineqordermainterm}
&-\Delta_0 r^{-m}=-m(m+2-n)r^{-m-2}>0\\
&\Delta_0^2r^{-m}=m(m+2-n)(m+2)(m+4-n)r^{-m-4}>0.\end{aligned}$$ But for $p\in \Omega_j$, by $(\ref{ineqdecreasing})$ $$\begin{aligned}
\frac{n-4}{2}\bar{Q}u_j(p)^{\frac{8}{n-4}}r^{-m}\leq \frac{n-4}{2}\bar{Q}C R_j^{-4}r^{-m-4}\end{aligned}$$ Therefore, $$\begin{aligned}
L_j\varphi_j\geq 0\,\,\text{in}\,\,\Omega_j\end{aligned}$$ for $j$ large. By $(\ref{ineqoutsidelim})$, for $A>1$, $$\begin{aligned}
\label{ineqinnerboundary}
u_j(p)< \varphi_j(p),\,\,\text{for}\,\,d_g(p, x_j)=R_jM_j^{-\frac{2}{n-4}}.\end{aligned}$$ Also, for $B>1$, $$\begin{aligned}
\label{ineqouterboundary}
u_j(p)< \varphi_j(p),\,\,\text{for}\,\,d_g(p, x_j)=\delta_2.\end{aligned}$$ We now want to check sign of the scalar curvature $R_{(\varphi_j - u_j)^{\frac{4}{n-4}}g}$ near $\partial \Omega_j$. By conformal transformation formula, it has the same sign as $$\begin{aligned}
-\frac{4(n-1)}{(n-4)}\Delta_g (\varphi_j - u_j)- \frac{8(n-1)}{(n-4)^2}\frac{|\nabla_g(\varphi_j - u_j)|^2}{(\varphi_j - u_j)}+R_g(\varphi_j - u_j)\end{aligned}$$ Combining $(\ref{boundground})$ and standard interior estimate of $(\ref{equation1})$, we have for $k=1,\,2$, $$\begin{aligned}
\label{ineqboundary1}
|D_g^ku_j(p)|\leq C d_g(p, x_j)^{-\frac{n-4}{2}-k}\end{aligned}$$ for some constant independent of $j$, where $p\in \Omega_j$. It is easy to check that for $0<m<n-4$, $$\begin{aligned}
\label{ineqboundary2}
\Delta_0|x|^{-m}+\frac{2}{n-4} \frac{|\nabla_0 |x|^{-m}|^2}{|x|^{-m}}&=\big(m(m+2-n)\,+\,\frac{2m^2}{n-4})|x|^{-m-2}\\
&=\frac{m(n-2)(m-(n-4))}{n-4}|x|^{-m-2}<0.\end{aligned}$$ Also, note that for any positive functions $\phi_1,\,\phi_2\in C^2$, it holds that $$\begin{aligned}
\label{ineqboundary3}
\Delta_0(\phi_1+\phi_2)+\frac{2}{n-4} \frac{|\nabla_0 (\phi_1+\phi_2)|^2}{\phi_1+\phi_2}\leq (\Delta_0 \phi_1 +\frac{2}{n-4} \frac{|\nabla_0 (\phi_1)|^2}{\phi_1})+(\Delta_0 \phi_2 +\frac{2}{n-4} \frac{|\nabla_0 (\phi_2)|^2}{\phi_2}).\end{aligned}$$ Here we have used the fact that for $a,b,c,d>0$ $$\begin{aligned}
&\frac{2c\,d}{a+b}\leq \frac{b\,c^2}{a(a+b)}+\frac{a\,d^2}{b(a+b)},\,\,\text{so that}\\
&\frac{(c+d)^2}{a+b}=\frac{c^2+2c\,d+d^2}{a+b}\leq \frac{c^2}{a}+\frac{d^2}{b}.\end{aligned}$$ Using $(\ref{ineqinnerboundary})$-$(\ref{ineqboundary2})$ and $(\ref{ineqboundary3})$, we can choose $A, B>100^n(1+ C)$ independent of $j$ and $t$ with $C>0$ in $(\ref{ineqboundary1})$ so that $$\begin{aligned}
\label{ineqcurvatureboundary}
-\frac{4(n-1)}{(n-4)}\Delta_g (t\varphi_j - u_j)- \frac{8(n-1)}{(n-4)^2}\frac{|\nabla_g(t\varphi_j - u_j)|^2}{(t\varphi_j - u_j)}+R_g(t\varphi_j - u_j)>0\,\,\,\text{on}\,\,\partial \Omega_j,\end{aligned}$$ for all $t\geq 1$. For $t\geq 1$, we define $$\begin{aligned}
\phi_j^t(p)=t\varphi_j(p)-u_j(p),\,\,p\in \Omega_j.\end{aligned}$$ Then $$\begin{aligned}
\label{ineqlinearineqn}
0\leq L_j \phi_j^t=P_g\phi_j^t - \frac{n-4}{2}\bar{Q}\phi_j^t\,\,\,\,\text{in} \,\,\Omega_j.\end{aligned}$$ If $$\begin{aligned}
\label{inequpperbound1}
\phi_j^1=\varphi_j-u_j\geq 0\,\,\,\,\text{in}\,\,\Omega_j,\end{aligned}$$ then we are done. Else, since $\Omega_j$ is compact, we pick up the smallest number $t_j>1$ so that $\phi_j^{t_j}\geq 0$. Therefore, by $(\ref{ineqlinearineqn})$ $$\begin{aligned}
\label{ineqlinearineqn1}
P_g\phi_j^{t_j}\geq \frac{n-4}{2}\bar{Q}\phi_j^{t_j}\geq 0.
\end{aligned}$$ Combining with $(\ref{ineqinnerboundary})$, $(\ref{ineqouterboundary})$, $(\ref{ineqcurvatureboundary})$ and $(\ref{ineqlinearineqn1})$, the maximum principle in Lemma \[lemmaximumprinciple\] implies $$\begin{aligned}
\phi_j^{t_j}>0\,\,\text{in}\,\,\Omega_j,\end{aligned}$$ contradicting with the choice of $t_j$. Therefore, $(\ref{inequpperbound1})$ holds. Now for $p \in \Omega_j$, we use Lemma \[lemH\], monotonicity of $\hat{u}_j$, and apply $(\ref{inequpperbound1})$ at $p$ to obtain $$\begin{aligned}
\delta_2^{\frac{n-4}{2}}\bar{M}_j&\leq C \hat{u}_j(\delta_2)\leq C \hat{u}_j(d_g(p, x_j))\\
&\leq Cd_g(p, x_j)^{\frac{n-4}{2}}(B\bar{M}_j\delta_2^{\sigma}d_g(p, x_j)^{-\sigma}+A M_j^{-\lambda}d_g(p, x_j)^{4-n+\delta}).\end{aligned}$$ Here $\frac{n-4}{2}> \sigma$. We choose $p$ with $d_g(p, x_j)$ to be a small fixed number depending on $n, \sigma, \delta_2$ to obtain $$\begin{aligned}
\bar{M}_j\leq C(n, \sigma, \delta_2)M_j^{-\lambda}.\end{aligned}$$ Therefore, the inequality $(\ref{ineqlembound1})$ is then established from $(\ref{inequpperbound1})$, and based on standard interior estimates for derivatives of $u_j$, the lemma is proved.
\[lemupperboundsphere\] Under the assumption in Proposition \[propupperbound\], for any $0< \rho\leq \frac{\delta_2}{2}$ there exists a constant $C(\rho)>0$ such that $$\begin{aligned}
\limsup_{j\to \infty}\max_{p\in \partial B_{\rho}(x_j)} u_j(p) M_j \leq C(\rho).\end{aligned}$$ where $M_j=u_j(x_j)$.
By Lemma \[lemH\], it suffices to show the inequality for some fixed small constant $\rho>0$.
For any $p_{\rho}\in \partial B_{\rho}(x_j)$, we denote $\xi_j(p)=u_j(p_{\rho})^{-1}u_j(p)$. Then $\xi_j$ satisfies $$\begin{aligned}
P_g\xi_j(p)=\frac{n-4}{2}\bar{Q}u_j(p_{\rho})^{\frac{8}{n-4}}\xi_j(p)^{\frac{n+4}{n-4}}.\end{aligned}$$ For any compact subset $K \subseteq B_{\frac{\delta_2}{2}}(\bar{x})- \{\bar{x}\}$, there exists $C(K)>0$ such that for $j$ large $$\begin{aligned}
C(K)^{-1}\leq \xi_j\leq C(K)\,\,\text{in}\,\,K.\end{aligned}$$ Moreover, by Lemma \[lemH\], there exists $C>0$ independent of $0< r< \delta_2$ and $j$ so that $$\begin{aligned}
\label{ineqHsphere1}
\max_{B_r(x_j)- B_{\frac{r}{2}}(x_j)}u_j\leq C\inf_{B_r(x_j)- B_{\frac{r}{2}}(x_j)}u_j.\end{aligned}$$ By the estimates $(\ref{ineqlembound1})$, $u_j(p_{\rho})\to 0$ as $j\to \infty$. Therefore, by interior estimates of $\xi_j$ , up to a subsequence, $$\begin{aligned}
\xi_j\to \xi\,\,\text{in}\,\,C_{loc}^4(B_{\frac{\delta_2}{2}}(\bar{x})- \{\bar{x}\}),\end{aligned}$$ with $\xi>0$ such that $$\begin{aligned}
P_g\xi =0\,\,\text{in}\,\,B_{\frac{\delta_2}{2}}(\bar{x})- \{\bar{x}\},\end{aligned}$$ and $\xi$ satisfies $(\ref{ineqHsphere1})$ for $0<r<\frac{\delta_2}{2}$. Moreover, for $0<r<\rho$ and $\bar{\xi}(r)=|\partial B_r|^{-1}\int_{\partial B_r(\bar{x})}\xi ds_g$, $$\begin{aligned}
\lim_{j\to\infty}u_j(p_{\rho})^{-1}r^{\frac{n-4}{2}}\bar{u}_j(r)=r^{\frac{n-4}{2}}\bar{\xi}(r).\end{aligned}$$ Since $x_j\to \bar{x}$ is an isolated simple blowup point, $r^{\frac{n-4}{2}}\bar{\xi}(r)$ is non-increasing in $0<r<\rho$. Therefore, $\bar{x}$ is not a regular point of $\xi$.
Recall that $$\begin{aligned}
-\frac{4(n-1)}{n-2}\Delta_gu_j^{\frac{n-2}{n-4}}+R_gu_j^{\frac{n-2}{n-4}}=R_{u_j^{\frac{4}{n-4}}g}u_j^{\frac{n+2}{n-4}}\geq 0.\end{aligned}$$ Passing to the limit, we have $$\begin{aligned}
-\frac{4(n-1)}{n-2}\Delta_g\xi^{\frac{n-2}{n-4}}+R_g\xi^{\frac{n-2}{n-4}}\geq 0,\end{aligned}$$ in $B_{\frac{\delta_2}{2}}(\bar{x})- \{\bar{x}\}$.
For later use, if $Q_g$ is not pointwisely non-negative in $B_{\rho}(\bar{x})$, by Theorem \[thm1\] we choose $\tilde{g}=\phi^{-\frac{4}{n-4}}g$ so that $R_{\tilde{g}}>0$ and $Q_{\tilde{g}}\geq 0$ in $M$. Then $\tilde{\xi}=\phi \xi$ is still singular at $\bar{x}$ and all above information for the limit holds with $\xi$ and $g$ replaced by $\tilde{\xi}$ and $\tilde{g}$. So from now on we assume that $R_{g}>0$ and $Q_g\geq 0$.
From Corollary \[cor9.1\], for $\rho>0$ small, there exists $m>0$ independent of $j$ such that for $j$ large $$\begin{aligned}
\label{ineqint1}
\int_{B_{\rho}(x_j)}(P_g\xi_j-\frac{n-4}{2}Q_g\xi_j) d V_g&=\int_{\partial B_{\rho}(x_j)}\big(\frac{\partial}{\partial \nu}\Delta_g \xi_j-(a_nR_g\frac{\partial}{\partial \nu} \xi_j-b_nRic_g(\nabla_g \xi_j, \nu))\big) d s_g\\
&=\int_{\partial B_{\rho}(x_j)}\big(\frac{\partial}{\partial \nu}\Delta_g \xi-(a_nR_g\frac{\partial}{\partial \nu} \xi-b_nRic_g(\nabla_g \xi, \nu))\big) d s_g+o(1)>m.\end{aligned}$$ On the other hand, nonnegativity of $Q_g$ implies $$\begin{aligned}
\label{ineqint2}
\int_{B_{\rho}(x_j)}(P_g\xi_j-\frac{n-4}{2}Q_g\xi_j) d V_g&=\int_{B_{\rho}(x_j)}(\frac{n-4}{2}\bar{Q}u_j(p_{\rho})^{-1}u_j(p)^{\frac{n+4}{n-4}} - \frac{n-4}{2}Q_g\xi_j)\,d V_g\\
&\leq \frac{n-4}{2}\bar{Q}\int_{B_{\rho}(x_j)}u_j(p_{\rho})^{-1}u_j(p)^{\frac{n+4}{n-4}} \,d V_g.\end{aligned}$$ Using $(\ref{ineqmeasurelimit})$ and $\epsilon_j\leq e^{-R_j}$, we have $$\begin{aligned}
\int_{B_{R_jM_j^{-\frac{2}{n-4}}}(x_j)}u_j^{\frac{n+4}{n-4}}dV_g\leq C M_j^{-1},\end{aligned}$$ while by $(\ref{ineqlembound1})$ we have $$\begin{aligned}
\int_{B_{\rho}(x_j)-B_{R_jM_j^{-\frac{2}{n-4}}}(x_j)}u_j^{\frac{n+4}{n-4}} d V_g &\leq C \int_{B_{\rho}(x_j)-B_{R_jM_j^{-\frac{2}{n-4}}}(x_j)}(M_j^{-\lambda} d_g(p, x_j)^{4-n+\sigma})^{\frac{n+4}{n-4}}\\
&\leq C (R_jM_j^{-\frac{2}{n-4}})^{-4+\frac{n+4}{n-4}\sigma}M_j^{-\lambda\frac{n+4}{n-4}}\\
&=R_j^{-4+\frac{n+4}{n-4}\sigma}M_j^{-1}=o(1)M_j^{-1}.\end{aligned}$$ Therefore, $$\begin{aligned}
\label{ineqint3}
\int_{B_{\rho}(x_j)}u_j^{\frac{n+4}{n-4}} d V_g\leq C M_j^{-1}.\end{aligned}$$ Lemma \[lemupperboundsphere\] follows from the inequalities $(\ref{ineqint1})$-$(\ref{ineqint3})$.
[*Proof of Proposition \[propupperbound\].*]{}Suppose $(\ref{inequpperboundlevel})$ fails. Let $M_j=u_j(x_j)$. Then there exists a subsequence $u_j$ and $\{p_j\}$ with $d_g(p_j, x_j)\leq \frac{\delta_2}{2}$ with $\delta_2$ in Lemma \[lemupperboundestimates1\] such that $$\begin{aligned}
\label{ineqdivergence}
u_j(p_j)M_jd_g(p_j, x_j)^{n-4}\to \infty.\end{aligned}$$ If $Q_g\leq 0$ does not hold in $B_{\delta_2}(\bar{x})$, let $g_0=\phi^{-\frac{4}{n-4}}g$ be the metric so that $Q_{g_0}\geq 0$ and $R_{g_0}>0$ on $M$. Then $(\ref{ineqdivergence})$ holds for $g$ and $u_j$ replaced by $g_0$ and $\tilde{u}_j=\phi u_j$. Therefore, from now on we assume that $Q_g\geq 0$ and $R_g>0$ on $M$. By Lemma \[lemlimitmodel\] and $0<\epsilon_j\leq e^{-R_j}$, $$\begin{aligned}
R_jM_j^{-\frac{2}{n-4}}\leq d_g(p_j, x_j) \leq \frac{\delta_2}{2}.
\end{aligned}$$ Let $x=(x^1,...,x^n)$ be the geodesic normal coordinates centered at $x_j$. Denote $y=d_j^{-1}x$ where $d_j=d_g(p_j, x_j)$. We do rescaling $$\begin{aligned}
v_j(y)=d_j^{\frac{n-4}{2}}u_j(\exp_{x_j}(d_jy)),\,\,|y|\leq 2.
\end{aligned}$$ Then $v_j$ satisfies $$\begin{aligned}
P_{h_j}v_j(y)=\frac{n-4}{2}\bar{Q}v_j(y)^{\frac{n+4}{n-4}},\,\,|y|\leq 2,
\end{aligned}$$ where $h_j=d_j^{-2}g_j$ so that $(h_j)_{pq}(y)=(g)_{pq}(d_jy)$. The metrics $h_j$ depend on $j$. But since $d_j$ has uniform upper bound, the sequence of metrics stays in compact sets with strong norms and all the results in Lemma \[lemupperboundsphere\] hold uniformly for $j$. Also, the conclusion of Lemma \[lemupperboundestimates1\] is scaling invariant. Note that as the metrics $h_j$ converge to $h$, Green’s functions of Paneitz operators $P_{h_j}$ converge to Green’s functions of Paneitz operators $P_{h}$ uniformly away from the singularity. In particular, if $d_j\to 0$ then $h_j$ converges to a flat metric on $B_2(0)$ so that in proof of Proposition \[propsingularity\], $G(p, \bar{x})$ will be replaced by $c_n|y|^{4-n}$ in Euclidean balls with $c_n$ in $(\ref{expansion1})$. Therefore, Lemma \[lemupperboundsphere\] holds for $v_j$ so that $$\begin{aligned}
\max_{|x|=1}v_j(0)v_j(x)\leq C,
\end{aligned}$$ which shows that $$\begin{aligned}
M_ju_j(p_j)d_g(p_j, x_j)^{4-n}\leq C,\end{aligned}$$ contradicting with $(\ref{ineqdivergence})$. We have proved $(\ref{inequpperboundlevel})$ in $B_{\frac{\delta_2}{2}}(\bar{x})$. By Lemma \[lemH\] the inequality $(\ref{inequpperboundlevel})$ holds in $B_{\delta_1}(\bar{x})$.0.2cm
The same properties for $\xi_j$ in Lemma \[lemupperboundsphere\] now hold for $M_j u_j$ in $B_{\frac{\delta_2}{2}}(\bar{x})$. Up to a subsequence $$\begin{aligned}
M_ju_j\to v\,\,\text{in}\,\,C_{loc}^4(B_{\frac{\delta_2}{2}}(\bar{x}))\end{aligned}$$ where $$\begin{aligned}
P_gv=0\,\,\text{in}\,\,B_{\frac{\delta_2}{2}}(\bar{x}).\end{aligned}$$ By Remark \[remark01\], $v>0$ in $B_{\frac{\delta_2}{2}}(\bar{x})$. Since $\bar{x}$ is an isolated simple blowup point, the same argument in Lemma \[lemupperboundsphere\] shows that $r^{\frac{n-4}{2}}\bar{v}(r)$ is non-increasing for $0<r<\frac{\delta_2}{2}$, where $\bar{v}(r)=|\partial B_r(\bar{x})|^{-1}\int_{\partial B_r(\bar{x})}v ds_g$. Combining with the Harnack inequality, it implies that $v$ is not regular at $\bar{x}$. Also, $v$ satisfies the condition in Proposition \[propsingularity\]. By Proposition \[propsingularity\], we obtain $(\ref{inequpperboundbehavior})$. This completes the proof of Proposition \[propupperbound\].
0.2cm As an easy consequence of Proposition \[propupperbound\] and by standard interior estimates of the elliptic equation $(\ref{equation1})$, we have
\[corerrorterms\] Under the condition in Lemma \[lemupperboundestimates1\], there exists $\delta_2>0$ independent of $j$ such that for $R_jM_j^{-\frac{2}{n-4}}\leq d_g(p, x_j)\leq \delta_2$ $$\begin{aligned}
\label{ineqboundout1}
|\nabla_g^k u_j(p)|\leq\,\,C\,M_j^{-1}d_g(p, x_j)^{4-n-k}\,\,\,\text{for}\,\,0\leq k\leq 4,\end{aligned}$$ where $M_j=u_j(x_j)$, and $C$ is a constant independent of $j$. Let $x$ be geodesic normal coordinates of $(\Omega ,g)$ centered at $x_j$. Then for any fixed $r\leq \delta_2$, there exists $C>0$ depending on $|g|_{C^3(\Omega)}$ such that
$$\begin{aligned}
\label{ineqbounderrorterm2}
|\int_{d_g(p, x_j)\leq r}(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u dx|\leq CM_j^{-\frac{4}{n-4}+o(1)}\end{aligned}$$
with the term $o(1)\to 0$ as $j\to \infty$.
Inequality $(\ref{ineqboundout1})$ is a direct consequence of Proposition \[propupperbound\] and standard interior estimates of the elliptic equation $(\ref{equation1})$. We will next establish $(\ref{ineqbounderrorterm2})$. Note that $0<\epsilon_j \leq e^{-R_j}$. Using the estimates $(\ref{ineqboundout1})$, $(\ref{ineqmeasurelimit})$ and $(\ref{ineqrulerlimit})$, and recall the error bound $(\ref{ineqbounderrorterms})$, we have $$\begin{aligned}
&\int_{|x|\leq R_jM_j^{-\frac{2}{n-4}}}\,|(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u|\,dx\\
&\leq \int_{|x|\leq R_jM_j^{-\frac{2}{n-4}}}\,C(|x||Du(x)|+ u(x))(|x|^2|D^4u(x)|+|x| \,|D^3 u(x)|+|D^2u(x)|+|D u(x)|+u(x)) dx\\
&\leq C\int_{|y|\leq R_j} M_j(1+4^{-1}|y|^2)^{-\frac{n-4}{2}}M_j(1+4^{-1}|y|^2)^{-\frac{n-4}{2}-1}M_j^{\frac{4}{n-4}} M_j^{-\frac{2n}{n-4}} dy\\
&=CM_j^{-\frac{4}{n-4}}\int_{|y|\leq R_j}(1+4^{-1}|y|^2)^{3-n}dy=CM_j^{-\frac{4}{n-4}+o(1)},\,\,\,\,\text{and}\\
&\int_{R_jM_j^{-\frac{2}{n-4}} \leq |x|\leq r}\,|(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u|\,dx\\
&\leq \int_{R_jM_j^{-\frac{2}{n-4}} \leq |x|\leq r}C(|x||Du(x)|+ u(x))(|x|^2|D^4u(x)|+|x| \,|D^3 u(x)|+|D^2u(x)|+|D u(x)|+u(x)) dx\\
&\leq C\, \int_{R_jM_j^{-\frac{2}{n-4}} \leq |x|\leq r}\,M_j^{-2}|x|^{6-2n}\,dx\\
&\leq CM_j^{-\frac{4}{n-4}+o(1)},$$ where the term $o(1)\to 0$ as $j\to \infty$ and $C>0$ is a constant depending on $|g|_{C^3(\Omega)}$. Therefore, $$\begin{aligned}
\int_{|x|\leq r}\,|(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u|\,dx\leq CM_j^{-\frac{4}{n-4}+o(1)}\,\,\text{for}\,\,R_jM_j^{-\frac{2}{n-4}}\leq r,\end{aligned}$$ where $C>0$ is a constant independent of $j$ and the term $o(1)\to 0$ as $j\to \infty$.
For $n=5$, it is good for the discussion to come. For $n\geq 6$, better estimate is needed in order to cancel the error terms in the Pohozaev identity. By $(\ref{ineqmeasurelimit})$, $$\begin{aligned}
u_j(\exp_{x_j}(x))\leq 2M_j(1+4^{-1}M_j^{\frac{4}{n-4}}|x|^2)^{-\frac{n-4}{2}},\,\,\text{for}\,\,|x|\,\leq\,R_jM_j^{-\frac{2}{n-4}}.\end{aligned}$$ Combining with Proposition \[propupperbound\], we have $$\begin{aligned}
u_j(\exp_{x_j}(x))&\leq C \min\{ M_j(1+4^{-1}M_j^{\frac{4}{n-4}}|x|^2)^{-\frac{n-4}{2}},\,CM_j^{-1}|x|^{4-n}\}\\
&\leq C\,M_j(1+4^{-1}M_j^{\frac{4}{n-4}}|x|^2)^{-\frac{n-4}{2}},\,\,\,\text{for}\,\,|x|\,\leq\,\delta_2.\end{aligned}$$ For $n=6$, $$\begin{aligned}
&\int_{|x|\leq r}\,|(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u|\,dx\\
&\leq C\int_1^{M_j^{\frac{2}{n-4}}r}M_j^{-2}M_j^{\frac{2(n-6)}{n-4}}|y|^{5-n}d |y|\\
&\leq CM_j^{-\frac{4}{n-4}}\ln(M_j^{\frac{2}{n-4}}r),\,\,\text{for}\,\,R_jM_j^{-\frac{2}{n-4}}\leq r,\end{aligned}$$ For $n\geq 7$, $$\begin{aligned}
&\int_{|x|\leq r}\,|(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u|\,dx\\
&\leq C\int_1^{M_j^{\frac{2}{n-4}}r}M_j^{-2}M_j^{\frac{2(n-6)}{n-4}}|y|^{5-n}d |y|\\
&\leq CM_j^{-\frac{4}{n-4}},\,\,\text{for}\,\,R_jM_j^{-\frac{2}{n-4}}\leq r,\end{aligned}$$
For the term $M_j^2\int_{|x|\leq r}|Q_g|\,(u_j^2+|x|\,|Du_j|\,u_j) d x$ with $r>0$ fixed, $$\begin{aligned}
M_j^2\int_{|x|\leq r}|Q_g|\,(u_j^2+|x|\,|Du_j|\,u_j)\, d x&\leq C\,M_j^2\int_0^{rM_j^{\frac{2}{n-4}}}M_j^2(1+|y|)^{8-2n}M_j^{-\frac{2n}{n-4}}|y|^{n-1}d|y|\\
&\leq C\,M_j^{2-\frac{8}{n-4}}\int_0^{rM_j^{\frac{2}{n-4}}}\,(1+|y|)^{7-n}d|y|\end{aligned}$$ For $n=6$, $$\begin{aligned}
M_j^2\int_{|x|\leq r}|Q_g|\,(u_j^2+|x|\,|Du_j|\,u_j)\, d x\leq \,C\,r^2.\end{aligned}$$ For $n=7$, $$\begin{aligned}
M_j^2\int_{|x|\leq r}|Q_g|\,(u_j^2+|x|\,|Du_j|\,u_j)\, d x\leq \,C\,r.\end{aligned}$$ These are good terms. For later use, estimates on the term $M_j^2\int_{|x|\leq r}\,(u_j+|x|\,|Du_j|)\,|D^2u_j|\, d x$ will be needed for $n=6$; while for $n=7$, estimates on the term $M_j^2\int_{|x|\leq r}\,(u_j+|x|\,|Du_j|)\,(|Du_j|+|D^2u|)\, d x$ is needed.
\[propinequality\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n=5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $\{u_j\}$ be a sequence of positive solutions to $(\ref{equation1})$ and $x_j\to \bar{x}$ be an isolated simple blowup point so that $$\begin{aligned}
u_j(x_j)u_j(p)\to h(p)\,\,\,\,\text{in}\,\,C_{loc}^{4,\alpha}(B_{\delta_2}(\bar{x})-\{\bar{x}\}),\end{aligned}$$ for some $0< \alpha <1$. Assume that for some constants $a>0$ and $A$, $$\begin{aligned}
h(p)=\frac{a}{d_g(p, \bar{x})^{n-4}}+A+o(1)\,\,\text{as}\,\,d_g(p, \bar{x})\to 0.\end{aligned}$$ Then $A= 0$.
Let $x=(x^1,...,x^n)$ be geodesic normal coordinates at $x_j$. Denote $\Omega_{\gamma, j}=B_{\gamma}(x_j)$ for $\gamma< \frac{\delta_2}{2}$. Then $\Omega_{\gamma, j}\to \Omega_{\gamma}=B_{\gamma}(\bar{x})$. By the Pohozaev identity, $$\begin{aligned}
&\int_{\partial \Omega_{\gamma, j}} \,\frac{n-4}{2}(u_j\frac{\partial}{\partial \nu}(\Delta u_j)-\Delta u_j \frac{\partial}{\partial \nu}u_j)+ ((x\cdot\nabla u_j)\frac{\partial}{\partial \nu}(\Delta u_j)- \frac{\partial}{\partial \nu} (x\cdot\nabla u_j)\Delta u_j + \frac{1}{2} (\Delta u_j)^2x\cdot \nu) ds\\
&=\int_{\Omega_{\gamma, j}}(x\cdot \nabla u_j+ \frac{n-4}{2}u_j)(\Delta^2-P_g)u_j\,dx\,+\,\frac{(n-4)^2}{4n}\bar{Q}\int_{\partial \Omega_{\gamma, j}}(x\cdot \nu)u_j^{\frac{2n}{n-4}}\, dx.\end{aligned}$$ Multiplying $M_j^2=u_j(x_j)^2$ on both sides of the identity and taking limit $\lim_{\gamma\to 0^+}\,\limsup_{j\to \infty}$ on both sides, we have that by Corollary \[corerrorterms\], $$\begin{aligned}
\lim_{\gamma\to 0}\limsup_{j\to \infty} M_j^2 \int_{\Omega_{\gamma, j}}(x\cdot \nabla u_j+ \frac{n-4}{2}u_j)(\Delta^2-P_g)u_j\,dx=0,\end{aligned}$$ and $$\begin{aligned}
&\lim_{\gamma\to 0}[\int_{\partial \Omega_{\gamma}} \,\frac{n-4}{2}(h\frac{\partial}{\partial \nu}(\Delta h)-\Delta h \frac{\partial}{\partial \nu}h)+ ((x\cdot\nabla h)\frac{\partial}{\partial \nu}(\Delta h)- \frac{\partial}{\partial \nu} (x\cdot\nabla h)\Delta h + \frac{1}{2} (\Delta h)^2x\cdot \nu) ds]\\
&=\lim_{\gamma\to 0}\limsup_{j\to \infty}\,\,M_j^2\int_{\partial \Omega_{\gamma, j}} \,[\frac{n-4}{2}(u_j\frac{\partial}{\partial \nu}(\Delta u_j)-\Delta u_j \frac{\partial}{\partial \nu}u_j)+ ((x\cdot\nabla u_j)\frac{\partial}{\partial \nu}(\Delta u_j)\\
&- \frac{\partial}{\partial \nu} (x\cdot\nabla u_j)\Delta u_j + \frac{1}{2} (\Delta u_j)^2x\cdot \nu)] ds\\
&=\lim_{\gamma\to 0}\limsup_{j\to \infty}\frac{(n-4)^2}{4n}\bar{Q} M_j^{-\frac{8}{n-4}} \int_{\partial \Omega_{\gamma, j}}(x\cdot \nu)(M_j\,u_j)^{\frac{2n}{n-4}}\, dx=\,0.\end{aligned}$$ By assumption, $$\begin{aligned}
&\lim_{\gamma\to 0}[\int_{\partial \Omega_{\gamma}} \,\frac{n-4}{2}(h\frac{\partial}{\partial \nu}(\Delta h)-\Delta h \frac{\partial}{\partial \nu}h)+ ((x\cdot\nabla h)\frac{\partial}{\partial \nu}(\Delta h)- \frac{\partial}{\partial \nu} (x\cdot\nabla h)\Delta h + \frac{1}{2} (\Delta h)^2x\cdot \nu) ds]\\
&=\lim_{\gamma\to 0}\,\int_{\partial \Omega_{\gamma}}(n-4)^2(n-2)a A |x|^{1-n} d s\\
&=(n-4)^2(n-2)a A |\mathbb{S}^{n-1}|,\end{aligned}$$ where $|\mathbb{S}^{n-1}|$ is area of $(n-1)-$dimensional round sphere. Therefore, $$\begin{aligned}
A= 0.\end{aligned}$$
\[remark6\] Corollary \[corerrorterms\] is not enough to prove Proposition \[propinequality\] for manifolds of dimension $n=6$ and $n=7$. Let $U_0(r)=(1+4^{-1}r^2)^{-\frac{n-4}{2}}$ be a solution to $$\begin{aligned}
\label{equation123}
\Delta^2U_0=\frac{n-4}{2}\bar{Q}U_0^{\frac{n+4}{n-4}}\end{aligned}$$ on $\mathbb{R}^n$ with dimension $n=6$ or $7$. The linearized equation of $(\ref{equation123})$ is $$\begin{aligned}
\label{equation1231}
\Delta^2\phi(y)=\frac{n+4}{2}\bar{Q}U_0(y)^{\frac{8}{n-4}}\phi(y)\end{aligned}$$ for $y\in \mathbb{R}^n$. As in [@Marques], if we can show that for any solution $\phi$ to $(\ref{equation1231})$ with $\phi(y)\to 0$ as $y\to \infty$ it holds that $$\begin{aligned}
\phi(z)=c_0(z\cdot \nabla U_0(z) + \frac{n-4}{2}U_0(z))+ \sum_{j=1}^nc_j\partial_{z_j}U_0(z),\,\,z\in \mathbb{R}^n,\end{aligned}$$ with $c_0,\,...,\, c_n$ some constant, then we can prove that $|v_j(y)- U_0(y)|\leq C M_j^{-2}$ for $|y|\leq rM_j^{\frac{2}{n-4}}$ where $v_j(y)=M_j^{-1}u_j(M_j^{-\frac{2}{n-4}}y)$ and $C>0$ is a constant independent of $j$. This combining with Green’s representation leads to the better estimate $$\begin{aligned}
|\int_{d_g(p, x_j)\leq r}(x\cdot \nabla u +\frac{n-4}{2}u)(\Delta^2 - P_g)u dx|= o(1)M_j^{-2},\end{aligned}$$ in conformal normal coordinates and the corresponding conformal metric $g$. Then Proposition \[propinequality\] still holds for $n=6$ and $n=7$ in conformal normal coordinates and the corresponding conformal metric $g$.
From isolated blowup points to isolated simple blowup points
============================================================
In this section we show that an isolated blowup point is an isolated simple blowup point.
\[propisolatedsingularpoints\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n=5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. Let $\{u_j\}$ be a sequence of positive solutions to $(\ref{equation1})$ and $x_j\to \bar{x}$ be an isolated blowup point. Let $M_j=u_j(x_j)$. Then $\bar{x}$ is an isolated simple blow up point.
We prove the Proposition by contradiction argument. Assume that $\bar{x}$ is not an isolated simple blow up point. Then there exists two critical point of $r^{\frac{n-4}{2}}\bar{u}_j(r)$ in $(0, \mu_j)$ with $\mu_j\to 0$ up to as subsequence as $j\to \infty$. By Lemma \[lemlimitmodel\] and let $0<\epsilon_j<e^{-R_j}$, $r^{\frac{n-4}{2}}\bar{u}_j(r)$ has precisely one critical point in $(0, R_jM_j^{-\frac{2}{n-4}})$. We choose $\mu_j$ to be the second critical point of $r^{\frac{n-4}{2}}\bar{u}_j(r)$ so that $\mu_j\geq R_jM_j^{-\frac{2}{n-4}}$ and by assumption $\mu_j\to 0$.
Let $x=(x^1,...,x^n)$ be the geodesic normal coordinates centered at $x_j$, and let $y=\mu_j^{-1}x$. For simple notations, we assume $\delta_2=1$. We define the scaled metric $h_j=\mu_j^{-2}g$ so that $(h_j)_{pq}(\mu_j^{-1} x)dx^pdx^q=g_{pq}(x)dx^pdx^q$, and $$\begin{aligned}
\xi_j(y)=\mu_j^{\frac{n-4}{2}}u_j(\exp_{x_j}(\mu_j y)),\,\,\,\,\text{for}\,\,|y|<\mu_j^{-1}.\end{aligned}$$ We denote $\bar{\xi}_j$ as spherical average of $\xi_j$ in the usual way. Then we have $$\begin{aligned}
&P_{h_j}\xi_j(y)=\frac{n-4}{2}\bar{Q}\xi_j(y)^{\frac{n+4}{n-4}},\,\,\,\,|y|<\,\mu_j^{-1},\\
&|y|^{\frac{n-4}{2}}\xi_j(y)\leq C,\,\,\,\,|y|<\,\mu_j^{-1},\\
&\lim_{j\to\infty}\xi_j(0)=\infty,\\
&-\frac{4(n-1)}{n-2}\Delta_{h_j}\xi_j^{\frac{n-2}{n-4}}+R_{h_j}\xi_j^{\frac{n-2}{n-4}}\geq 0, \,\,\,\,|y|<\,\mu_j^{-1},\\
&r^{\frac{n-4}{2}}\bar{\xi}_j(r)\,\,\text{has precisely one critical point in}\,\,0< r <1,\\
&\label{criticalpoint}\frac{d}{d r}(r^{\frac{n-4}{2}}\bar{\xi}_j(r))=0\,\,\text{at}\,\,r=1.\end{aligned}$$ Therefore $\{0\}$ is an isolated simple blowup point of the sequence $\{\xi_j\}$. Note that the Remark \[remark01\] holds for $u_j$ so that $$\begin{aligned}
\label{ineqlowerbound1}
\xi_j(0)\xi_j(y)\geq C|y|^{4-n}\,\,\,\,\text{for}\,\,|y|\geq \mu_j^{-1}R_jM_j^{-\frac{2}{n-4}},\end{aligned}$$ where $\mu_j^{-1}R_jM_j^{-\frac{2}{n-4}}\leq 1$. By Lemma \[lemH\], there exists $C>0$ independent of $j$ and $k$ so that for any $k\in \mathbb{R}$, $$\begin{aligned}
\label{ineqpolynomialgrowth}
\max_{2^k\leq |y|\leq 2^{k+1}}\xi_j(0)\xi_j(y)\,\,\leq\, C \min_{2^k\leq |y|\leq 2^{k+1}}\xi_j(0)\xi_j(y),\,\,\text{when}\,\,2^{k+1}<\mu_j^{-1}\frac{\delta_2}{3}.\end{aligned}$$ Note that $Q_{h_j}\geq 0$ and $R_{h_j}>0$ in $M$. Also the metrics $h_j$ are all well controlled in $|y|\leq 1$. In proof of Lemma \[lemupperboundestimates1\] the maximum principle holds for $h_j$ and the coefficients of the test function are still uniformly chosen for $h_j$ so that the estimate in Lemma \[lemupperboundestimates1\] holds for each $\xi_j$ in $|y|\leq \tilde{\delta}_2$ for some $\tilde{\delta}_2<1$ independent of $j$. Similarly Proposition \[propupperbound\] holds for $\xi_j$ in $|y|\leq\tilde{\delta}_2$. This combining with $(\ref{ineqlowerbound1})$ and $(\ref{ineqpolynomialgrowth})$ implies $$\begin{aligned}
C(K)^{-1}\leq \xi_j(0)\xi_j(y) \leq C(K)\end{aligned}$$ for $K\subset\subset \mathbb{R}^n-\{0\}$ when $j$ is large, $h_j$ converges to the flat metric and there exists $a>0$ so that $\xi_j(0)\xi_j(y)$ converges to $$\begin{aligned}
h(y)=a|y|^{4-n}+b(y)\,\,\text{in}\,\,C_{loc}^4(\mathbb{R}^n-\{0\}),\end{aligned}$$ where $b(y)\in C^4(\mathbb{R}^n)$ satisfies $$\begin{aligned}
\Delta^2 b =0\end{aligned}$$ in $\mathbb{R}^n$. Here $h>0$ in $\mathbb{R}^n-\{0\}$. Also, $$\begin{aligned}
\label{ineqscalarcurvature1}
-\Delta h(y)^{\frac{n-2}{n-4}}\geq 0,\,\,|y|>0.\end{aligned}$$ Moreover, for a fixed point $y_0$ in $|y|=1$, by $(\ref{ineqpolynomialgrowth})$, $$\begin{aligned}
h(y)\leq |y|^{2+\frac{\ln C}{\ln 2}} h(y_0),\end{aligned}$$ for $|y|\geq 1$. Since $h>0$ for $|y|>0$, it follows that $b(y)$ is a polyharmonic function of polynomial growth on $\mathbb{R}^n$. Therefore, $b(y)$ must be a polynomial in $\mathbb{R}^n$, see [@Armitage]. Non-negativity of $h$ near infinity implies that $b(y)$ is of even order. Then either $b(y)$ is a non-negative constant or $b(y)$ is a polynomial of even order with order at least two and $b(y)$ is non-negative at infinity. The later case contradicts with $(\ref{ineqscalarcurvature1})$ for $y$ near infinity. Therefore, $b(y)$ must be a non-negative constant on $\mathbb{R}^n$ and $$\begin{aligned}
h(y)=a |y|^{4-n}+b\end{aligned}$$ with a constant $a>0$ and a constant $b$.
By $(\ref{criticalpoint})$, $$\begin{aligned}
\frac{d}{d r}(r^{\frac{n-4}{2}}h(r))=0\,\,\text{at}\,\,r=1.\end{aligned}$$ We then have $b=a>0$, which contradicts with Proposition \[propinequality\]. In fact, Proposition \[propinequality\] applies to isolated simple blowup points with respect the sequence of metrics $\{h_j\}$ with uniform curvature bound and uniform bound of injectivity radius with the property that $Q_{h_j}>0$ and $R_{h_j}>0$.
Compactness of solutions to the constant $Q$-curvature equations
================================================================
Based on Proposition \[propupperbound\] and Proposition \[propisolatedsingularpoints\], proof of compactness of the solutions is more or less standard, see in [@Li-Zhu]. But again we need to deal with the limit of the blowup argument carefully, see Lemma \[lembehaviornearlargepoint\] and Proposition \[propdistanceofsingularities\].
We first show that there are no bubble accumulations.
\[lembehaviornearlargepoint\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $5\leq n \leq 9$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. For any given $\epsilon>0$ and large constant $R>1$, there exist some constant $C_1>0$ depending on $M,\,g,\,\epsilon,\,R$, $\|Q_g\|_{C^1(M)}$ such that for any solution $u$ to $(\ref{equation1})$ and any compact subset $K \subset M$ satisfying $$\begin{aligned}
&\max_{p\in M-K}d(p, K)^{\frac{n-4}{2}}u(p)\geq C_1,\,\,\,\text{if}\,\,K\neq\emptyset,\,\,\text{and}\\
&\max_{p\in M}u(p)\geq C_1,\,\,\,\text{if}\,\,K=\emptyset,\end{aligned}$$ we have that there exists some local maximum point $p'$ of $u$ in $M-K$ with $B_{R\,u(p')^{-\frac{2}{n-4}}}(p') \subset M-K$ satisfying $$\begin{aligned}
\label{ineqlimitbehavior1}
\|u(p')^{-1}u(\exp_{p'}(u(p')^{-\frac{2}{n-4}}y))\,-\,(1+4^{-1}|y|^2)^{-\frac{n-4}{2}}\|_{C^4(|y|\leq 2R)}<\epsilon.\end{aligned}$$
We argue by contradiction. That is to say, there exist a sequence of compact subsets $K_j$ and a sequence of solutions $u_j$ to $(\ref{equation1})$ on $M$ such that $$\begin{aligned}
\max_{p\in M-K_j}d(p, K_j)^{\frac{n-4}{2}}u(p)\geq j,\end{aligned}$$ with $d(p, K_j)=1$ when $K_j=\emptyset$, but no point satisfies $(\ref{ineqlimitbehavior1})$. We choose $x_j \in M-K_j$ satisfying $$\begin{aligned}
d_g(x_j, K_j)^{\frac{n-4}{2}}u_j(x_j)=\max_{p\in M- K_j}d_g(p, K_j)^{\frac{n-4}{2}}u_j(p).\end{aligned}$$ We then define $$\begin{aligned}
v_j(y)=u_j(x_j)^{-1}u_j(\exp_{x_j}(u_j(x_j)^{-\frac{2}{n-4}}y)),\,\,\text{for}\,\,|y|\leq R_j=\frac{1}{4}u_j(x_j)^{\frac{2}{n-4}}d_g(x_j, K_j).\end{aligned}$$ Let $h_j=u_j(x_j)^{\frac{4}{n-4}}g$. The resecaled function $v_j$ satisfies $$\begin{aligned}
P_{h_j}v_j=\frac{n-4}{2}\bar{Q}v_j^{\frac{n+4}{n-4}},\end{aligned}$$ and by Theorem \[thm1\], $$\begin{aligned}
\Delta_{h_j}v_j\leq \frac{(n-4)}{4(n-1)}R_{h_j}v_j.\end{aligned}$$ We will analyze limit of the sequence $\{v_j\}$ as in Theorem \[thmlowerbound\] and conclude that $(\ref{ineqlimitbehavior1})$ indeed holds. By assumption, $$\begin{aligned}
R_j=\frac{1}{4}u_j(x_j)^{\frac{2}{n-4}}d_g(y_j, K_j)\geq \frac{1}{4}j^{\frac{2}{n-4}},\end{aligned}$$ and $$\begin{aligned}
d_g(\exp_{x_j}(u_j(x_j)^{-\frac{2}{n-4}}y), K_j)\geq \frac{1}{2}d_g(x_j, K_j),\,\,\text{for}\,\,|y|\leq R_j.\end{aligned}$$ It follows that $$\begin{aligned}
0< v_j(y)&=u_j(x_j)^{-1}u_j(\exp_{x_j}(u_j(x_j)^{-\frac{2}{n-4}}y))\\
&\leq u_j(x_j)^{-1} d_g(\exp_{x_j}(u_j(x_j)^{-\frac{2}{n-4}}y), K_j)^{-\frac{n-4}{2}}d_g(x_j, K_j)^{\frac{n-4}{2}}u_j(x_j)\\
&\leq 2^{\frac{n-4}{2}},\,\,\text{for}\,\,|y|\leq R_j.\end{aligned}$$ Standard elliptic estimates imply that up to a subsequence, $$\begin{aligned}
v_j\to v\,\,\text{in}\,\,C_{loc}^4(\mathbb{R}^n),\end{aligned}$$ with $v$ satisfying $$\begin{aligned}
&\Delta^2v=\frac{n-4}{2}\bar{Q}v^{\frac{n+4}{n-4}}\,\,\text{in}\,\,\mathbb{R}^n,\\
&v(0)=1,\,\,0\leq v \leq \,\,2^{\frac{n-4}{2}}\,\,\text{in}\,\,\mathbb{R}^n,\\
&\Delta v\leq 0,\,\,\text{in}\,\,\mathbb{R}^n.\end{aligned}$$ By strong maximum principle, $v>0$ in $\mathbb{R}^n$. Then by the classification theorem of C.S. Lin ([@Lin]), $$\begin{aligned}
v(y)=\big(\frac{\lambda}{1+4^{-1}\lambda^2|y-\bar{y}|^2}\big)^{\frac{n-4}{2}}\,\,\text{in}\,\,\mathbb{R}^n,\end{aligned}$$ with $v(0)=1$ and $v(y)\leq \lambda^{\frac{n-4}{2}}\leq 2^{\frac{n-4}{2}}$. Therefore, $|\bar{y}|\leq C(n)$ with $C(n)>0$ only depending on $n$. We choose $y_j$ to be the local maximum point of $v_j$ converging to $\bar{y}$. Then $p_j=\exp_{x_j}(u_j(x_j)^{-\frac{2}{n-4}}y_j)\in M-K_j$ is a local maximum point of $u_j$. We now repeat the blowup argument with $x_j$ replaced by $p_j$ and $u_j(x_j)$ replaced by $u_j(p_j)$ and obtain the limit $$\begin{aligned}
v(y)=(1+4^{-1}|y|^2)^{-\frac{n-4}{2}}\,\,\text{in}\,\,\mathbb{R}^n.\end{aligned}$$ Therefore, for large $j$, there exists $p_j\in M- K_j$ so that $(\ref{ineqlimitbehavior1})$ holds. This contradicts with the assumption. Therefore, the proof of the lemma is completed.
\[lemalargefunction\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $5\leq n \leq 9$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. For any given $\epsilon>0$ and a large constant $R>1$, there exist some constants $C_1>0$ and $C_2>0$ depending on $M,\,g,\,\epsilon,\,R$, $\|Q_g\|_{C^1(M)}$ such that for any solution $u$ to $(\ref{equation1})$ with $$\begin{aligned}
\max_{p\in M}u(p)> C_1,\end{aligned}$$ there exists some integer $N=N(u)$ depending on $u$ and $N$ local maximum points $\{p_1,...,p_N\}$ of $u$ such that
1. for $i\neq j$, $$\begin{aligned}
\overline{B_{\gamma_i}(p_i)}\bigcap \overline{B_{\gamma_j}(p_j)}=\emptyset,\end{aligned}$$ with $\gamma_j=Ru(p_j)^{-\frac{2}{n-4}}$ and $B_{\gamma_j}(p_j)$ the geodesic $\gamma_j$-ball centered at $p_j$, and $$\begin{aligned}
\label{ineqlimitbehavior2}
\|u(p_j)^{-1}u(\exp_{p_j}(u(p_j)^{-\frac{2}{n-4}}y))\,-\,(1+4^{-1}|y|^2)^{-\frac{n-4}{2}}\|_{C^4(|y|\leq 2R)}<\epsilon,\end{aligned}$$ where $y=u(p_j)^{\frac{2}{n-4}}x$, with $x$ geodesic normal coordinates centered at $p_j$, and $|y|=\sqrt{(y^1)^2+..+(y^n)^2}$.
2. for $i < j$, $d_g(p_i, p_j)^{\frac{n-4}{2}}u(p_j)\geq C_1$, while for $p\in M$ $$\begin{aligned}
d_g(p, \{p_1,..,p_n\})^{\frac{n-4}{2}}u(p)\leq C_2.\end{aligned}$$
We will use Lemma \[lembehaviornearlargepoint\] and prove the lemma by induction. To start, we apply Lemma \[lembehaviornearlargepoint\] with $K=\emptyset$. We choose $p_1$ to be a maximum point of $u$ and $(\ref{ineqlimitbehavior2})$ holds. Next we let $K=\overline{B_{\gamma_1}(p_1)}$.
Assume that for some $i_0\geq 1$, $i)$ in the lemma holds for $1\leq j\leq i_0$ and $1\leq i<j$, and also $d_g(p_i, p_j)^{\frac{n-4}{2}}u(p_j)\geq C_1$ with $p_j$ chosen as in Lemma \[lembehaviornearlargepoint\] by induction.( This holds for $i_0=1$.) Then we let $K=\bigcup_{j=1}^{i_0}\overline{B_{\gamma_j}(p_j)}$. It follows that for $\epsilon>0$ small, for any $p$ such that $d_g(p, p_j)\leq 2\gamma_j$ with $1\leq j \leq i_0$, we have $$\begin{aligned}
d_g(p, \{p_1,..,p_{i_0}\})^{\frac{n-4}{2}}u(p)&\leq d_g(p, p_j)^{\frac{n-4}{2}}u(p)\leq 2 d_g(p, p_j)^{\frac{n-4}{2}} u(p_j)\\
&\leq 2 ( 2R u(p_j)^{-\frac{2}{n-4}})^{\frac{n-4}{2}}u(p_j)=2^{\frac{n-2}{2}}R^{\frac{n-4}{2}},\end{aligned}$$ and therefore, for $p\in \bigcup_{j=1}^{i_0}\overline{B_{2\gamma_j}(p_j)}$, $$\begin{aligned}
\label{ineqinball}
d_g(p, \{p_1,..,p_{i_0}\})^{\frac{n-4}{2}}u(p)\leq 2^{\frac{n-2}{2}}R^{\frac{n-4}{2}}.\end{aligned}$$
If for all $p\in M$ $$\begin{aligned}
d_g(p, \{p_1,..,p_{i_0}\})^{\frac{n-4}{2}}u(p)\leq C_1,\end{aligned}$$ the induction stops. Else, we apply Lemma \[lembehaviornearlargepoint\], and we denote $p_{i_0+1}$ as the local maximum point $y_0$ obtained in Lemma \[lembehaviornearlargepoint\] so that $$\begin{aligned}
B_{R\,u(p_{i_0+1})^{-\frac{2}{n-4}}}(p_{i_0+1}) \subset M-K.\end{aligned}$$ Therefore, $i)$ in the lemma holds for $i_0+1$. Also, by assumption, $d_g(p_j, p_{i_0+1})^{\frac{n-4}{2}}u(p_{i_0+1})> C_1$. By the same argument, $(\ref{ineqinball})$ holds for $i_0$ replaced by $i_0+1$. The induction must stop in a finite time $N=N(u)$, since $\int_M u^{\frac{2n}{n-4}}dV_g$ is bounded and that $$\begin{aligned}
\int_{B_{\gamma_j}(p_j)}u^{\frac{2n}{n-4}}dV_g\end{aligned}$$ is bounded below by a uniform positive constant. It is clear now that for $p \in M - \bigcup_{j=1}^NB_{\gamma_j}(p_j)$, $$\begin{aligned}
d(p, \{p_1,..,p_N\})^{\frac{n-4}{2}}u(p)\leq 2^{\frac{n-4}{2}}d(p, \bigcup_{j=1}^NB_{\gamma_j}(p_j))^{\frac{n-4}{2}}u(p)\leq 2^{\frac{n-4}{2}}C_1.\end{aligned}$$ By induction, $(\ref{ineqinball})$ holds for $i_0$ replaced by $N$. We set $C_2=2^{\frac{n-2}{2}}R^{\frac{n-4}{2}}+2^{\frac{n-4}{2}}C_1$. This proves the lemma.
The next proposition rules out the bubble accumulations.
\[propdistanceofsingularities\] Let $(M^n, g)$ be a closed Riemannian manifold of dimension $n=5$ with $R_g\geq 0$, and also $Q_g\geq 0$ with $Q_g(p_0)>0$ for some point $p_0\in M$. For $\epsilon>0$ small enough and a constant $R>1$ large enough, there exists $\gamma>0$ depending on $M, g, \epsilon, R,$ $\|R_g\|_{C^1(M)}$ and $\|Q_g\|_{C^1(M)}$ such that for any solution $u$ to $(\ref{equation1})$ with $\max_{p\in M}u(p)> C_1$, we have $$\begin{aligned}
d(p_i, p_j)\geq \gamma,\end{aligned}$$ for $1\leq i,\,j\leq N$ and $i\neq j$, where $N=N(u)$, $p_j=p_j(u)$, $p_i=p_i(u)$ and $C_1$ are defined in Lemma \[lemalargefunction\].
Suppose the proposition fails, which implies that there exist $\epsilon>0$ small and $R>0$ large and a sequence of solutions $u_j$ to $(\ref{equation1})$ such that $\max_{p\in M}u_j(p)> C_1$ and $$\begin{aligned}
\lim_{j\to\infty}\min_{i\neq k}d(p_i(u_j), p_k(u_j))=0.\end{aligned}$$ We denote $p_{j,1}$ and $p_{j,2}$ to be the two points realizing minimum distance in $\{p_1(u_j),..,p_N(u_j)\}$ of $u_j$ constructed in Lemma \[lemalargefunction\]. Let $\bar{\gamma}_j=d_g(p_{j,1}, p_{j,2})$. Since $$\begin{aligned}
B_{Ru_j(p_{1,j})^{-\frac{2}{n-4}}}(p_{1,j})\bigcap B_{Ru_j(p_{2,j})^{-\frac{2}{n-4}}}(p_{2,j})=\emptyset,\end{aligned}$$ we have that $u_j(p_{1,j})\to \infty$ and $u_j(p_{2,j})\to \infty$.
Let $x=(x^1,..,x^n)$ be geodesic normal coordinates centered at $p_{1, j}$, $y=\bar{\gamma}_j^{-1} x$, $\exp_{p_{1,j}}(x)$ be exponential map under the metric $g$. We define the scaled metric $h_j=\bar{\gamma}_j^{\frac{4}{n-4}}g$, and the rescaled function $$\begin{aligned}
v_j(y)=\bar{\gamma}_j^{\frac{2}{n-4}}u_j(\exp_{p_{1,j}}(\bar{\gamma}_jy)).\end{aligned}$$ It follows that $v_j$ satisfies $v_j>0$ in $|y|\leq \bar{\gamma}_j^{-1}r_0$ and that $$\begin{aligned}
&\label{ineq7.1}P_{h_j}v_j(y)=\frac{n-4}{2}\bar{Q}v_j(y)^{\frac{n+4}{n-4}},\,\,\text{for}\,\,|y|\leq \bar{\gamma}_j^{-1}r_0,\\
&\label{ineq7.2}\Delta_{h_j}v_j\leq \frac{(n-4)}{4(n-1)}R_{h_j}v_j,\,\,\text{for}\,\,|y|\leq \bar{\gamma}_j^{-1}r_0,\end{aligned}$$ where $r_0$ is half of the injectivity radius of $(M, g)$. We define $y_k=y_k(u_j)\in \mathbb{R}^n$ such that $\exp_{p_{1,j}}(\bar{\gamma}_jy_k)=p_k$ for the points $p_k(u_j)$. It follows that for $p_k\neq p_{1, j}$, $$\begin{aligned}
|y_k|\geq 1+o(1)\end{aligned}$$ with $o(1)\to 0$ as $j\to \infty$. Let $y_{2, j}\in \mathbb{R}^n$ be so that $p_{2,j}=\exp_{p_{1, j}}(\bar{\gamma}_jy_{2, j})$. Then $$\begin{aligned}
|y_{2, j}|\to 1\,\,\text{as}\,\,j\to \infty.\end{aligned}$$ It follows that there exists $\bar{y} \in \mathbb{R}^n$ with $|\bar{y}|=1$ such that up to a subsequence, $$\begin{aligned}
\bar{y}=\lim_{j\to \infty}y_{2,j}.\end{aligned}$$ By Lemma \[lemalargefunction\], $$\begin{aligned}
\bar{\gamma}_j\geq C \max\{Ru_j(p_{1,j})^{-\frac{2}{n-4}},\,Ru_j(p_{2,j})^{-\frac{2}{n-4}}\}.\end{aligned}$$ Therefore, we have $$\begin{aligned}
&v_j(0)\geq C_3,\,\,v_j(y_{2,j})\geq C_3\,\,\text{for some}\,\,C_3>0\,\,\text{independent of}\,\,j,\\
&y_k\,\,\text{is a local maximum point of }\,\,v_j\,\,\text{for all}\,\,1\leq k\leq N(u_j),\\
&\min_{1\leq k\leq N(u_j)}|y-y_k|^{\frac{n-4}{2}}v_j(y)\leq C_2\,\,\text{for all}\,\,|y|\leq \bar{\gamma}_j^{-1}.\end{aligned}$$ We [**claim**]{} that $$\begin{aligned}
v_j(0)\to \infty,\,\,\text{and}\,\,v_j(y_{2,j})\to \infty.\end{aligned}$$ To see this, we first assume that one of them tends to infinity up to a subsequence, say $v_j(0)\to \infty$ for instance. It is clear that $0$ is an isolated blowup point, and by Proposition \[propisolatedsingularpoints\] it is an isolated simple blowup point. Then $v_j(y_{2,j})\to \infty$ in this subsequence since otherwise, by the control $(\ref{ineqlimitbehavior2})$ at $p_{2, j}$ in Lemma \[lemalargefunction\] and the rescaling, $v_j$ is uniformly bounded in a uniform neighborhood of $y_{2,j}$ and therefore by Harnack inequality $(\ref{ineqH})$ and Proposition \[propupperbound\], $v_j\to 0$ near $p_{2,j}$, contradicting with $v_j(y_{2,j})\geq C_3$. If both $v_j(0)$ and $v_j(y_{2,j})$ are uniformly bounded, similar argument shows that $v_j$ is uniformly bounded on any fixed compact subset of $\mathbb{R}^n$. Then as discussed in Lemma \[lembehaviornearlargepoint\], $v_j\to v$ in $C_{loc}^4(\mathbb{R}^n)$ with $v>0$ and $$\begin{aligned}
\Delta^2v=\frac{n-4}{2}\bar{Q}v^{\frac{n+4}{n-4}}\end{aligned}$$ in $\mathbb{R}^n$. Also, $0$ and $\bar{y}$ are local maximum points of $v$. That contradicts with the classification theorem in [@Lin]. The [**claim**]{} is established. Therefore, both $0$ and $\bar{y}$ are isolated simple blowup points of $v_j$. Let $K_0$ be the set of blowup points of $\{v_j\}$ after passing to a subsequence. It is clear that $0, \bar{y}\in K_0$ and for any two distinct points $y, z \in K$, $d_g(y, z)\geq 1$. By Proposition \[propupperbound\], $v_j(0)v_j$ is uniformly bounded in any fixed compact subset of $\mathbb{R}^n-K_0$. Multiplying $v_j(0)$ on both sides of $(\ref{ineq7.1})$ and $(\ref{ineq7.2})$, we have that up to a subsequence, $$\begin{aligned}
\lim_jv_j(0)v_j\to F\geq 0\,\,\text{in}\,\,C_{loc}^4(\mathbb{R}^n-K_0),
\end{aligned}$$ such that $$\begin{aligned}
&\label{ineq7-a}\Delta^2F=0,\,\,\text{in}\,\,\mathbb{R}^n-K_0,\\
&\label{ineq7-b}\Delta F\leq 0,\,\,\text{in}\,\,\mathbb{R}^n-K_0.
\end{aligned}$$ Since all the blowup points in $K_0$ are isolated simple blowup points, by Proposition \[propupperbound\], $$\begin{aligned}
F(y)=a_1|y|^{4-n}+\Phi_1(y)=a_1|y|^{4-n}+a_2|y-\bar{y}|^{4-n}+\Phi_2(y)
\end{aligned}$$ for $y \in \mathbb{R}^n-K_0$ with the constants $a_1, a_2>0$. Moreover, $\Phi_2\in C^4(\mathbb{R}^n-(K_0-\{0, \bar{y}\}))$ and $\Phi_2$ satisfies $(\ref{ineq7-a})$ in $\mathbb{R}^n-(K_0-\{0, \bar{y}\})$. We define $\xi= \Delta \Phi_1$ in $\mathbb{R}^n-(K_0-\{0\})$. By $(\ref{ineq7-b})$, $F>0$ in $\mathbb{R}^n-K_0$. Therefore, $$\begin{aligned}
&\label{ineq71a}\liminf_{|y|\to \infty}\Phi_1(y)=\liminf_{|y|\to \infty} (F(y)-a_1|y|^{4-n})\geq 0,\\
&\liminf_{|y|\to \infty}\xi(y)=\liminf_{|y|\to \infty}\Delta (F(y)-a_1|y|^{4-n})\leq 0.
\end{aligned}$$ Moreover, $\xi<0$ near any isolated point in $\mathbb{R}^n-(K_0-\{0\})$ by Proposition \[propupperbound\]. Applying strong maximum principle to $\xi$ and the equation $$\begin{aligned}
\Delta \xi=\Delta^2(F-a_1|y|^{4-n})=0
\end{aligned}$$ in $\mathbb{R}^n-(K_0-\{0\})$, we have that $$\begin{aligned}
\xi=\Delta \Phi_1<0
\end{aligned}$$ in $\mathbb{R}^n-(K_0-\{0\})$. Since $\Phi_1>0$ near any isolated point in $\mathbb{R}^n-(K_0-\{0\})$ by Proposition \[propupperbound\], and also $(\ref{ineq71a})$ holds, applying strong maximum principle to $\Phi_1$ and $\Delta \Phi_1<0$ in $\mathbb{R}^n-(K_0-\{0\})$, we have that $\Phi_1>0$ in $\mathbb{R}^n-(K_0-\{0\})$. It follows that $$\begin{aligned}
F(y)=a_1|y|^{4-n}+\Phi_1(0)+O(|y|)\,\,\text{with}\,\,\Phi_1(0)>0\,\,\text{near}\,\,y=0,
\end{aligned}$$ contradicting with Proposition \[propinequality\].(It is easy to check that Proposition \[propinequality\] applies for the scaled metrics $h_j$ instead of $g$.) Proposition \[propdistanceofsingularities\] is then established.
We are now ready to prove the compactness theorem of positive solutions to the equation $(\ref{equation1})$.
By Lemma \[lemlowerbound1\] and ellipticity theorem for $(\ref{equation1})$, we only need to show that there is a constant $C>0$ depending on $M$ and $g$ such that $$\begin{aligned}
u\leq C.\end{aligned}$$ Suppose the contrary, then there exists a sequence of positive solutions $u_j$ to $(\ref{equation1})$ such that $$\begin{aligned}
\max_{p\in M}u_j\to \infty\end{aligned}$$ as $j\to \infty$. By Proposition \[propdistanceofsingularities\], after passing to a subsequence, there exists $N$ isolated simple blowup points $p_{1, j}\to p_1,\,...,\,p_{N, j}\to p_N$ with $N\geq 1$ independent of $j$. Applying Proposition \[propupperbound\], we have that up to a subsequence, $$\begin{aligned}
u_j(p_{1,j})u_j(p)\to F(p)=\sum_{k=1}^Na_kG_g(p_k, p)+b(p)\,\,\text{in}\,\,C_{loc}^4(M-\{p_1,..,p_N\}),\end{aligned}$$ where $a_1>0,\, ... ,\,a_N>0$ are some constants, $G_g$ is Green’s function of $P_g$ under the metric $g$ and $b(p)\in C^4(M)$ satisfying $$\begin{aligned}
P_gb=0\end{aligned}$$ in $M$. Since $Q_g\geq 0$ on $M$ with $Q_g>0$ at some point, by the strong maximum principle of $P_g$, $b=0$ in $M$. We know that $G_g(p_k, p)>0$ for $1\leq k\leq N$ by Theorem \[thm1\]. Let $x=(x^1,..,x^n)$ be conformal normal coordinates( see [@Lee-Parker]) centered at $p_{1,j}$( resp. $p_1$) with respect to the conformal metric $h_j=\phi_j^{-\frac{4}{n-4}}g$(resp. $h=\phi^{-\frac{4}{n-4}}g$) such that $$\begin{aligned}
\det(h_{ij})=1+O(|x|^{10n}).\end{aligned}$$ Then there exists $C_1>0$ independent of $j$ such that $$\begin{aligned}
C_1^{-1}\leq \phi_j\leq C_1,\end{aligned}$$ and $$\begin{aligned}
\|\phi_j-\phi\|_{C^5(M)}\to 0\,\,\text{as}\,\,j\to \infty.\end{aligned}$$ As shown in Theorem \[thm1\], under the conformal normal coordinates $x$ centered at $p_1$, the Green’s function under metric $h$ satisfies $$\begin{aligned}
G_h(p_1, p)=\phi^2(p)G_g(p_1, p)=d_h(p_1, p)^{4-n}+A+o(1)\end{aligned}$$ near $p_1$ with the constant $A>0$ and $o(1)\to 0$ as $p\to p_1$. Therefore, $$\begin{aligned}
\phi(p)^2F(p)=a_1d_h(p_1, p)^{4-n}+B+o(1)
\end{aligned}$$ with $B=a_1A+\sum_{k=2}^Na_k\phi(p_1)^2G_g(p_k, p_1)>0$ and $o(1)\to 0$ as $p\to p_1$. Note that since $\phi_j$ are uniformly controlled in the construction of conformal normal coordinates and the corresponding metrics, the conclusions in Corollary \[corerrorterms\] and consequently in Proposition \[propinequality\] still hold for $g$ replaced by the conformal metrics $h_j$ and $u_j$ replaced by $\tilde{u}_j=\phi_j u_j$. This leads to a contradiction. Therefore, Theorem \[thm27\] is established.
Appendix: Positive solutions of certain linear fourth order elliptic equations in punctured balls
=================================================================================================
Assume $B_{\delta}(\bar{x})$ is a geodesic $\delta$-ball on $\mathbb{R}^n$ under the metric $g$ with $2\delta$ less than the injectivity radius. For application, for $5\leq n \leq 9$ it could sometime be assumed as a geodesic $\delta$-ball embedded in a closed Riemannian manifold $(M^n, g)$, where $(M, g)$ is as in Proposition \[propupperbound\].
\[lemlinear9.1\] Let $u \in C^4(B_{\delta}(\bar{x})-\{\bar{x}\})$ be a solution to $$\begin{aligned}
\label{equlinear}
P_g u=0\,\,\text{in}\,\,B_{\delta}(\bar{x})-\{\bar{x}\}.\end{aligned}$$ If $u(p)=o(d_g(p, \bar{x}))^{4-n}$ as $p\to \bar{x}$, then $u\in C_{loc}^{4, \alpha}(B_{\delta}(\bar{x}))$ for $0<\alpha <1$.
The proof is standard.
Step 1. We show that $(\ref{equlinear})$ holds in $B_{\delta}(\bar{x})$ in distribution sense.
To see this, given any small $\epsilon>0$, we define the cutoff function $\eta_{\epsilon}$ on $B_{\delta}(\bar{x})$ with $0<\eta_{\epsilon}<1$ so that $$\begin{aligned}
&\eta_{\epsilon}(p)=1\,\,\,\,\,\,\text{for}\,\,d_g(p, \bar{x})\leq \epsilon,\\
&\eta_{\epsilon}(p)=0\,\,\,\,\,\,\text{for}\,\,d_g(p, \bar{x})\geq 2\epsilon,\\
&|\nabla\eta_{\epsilon}(p)|\leq C\epsilon^{-1}\,\,\text{for}\,\,\epsilon\leq d_g(p, \bar{x})\leq 2\epsilon.\end{aligned}$$ For any given $\phi \in C_c^{\infty}(B_{\delta}(\bar{x}))$ we multiply $\phi (1-\eta_{\epsilon})$ on both side of $(\ref{equlinear})$ and do integration by parts, $$\begin{aligned}
\int_{B_{\delta}(\bar{x})}P_g(\phi (1-\eta_{\epsilon}))u dV_g=0.\end{aligned}$$ Let $\epsilon\to 0$, then $$\begin{aligned}
\int_{B_{\delta}(\bar{x})}P_g\phi (1-\eta_{\epsilon})u dV_g=O(1)(C\epsilon^{-4}\int_{B_{2\epsilon}(\bar{x})-B_{\epsilon}(\bar{x})}|u|)+C\int_{B_{\epsilon}(\bar{x})}|u|\to 0,\end{aligned}$$ where in the last step we have used $u(p)=o(d_g(p, \bar{x}))^{4-n}$. Therefore, Step 1 is established.
Step 2. The assumption of $u$ near $\bar{x}$ implies that $u\in L^p_{loc}(B_{\delta}(\bar{x}))$ for any $1<p<\frac{n}{n-4}$. By $W^{4, p}$ estimates of the elliptic equation we obtain that $u\in W_{loc}^{4,p}(B_{\delta}(\bar{x}))$, see [@S.Agmon] for instance. Then standard bootstrap argument gives $u\in C_{loc}^{4, \alpha}(B_{\delta}(\bar{x}))$.
For later use, we now employ Lemma 9.2 from [@Li-Zhu] without proof.
\[lemGreenfunction\] There exists some constant $0<\delta_0\leq \delta$ depending on $n,\,\|g_{ij}\|_{C^2(B_{\delta}(\bar{x}))}$ and $\|R_g\|_{L^{\infty}(B_{\delta}(\bar{x}))}$ such that the maximum principle for $-\frac{4(n-1)}{n-2}\Delta_g+R_g$ holds on $B_{\delta_0}(\bar{x})$, and there exists a unique $G_1(p)\in C^2(B_{\delta_0}(\bar{x})-\{\bar{x}\})$ satisfying $$\begin{aligned}
&-\frac{4(n-1)}{n-2}\Delta_gG_1+R_gG_1=0\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\},\\
&G_1=0\,\,\text{on}\,\,\partial B_{\delta_0}(\bar{x}),\\
&\lim_{p\to \bar{x}}d_g(p, \bar{x})^{n-2}G_1(p)=1.\end{aligned}$$ Furthermore, $G_1(p)=d_g(p, \bar{x})^{2-n}+\mathcal {R}(p)$ where $\mathcal {R}(p)$ satisfies for all $0<\epsilon <1$ that $$\begin{aligned}
d_g(p, \bar{x})^{n-4+\epsilon}|\mathcal {R}(p)|+ d_g(p, \bar{x})^{n-3+\epsilon}|\nabla \mathcal {R}(p)|\leq C(\epsilon),\,\,p\in\,B_{\delta_0}(\bar{x}),\,\,n\geq 4,\end{aligned}$$ where $C(\epsilon)$ depends on $\epsilon$, $n$, $\|g_{ij}\|_{C^2(B_{\delta}(\bar{x}))}$ and $\|R_g\|_{L^{\infty}(B_{\delta}(\bar{x}))}$.
\[lem9.3\] Suppose a positive function $u\in C^4(B_{\delta}(\bar{x})-\{\bar{x}\})$ satisfies $(\ref{equlinear})$ in $B_{\delta}(\bar{x})-\{\bar{x}\}$, and assume that there exists a constant $C>0$ such that for $0< r < \delta$, the Harnack inequality holds: $$\begin{aligned}
\max_{d_g(p, \bar{x})=r}u(p)\leq C \min_{d_g(p, \bar{x})=r}u(p).\end{aligned}$$ If moreover, $$\begin{aligned}
-\frac{4(n-1)}{n-2}\Delta_gu^{\frac{n-2}{n-4}}+R_gu^{\frac{n-2}{n-4}}\geq 0\,\,\text{in}\,\,B_{\delta}(\bar{x})-\{\bar{x}\},\end{aligned}$$ then $$\begin{aligned}
a=\limsup_{p\to \bar{x}} d_g(p, \bar{x})^{n-4}u(p)<+\infty.\end{aligned}$$
If the lemma is not true, then for any $A>0$, there exists $r_i\to 0^+$ satisfying $$\begin{aligned}
u(p)>\,A\,r_i^{4-n},\,\,\text{for all}\,\,d_g(p, \bar{x})=r_i.\end{aligned}$$ Let $v_A=\frac{A^{\frac{n-2}{n-4}}}{2}G_1$ with $G_1$ in Lemma \[lemGreenfunction\]. For $i$ large, by maximum principle, $$\begin{aligned}
u(p)^{\frac{n-2}{n-4}}\geq v_A(p)\,\,\text{for}\,\,r_i <d_g(p, \bar{x})<\delta_0.\end{aligned}$$ As $i\to \infty$, it holds that $$\begin{aligned}
u(p)^{\frac{n-2}{n-4}}\geq v_A(p)\,\,\,\,\text{for}\,\,0<d_g(p, \bar{x})<\delta_0.\end{aligned}$$ Since $A$ can be arbitrarily large, $u(p)=\infty$ in $0<d_g(p, \bar{x})<\delta_0$, which is a contradiction.
\[propsingularity\] Let $u$ be as in Lemma \[lem9.3\]. Then there exists a constant $b\geq 0$ such that $$\begin{aligned}
\label{equsingularity9.1}
u(p)=bG(p, \bar{x})+E(p)\,\,\text{for}\,\,p\in B_{\delta_0}(\bar{x})-\{\bar{x}\},\end{aligned}$$ where $G$ is Green’s function of $P_g$, ( for the existence of the Green’s function in our application, it is limit of Green’s function of Paneitz operator of a sequence of metrics on $M$ restricted to certain domains, and when $g$ is the flat metric, let $G(x,y)=c_n|x-y|^{4-n}$) and $\delta_0$ is defined in Lemma \[lemGreenfunction\]. Here $E\in C^4(B_{\delta_0}(\bar{x}))$ satisfies $P_g E=0$ in $B_{\delta_0}(\bar{x})$.
We rewrite $(\ref{equlinear})$ as $$\begin{aligned}
\Delta_g(\Delta_g u)= \text{div}_g(a_nR_g g -b_nRic_g)\nabla_gu - \frac{n-4}{2}Q_gu.\end{aligned}$$ By Lemma \[lem9.3\], $0 < u(p) \leq a_1 G(p, \bar{x})$ with some constant $a_1>a$ in $B_{\delta_0}(\bar{x})-\{\bar{x}\}$ with $\delta_0>0$ in Lemma \[lemGreenfunction\]. Combining with the interior estimates, there exists a constant $C>0$ such that $$\begin{aligned}
\label{ineqboundlowerorderterms}
&|\text{div}_g(a_nR_g g -b_nRic_g)\nabla_gu - \frac{n-4}{2}Q_gu|\leq C d_g^{2-n}(p, \bar{x}),\,\,\text{and}\\
&\label{ineqLaplacian}|\Delta_g u(p)|\leq C\,d_g^{2-n}(p, \bar{x}),\end{aligned}$$ for $p\in \overline{B}_{\delta_0}(\bar{x})-\{0\}$. We define $G_2$ to be a Green’s function of $\Delta_g$ on $\overline{B}_{\delta_0}(\bar{x})$ such that $$\begin{aligned}
\label{inequkernel0}
0< G_2(p, q)\leq C d_g(p, q)^{2-n},\end{aligned}$$ for some constant $C>0$ and any two distinct points $p$ and $q$ in $B_{\delta_0}(\bar{x})$. Then $$\begin{aligned}
\phi_1(p)=\int_{B_{\delta_0}(\bar{x})}G_2(p, q)(\text{div}_g(a_nR_g g -b_nRic_g)\nabla_gu(q) - \frac{n-4}{2}Q_gu(q)) d V_g(q)\end{aligned}$$ is a special solution to the equation $$\begin{aligned}
\Delta_g \phi= \text{div}_g(a_nR_g g -b_nRic_g)\nabla_gu - \frac{n-4}{2}Q_gu,\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\}.\end{aligned}$$ Combining $(\ref{ineqboundlowerorderterms})$ and $(\ref{inequkernel0})$, we have that there exists a constant $C>0$ such that $$\begin{aligned}
|\phi_1(p)|\leq C d_g(p, \bar{x})^{4-n},\end{aligned}$$ for $p \in B_{\delta_0}(\bar{x})-\{\bar{x}\}$. Therefore, $$\begin{aligned}
\Delta_g(\Delta_g u- \phi_1)=0,\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\}.\end{aligned}$$ Since we also have $(\ref{ineqLaplacian})$, proof of Proposition 9.1 in [@Li-Zhu] applies and there exists a constant $-C\leq b_2\leq C$ such that $$\begin{aligned}
(\Delta_g u(p)- \phi_1(p))=b_2 G_1(p)+\varphi_1(p),\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\},\end{aligned}$$ with $G_1$ as in Lemma \[lemGreenfunction\] and $\varphi_1$ a harmonic function on $\overline{B}_{\delta_0}(\bar{x})$. Therefore, $$\begin{aligned}
\Delta_g u(p)=b_2 G_1(p)+\phi_1(p)+\varphi_1(p),\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\}.\end{aligned}$$ By the same argument, there exists $ b_3 \in \mathbb{R}$ such that $$\begin{aligned}
u(p)&=b_3G_1(p)+\phi_2(p)+\int_{B_{\delta_0}(\bar{x})}G_2(p, q)[b_2 G_1(q)+\phi_1(q)+\varphi_1(q)] d V_g(q)\\
&=b_3G_1(p)+\phi_2(p)+O(d_g(p, \bar{x})^{4-n})\end{aligned}$$ in $B_{\delta_0}(\bar{x})-\{\bar{x}\}$, with $\varphi_2$ a harmonic function on $B_{\delta_0}(\bar{x})$. But since $0 < u(p) \leq a_1 G(p, \bar{x})$, we have $b_3=0$ and $$\begin{aligned}
u(p)=b_2\int_{B_{\delta_0}(\bar{x})}G_2(p, q) G_1(q) d V_g(q)+o(d_g(p, \bar{x})^{4-n})\end{aligned}$$ in $B_{\delta_0}(\bar{x})-\{\bar{x}\}$. Therefore, there exists a constant $b\geq 0$ such that $$\begin{aligned}
u(p)&=bd_g(p, \bar{x})^{4-n}+o(d_g(p, \bar{x})^{4-n})\\
&=bG(p, \bar{x})^{4-n}+o(d_g(p, \bar{x})^{4-n}).\end{aligned}$$ Then by Lemma \[lemlinear9.1\], there exists a function $E \in C^4(B_{\delta_0}(\bar{x}))$ satisfying $(\ref{equlinear})$ and $$\begin{aligned}
u(p)=bG(p, \bar{x})^{4-n}+E(p)\end{aligned}$$ for $p\in B_{\delta_0}(\bar{x})-\{\bar{x}\}$.
The proof is a modification of Proposition 9.1 in [@Li-Zhu]. Let $G_1$ be as in Lemma \[lemGreenfunction\]. We will use maximum principle of the second order equation and $G_1$ to determine $b$. Define $$\begin{aligned}
b_1=b_1(u)=\sup\{\tau\geq 0,\,\tau G_1\leq u\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-\{\bar{x}\}\,\}.\end{aligned}$$ By Lemma \[lem9.3\], we know that $0\leq b_1 \leq c_na$.
[**Case I.**]{} $b_1=0$.
We [**claim**]{} that for each $\epsilon>0$, there exists $0< r_{\epsilon} < \delta_0$ such that for $0<r<r_{\epsilon}$, $$\begin{aligned}
\inf_{d_g(p, \bar{x})=r}\big(u(p)^{\frac{n-2}{n-4}}-\epsilon G_1(p)\big)\leq 0.\end{aligned}$$ Else, there exist $\epsilon_1>0$ and $r_j\to 0^+$ such that $$\begin{aligned}
\inf_{d_g(p, \bar{x})=r_j}\big(u(p)^{\frac{n-2}{n-4}}-\epsilon_1 G_1(p)\big)>0.\end{aligned}$$ Since $G(p)=0$ for $d_g(p, \bar{x})=\delta_0$, by maximum principle, $$\begin{aligned}
u(p)^{\frac{n-2}{n-4}}-\epsilon_1 G_1(p)\geq 0\,\,\text{in}\,\,B_{\delta_0}(\bar{x})-B_{r_i}(\bar{x})\end{aligned}$$ for all $i>0$. Therefore, the inequality holds for $p\in B_{\delta_0}(\bar{x})-\{\bar{x}\}$. Then $b_1\geq \epsilon_1>0$, a contradiction. This proves the [**claim**]{}.
Therefore, for any $\epsilon>0$ and $0< r <r_{\epsilon}$ there exists $p_{\epsilon, r}$ on $\partial B_r(\bar{x})$ such that $u(p_{\epsilon, r})^{\frac{n-2}{n-4}}\leq \epsilon G_1(p_{\epsilon,r})$. The Harnack inequality then implies $$\begin{aligned}
\sup_{\partial B_r(\bar{x})}u \leq C u(p_{\epsilon, r}) \leq C (\epsilon G_1(p_{\epsilon,r}))^{\frac{n-4}{n-2}}.\end{aligned}$$ Consequently, $$\begin{aligned}
u(p)=o(d_g(p, \bar{x})^{4-n})\end{aligned}$$ as $p\to \bar{x}$. By Lemma \[lemlinear9.1\], $u\in C_{loc}^{4, \alpha}(B_{\delta_0}(\bar{x}))$. Now let $b=0$. This proves the lemma for Case I.
[**Case II.**]{} $b_1>0$.
Let $v=u^{\frac{n-2}{n-4}} -b_1 G_1$. Therefore, $v\geq 0$ in $B_{\delta_0}(\bar{x})-\{\bar{x}\}$. By strong maximum principle, either $v=0$ in $B_{\delta_0}(\bar{x})-\{\bar{x}\}$, or $v>0$. For later case, we try the argument in Case I for $v$. By definition of $v$, $b_1(v)=0$ and as the claim in Case I, for each $\epsilon>0$, there exists $0< r_{\epsilon} < \delta_0$ such that for $0<r<r_{\epsilon}$ $$\begin{aligned}
\inf_{d_g(p, \bar{x})=r}\big(v(p) -\epsilon G_1(p)\big)\leq 0.\end{aligned}$$ Therefore, $v(p)=o(d_g(p, \bar{x})^{2-n})$?( In lack of Harnack inequality for $v$)
Therefore, for Case II, we have $v(p)=o(d_g(p, \bar{x})^{2-n})$ as $p\to \bar{x}$,(??) which implies that $$\begin{aligned}
&u(p)^{\frac{n-2}{n-4}}=b_1(u) \,d_g(p, \bar{x})^{2-n} (1+o(1)),\,\,\text{so that}\\
&u(p)=b_1(u)^{\frac{n-4}{n-2}}d_g(p, \bar{x})^{4-n} (1+o(1)),\end{aligned}$$ as $p\to \bar{x}$.
By $(\ref{expansion1})$, $$\begin{aligned}
c_nd_g(p, \bar{x})^{4-n}- G(p, \bar{x})=o(d_g(p, \bar{x})^{4-n})\end{aligned}$$ as $p\to \bar{x}$. It follows that $$\begin{aligned}
u(p) - c_n^{-1}b_1(u)^{\frac{n-4}{n-2}} G(p, \bar{x})&= o(1)\,d_g(p, \bar{x})^{4-n},\end{aligned}$$ as $p\to \bar{x}$. Denote $$\begin{aligned}
b=c_n^{-1}b_1(u)^{\frac{n-4}{n-2}}.\end{aligned}$$ By Lemma \[lemlinear9.1\], $E(p)= u(p) - b G(p, \bar{x})\in C_{loc}^{4, \alpha}(B_{\delta_0}(\bar{x}))$ and $E(p)$ satisfies $(\ref{equlinear})$ in $B_{\delta_0}(\bar{x})$.
This completes the proof of the proposition.
Using Proposition \[propsingularity\], we immediately conclude the following corollary.
\[cor9.1\] For $n\geq 5$, assume that $u\in C^4(B_{\delta_0}(\bar{x})-\{\bar{x}\})$ is a positive solution of $(\ref{equlinear})$ with $\bar{x}$ a singular point, and also the assumptions in Lemma \[lem9.3\] holds for $u$. Then $$\begin{aligned}
\lim_{r\to 0}\int_{B_r(\bar{x})}(P_g u- \frac{n-4}{2}\bar{Q} u) d V_g&=\lim_{r\to 0}\int_{\partial B_r(\bar{x})}\big(\frac{\partial}{\partial \nu}\Delta_g u-(a_nR_g\frac{\partial}{\partial \nu} u-b_nRic_g(\nabla_g u, \nu))\big) ds_g\\
&=b\,\lim_{r\to 0}\int_{\partial B_r(\bar{x})}\frac{\partial}{\partial \nu}\Delta_gG(p, \bar{x}) ds_g(p)=2(n-2)(n-4)|\mathbb{S}^{n-1}|\,b>0,\end{aligned}$$ where $\nu$ is the outer unit normal and $b>0$ is as in $(\ref{equsingularity9.1})$.
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Experience a happy blending of cosmopolitan and rustic lifestyle in Negros Occidental, the sugar capital of the Philippines and aiming to be the Organic Food Bowl of Asia.
Celebrate nature’s blessings, go hiking and camping, fishing and angling, scuba diving and snorkeling and mountain biking. Travel centuries back in time by visiting stately mansions in Silay City, one of the Top 25 destinations of the Philippines or go on Iron Dinosaurs (steam locomotives) and other specialty tours. | https://nro6.neda.gov.ph/province-of-negros-occidental/ |
Customer churn and its role in developing customer lifetime value models have been previously discussed. Churn rate - or its complement, retention rate is one of the key factors that modifies the customer lifetime value formula to properly account for future cash flows.
In this article we will show how to predict retention rates assuming we have the data in the right format. The process of getting the data in this format involves some amount of number crunching on the transactional data which will not be covered here. Typical transactional datasets will contain multiple rows for the same customer id, while the columns will include data such as date of purchase, purchase flag (yes or no) or purchase amount, etc.
The input data for churn prediction however is somewhat unique. Each row will contain the purchase amounts (or flags) for the last few transactions. For example, in the case of cell phone customer transactions, we could have a few hundred rows of data for one customer where each row corresponds to the number of minutes used by the customer in prior weeks. Therefore the columns of this row would contain minutes used 4 weeks earlier, 3 weeks earlier, 2 weeks earlier and the current week. A final column would be the label or target value which would contain a true/false statement corresponding to whether the customer is likely to churn next week or not. The input to our churn prediction model will be formatted as shown below.
In the above case, row 16 could correspond to the current week and we know for a fact that this customer (id= c_1) churned. Thus the last row of the dataset is now changed to "true". Note that what was prior week in row 16 is the current week for row 15 (observe the "489" entry for the two cells).
Once we have the data in this format we can easily train any learning algorithm to predict customer churn. For example, we can use Linear Regression and get a class recall of about 57% for all the true cases. A better technique would be Logistic Regression and we can get a class recall of about 67% for the true cases. In either case, the class recall for "false" cases range in the 90+% giving an overall accuracy of around 90%. The reason for this lower class recall for the "true" is the lower sample size of "true" cases. (Read how to address this data imbalance issue and get a better performance).
This is one of the unique advantages of a program like RapidMiner which would allow you to easily switch back and forth between various algorithms and choose the best fit for your data. Once you have a customer churn prediction model that you are happy with, the next step is to actually deploy this model on "live" data. For example, every week we can select a cohort of customers and examine who is likely to churn the next week based on their usage behavior.
The above example shows how this is done for one customer. The confidence (false) value can be plugged into the retention rate factor of the customer lifetime value formula to complete the calculation of CLV.
Download our free whitepaper on components of the CLV formula for more details. | http://www.simafore.com/blog/bid/113089/Predicting-customer-churn-for-use-in-customer-lifetime-value-formula |
I had pets in my younger life. In fact we had a pet when Emma was a baby. It was a rabbit named Abby, and it was a boy. He was awesome and super cute! But I don’t want any more pets. And I’m trying to teach my kids how much work a pet really is. I’d love for them to have the responsibility of looking after a living thing, but I don’t want to get stuck doing the work. Here are some other reasons we won’t get a pet.
The number one reason why we won’t get a pet is the time commitment. Not only the years of my life that a pet is alive for, but also the hours of the day. Both my husband and I work outside the home so we are not home during the day. And sometimes we’re not home at night either, when the kids are at extra-curriculars or birthday parties, or if we decide to go out for a family evening. I’m not planning my life around how many times I have to go home in a day so I can feed/walk/check on a pet.
My family has pets – if the kids want to see a dog, they’ll go to Nanny’s; a cat – that’s at Grandma and Grandpa’s (actually 3 cats but you know what I’m saying), and even at daycare there are dogs. I love the family pets, but what I don’t love is critter claws on my bare legs in the summer, or when the little dog licks my legs for no reason. And the pet hair – ick! I have enough trouble staying on top of the cleaning in my home without adding pet messes to it!
Now that my kids are out of diapers and baby stuff the last thing I want to be doing is still dealing with poop. When you get a pet you’re signing up for that critter’s lifetime of cleaning up its excrement. Strangely I don’t find myself wanting to do that at all. I’ve been training my kids for the last 7+ years and they’re getting it now – I don’t want to start again.
Don’t get me wrong, pets are cute and great fun. They can be wonderful companions and learning tools for kids, but they can also be neglected. We want to be able to travel whenever we want, without making concessions for our pet. I’ve done the pet-sitter thing, as both a pet sitter and a pet owner, and it’s not great on either side.
Lots of families choose to get a pet when their children are old enough to help look after one. I don’t think we’re going to be one of those families. | https://www.jessicafoley.ca/mommusings/we-wont-get-a-pet/ |
Planning the Garden
- Decide what type of garden to grow. What purpose do you want your garden to serve? Some gardens are functional, and produce fruit and vegetables you can use to feed your family or give away to neighbors. Others are more ornamental in purpose, serving to beautify your property and provide a pleasing sight to people passing by. If you’re not sure what type of garden you want, consider the following options:
- Vegetable gardens can include peppers, tomatoes, cabbages and lettuces, potatoes, squash, carrots, and many other vegetables. If a vegetable is able to grow in your region, you can find a way to grow it in your yard.
- In flower gardens, different types of flowers may be strategically planted so that something is in bloom almost all year long. Some flower gardens are structured, with flowers planted in neat rows and patterns; others are wilder in appearance. Your personal style and yard type will determine what type of flower garden you might plant.
- Herb gardens often complement both flower and vegetable gardens, since they tend to flower beautifully and also serve the functional purpose of adding flavor to your food. Herb gardens might include rosemary, thyme, dill, cilantro and a variety of other herbs you may want to use to make dried spices and teas.
- In general, vegetable gardens have the highest soil and maintenance requirements. Flowers and herbs can withstand more neglect than vegetable plants.
- Decide what specific plants to include in your garden. Find out what grows well in your region garden zone by using this zone finder to determine what zone you are in, then research which plants do well in your area. As you find out more about your options, make a list of the plants you want to buy and the best time of year to plant them.
- Some plants don’t grow as well in certain zones. If you live in a place with mild winters and hot summers, you may have trouble growing plants that require a cold snap to grow properly.
- Unless you plan to make your garden relatively large, try to choose plants that need similar growing conditions. Do they need the same soil type and sun exposure? If not, you may have to create a garden with several types of growing conditions, which can be complicated in a small garden.
- Visit a farmers market or plant sale in the spring. Often, you can learn useful information from vendors and buy healthy plants that grow well in your area. GO EXPERT ADVICE Consider the season. According to the owners of Grow it Organically, a gardening company in California: “The best time to start a garden is after the last frost date in the spring. You can usually find that information out from the agricultural tables for the county where you live. And if you have a long season, you may be able to plant again in August or September.”
- Choose a spot for your garden. Take a look around your yard to assess the where you want the garden to be located. The location you choose should both help the garden serve the function you want it to have and also be in a spot that is conducive to growing strong, healthy plants.
- No matter what type of garden you’re planting, most plants will grow better in rich, well-drained soil. Avoid spots in your yard where water seems to stand for awhile after a heavy rain, as this could indicate the soil there is too soggy or clay-based for healthy plant growth.
- Most vegetables grow best with a lot of sunlight, so if you’re planting a vegetable garden choose a spot that isn’t shaded by trees or your house. Flowers are more versatile, and if you’d like a flower plot next to your house you can choose flowers that grow best in partial or full shade.
- If your soil isn’t high quality, you can make a raised bed and grow flowers or vegetables there. Raised beds are planting beds that are built on top of the ground within wood frames that are filled with soil.
- If you don’t have a yard, you can still have a garden. Plant flowers, herbs and certain vegetables in large pots on your patio. You can move them around according to the amount of sun the plants need.
- Make a garden design. Draw an outline of your garden or yard space. Map out different options where you want to plant various items in the location you chose. Tailor the design to fit the needs of your plants, making sure the ones that need shade will be planted in shade, and the ones that need full sun are in an area that isn’t cast in shadows during the day.
- Take into account the space each plant will need, both at planting time and after it starts to mature. Make sure everything you want to plant will fit in your garden and have enough space to spread out.
- Take the timing into account. Many plants need to be planted at different times in different zones. For example, if you live in a region with mild winters and hot summers, you may need to plant your flowers earlier than you would if you lived in a region with cold winters and shorter summers.
- If you’re planting a vegetable garden, design it so that it’s convenient for you to walk into the garden and harvest vegetables as they ripen. You may want to make a path through the garden for this purpose.
- Flower gardens should be designed with aesthetics in mind. Choose colors that look pretty together, and make patterns that are pleasing to the eye. Keep in mind when different flowers will begin to bloom.
- Also take your lifestyle into account. Do you have children or pets who might run through the area? Is the garden within reach of your hose? Is it too close or too far away from your living space?
Getting Ready to Plant
- Buy gardening supplies. It takes a lot of equipment to plant a garden, but once you buy most of the supplies they will last through many gardening seasons. You’ll find the best selection at a home and garden store or a nursery. Gather the following supplies:
- Seeds or young plants. You can choose to either start your garden from seeds or buy young sprouted plants that already have a head start. Check the list of plants you intend to grow and buy as many seeds or young plants as you need for the different components of your garden.
- Soil fertilizer and topsoil. Bone meal, blood meal, and other fertilizers help your plants grow healthy, and a layer of topsoil is useful to have in case you’re planting something that needs extra protection.
- Compost. You can mix compost into the soil to improve its moisture retention, buffer pH, and provide micro nutrients. You can buy compost or make your own.
- Mulch. Many plants require a layer of mulch to protect them from inclement weather and extreme temperatures while they’re in the early stages of growth. Mulch should be spread on top of the soil to help retain moisture and reduce weeds.
- Soil tilling equipment. If you’re planning to create a large garden, you might want to buy or rent a soil tiller, which is wheeled over the ground to break up the soil and make it into a soft plant bed. For smaller areas, a garden rake and hoe should be sufficient.
- A shovel and spade. This equipment makes it much easier to dig the proper-sized holes for seeds or young sprouts.
- A garden hose. Get a hose with a fixture that allows you to either lightly mist or fully spray plants, depending on what each one needs. Alternatively, if you are planting a large area, a sprinkler and even an automatic timer will save you time.
- Fencing materials. If you’re planting a vegetable garden, you may need to construct a fence around it to keep bunnies, squirrels, deer, and the neighborhood pets from taking ripe vegetables.
- Prepare the soil. Use the soil tiller or garden rake to break the soil in the area you mapped out for your garden. Work the soil to a depth of about 12 inches (30.5 cm), making sure it is loose and does not have large clumps. Remove rocks, roots, and other solid objects from the garden bed, then fertilize it and work in compost to prepare it for planting.
- How your plants grow depends on the quality of the dirt. You can buy a soil testing kit to find out the amount of organic matter, the amount of nutrients, and the pH level of the soil. Use the results to determine how much fertilizer and other ingredients to add. Alternatively, you can take a soil sample to your county extension office and they will test the soil for free or a small fee.
- Don’t add more fertilizer than the directions tell you to. Extra fertilizer can be toxic to plants. Note that not all plants like very fertilized soil; some would actually benefit from poor soil, so remember to find out the soil preferences of the plants you choose.
- If your soil test shows a pH level that is too acidic, add limestone to raise the pH. If your soil is alkaline, you can add cottonseed meal, sulfur, pine bark, compost, or pine needles to make it more acidic.
Growing the Garden
- Plant the seeds or young plants according to your design. Use the spade to dig holes spaced a few inches apart, or as indicated on the packaging of the seeds or young plants you bought. Make sure the holes are as deep and wide as they need to be. Place the seeds or plants in the holes and cover them with soil. Pat the soil gently into place.
- Fertilize as necessary. Depending on the plants you choose to grow, you may need to fertilize the garden again after planting. Some plants may need more fertilizer than others, so make sure you only use it in the areas that require it.
- Add compost, mulch or topsoil as necessary. Some types of plants require a thin covering of compost, mulch or topsoil to protect them during seed germination and while the plant is young and fragile. Spread the material by hand, or use a soil spreader to cover a larger area.
- Some types of compost or mulch aren’t appropriate for certain plants. Conduct research on the produce you’re growing to make sure you use the right ground cover.
- Too thick a layer could in inhibit growth, so make sure you add only as much as each type of plant needs.
- Water the garden. When you’re finished planting and treating the soil, use the garden hose’s “sprinkler” setting to thoroughly dampen the garden. Continue watering the garden every day, adding more or less water to different areas according to the plants’ needs, for the first few weeks after planting.
- Over saturating the soil could drown the seeds and prevent the plants from growing. Don’t water to the point where streams of water run through the garden.
- Never let the soil completely dry out. Watering once a day is sufficient.
- Once the plants have sprouted, water in the morning, rather than at night. Water sitting on the leaves and stems all night can lead to the production of mold and other plant diseases.
- After a few weeks, reduce how often you water the plants. Give the garden deep watering two or three times a week or as needed.
- Weed the garden. Sprouting weeds take nutrients from the soil, leaving less for your vegetables or flowers. Weed the garden every few days to make sure your plants get the nutrients they need – just be careful not to pull up sprouting garden plants.
- A stirrup hoe will help remove weeds before they get too big. You can run the hoe along the soil beside plants and knock the weeds down.
- Consider erecting a small fence. If you see small animals such as rabbits, squirrels, deer, and gophers or voles in your garden or in your neighborhood, you may want to go ahead and put a fence around the garden to protect it. A two or three foot tall fence should be tall enough to keep small creatures out. If you have deer in your area, the fence may need to be as high as eight feet. | http://homeyardandgardendepot.com/6/ |
69% of the survey respondents mentioned that they sometimes dealt with private matters during working hours. Male respondents reflected this trend a little higher (70%) than female respondents (67%).
76% of the employees mentioned that they do not mind handling work-related matters in their private time and 75% from India indicated that they do this because they would like to stay involved. An almost equivalent number of male and female respondents seemed to share the same views.
84% of the respondents mentioned that they prefer having an option to choose if they get their holidays in time off or paid money. 82% said they respond to work-related calls and emails immediately after the regular office hours.
Moorthy K. Uppaluri, MD and CEO, Randstad India said: “Today, the way India Inc. works has undergone a paradigm shift. Going to work is all about getting work done regardless of where one is. Employers should be cognizant of this scenario where the boundaries of work and life are getting blurred and hence offer their workforce with the much needed flexibility and freedom at work to keep them motivated”.
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Under the lens
Asian Paints’ related-party transactions: what’s worrying the whistleblower and the proxy advisor? | https://economictimes.indiatimes.com/articleshow/49076220.cms?utm_source=contentofinterest&utm_medium=text&utm_campaign=cppst |
We couldn't help ourselves. After our blackberry picking session on the way home from school we wanted more. We had the taste for these dark jewels with the blackberry crumble traybake and it wasn't enough. We waited a couple of days and then took our chance on a sunny afternoon. Off we went on our bikes to a new spot where the blackberries had grown even bigger and there was something else hiding in the trees – apples. In total we found three trees with different varieties of apple. I can only presume these have all grown from discarded apple cores. So we came home laden with blackberries and apples. What a beautiful combination.
One crumble was made to have that night and and another to be put away in the freezer for a treat to savour another day. With a bowl of blackberries left and apples aplenty I decided I had enough to make a jar of jelly. The great thing about making a jelly with blackberries is that all the seeds are eliminated leaving a smooth preserve. Blackberries are naturally low in pectin so normally you would need to add something to help the jelly set. Apples on the other hand are packed full of pectin. With them both ripe at the same time it's obviously a pairing nature approves of.
Equipment: Large high sided saucepan, jelly strainer stand, jelly bag or muslin square, glass jug, 2 small plates/saucers, freezer, 1 standard sized jam jar and lid.
1. Put the fruit into the saucepan and add the water.
2. Bring to the boil and squash the fruit (I use a potato masher) to help extract the juices. Reduce to a simmer.
3. Cook for about 30-45 minutes making sure the fruit doesn't burn.
4. Set up your jelly strainer and put a glass jug underneath.
5. Spoon some the fruit and juices into the jelly strainer. Add small amounts until all the mixture is in the strainer.
6. Leave to strain for about an hour. Do not be tempted to squeeze the bag as this will result in a cloudy jelly.
7. Check the volume of liquid and calculate the amount of sugar required. 600ml requires 450g. 500ml requires 375g of sugar.
8. Put the plates or saucers into the freezer.
9. Ensure the jar is clean and dry. Put into the oven to sterilize at 120°C/Gas mark ½-1.
10. Pour the jelly liquid back in the large saucepan (ensure no bits remaining in the pan) and bring to the boil. Add the sugar and stir gently to dissolve.
11. Keep at a rolling boil for about 10-12 minutes and then try the setting point.
12. Take one of the plates out of the freezer and drop a small amount of the liquid onto the plate. Push it with the spoon and if it wrinkles and moves it is ready to bottle. If not keep boiling and try again in a minute.
13.Quickly take the pan off the heat and take the jar out of the oven (remember it will be hot!).
14. Pour jelly into the jar. Skim any scum off the top. Put the lid on immediately to ensure that 'pop' when first opened.
I love crumble, and blackberries, but have never made jelly before. Think I am going to try this. Thank you!
I love jellies over jam because I'm not a fan of the pips. I bet this is gorgeous on a nice warm scone.
My kids prefer the jelly consistency to jam. The only reason I don't make my own is because of the high sugar content required.
Oh I might give this a try, every time I have tried jams they fail me!
yum! i just opened the last jar of blackberry jam I made last year - so it must be time to make more!
This sounds amazing! I have never had a go at making jam but would love to.
This is perfect for me as we've loads of apples and blackberries at the moment. Thank you!
Great timing - I've just nobbled so windfall apples from a neighbour and plan to do something yummy with them. I've never made jam before - now might be the perfect time to try!
I was so happy that I could give my son a berry picking experience when he was four years old and we visited my family in Finland. It's such a beautiful childhood memory for me.
Noooooo!!!! I made blackberry and apple jam this week too! We must be living in the same head right now!
Lovely. I'm a big fan of jellies and there's always room for one more. I must pick more blackberries - the last lot went into ketchup.
This sounds perfect as we have both growing in the garden. Will certainly give it a try.
Oh my goodness that sounds delicious. We have so many growing near here too so that's a perfect way to deal with them! | https://www.jibberjabberuk.co.uk/2014/09/blackberry-and-apple-jelly.html |
, moon, moon, boom, boom, boom even brighter than the moon, moon, moon ♪ >> oh, yeah. fireworks in the sky over our nation's capitol. january 20, 2021. this has been a uniquely historic inauguration day. thanks to all of our performers, friends and neighbors who celebrated america with us. and once again, congratulations president biden and vice president harris. godspeed your efforts. from the lincoln memorial, good night, america. >> extraordinary fireworks display over washington, d.c. as the celebration of the inauguration of president biden comes to a close. welcome back to our continuing coverage, an extraordinary display not only of fireworks but of star power coming out for this new administration celebrating the united states and celebrating clearly what they hope will be a big difference from the last administration. van jones? >> shock and awesome. after the rain the rainbow. this is -- what you saw was a psa advertising the america that we can be, the america that biden is promising is available to us. and you got the chance to see it. and the thing about, | https://www.aps.org.wstub.archive.org/details/tv?and[]=publicdate:[2021-01-01+TO+2021-01-31]&q=moon&red=1 |
Cute cottage approximately .75 Acres (to be subdivided by seller) in the beautiful East Feliciana Countryside. Home is over 80 years old and has been partially restored. One bedroom has been converted to a master suite. Exterior has been restored with stucco and hardiplank siding. Needs some work but with a little imagination could be the opportunity of a lifetime. Schedule a private tour as soon as possible. Seller will be sub dividing the property to remove the rear portion of the lot where the barn & mobile home currently sites. Seller will complete re-subdivision prior to closing.
Features and Details
- Category Residential
- Type Single Family Residence
- Area/Neighborhood Rural Tract (No Subd)
- County East Feliciana
- Lot Size 210' x 312' x 84' x 300'
- Acreage 1.05 Acres
- MLS ID# 2018012453
- Beds Total 2
- Baths Total 2
- Baths Full 2
- Baths Half 0
- Levels 1
- Garage Capacity 0
- Living Area 1198 sqft
- Building
- Laminate Floor, VinylSheet Floor, Traditional Style
- Construction
- Asbestos Siding, Cement Fiber Siding, Stucco Siding, Asphalt Comp Shingle Roof, Frame Const, Laminate Floor, VinylSheet Floor
- Exterior Features
- Barn, Deck, Porch, Storage Shed/Bldg. | https://www.mlsbox.com/listing/2073-La-957-Hwy/5b541052de75bee430f78207 |
Do you have a request or concern; questions about your water, sewer, & garbage bill; maybe a question about zoning, floodplain and or building regulations?
No matter the question or concern, please feel free to use the contact form below. Simply select the appropriate department you wish to contact from the drop-down menu, fill out the remaining information and let us know how we can help you. Any email sent over the weekend or holiday may not be read immediately.
After Hours Village Emergency Contact Number (309) 792-3169.
The after hours emergency number should be used to report water main breaks, request emergency curb-box shut offs, report downed trees in the roadway, etc. DO NOT use this number to pay your bills! Please all (309) 277-9352 or visit our bill payment options page.
Rock Island County Sheriff's Office (309) 788-8988.
Office Hours: Monday - Friday: 9:00am to 5:00pm
E-mail correspondence is maintained in accordance with applicable provisions of State law and may be considered public records. Public disclosure of your communication may be required. | https://carbon-cliff.com/contact-us |
Solar Insolation Maps of the United States
On this page you will find a map of solar insolation values for the United States. Also, there are image links below the map that will take you to regional solar insolation maps for the different U.S. regions.
What is solar insolation?
According to SolarInsolation.org, "solar insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter (W/m2) or kilowatt-hours per square meter per day (kW•h/(m2•day))(or hours/day). In the case of photovoltaics it is commonly measured as kWh/(kWp•y)(kilowatt hours per year per kilowatt peak rating).
Solar insolation is very important in determining what size a solar power system needs to be in order to meet the expectations of the system's owner. For instance, in northern latitudes there is less solar insolation and therefore more solar panels will be required than are needed in southern latitudes to produce the same amount of electricity.
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You can also use these maps which include global solar insolation maps: | https://www.mrsolar.com/solar-insolation-maps-of-the-united-states/ |
What Is 30/75 as a Decimal + Solution With Free Steps
The fraction 30/75 as a decimal is equal to 0.4.
The fraction 30/75 can be written in decimal form by carrying out the division of the numerator by the denominator. The fraction 30/80 is a proper fraction. A proper fraction is a fraction that has its numerator value less than the denominator.
Here, we are more interested in the division types that result in a Decimal value, as this can be expressed as a Fraction. We see fractions as a way of showing two numbers having the operation of Division between them that result in a value that lies between two Integers.
Now, we introduce the method used to solve said fraction to decimal conversion, called Long Division, which we will discuss in detail moving forward. So, let’s go through the Solution of fraction 30/75.
Solution
First, we convert the fraction components, i.e., the numerator and the denominator, and transform them into the division constituents, i.e., the Dividend and the Divisor, respectively.
This can be done as follows:
Dividend = 30
Divisor = 75
Now, we introduce the most important quantity in our division process: the Quotient. The value represents the Solution to our division and can be expressed as having the following relationship with the Division constituents:
Quotient = Dividend $\div$ Divisor = 30 $\div$ 75
This is when we go through the Long Division solution to our problem. The following figure shows the solution for fraction 30/75.
30/75 Long Division Method
We start solving a problem using the Long Division Method by first taking apart the division’s components and comparing them. As we have 30 and 75, we can see how 30 is Smaller than 75, and to solve this division, we require that 30 be Bigger than 75.
This is done by multiplying the dividend by 10 and checking whether it is bigger than the divisor or not. If so, we calculate the Multiple of the divisor closest to the dividend and subtract it from the Dividend. This produces the Remainder, which we then use as the dividend later.
Now, we begin solving for our dividend 30, which after getting multiplied by 10 becomes 300.
We take this 300 and divide it by 75; this can be done as follows:
300 $\div$ 75 = 4
Where:
75 x 4 = 300
Finally, we have a Quotient generated after combining the one piece of it as 0.4, with a Remainder equal to 0.
Images/mathematical drawings are created with GeoGebra. | https://www.storyofmathematics.com/fractions-to-decimals/30-75-as-a-decimal/ |
It isfairly certain that the projected taking over of most jobs in the not too distant future by new technologies, like artificial intelligence, will occur.
Gitura Mwaura
It probably is, therefore, not merely just a matter of time before the post-work scenario, whose viability and moral justification has long been debated, will become a reality.
With this, there has been growing literature, beginning with Karl Marx in the 19th Century who saw virtue in less work in a communist ideal to the present by contemporary thinkers lamenting the crises of work— crises that have been characterised by rising inequality as witnessed in low pay and dismal rate of growth in wages, underemployment and unemployment, and the threat to work by the digital revolution and climate change.
Even as the push for technological Africa in an increasingly digital world continues, there are many who think post-work imminent. First, however, consider one of the more persuasive arguments discussed in The Guardian recently that, contrary to conventional wisdom, the idea of work as we know it today is neither natural nor very old.
It quotes, among others, Benjamin Kline Hunnicutt, an American professor of leisure studies who has extensively explored the history of movements for reduced working hours and shorter working days.
He argues that work is “an accident of history,” and that it is a recent construct. He identifies the main building blocks of our work culture as 16th-century Protestantism, which saw effortful labour as leading to a good afterlife; 19th-century industrial capitalism, which required disciplined workers and driven entrepreneurs; and the 20th-century desires for consumer goods and self-fulfillment.
Before then, Prof Hunnicutt says, all cultures thought of work as a means to an end, not an end in itself. From urban ancient Greece to agrarian societies, work was either something to be outsourced to others – often slaves (and serfs even in traditional African settings) – or something to be done as quickly as possible so that the rest of life could happen.
As one always on the grind and often without the time to engage in my more pleasurable, if artistic, pursuits, I am persuaded to agree with the post-work thesis. I want to believe that somewhere in the future humanity will have to dispense with the idea of work and free humanity to engage in passions as pleases each one’s fancy to pursue happiness and a fulfilled life.
It is, nevertheless, not a light thing to question the place of work; it’s not only the means people are able to put food on the table and afford their preferred lifestyles but among the most important sources of identity and purpose in many individuals’ lives.
Yet, looking at the negative human impact on global environment often due to work-related activities, many remain inclined to agree with other post-work advocates such as David Frayne who’s quoted to have quipped that “either automation or the environment, or both, will force the way society thinks about work to change.” His 2015 book, The Refusal of Work, has been touted as one of the most persuasive post-work volumes.
Still, if we are to buy the post-work argument and accept we are destined to forfeit most of our jobs to artificial intelligence, how then to put food on the table?
This, too, has long been anticipated with one of the possible options being the provision of universal basic income (UBI), already under experimentation around the world including in the region (see “Biggest basic income experiment slated for East Africa”, February 10, 2017, The New Times).
UBI assures every person – the employed and unemployed, rich or poor – a small income to cover basic living costs with no questions asked about what one does with the money.
And, aside from ongoing government-led UBI experiments in countries such as Finland, Canada and the Netherlands, it is probably no coincidence that the largest trial to test the viability of the concept is being led by tech companies, perhaps as a prick to their techie conscience and atonement for the concentration of riches in the hands of a few as artificial intelligence threatens to obliterate livelihoods and the dignity work has hitherto afforded.
In November last year, the charity organisation, GiveDirectly, launched in Kenya the largest trial of basic income to date costing US$30million. Underwritten by some of the major the tech companies, it is significant in terms of both size and duration.
The organisation explains that residents of about 120 rural Kenyan villages, comprising more than 16,000 people in total, will receive some type of unconditional cash transfers during the experiment.
Some of the villages will receive the universal basic income of about US$23per resident per month for twelve years.
The study is expected to come up with some of the most comprehensive data yet about what happens when people are given money for nothing.
It’ll help answer questions such as: Do people stop working? Do they start businesses? Are they more likely to spend money on drugs and alcohol — or education?
While the results are keenly anticipated, sustainability concerns notwithstanding, earlier small-scale experiments have suggested quite some encouraging outcomes.
As for post-work future, I think we should brace for the day of its coming.
Twitter: @gituram
The views expressed in this article are of the author.
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The invention relates to the technical field of rail traffic and particularly relates to an automatic correction method and device for kilometer posts in rail traffic. According to the method, a number of fixed signal sources are distributed on a rail traffic line, signals on the way are transmitted through a leaky coaxial cable, when a train provided with a signal receiving device is driven to the position of one signal source, the obtained signal intensity is highest, and meanwhile, the received signal intensity is correspondingly changed along the change of the distance from the position ofthe train to the signal sources and reaches the peak value again at the next signal source; a relation between the distance from a certain signal sampling point to the signal source and the signal weak attenuation intensity is analyzed by virtue of the stationarity of a driving line; and the kilometer posts are determined according to the fixed point positions of the signal sources, the highest signal intensity and the relation between the attenuation intensity and the distance. | |
Following a spirited discussion, the State Industrial Commission altered its rule governing travel expense reimbursement for injured workers seeking medical treatment.
The new rule allows injured workers to be paid 25 cents per mile or actual reasonable costs of practical transportation, above the 25 cents per mile, that may be required due to the nature of the disability.In proposing the new rule, Joyce Sewell, director of the commission's Industrial Accidents Division, said she has received complaints from injured workers that the old rule was arbitrary in that it allowed workers' compensation insurance carriers to pay the least expensive mode of travel when another mode was necessary.
But Richard Sumsion, an attorney for the Workers' Compensation Fund of Utah, said the new rule allows an injured worker to to be paid 25 cents per mile, take the bus and then pocket the difference.
He said the money the injured worker obtains is probably tied to the longer distance the person must travel to obtain medical treatment.
But Patrick J. O'Connor, president of the Injured Workers Association of Utah, said the new rule doesn't go far enough.
Rather than have workers' compensation insurance carriers pay for travel expenses for medical treatment, O'Connor wants injured workers to be reimbursed for travel in connection with rehabilitation; meetings with commission personnel, insurance adjusters or attorneys; to pick up a compensation check; meet with an employer; or pick up medical supplies or prescriptions.
O'Connor said when people come to his organization because they have a complaint about the workers' compensation system, he usually asks them if they are receiving travel expenses to receive medical treatment. Most of the people don't know they can be reimbursed for medical expenses.
The three commissioners believe the number of cases where people will make money off the travel reimbursement will be low.
Commissioner Thomas Carlson said if experience shows the rule should be changed, it is a relatively easy thing to do. "They aren't set in concrete," he said. | https://www.deseret.com/1991/3/14/18910286/panel-oks-rule-on-travel-reimbursement-for-injured-workers-seeking-treatment |
Helping hospitalized smokers in Hong Kong quit smoking by understanding their risk perception, behaviour, and attitudes related to smoking.
To understand the risk perceptions, behaviour, attitudes, and experiences related to smoking among hospitalized Chinese smokers. Understanding hospitalized smokers' perceptions of risks associated with smoking, along with their behaviour, attitudes, and smoking-related experiences, is essential prerequisite to design effective interventions to help them quit smoking. A phenomenological research design was adopted. A purposive sampling approach was used. Between May 2016-January 2017, 30 hospitalized smokers were invited for an interview. Four themes were generated: (a) associations between perception of illness and smoking; (b) perceived support from healthcare professionals to quit smoking; (c) impact of hospitalization on behaviour, attitudes, and experiences; and (d) perceived barriers to quitting smoking. Development of an innovative intervention that helps to demystify misconceptions about smoking through brief interventions and active referrals is recommended to enhance the effectiveness of healthcare professionals promoting smoking cessation for hospitalized smokers. To date, no study examining smoking behaviour among hospitalized patients in Hong Kong has been conducted. Misconceptions about smoking and health, barriers to quitting that outweighed perceived benefits, lack of support from healthcare professionals, and difficulty overcoming withdrawal symptoms or cigarette cravings precluded hospitalized smokers sustaining smoking abstinence after discharge. Smoking is detrimental to physical health. Smoking cessation has beneficial effects on treatment efficacy and prognosis and helps to reduce the economic burden on society from smoking-attributable diseases.
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How Many Server Instances should I run?
Web Connection supports and should be run in production with multiple server instances to allow for multiple requests to be processed simultaneously. Both File and COM modes support multiple server instances.
Multiple instances are necessary so requests can process at the same time. A single instance can only process one request at a time, so if two long requests come it at the same time one has to wait. Multiple instances solve this problem.
You can configure Web Connection to use a specific number of instances to run simultaneously. Web Connection can start a configured number of server instances for you and in the case of COM can monitor those instances and if they crash restart them.
Starting Multiple Instances
Depending on how you run Web Connection there are a number of ways that you can start new instances. For file mode in particular you have several options, while COM mode can only be launched through the Web interface or automatically.
File Mode - Run multiple Exes
For File Mode to run multiple instances you can simply start up multiple instances from Explorer or the Windows terminal.
You can also start new instances from the .NET Module Admin Console.
The
web.config also includes an
AutoStartServers flag which applies only to file mode. When
true it automatically starts servers if they are not already running when the application is first hit.
COM Mode
COM Mode can only be 'started' through configuration which is done by switching into COM mode and specifying a server count for the number of instances to fire up.
COM instances are autoamtically started when a hit comes in or you can explicitly launch the servers using the 'load servers' button in the admin form.
How Many Instances Should I Run?
How many instances you run depends on the kind of application you are running and also on the hardware that you are running on.
The base guideline is this:
For optimal CPU usage use 2 instances per CPU physical or virtual CPU Core. At minimum I recommend 2 total instances to allow for slow processing of 1 instance.
So if you have a 4 core computer you can easily run 8 instances of Web Connection simultaneous.
For most small applications, you probably don't need that many instances to run an effective application. However, I highly recommend running at least 2 instances for anything but the simplest, non-busy applications. Having at least 2 instances will ensure that if you have a slow request you have at least something to fall back to.
Personally I tend to run most of my applications with 2 instances - for example the message board and Weblog both run with 2 instances. While neither is very busy both have a few slow requests (searching in both cases) and having the extra instances frees up the application during searches.
Figuring out best load factor
If you have a busy application and you want to find the right amount of instances that you can load up, there's an good process you can use to check for load.
Note the following only works for COM mode
- Set the Web Connection Server into Load Based Server Loading
- Set the server count near the high end you expect
- Run a load test (you can quickly set one up with WebSurge)
- Go to the Web Connection Handler Page
- Watch the # of Hits
- Where the # trails off or goes to 0 is the sweet spot
Notice that after 8 instances the request count drastically drops off. This likely is because I didn't actually generate enough load to keep the server busy but this is what the simulated load emulates.
This doesn't automatically mean that that spot is where you should run your servers. If you load too much traffic and you are swamping the CPU with too much traffic, running at the max load where it trails off may very well mean that you are maxxed out on CPU resources. If that's the case, try making the server count number lower.
Essentially using this approach you're looking where your server is topping out or the requests that are processing are not actually enough to overload your server instances. You need to look at this number in combination with the CPU load of the server - if the server is maxing out CPU, then you have to try to dial down the number of instances and hope that you still handle traffic. In some cases you may simply have too much traffic for a machine to handle in which case you have to think about load balancing across machines.
IIS on Windows Client OS is limited to 10 simultaneous connections
If you're testing on a Desktop Windows Client Install, please note that IIS is limited to 10 simultaneous connections. This means that on a local machine you're likely to see a drop to see a bottleneck at 10 instances for high volumne requests. For realistic server load testing for rapid fire requests you may have to check load on a server.
Scenarios
There are a number of different scenarios for applications and how you can handle instance count.
Busy High CPU Application
Everything I've talked about above - 2 Instances per CPU specifically - applies mostly to CPU intensive applications. Basically if you're application is CPU bound it'll use as much resources as are available because FoxPro is not very good at sharing. You do not want to run 100 instances simultaneously chewing on CPU intensive operations as it will drag the CPU to 100% and leave it maxed out. Less can be more when it comes to scalability of FoxPro processing.
IO Heavy Application
If your application uses a remote SQL Service on a different machine, a lot of your heavy duty processing is done on another machine, while your application sits and waits on the remote server call to return. This doesn't tie up the CPU nearly as much as local data processing and it means you can run more instances locally without critical overhead.
Low Volumne Applications
If you have an application that isn't very busy I'd recommend running with 2 instances. Even if the application is very lowly loaded having that second instance tremendously helps responsiveness of the application.
Low Volume Applications with Long Requests If you have an application that's not very busy but has several long running requests you might want to run more instances than you normally would. Although the same CPU load issues mentinoned above apply, you'll want to have the extra instances to avoid locking up your requests, due to all servers being busy.
Use Single COM Mode for Long Requests
If you have a few very long requests in your applications you can use Single COM Mode to run these requests on COM servers that are started up to run that single task. Rather than using one of the server instances from the COM thread pool, a brand new instance is started in addition to the ones in the pool and run independently. The server is the same server, but it's run as a single request and then shut down. For more info see, Running long Requests in Single COM Mode.
Load Related Configuration Values
There are a number of settings related to server loading. The most critical one to instance counts is the
ServerCount property which is also set on the Module Admin form.
<add key="ServerCount" value="12" /> <!-- COM Server Settings --> <add key="ComServerProgId" value="wcdemo.wcdemoServer" /> <add key="ComServerLoadingMode" value="LoadBased" /> <!-- File Server Settings --> <add key="AutoStartServers" value="True" /> <add key="ExeFile" value="~\..\wcdemo.exe" />
For file based servers
AutoStartServers determines whether servers are automatically started if they are not already running. The
ExeFile points at the EXE that is launched which should be your Web Connection server EXE.
For COM Servers you provide the COM Server ProgId to identify the COM server that should be loaded.
The ServerCount is applied to either File or COM based messaging. | https://webconnection.west-wind.com/docs/_5hj000feu.htm |
---
abstract: 'In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable $\eta$ and then transform it using a change of variable $\rho=\sqrt{\eta_x}$. The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.'
address: 'Department of Mathematics, The Graduate Center, City University of New York, NY 11106, USA'
author:
- Jae Min Lee
title: 'Global Lagrangian Solutions of the Camassa-Holm equation'
---
Introduction
============
The Camassa-Holm(CH) equation $$u_t-u_{txx}+3uu_x-uu_{xxx}-2u_x u_{xx}=0,\;\;t\ \in {\mathbb{R}},\;x \in S^1.$$ was originally derived as a model for shallow water waves and has remarkable properties like infinitely many conservation laws via bi-hamiltonian structures [@AK1999; @CH1993; @Con2; @FF1981] and soliton-like solutions [@BSS2000; @ConM1999; @ConStra; @ConStra2]. Also, the solutions of this equation can be interpreted as geodesics of the right invariant Sobolev $H^1$ metric on the diffeomorphism group on the circle [@EM1970; @KM2003; @K2007; @Mis1998]. In fact, in terms of the diffeomorphism $\eta$ defined by the flow equation $$\label{flow}
\frac{\partial \eta}{\partial t}=u(t,\eta(t,x)),$$ the CH equation is rewritten as an ODE on the Banach space and the local well-posedness of the corresponding Cauchy problem is obtained by using the standard ODE technique.
It is known that some solution $u(t,x)$ of the CH equation develops a finite time singularity [@M1; @Con; @ConE2; @ConE3]. The mechanism for this breakdown is called a wave breaking; the solution remains bounded but its slope becomes unbounded at the breakdown time. Wave breaking can be nicely illustrated in terms of peakon-antipeakon interaction. When these two waves collide at some time, the combined wave forms an infinite slope. After this collision, there are two possible scenarios; either two waves pass through each other with total energy preserved, or annihilate each other with a lose of energy. The solutions in the former case is called conservative and the latter case is called dissipative. In this paper, we focus on the conservative solution case.
The continuation of the solutions after wave breaking has been studied extensively. There are different situations where such solutions are constructed; type of domain(periodic/non-periodic), vanishing/non-vanishing asymptotics in the case of non-periodic domain, energy preservation after the breakdown(conservative/dissipative), etc. We will list some previous known results on the global weak solutions by their distinct approaches. Bressan-Constantin [@BC1; @BC2] and Holden-Raynaud [@HR1; @HR2; @HR3; @HR4; @HR5] obtained global weak solutions by reformulating the CH equation into a semilinear system of ODEs after introducing a new set of independent and dependent variables. Another approach was taken by Xin-Zhang [@XZ1] using the limit of viscous approximation. Bressan-Fonte [@BF] defined a Lipschitz distance functional to extract global weak solutions as the uniform limit of multi-peakon solutions. Grunert-Holden-Raynaud [@GHR1; @GHR2] defined the new Lipschitz metric that is consistent with the construction of the solutions as in [@HR1] and [@HR4]. The authors also studied the aspects of global conservative solutions of the CH equation with nonvanishing asymptotics [@GHR3] and as a limit of vanishing density in the two-component CH system [@GHR4].
In this paper, we construct global weak conservative solutions of the CH equation by using a simple change of variables on the Lagrangian variable $\eta$: $$\label{rho}
\rho=\sqrt{\eta_x}.$$ This idea is motivated by Lenells [@L2] who constructed global weak conservative solutions of the Hnter-Saxton(HS) equation $$u_{txx}+2u_x u_{xx}+uu_{xxx}=0,\;\;t\ \in {\mathbb{R}},\;x \in S^1.$$ The author used the geometric interpretation that the HS equation describes the geodesic flow on the $L^2$ sphere via . In particular, the geodesic remains on the $L^2$ sphere for all time and this makes it possible to continue the geodesic in the weak sense after the blowup time. Since both CH and HS equations form singularities via wave breaking, it is natural to expect that the same kind of weak continuation holds true in the CH equation. We will show that the transformation removes the wave breaking singularities in the CH equation case as well, even though we don’t have exactly the same geometric picture and explicit formula as in HS equation case.
In the geometric perspective, wave breaking is described by the particle trajectory $\eta$ forming a horizontal tangent as it evolves in time. This means that the geodesic flow hits the boundary of the diffeomorphism group. The CH equation written in $\rho$ variables has a global solution from which we can reconstruct the flow $\eta$ defined by $$\label{eta}
\eta(t,x):=\int_0^x \rho^2(t,y) dy+c(t),\;\text{where $c(t)$ is some function of time,}$$ in the space of absolutely continuous functions. This flow is a weak geodesic for almost all time since the spatial derivative of $\eta$ vanishes on a set of measure zero for the most time. The idea is that $\rho$ can assume both signs and $\rho$ passes through the axis whenever it vanishes. The sign change of $\rho$ ensures that if $\rho$ vanishes at a point $x_0$ at some time $T$, $\rho(t,x_0)$, as a function of a time, does not vanish on the punctured neighborhood of $T$. It is certainly possible that $\rho$ vanishes at different places as it evolves, but the imporatant point is that the flow $\eta$ remains a homeomorphism for almost all time. This is precisely how the singularity of the CH equation is removed by introducing the new Lagrangian variable $\rho$.
Our construction of the global weak solution shows that the spatial smoothness of the Lagrangian trajectories $\eta$ in is completely determined by the smoothness of $\rho$, which is dependent on the smoothness of the initial condition(see Proposition 3). This is an interesting phenomenon of the CH equation observed by McKean [@M2]; the solution of the CH equation experiences the jump discontinuity of its slope even for the smooth initial data, but the Lagrangian trajectory $\eta$ is spatially smooth for all time. We will prove that the Lagrangian flow $\eta$ is in $C^k$ for all time whenever the initial condition $u_0$ is in $C^k$. Our result improves the McKean’s by showing the exact correspondence between the smoothness of the initial condition and the smoothness of the Lagrangian flow. Also, our approach does not use the complete integrability and explicit formula for the solutions of the CH equation.
The outline of the paper is following. In Section 2, we write the CH equation in $\rho$ variables defined by . Then we will regard the resulting equation as an abstract ODE independent of the derivation and prove that solution exists globally. Using this global solution in $\rho$ variables, in Section 3 we will construct global weak conservative solutions of the original CH equation. In particular, we obtain the same global results for the CH equation, which is due to Bressan-Constantin [@BC1], in much simpler way. Also, by using the estimates we already have, we can improve the result of McKean [@M2] on the persistence of the smoothness of Lagrangian trajectories. Finally, Section 4 contains some conclusions and remarks. The main theorems of this paper are the following:
The Cauchy problem for the periodic Camassa-Holm equation $$\label{wCH}
\left\{
\begin{array}{l l}
u_t+uu_x=-(1-\partial_x^2)^{-1}\partial_x\left(u^2+\frac{u_x^2}{2}\right)\\
u(0,x)=u_0(x)
\end{array} \right.$$ has a global solution $u \in C({\mathbb{R}}_+, H^1(S^1)) \cap \mathrm{Lip}({\mathbb{R}}_+,L^2(S^1))$. The solution is weak in the sense that the equality in the equation is satisfied in the distributional sense. Also, the solution $u$ is conservative; ${\left\Vertu(t)\right\Vert}_{H^1}={\left\Vertu_0\right\Vert}_{H^1}$ for $t \in {\mathbb{R}}_+$ almost everywhere.
Let $\eta(t,x)$ be the Lagrangian flow of the weak solution $u$ of the CH equation . Then $\eta$ is spatially absolutely continuous for all time. Furthermore, if $\eta$ is initially $C^k$, then $\eta$ remains $C^k$ for all time.
The Setup
=========
In terms of the diffeomorphism $\eta$ defined by the flow equation , the Cauchy problem can be written in $(\eta,\eta_t)$ variables as following: $$\label{CH2}
\left\{
\begin{array}{l l}
\eta_{tt}=-\left\{\Lambda^{-1} \partial_x \left[\left(\eta_t \circ \eta^{-1}\right)^2+\frac{1}{2}\left(\frac{\eta_{tx}}{\eta_x}\circ \eta^{-1}\right)^2\right]\right\}\circ \eta\\
\eta(0,x)=x,\;\;\eta_t(0,x)=u_0(x)
\end{array} \right.$$ Here, $\Lambda^{-1}=(1-\partial_x^2)^{-1}$ is the operator defined by $$\Lambda^{-1}u(x)=\int_{S^1}g(x-y)u(y)dy,$$ where $g(x)=\frac{\cosh(|x|-1/2)}{2\sinh(1/2)}$. We want to write this equation in terms of the new variable $\rho$ defined by . To do this, we use the conserved quantities of the CH equation to redefine quantities that appear in the equation. We will first assume that ${\left\Vert\rho\right\Vert}_{L^2}=1$ to derive the equation. Later, we will prove the local existence and uniqueness of the solution for $\rho \in L^2(S^1)$ without the constraint on the norm. In order to show that the solution is global, however, we will restrict $\rho$ back to the unit sphere.
First, the mean of the velocity $u$ of the CH equation is conserved, so $\mu=\int_0^1 u(t,x) dx=\int_0^1 u_0(x) dx$ is a constant determined by the initial condition $u_0$. By changing variables, we have $$\mu=\int_0^1 (u \circ \eta) \cdot \eta_x dx=\int_0^1 \eta_t \eta_x\;dx=\int_0^1 \eta_t \rho^2 dx.$$ So $\eta_t$ is a function of $\mu$ and $\rho$. By integrating $\eta_{tx}=2\rho\rho_t$ in the spatial variable $x$, we get $\eta_t=\int_0^x 2\rho\rho_t dy+c(t)$ for some function of time $c(t)$. We can substitute this expression into the $\mu$ equation, and determine $c(t)=\mu-\int_0^1 \int_0^y 2\rho\rho_t dz \rho^2 dy$. Hence, we have determined $\eta_t$ completely in terms of $\mu$, $\rho$, and $\rho_t$. Denoting $\eta_t$ in new variable $G$, we have $$G(\mu,\rho,\rho_t)(t,x):=\int_0^x 2\rho\rho_t dy+\mu-\int_0^1 \int_0^y 2\rho\rho_t dz \rho^2 dy.$$ Next, we can write $\frac{\eta_{tx}^2}{2\eta_x}=2\rho_t^2$ since $\rho_t=\frac{\eta_{tx}}{2\sqrt{\eta_x}}$.
We are now ready to rewrite the equation in $\rho$ variables. By differentiating the equation with respect to $x$, we get $$\label{xxt}
\eta_{xtt}=\left(\eta_t^2\eta_x +\frac{\eta_{tx}^2}{2\eta_x}\right)-\Lambda^{-1}\left[\left(\eta_t \circ \eta^{-1}\right)^2+\frac{1}{2}\left(\frac{\eta_{tx}}{\eta_x} \circ \eta^{-1}\right)^2\right]\circ \eta \cdot \eta_x.$$ Here, we used the identity $\Lambda^{-1}\left(-\partial_x^2\right)=1-\Lambda^{-1}$ and the chain rule. By using the explicit formula for $\Lambda^{-1}$ and changing variables in the integration, we have $$\begin{aligned}
\Lambda^{-1}\left[\left(\eta_t \circ \eta^{-1}\right)^2+\frac{1}{2}\left(\frac{\eta_{tx}}{\eta_x}\circ \eta^{-1}\right)^2\right] \circ \eta=&\int_0^1 \frac{\cosh\left(|\eta(x)-y|-\frac{1}{2}\right)}{2\sinh(1/2)}\left[\left(\eta_t \circ \eta^{-1}\right)^2+\frac{1}{2}\left(\frac{\eta_{tx}}{\eta_x}\circ \eta^{-1}\right)^2\right]dy.\\
=&\int_0^1 \frac{\cosh\left(|\eta(x)-\eta(y)|-\frac{1}{2}\right)}{2\sinh(1/2)}\left(\eta_t^2\eta_y+\frac{\eta_{ty}^2}{2\eta_y}\right)dy.\end{aligned}$$ Since $\eta(x)-\eta(y)=\int_y^x \eta_z dz=\int_y^s \rho^2 dz$, we can determine this integral completely in terms of $\rho$, $\rho_t$, and $G$. Denoting it in new variable $F$, we have $$\begin{aligned}
\label{F}
F(\mu, \rho,\rho_t)(t,x):=\int_0^1 \frac{\cosh\left(\left\vert\int_y^x \rho^2 dz\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2G^2+2\rho_t^2)dy\end{aligned}$$ Since $\eta_{xtt}-\frac{\eta_{tx}^2}{2\eta_x}=2\rho\rho_{tt}$, the equation becomes $$2\rho\rho_{tt}=G^2 \rho^2-\rho^2 F.$$ Dividing by $2\rho$ on both sides, we get the following Cauchy problem in $\rho$ variables: $$\label{CH3}
\left\{
\begin{array}{l l}
\rho_{tt}=\frac{1}{2}\rho\left(G^2-F\right)\\
\rho(0,x)=1(\text{constant function}),\;\;\rho_t(0,x)=\frac{1}{2}u_0'(x)
\end{array} \right.$$ Now, we want to solve this equation by viewing it as an abstract ODE in $(\rho, \rho_t)$ variables independent of the above derivation. That is, we assume that $\rho$ and $\rho_t$ are just functions in $L^2(S^1)$ satisfying the equation . Note that this second order equation describes the integral curve of the vector field $(\rho,\rho_t) \mapsto f(\rho,\rho_t):=\frac{1}{2}\rho\left(G^2(\rho,\rho_t)-F(\rho,\rho_t)\right)$. We can easily show that $f$ is smooth in $(\rho,\rho_t)$ since $f$ is essentially a polynomial in $\rho$ and $\rho_t$, and this implies the local existence and uniqueness of the solution.
The system $$\left\{
\begin{array}{l l}
\frac{d \rho}{d t}= \rho_t\\
\frac{d \rho_t}{d t}=f(\rho,\rho_t)
\end{array} \right.$$ with initial conditions $\rho(0,\cdot)=1$(constant function) and $\rho_t(0,\cdot)=\frac{1}{2}u_0'$ describes the flow of a $C^{\infty}$ vector field on $TL^{2}(S^1)=L^2(S^1) \times L^2(S^1)$ and the curve $(\rho,\rho_t)$ exists for some time $T>0$.
We first compute the variational derivative in $\rho_t$. $$\frac{\partial f}{\partial \rho_t}=\frac{1}{2}\rho\left(2G\frac{\partial G}{\partial \rho_t}-\frac{\partial F}{\partial \rho_t}\right).$$ Note that $G$ is linear in $\rho_t$ so it is smooth with respect to $\rho_t$. Also, for $\psi \in L^2(S^1)$, $$\frac{\partial F}{\partial \rho_t}(\psi)=\int_0^1 \frac{\cosh\left(\left\vert \int_y^x \rho^2 dz\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}\left(2G\frac{\partial G}{\partial \rho_t}+2\rho_t \psi\right)dy,$$ and we can check that $\frac{\partial F}{\partial \rho_t}$ is in $L^2$ and continuous with respect to $\rho_t$. By repeating differentiation, we can see that $F$ is also smooth with respect to $\rho_t$. Similarly, we can show that $f$ is smooth with respect to $\rho$.
In order to show that the smooth curve desribed by the equation exists for all time, we need to restrict the base point $\rho$ to be on the unit sphere $U:=\{\rho \in L^2(S^1)\;:\;{\left\Vert\rho\right\Vert}_{L^2}=1\}$. The resaon why we need this constraint is because $F$ and $G$ are periodic on $S^1$ only if $\int_0^1 \rho^2 dx=1$. Also, this condition is necessary for the weak geodesic flow for the CH equation defined spatially on $S^1$. We first prove that $\rho_t$ remains perpendicular to $\rho$ for all time when $\rho$ is constrained on the sphere.
Let $f(\rho,\rho_t)$ be the vector field on $TL^2(S^1)$ defined by the equation . If we restrict $\rho$ to be on the unit sphere $U$, then $f$ restricts on the tangent bundle $TU$.
We want to show that $\int_0^1 \rho \rho_t dx=0$ for all time. When $t=0$, we have $\int_0^1 \rho(0,x)\rho_t(0,x)dx=\int_0^1 \frac{1}{2}u_0'(x)dx=0$ since $u_0$ is periodic. Next, we can compute $$\frac{d}{dt}\int_0^1 \rho\rho_t dx=\int_0^1 \rho_t^2+\rho\rho_{tt}dx=\int_0^1 \rho_t^2+\frac{1}{2}\rho^2(G^2-F)dx.$$ We claim that $\int_0^1 \rho^2 F dx=2\rho_t^2+\rho^2 G^2$. Note that $$\begin{aligned}
\int_0^1 \rho^2 F dx=&\int_0^1 \rho^2 \int_0^x \frac{\cosh\left(\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2 G^2+2\rho_t^2)dydx\\
&+\int_0^1 \rho^2 \int_x^1\frac{\cosh\left(-\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2G^2+2\rho_t^2)dy dx\\
=&(I)+(II).\end{aligned}$$ By changing the order of integration, we have $$\begin{aligned}
(I)=&\int_0^1 (\rho^2 G^2+2\rho_t^2)\int_y^1 \rho^2 \frac{\cosh\left(\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)} dx dy\\
(II)=&\int_0^1 (\rho^2 G^2+2\rho_t^2)\int_0^y \rho^2 \frac{\cosh\left(-\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)} dx dy.\end{aligned}$$ Note that $\frac{d}{dx}\int_y^x \rho^2 dx=\rho^2$, so we can compute the inner integrals explicitly by FTC: $$\begin{aligned}
&\int_y^1 \rho^2 \frac{\cosh\left(\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)} dx+\int_0^y \rho^2 \frac{\cosh\left(-\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)} dx\\
=&\frac{1}{2\sinh(1/2)}\left\{\sinh\left(\int_y^x \rho^2 dz-\frac{1}{2}\right)\Big|_y^1-\sinh\left(-\int_y^x \rho^2 dz-\frac{1}{2}\right)\Big|_0^y\right\}\\
=&\frac{1}{2\sinh(1/2)}\left\{\sinh\left(\int_y^1 \rho^2 dz-\frac{1}{2}\right)-\sinh\left(-1/2\right)-\sinh\left(-1/2\right)+\sinh\left(-\int_y^0 \rho^2 dz-\frac{1}{2}\right)\right\}=1,\end{aligned}$$ since $$\sinh\left(\int_y^1 \rho^2 dz-\frac{1}{2}\right)=\sinh\left(-\int_0^y \rho^2 dz+\frac{1}{2}\right)=\sinh\left(\int_y^0 \rho^2 dz+\frac{1}{2}\right).$$ Here, we used the assumption that $\int_0^1 \rho^2 dz=1$. Hence, $(I)+(II)=\int_0^1 (\rho^2 G^2+2\rho_t^2) dy$ and by substituting this back, we can conclude that $\frac{d}{dt}\int_0^1 \rho \rho_t dx=0$ for all time. Since the integral is initially zero, this shows that it remains to be zero for all time.
Next, we want to show that the conservation of the $H^1$ energy in the original CH equation holds true in $\rho$ variables as well. This energy conservation will ensure the uniform boundedness of the estimates that we will need later. We first prove the following lemma.
Let $G$ be defined as above. Then $G_t=-H$, where $$H(t,x)=\int_0^x \frac{\sinh\left(\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2G^2+2\rho_t^2)dy+\int_x^1\frac{\sinh\left(-\int_y^x \rho^2 dz-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2G^2+2\rho_t^2)dy.$$
Note that $$\frac{\partial H}{\partial x}=-(\rho^2G^2+2\rho_t^2)+\rho^2F.$$ By differentiating $G$ with respect to $t$, we have $$\begin{aligned}
G_t=&\int_0^x 2\rho_t^2+2\rho\rho_{tt} dy+c'(t)=\int_0^x 2\rho_t^2+\rho^2 G^2-\rho^2F dy+c'(t),\end{aligned}$$ where $c(t)=\mu-\int_0^1 \int_0^x 2\rho\rho_t\;dy \rho^2\;dx$. So $$G_t=\int_0^x \frac{\partial H}{\partial y}dy+c'(t)=-H(t,x)+H(t,0)+c'(t).$$ So we want to show that $H(t,0)+c'(t)=0$. We can compute $$\begin{aligned}
c'(t)=&-\int_0^1 \int_0^x 2\rho_t^2+2\rho\rho_{tt} dy \rho^2 dx\\
=&-\int_0^1 \int_0^x 2\rho_t^2+\rho^2 G^2-\rho^2 F dy \rho^2 dx\\
=&-\int_0^1 (2\rho_t^2+\rho^2 G^2-\rho^2 F)\int_y^1 \rho^2 dx dy\;(\because \text{change the order of integration})\\
=&\int_0^1 \frac{\partial H}{\partial y}\int_y^1 \rho^2 dx dy\\
=&-H(t,0)+\int_0^1 H \rho^2 dy\\
=&-H(t,0),\end{aligned}$$ since $H\rho^2=F_x$ and $F$ is periodic. Hence, $G_t=-H$ as desired.
The ‘$H^1$ energy’ of the $\rho$ equation is conserved: $$\label{conservation}
\frac{d}{dt}\int_0^1 \rho^2 G^2+4\rho_t^2\;dx=0.$$
Recall that we have
$G_x=2\rho\rho_t$ and $G_t \rho^2=-F_x$.
Then $$\begin{aligned}
\frac{d}{dt}\int_0^1 \rho^2 G^2+4\rho_t^2\;dx=&\int_0^1 2\rho\rho_t G^2+2\rho^2 GG_t+8\rho_t\rho_{tt}\;dx\\
=&\int_0^1 2\rho\rho_t G^2-2GF_x+4\rho\rho_t(G^2-F)\;dx\\
=&\int_0^1 6\rho\rho_t G^2\;dx-2G(1)F(1)+2G(0)F(0)\\
=&\int_0^1 \frac{d}{dx}\left[G^3\right] dx=0,\end{aligned}$$ since $F$ and $G$ are periodic.
Now, we are ready to prove that the solution of the equation is global when $\rho$ is restricted on the unit sphere. The idea is that all estimates are bounded in terms of ${\left\Vert\rho\right\Vert}_{L^2}$ and ${\left\Vert\rho_t\right\Vert}_{L^2}$ and restriction on the unit sphere guarantees that the two norms are uniformly bounded.
The flow described by the equation , when $\rho$ is restricted on the unite sphere, exists for all time.
We claim that the RHS of the equation is uniformly bounded. Note that we have $$\label{estimate}
|G^2-F| \le |G^2|+|F|\le2 {\left\Vert\rho\right\Vert}_{L^2}{\left\Vert\rho_t\right\Vert}_{L^2}+\frac{1}{4\sinh(1/2)}\left(2{\left\Vert\rho\right\Vert}_{L^2}^3 {\left\Vert\rho_t\right\Vert}_{L^2}+{\left\Vert\rho_t\right\Vert}_{L^2}^2\right).$$ Since ${\left\Vert\rho\right\Vert}_{L^2}=1$ and ${\left\Vert\rho_t\right\Vert}_{L^2}$ is uniformly bounded by the energy conservation, $|G^2-F|$ is uniformly bounded as well. Then the RHS of the equation is smooth and uniformly bounded in $(\rho,\rho_t)$. Thus, the solution of the Cauchy problem can be extended for all time by Wintner’s Theorem from the ODE theory(see [@H]).
Global Weak Solution of the CH equation in $u$ variable
=======================================================
From the global solution $\rho$, we can readily construct global weak solutions for the original CH equation . We first introduce new Lagrangian variables. Define $$\begin{aligned}
\label{Lagvar}
K(t,x):=&\int_0^x \rho^2 dy+tu_0(0)-\int_0^t \int_0^\tau H(s,0)\;ds d\tau,\\
G(t,x):=&\int_0^x 2\rho\rho_t dy+\mu-\int_0^1 \int_0^x 2\rho\rho_t dy \rho^2 dx.\end{aligned}$$ As we have seen in the Lemma 5, $-H(s,0)=c'(t)$ where $c(t)=\mu-\int_0^1 \int_0^x 2\rho\rho_t dy \rho^2 dx$, and so $K$ satisfies the first order equation $\frac{\partial K}{\partial t}=G$. Next, we claim that $(K,G)$ solves the second order equation . As in Lenells’s paper [@L2], we can decompose $S^1$ by $S^1=N \cup A \cup Z$ where $$\begin{aligned}
N:=&\{x \in S^1\;:\;\text{$K_x$ exists and equals 0, i.e., $\rho(t,x)=0$}\},\\
A:=&\{x \in S^1\;:\;\text{$K_x$ exists and $K_x(x)>0$, i.e., $\rho(t,x)>0$}\},\;\text{and}\end{aligned}$$ $Z$ is a set of measure zero. We first prove the following lemma.
For almost all time $t \in {\mathbb{R}}_+$, we have $\int_{N}\rho_t^2dy=0$.
It suffices to show that the set $N$ has a measure zero for almost all time. As in [@L2], the Fubini theorem gives $$\int_0^T m(N)dt=\int_{S^1}\int_0^T \chi_{\{\rho^{-1}(0)\}}dtdx,$$ for $T<\infty$ where $\chi_{\{\rho^{-1}(0)\}}:[0,\infty)\times S^1 \to \{0,1\}$ is the characteristic function. Hence, we want to show that the RHS of the equation vanishes. Let $x_0 \in S^1$.Then the following set $$N'(x_0)=\{0 \le t \le T\;:\;\rho(t,x_0)=0\}$$ has the Lebesgue measure zero. This is because $\rho_t(t,x_0)$, as a function of time $t$, cannot vanish on this set. If not, there is a time $t_0$ such that $\rho(t_0,x_0)=0=\rho_t(t_0,x_0)$. Note that we have the following differential inequality satisfied by the solution of : $$\begin{aligned}
\frac{d}{dt}\left[\rho^2(t,x_0)+\rho_t^2(t,x_0)\right]=&2\rho\rho_t+2\rho_t\rho_{tt}\\
=&2\rho\rho_t+\rho\rho_t(G^2-F)\\
=&2\rho\rho_t\left(1+\frac{G^2-F}{2}\right)\\
\le&C(\rho^2+\rho_t^2),\end{aligned}$$ where $C=\max \left\{1+\frac{G^2-F}{2}\right\}$ is the uniform constant for the solution of the equation restricted on the unit sphere(see the Proposition 7.) Hence, by the Gronwall’s lemma, we have $$\rho^2(t,x_0)+\rho_t^2(t,x_0) \le \left(\rho^2(t_0,x_0)+\rho_t^2(t_0,x_0)\right)e^{Ct}.$$ If $\rho(t_0,x_0)=0=\rho_t(t_0,x_0)$, the RHS vanishes and this implies that $\rho(t,x_0)=0=\rho_t(t,x_0)$ for all time $t$. In particular, $\rho(0,x_0)=0$ and this contradicts the initial condition of $\rho$. Hence, $\rho_t(t,x_0)$ must be nonzero on $N'(x_0)$. Since $\rho(t,x_0)$ is continuous as a function of a time, every neighborhood of $t_0$ contains a point where $\rho$ is nonzero, i.e., the set $N'(x_0)$ is isolated. Hence the set $N'(x_0)$ must be finite and it has a measure zero.
It is possible that $\rho$ vanishes on a set of positive measure, e.g., on an interval, at some time. However, the proof of the lemma shows that these appearances are rare and $\rho$ can vanish only on a set of measure zero for almost all time. Since $K_x=\rho^2$, the slope of the particle trajectories looks like a parabola. As a result, $K$ is generically a homeomorphism whenever $K_x$ vanishes on a set of measure zero. This implies that $K^{-1}$ is well-defined for almost all time.
For almost all time $t$, $(K, G)$ satisfies the CH equation $$\label{GKeqn}
G_t=-\left\{\partial_x \Lambda^{-1}\left[(G \circ K^{-1})^2+\frac{1}{2}\left(\frac{G_x}{K_x} \circ K^{-1}\right)^2\right]\right\}\circ K.$$ Note that this solution is weak since $K$ is only absolutely continuous in the spatial variable $x$ and the equation is satisfied for almost all time $t$.
We want to show that the equation $G_t=-H$ is equivalent to the equation in the weak sense. Since $K$ is a diffeomorphism on $A$, we can change variables via $K$ and get $$\begin{aligned}
G_t=&-\int_{S^1} \frac{\sinh\left(\left\vert\int_y^x \rho^2 dz\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}(\rho^2G^2+2\rho_t^2)dy\\
=&-\int_{A}\frac{\sinh\left(\left\vert K(x)-K(y)\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}\left(K_x G^2+\frac{G_x^2}{2K_x}\right)dy+\int_{N}\rho_t^2dy\\
=&-\int_{K(A)}\frac{\sinh\left(\left\vert K(x)-y\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}\left(\left(G \circ K^{-1}\right)^2+\frac{1}{2}\left(\frac{G_x}{K_x} \circ K^{-1}\right)^2\right)dy+\int_{N}\rho_t^2dy\\
=&-\partial_x \left[\int_{K(A)}\frac{\cosh\left(\left\vert x-y\right\vert-\frac{1}{2}\right)}{2\sinh(1/2)}\left(\left(G \circ K^{-1}\right)^2+\frac{1}{2}\left(\frac{G_x}{K_x} \circ K^{-1}\right)^2\right)dy\right]\circ K+\int_{N}\rho_t^2dy.\end{aligned}$$ Since the Lebesgue measure of the set $K(A)$ is 1, the first integral on the RHS is equivalent to the RHS of the equation . Also, $\int_{N}\rho_t^2dy=0$ for almost all time from the previous lemma. Thus, for almost all time $t \in {\mathbb{R}}_+$, $(K,G)$ satisfies the equation as desired.
Now, we prove the main theorems of the paper. We first check that the velocity field $u$ satifying the flow equation is well-defined in $H^1$ for all time.
Let $(K,G)$ be a weak solution of the CH equation in the Lagrangian form. Then the velocity field $u \in C({\mathbb{R}}_+,H^1(S^1)) \cap \mathrm{Lip}({\mathbb{R}}_+,L^2(S^1))$ is well-defined by the formula $$u(t,K(t,x))=G(t,x),\;\;(t,x) \in [0,\infty) \times S^1.$$
First, note that the flow $K$ is a bijection from $A \to K(A)$, so we can define $u$ a.e. on $S^1$ by
$u(y):=G(K^{-1}(y))$ for $y \in K(A)$.
It remains to show that $u$ is well-defined by this formula on $N$, as well. The idea is that the Lagrangian velocity $G$ is well-defined for all time, so we can define the velocity $u$ from it.
Suppose that $K_x$ vanishes at some time $T$. Since $K$ is a nondecreasing absolutely continuous funciton, the set $N$ must be the union of isolated points and intervals. In the case where $x \in N$ is an isolated point, $K$ is still a bijection around a neighborhood of $x$ so we can defined the velocity $u$ in the same way as above. Suppose that $K_x$ vanishes on an interval $I \subset N$, i.e., $K_x(T,x)=0$ for $x \in I$. Then we have $K(T,x)=x^\ast$ for some $x^\ast \in S^1$ on $I$. Also, $\rho(T,x)=0$ on that interval and this implies that $G_x(T,x)=0$ since $G_x=2\rho\rho_t$ by differentiating in . Then $G(T,x^\ast)$ is finite and is a constant on $I$, so we can define $u(T,x):=G(T,x^\ast)$ for $x \in I$. This completes the proof for $u$ being well-defined.
Once we define $u=G \circ K^{-1}$, we can show that its distributional derivative is $u_x=\frac{G_x}{K_x} \circ K^{-1}$. We omit the proofs for these and their regularity since it is exactaly the same argument as in [@L2].
When $K(T,x)=x^\ast$ on an interval $I$, we can think of this as a set of particles concentrating at one point $x^\ast$. This is consistent with the physical interpretation of the CH equation since it describes a compressible fluid motion. In this case, a set of particles concentrate at a point and moves in a same velocity.
Proof of Theorem 1
------------------
We claim that $u$ is a weak solution of the equation . We want to show that
${\displaystyle \int_{S^1 \times {\mathbb{R}}_+}(u_t+u u_x) \phi\;dxdt=\int_{S^1 \times {\mathbb{R}}_+}-p_x\phi\;dxdt}$ for all $\phi \in C_c^\infty(S^1 \times {\mathbb{R}}_+)$,
where $p=\Lambda^{-1}(u^2+\frac{1}{2}u_x^2)$. We have $$\begin{aligned}
\int_{S^1 \times {\mathbb{R}}_+}(u_t+u u_x) \phi\;dxdt=&\int_{S^1 \times {\mathbb{R}}_+} -u\phi_t+uu_x\phi\;dxdt\\
=&\int_{S^1 \times {\mathbb{R}}_+}-U K_x \phi_t \circ K+UU_x \phi \circ K\;dxdt,\;\text{where $U=u \circ K$}\\
=&\int_{S^1 \times {\mathbb{R}}_+} U_t K_x \phi \circ K\;dxdt,\end{aligned}$$ since $$\begin{aligned}
\left(U K_x \phi \circ K\right)_t-\left(U^2 \phi \circ K\right)_x=&U_t K_x \phi \circ K-UU_x \phi \circ K+U K_x \phi_t \circ K,\end{aligned}$$ and $\phi$ has a compact support. Since $U_t=-\partial_x\Lambda^{-1}\left[u^2+\frac{1}{2}u_x^2\right] \circ K$, we get $$\int_{S^1 \times {\mathbb{R}}_+} U_t K_x \phi \circ K\;dxdt=\int_{S^1 \times {\mathbb{R}}_+}-p_x\phi\;dxdt,$$ by another change of variables. Finally, the conservation of $H^1$ energy of the weak solution comes from rewriting all quantities appearing in in $u$ variables. That is, $$\begin{aligned}
\int_{S^1} u^2+u_x^2 dx=&\int_{K(A)}\left(G \circ K^{-1}\right)^2+\left(\frac{G_x}{K_x}\circ K^{-1}\right)^2 dx\\
=&\int_{A} G^2 K_x+\frac{G_x^2}{K_x}dx\\
=&\int_{S^1} G^2 \rho^2+4\rho_t^2 dx\end{aligned}$$ is conserved for almost all time $t$. This completes the proof of the Theorem 1.
Proof of Theorem 2
------------------
We can observe that by providing extra smoothness on $\rho$ and $\rho_t$ variables, we can improve the smoothness of the $G$. In fact, the same estimate will continue to work with $L^2$ norms replaced by $C^k$ norms since the spatial domain is compact. Hence, the solutions for the Cauchy problem in $\rho$ variables will be global in $C^k$ spaces. Consequently, the Lagrangian variables $K$ and $G$ will be $C^{k+1}$ on $S^1$ for all time whenever $\rho$ and $\rho_t$ is $C^k$. This completes the proof of the Theorem 2.
Theorem 2 improves the result of McKean [@M2]. The difference between the current work and Mckean’s work is that we don’t need the assumption for the momentum $m=u-u_{xx}$ to satisfy $m, m_x \in H^1$ initially. Our result shows that that for each $k$, $K$ is exactly in $C^k$ whenever the initial condition $u_0$ is in $C^k$.
Future Research
===============
The global weak continuation of the CH equation in this research suggests that there might be a general theory that explains why it works. We can first suspect that the metrics corresponding to the HS and CH equations are close. That is, in the space of all Riemannian metrics on the group of diffeomorphisms, the Sobolev $H^1$ metric can be regarded as a nonlinear perturbation of $\dot{H}^1$ metric, where the global weak continuation property is a consequence of the robustness of the perturbation. However, the geometry of two diffeomorphism groups are different since the sectional curvature of $\dot{H}^1$ metric is a positive constant, whereas the $H^1$ metric has sectional curvature positive in ‘most directions’ but also assumes negative sign(see [@L1; @Mis1998])
We can apply the change of variable technique in this paper to other Euler-Arnold equations or generalize this Lagrangian change of variable technique as well. In particular, we can consider the Wunsch equation which is the Euler-Arnold equation with a right invariant $\dot{H}^{1/2}$ metric: $$\label{wunsch}
\left\{
\begin{array}{l l}
\omega_t+u \omega_x+2\omega u_x=0,\\
\omega=Hu_x,\;\text{$H$ is the Hilbert transform,}
\end{array} \right.$$ It was studied by Bauer-Kolev-Preston [@BKP] as a 1 dimensional vorticity equation for the 3 dimensional Euler’s equation. The solution of Wunsch equation forms a finite time singularity along a particle trajectory due to wave breaking and the blow up result was further extended in the framework of Teichmüller theory by Preston-Washabaugh [@PW]. In this case, we can instroduce a new variable $Z=u_x+ i\omega$ to get $$Z_t+uZ_x+Z^2=-F,$$ where $F$ is some positive function. In this framework, we can study the geodesic equation for the Sobolev $\dot{H}^s$ metric for $\frac{1}{2}<s<1$ in a uniform way. Since the sign change of the vorticity is the crucial assumption to get the blow up in all known cases, we can analyze the second-order ODE in this new Lagrangian variable $Z$ to find a direct proof of the blow up.
Lastly, it would be interesting to interpret the global weak solution of the CH equation constructed in this paper in the context of optimal transport. Recently, Gallouët-Vialard formulated generalized CH equation as an Euler-Arnold equation which can be identified as a particular solution of the incompressible Euler’s equation on the group of homeomorphisms on ${\mathbb{R}}^2 \setminus \{0\}$. We can investigate how the blow up and the global weak continuation of the CH equation can be described in the language of optimal transport, and understand the role of the change of variables .
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Little is known about how recreational triathletes prepare for an Olympic distance event. The aim of this study was to identify the training characteristics of recreational-level triathletes within the competition period and assess how their preparation for a triathlon influences their health and their levels of fatigue. During the 6 weeks prior to an Olympic distance triathlon, and the 2 weeks after, 9 recreational athletes (5 males, 4 females) completed a daily training log. Participants answered the Daily Analysis of Life Demands Questionnaire (DALDA), the Training Distress Scale (TDS), and the Alberta Swim Fatigue and Health Questionnaire weekly. The Recovery-Stress Questionnaire (REST-Q) was completed at the beginning of the study, on the day before the competition, and at the end of week 8. Training loads were calculated using session-based rating of perceived exertion (sRPE). The data from every week of training was compared to week 1 to determine how athletes’ training and health changed throughout the study. No changes in training loads, duration, or training intensity distribution were seen in the weeks leading up to the competition. Training duration was significantly reduced in week 6 (p=0.041, d = 1.58, 95% CI = 6.9, 421.9), while the number of sessions was reduced in week 6 (Z=2.32, p=0.02, ES = 0.88) and week 7 (Z = 2.31, p=0.02, ES = 0.87). Training was characterized by large weekly variations in training loads and a high training intensity. No significant changes were seen in the DALDA, TDS, or REST-Q questionnaire scores throughout the 8 weeks. Despite large spikes in training load and a high overall training intensity, these recreational-level triathletes were able to maintain their health in the 6 weeks of training prior an Olympic distance triathlon.
ARTICLE | doi:10.20944/preprints202109.0078.v1
Subject: Medicine & Pharmacology, Sport Sciences & Therapy Keywords: Functional Fitness; High intensity Functional training; Periodization; Overreaching; Muscle recovery.
Online: 6 September 2021 (07:19:09 CEST)
The study describes the acute and delayed time course of recovery following the CrossFit® Benchmark Workout Karen. Eight trained men (28.4±6.4 years; 1RM back squat 139.1±26.0 kg) undertook the Karen protocol. The protocol consists of 150 Wall Balls, aiming to hit a target 3 meters high. Countermovement jump height (CMJ), creatine kinase (CK), and perceived recovery status scale (PRS) (general, lower and upper limbs) were assessed pre, post-0h, 24h, 48h and 72h after the session. The CK concentration 24h after was higher than pre-exercise (338.4 U/L vs. 143.3 U/L; effect size: 0.74; p≤0.05). At 48h and 72h following exercise, CK concentration had returned to baseline levels. The PRS general and of the lower limbs were lower in the 24-hours post-exercise compared to pre-exercise (PRS general: 4.7 ±1.5 and 7.9 ±1.7 mmol/L; and PRS of the lower limbs: 4.0 ±2.5 and 7.9 ±0.8, respectively). The PRS general, lower, and upper limbs were reduced at 48-post exercise compared to 72-hours post-exercise scores. Our findings provide insights into the fatigue profile and recovery in acute CrossFit® and can be useful to coaches effectively design the daily session. | https://www.preprints.org/search?search1=Jo%C3%A3o%20Henrique%20Falk%20Neto&field1=authors |
Will Changed Workplace Laws Increase Competitiveness for Ontario Businesses?
“We’ve heard loud and clear from businesses across Ontario that job growth starts with cutting the burdensome, job-killing red tape that is driving jobs and investment out of our province … We are making Ontario open for business.”
Premier Ford in his closing speech this October at the annual Ontario Economic Summit.
Less than 2 months later, the Ontario government tabled Bill 66, Restoring Ontario’s Competitiveness Act, 2018.
Bill 66, an omnibus bill, is part of the Ontario Open for Business Action Plan and announces over 30 actions to make it easier for businesses to create jobs.
Bill 66 at Schedule 9 outlines, among other things, the proposed changes to Ontario’s Employment Standards Act, 2000 (“ESA”).
How Bill 66 Proposes to Amend the ESA
The changes Bill 66 makes to the ESA are intended to reduce regulatory burdens on businesses. Firstly, Bill 66 amends section 2 of the ESA so that employers no longer must display a poster that provides information about the ESA and its regulations in the workplace. However, the requirement to provide a copy of the most recent version of this ESA poster to each employee is retained.
Second, Bill 66 would remove the requirement for employers to obtain approval from the Director of Employment Standards for excess hours of work and overtime averaging. Specifically, Bill 66 would amend Part VII of the ESA so that a Director’s approval would no longer be required for employers in order to make an agreement that allows their employees to exceed 48 hours of work in a work week.
Further, Bill 66 proposes to amend Part VIII of the ESA to remove the requirement to obtain the Director’s approval for employers to make agreements which allow them to average their employees’ hours of work for the purpose of determining the employees’ entitlement to overtime pay. This means that for the purposes of overtime entitlement, an employee’s hours may be averaged over a period that does not exceed 4 weeks, in accordance with the terms of an averaging agreement between the parties.
Bill 66 also proposes that existing averaging agreements be deemed to have met the requirements set out in the ESA. Therefore, such agreements would continue to be valid until it is revoked by the employer, employee, or the Director.
As a result of these changes, employers would no longer be required to apply to the Ministry of Labour for approval of their employees’ excess weekly hours of work and overtime averaging. Employers are for these changes because it provides for increased flexibility to manage employee shifts. Employee groups oppose these changes, believing that they could result in more hours and less overtime pay for workers.
Lesson To Be Learned
If Bill 66 becomes law employers should review their employment contracts to make sure employees agree to work excess hours if requested, and also make sure that employees agree that their hours can be averaged over 2 or more weeks for the purpose of calculating overtime pay. This will ensure that there are enough employees available to address surges in business, and reduce payroll costs in workplaces where there are ebbs and flows in hours of work. We would would be pleased to assist with this contractual review.
Although Bill 66 is not yet law, it is expected to proceed quickly through legislature, just as Bill 47 did. For more information on the impact Bill 66 will have on your business, contact an employment lawyer at MacLeod Law Firm. You can reach us at [email protected] or 647-204-8107.
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