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# Democracy Actually Is Good at Governing Infrastructure, and Infrastructure Is Exactly What Democracy Should Govern

Matt Prewitt

September 7, 2021

In this essay I will link together several arguments that point toward a case for expanding – and doubling down on – multi-level democracy as an approach to managing infrastructure.

Flourishing societies have always depended integrally on infrastructure. Today this is truer than ever. But we’ve lost the plot regarding what infrastructure is. Therefore we’re putting infrastructure under the misrule of a set of institutions called “capitalism”, which were never meant to steward infrastructure wisely, and which in fact systematically mismanage it. For different reasons, technocratic state apparatuses are also absolutely terrible at managing infrastructure. So revitalized and expanded democracy is our only serious option.

To make democracy work better, we need to make democracy better. We need to make it something that we engage with more intimately, more locally, and more often – something that penetrates our lives more deeply than once-every-few-years ballots, and that governs more than just the occupants of a few sclerotic state offices. The challenges are real as rain, but there are also reasons to be very optimistic about this.

The argument proceeds like this.

First: infrastructure is best interpreted not as roads or power lines or anything else in particular, but as scarce resources upon which self-reinforcing processes depend.

Second: whoever owns (or has absolute authority over) infrastructure has immense power, because exponentially-evolving processes depend on it.

Third: capitalists should never own infrastructure. Capitalism isn’t built for that. The justifications of profit and private ownership end wherever infrastructure begins.

Fourth: technocratic management of infrastructure fails too, because the responsible stewardship of infrastructure depends on answering hypercomplex questions, the answers to which depend on information locked away in the minds of the people likely to be impacted by infrastructure.

Fifth: improved and expanded democratic practices, at multiple levels of society, are the way to elicit that information.

Finally we could do worse than to understand the wise stewardship of infrastructure, broadly construed, as the essential task of democracy.

Now I’ll try to flesh it out.

## Abundance that Depends on Limited Resources

Many patterns of human activity strengthen themselves, tending toward exponential growth, resulting in limitless proliferation of wealth, culture, and other goods. Consider cities.

People go to cities in search of beneficial interactions with other people: culture, trade, conversations, and love.

This connection-seeking pattern strengthens itself. The more people move to the city, the more culture, trade, conversations, and love the city contains, and the more strongly it therefore attracts others. People meet one another serendipitously in parks, metros, restaurants, and offices, striking up friendships, starting businesses and making art together, sharing ideas and inspiration. Diversity, wealth, and dynamism beget one another in virtuous circles. It’s like a big snowball getting bigger as it rolls downhill. Let’s call this an “abundant process”.

Not every pattern of activity is an abundant process. Some patterns are self-depleting. Consider roads.

Drivers use roads to go places fast. But each new driver on the road harms the usefulness of the road for everyone else by adding congestion and slowing traffic. The road is a scarce resource.

Eventually, the road grows so gridlocked that using it is barely faster than walking.

This is exactly opposite to how cities work. Adding more people makes cities more useful, but it makes roads less useful.

Notice, however, that abundant processes like cities depend intricately on finite, scarce resources like roads.

Without roads, people wouldn’t be able to live in big urban areas receiving shipments of distantly-produced essentials like food and medical supplies. The abundant magic of cities could never occur.

This is true of all abundant processes. Universities get better the more great scholars they attract – that’s an abundant process. But they can’t grow without adding new, all-too-finite buildings and equipment, or else scientists will start fighting over microscopes. Natural ecosystems, too, are abundant processes: the trees, bees, and other species support each other in virtuous circles. But whole ecosystems still depend on scarce resources like water. When a necessary resource runs low, it curtails the whole system’s exponential development.

Resource bottlenecks can slow or stop abundant processes.

## Scarce Resources Are Infrastructure

It’s useful to see the similarity between human-made scarce resources like roads, and natural scarce resources like water. Everything we conventionally call infrastructure – roads, electrical grids, broadband, medical services, police protection, and so on – is basically an attempt to mitigate some kind of resource scarcity, in order to keep abundant processes rolling. Like natural ecosystems, human civilization is an abundant process that slows or stops when it reaches bottlenecks. Scarce resources are infrastructure, and vice versa.

Human systems depend on many of the same scarce resources as natural ones, such as water. Without water, cities don’t grow any more than oases do. Infrastructure like protected watersheds, reservoirs, and water mains help cities deal with that.

We also use ingenuity and technology to address more modern bottlenecks. For example, sewage accumulation limits residential density by spreading disease, so we have sewers. And the difficulty of communicating across distance slows collaboration and exchange, so we have telecom networks. Macro-scale bottlenecks sometimes form for which we do not yet have any effective infrastructure at all, such as the destruction of the atmosphere with CO2 emissions, and the acceleration of polarization on social media. But that’s for another essay. For now, I want to tell a clear story about the relationship between infrastructure and power.

## Infrastructure Is Power

Anyone who controls infrastructure wields total power over the complex abundant processes thriving atop it. And whenever this power is wielded for self interest (instead of for the interests of the whole ecosystem), it curtails that thriving.

This is fairly obvious in the case of vital natural resources. Suppose you are the private owner of an aquifer feeding an oasis. You can then demand “rent” from the birds and palm trees. In fact, you can push those unfortunate creatures to within an inch of their lives. For if you shut off the water, their world ends.

The same goes for human-made infrastructure. If you – a private citizen, or a company, or a government – control the roads of a city, you can charge tolls upon them. And you have total bargaining power. For without the roads, the whole complexity of the city will be erased. Power over infrastructure is power to hold vast, thriving systems hostage.

## Owning Infrastructure For Profit Is Bad

Understanding the power of infrastructure is the key to seeing the limits of capitalism, and also grasping the direction in which democracy must grow and evolve. Here’s why.

While the profit motive drives social efficiency in certain contexts, the exercise of authority over infrastructure is never one of those contexts. For whenever an owner of infrastructure takes a profit, they destroy value for society that exceeds the amount of the profit they take. The profit imperative makes them systematically bad stewards.

Before I explain this in more detail, simply observe that it conforms to conventional wisdom and common sense. It is the reason that utility companies are regulated, and the reason the state owns roads.

Here’s a quick explanation of the too-often-overlooked economic logic.

When Is Profit Socially Efficient?

The canonical justification of profit goes as follows. Entrepreneurs are operating in a competitive environment, and managing resource-constrained operations, such as factories where only so many laborers can work, and apartments where only so many people can live. Therefore, the value they generate diminishes with scale, like so:

These resource constraints explain why capitalists can pay suppliers and workers their true marginal value – that is, a fair wage rate – while still making a profit. For this “diminishing marginal returns” structure of productivity implies that the marginal value of any input (such as labor) will be less than the average value of that input. So the difference between the marginal and average productivity may then be taken as profit.

Let me repeat this important point: The social justification of profit depends on the assumption that the profitable activity has diminishing marginal returns – in other words, that it is a self-depleting, resource-constrained activity. This is a little bit subtle. I am including a footnote illustrating it further.[1]

And here’s the thing: this assumption is not always true!

When Is Profit Not Socially Efficient?

The logic that justifies profit is completely flipped with infrastructure. Because infrastructure doesn’t feed into self-depleting processes, but rather self-reinforcing, abundant ones: the serendipitous connections in a city, and the complex mutualism of an ecosystem. The value produced by such abundant processes scales instead like this:

When value scales like that (i.e., exponentially), taking any kind of profit hurts society. For such extractions of value reduce output, effectively depriving society of the most vertical, “hockey stick” parts of the exponential, abundant processes.

Moreover, the payment of a fair wage – that is, payment equal to the marginal value added by each contributor to the process – becomes impossible, because the marginal value added by each contributor exceeds the average value added by all of them. If an owner of infrastructure tried to pay everyone for their marginal contribution of value to the system, it would bleed money and bankrupt itself.

This is why the governors of infrastructure, for reasons of social efficiency, should never be profit-seeking enterprises.

## Why Technocrats Also Fail

So if we don’t want capitalists running our infrastructure, how about decisive, expert, public-oriented state actors? It turns out they have problems too. Let’s see why.

In order to be efficient, stewards of infrastructure need to be accountable to all the parties affected by the complex processes that depend on the infrastructure – that is, the actual societies in which these abundant processes unfold. This means that three groups of people should be heavily-overlapping, indeed almost coincident:

a) the participants in abundant processes (i.e., the people living in the societies where they are happening);

b) the stewards of infrastructure (i.e., authorities); and

c) the exercisers of democratic power (i.e., voters).

Maximizing the flow of information and accountability between these groups is the only serious way of ensuring that infrastructure stewards don’t destroy value by either incompetence or rent-seeking.

Here’s another way of looking at this: Capitalism fails at infrastructure stewardship because owners of infrastructure are systematically misaligned with the participants in abundant processes that depend on the infrastructure.

And, technocratic stewards have exactly the same problem.

There is a real difference, though, between government decision-makers who listen thoughtfully and rigorously to democratic input, and technocrats who don’t. Both can be well-intentioned. But technocrats who dismiss democratic input deprive themselves of necessary information. They’re trying to govern society without the benefit of context. It’s like flying blind. Proceeding this way, they can appear to get a lot done, quickly. But it’s a short-run illusion: In the longer run, they crash. Or worse: they fly somewhere they didn’t mean to go.

Technocrats inevitably overlook important interests. They have mental maps that don’t match the territory. They understand the communities they come from (usually privileged ones) better than other communities. Their unilateral stewardship of infrastructure thus introduces distortions that reflect their personal, biographical, and psychological limitations.

Obvious failure modes include investing in infrastructure nobody wants, like “bridges to nowhere”. But failure can also take subtler forms, like building highways in places that serve the wealthy or connected, while immiserating the poor or marginalized. Think, for example, of Robert Moses, New York’s famously imperious 20th-century urban planner. Exercising unilateral power, he got bridges and highways built. It wasn’t all bad. But he destroyed and marginalized Black and poor communities to do it. Those harms plague society in less visible ways, decades later.

Thus, technocratic governance can take societies full steam ahead toward an unattractive, dysfunctional future. Years of apparent progress are reevaluated in the fullness of time as decades of folly.

The subtler issues are hard for technocrats to foresee in their planning spreadsheets. But they are at the very heart of normative political conversations. Thus, broad-based participation in decision-making is our best way of understanding them.

Smart 21st-century governments with a democratic mindset try to get this right. They try to achieve coincidence between the governor and the governed, and they generally do a much better job representing the public interest than private parties would. But they often fail nonetheless. Sometimes they lack the tools or the mandate to incorporate rich and legitimate democratic feedback into their decisions. Sometimes they use problematic institutions like town hall or public planning meetings. They need to do better, and many are working on it.[2]

## And Why Is Democracy Good At Infrastructure Management?

Infrastructure doesn’t fully determine the kind of activities that thrive on top of it. A phone line doesn’t dictate what people are going to talk about. The Brooklyn Bridge works just as well no matter why you want to cross the East River.

For this reason, questions like where infrastructure is needed, what it’s needed for, and how it ought to be managed, are too complex for experts alone to answer. Every theory about the consequences of a particular infrastructure investment is only that, a theory. And when theories deal with very complex systems, their predictive power fades. Having lots of data becomes more important.

The thing is, everybody impacted by infrastructure holds in their mind precious data about what that infrastructure is good or bad for, and how it therefore ought to be stewarded. In this sense, impacted parties are always qualified to contribute critical information to meaningful decisions about infrastructure. These are the people who will (or won’t) build social structures atop it, whose lives will be altered by it. They have to have a say.

Moreover, giving people genuine democratic authority is the only way to incorporate their insights into decisions. You cannot just surveill people and interpret that as a signal about what they need, because people’s actions don’t express their ideals.[3] What they do isn’t necessarily what they wish to do, or hope to do, or would do if they were differently resource-constrained. Passive, surveillance-based citizen feedback is not democratic participation at all: it’s just a weird, counterproductive reification of society’s present patterns.

Yes, democratic infrastructure governance is egalitarian. But it need not be motivated by egalitarianism, because wise and efficient long-term governance simply can’t happen without egalitarian and democratic decision-making. This is turtles all the way down.

In a sense, infrastructure governance is exactly what legitimate democratic power should do – no more and no less. Most literature on government legitimacy concerns the legitimacy of coercion. But that whole literature fits cleanly with this framework. After all, private ownership of infrastructure depends upon government coercion (e.g., police protecting property). And government coercion depends on its control of scarce resources (e.g., the biggest guns, fastest helicopters, and deepest surveillance). Thus, scarce resource control (which is to say, infrastructure governance) is simply the inseparable flipside of coercion’s coin.

## How We Can Get Better At This

Glen Weyl has written about mismatches between “actual polities” and “natural polities”. This gets at the same idea. Infrastructure always has a “natural” constituency, a set of people whose interests it de facto affects. When the people who “actually” govern the infrastructure do not match that group, the stage is set for exploitation and waste on a systemic level. Actual polities, when misaligned with natural polities, govern infrastructure without full information or attention regarding the complex systems it feeds or strangles.

Perhaps the hardest problem for democracy, then, is determining who belongs in that “natural polity” – that set of people that should jointly make a particular infrastructural decision. It isn’t known in advance.

But getting better at finding those “natural polity” sets, and inviting them into democratic processes, might be the key to nimbler and healthier societies. We are taking steps in that direction with the experimental processes in RadicalxChange Voice, where groups of people can invite participants into group deliberative processes and distribute voice credits amongst themselves (matched by quadratic funding). Processes of “finding the right polity” for public decisions also informed the thinking behind the Data Freedom Act.

Infrastructure is all around us. We interact with it every day, in countless ways. Democracy should be no different. We hope you’ll help us work toward making that happen.

A big thanks to Glen Weyl for major contributions to this piece. Many thanks also to Paula Berman, Albert Wenger, Alex Randaccio, and Angela Corpus.

Notes

1. Imagine an apple orchard full of 100 workers harvesting apples. The hundredth harvester in the orchard picks apples worth about $15 each hour; this is the “marginal value of labor”. Consequently, all hundred employees in the factory are paid$15 per hour: after all, any one of them could be fired at any time; and $15 per hour is all the owner would lose. However, the first few employees entering the orchard can harvest apples at a much faster clip than that. This might seem mysterious but it isn’t. It’s simply that, in a nearly-empty orchard, they can run around picking the low-hanging fruits. The last employees added to an orchard full of workers, on the other hand, have to go after the hard-to-reach apples near the tops of the trees. Because the first few harvesters can produce faster than the last few harvesters, all hundred employees taken together can produce at a rate that exceeds$15/hour times 100: average productivity exceeds the marginal productivity. The owner pays out $1500 per hour, an honest recognition of each worker’s marginal contribution to the enterprise, but the workers are producing more than that in aggregate. If they produce$2000 per hour, the owner keeps \$500 per hour as profit. Competitors, seeing that sweet, juicy profit, will enter the industry. As the apple orchard game grows more competitive, apple prices will go down or wages will go up; and profit will become harder to come by. Thus entrepreneurs pursuing profit make workers and consumers better off. ↩

2. Important work is happening along these lines, partly inspired by Taiwan’s successes with digital democracy. RadicalxChange’s Voice project is a key example. ↩

3. For example, you can’t just look at traffic patterns to determine where people want to go. People might want to go to different places if new roads were built, or old ones were torn up. ↩


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## Monday, November 4, 2013

### Argentine Inflation Saga Approaches Critical Moment

Argentina's time is up. In February, the IMF censured Argentina on account of its notoriously dubious inflation statistics. Under terms of the censure, Argentina was given until September 29, 2013, to improve the flawed data. The deadline has passed, and IMF Chief Christine Lagarde will report to the IMF Executive Board by November 13 on Argentina's progress (or, likely, lack thereof.)

Lagarde's impending report follows a long and nearly unbelievable saga (see timeline at the end of this post.) The gist of the matter is that officially-reported inflation has been in the vicinity of 10% for the past few years, while a variety of private estimates place the true value at 25% or higher. Oh, and the independent economists who publish these private estimates are threatened with criminal prosecution. This September, economist Orlando Ferreres published his estimate that June monthly inflation was 1.9%, compared to the official estimate of 0.8% (with compounding, that's a big difference) and could face a two-year prison sentence. I searched for Ferreres' website on November 3, and it appears to have been hacked (see image below.) I could not find Ferreres' independent inflation estimates on his site; only official estimates appear.

Red flags were raised in earnest in 2007, when then-President Néstor Kirchner dismissed several staff statisticians at INDEC, the statistics institute that publishes the official consumer-price index (CPI) for Argentina. The government takeover of INDEC included the demotion of Graciela Bevacqua, director of the inflation index, who had prepared the inflation data for six years. In 2005, Bevacqua estimated that inflation was over 12% and rising. Bevacqua says that Interior Commerce Secretary Guillermo Moreno began pressuring her to underreport inflation data in May 2006. From then until her demotion, Moreno harassed her unrelentlessly, challenging her data and methodology and demanding more "favorable" estimates. Bevacqua refused to comply, resulting in her demotion and eventual resignation from government.

The 2007 Presidential elections certainly contributed to the pressure for more "favorable" inflation data. With Kirchner's appointees in place at INDEC, end-of-year inflation "fell" from 12.3% in 2005 to 9.8% in 2006 to 8.5% in 2007. Christina Fernandez de Kirchner, wife of Néstor Kirchner, was elected President. After the elections, the data manipulation had to continue, to cover the previous manipulation.

In Argentina, inflation statistics have political significance that extends even deeper than their economic significance, as is common in countries with a history of hyperinflation. In the 1970s, like several other Latin American countries, Argentina borrowed heavily to fund its industrialization efforts. A variety of factors combined to exacerbate a debt crisis in the 1980s, which was accompanied by high inflation in Argentina and neighboring countries. Argentina faced quadruple-digit inflation in 1989 and 1990, with a hyperinflationary peak in March 1990. The disastrous consequences of hyperinflation, including stagnation of growth and capital flight, prompted structural reforms in the 1990s, under direction of the IMF, which restored inflation to low levels. For a time, Argentina was viewed as a model of success.

The situation in Argentina took a turn for the worse around the time of the Brazilian devaluation in 1999. A protracted recession erased most of the decade's gains in poverty reduction. As the federal government deficit widened to 2.5% of GDP in 1999, the IMF advised the De la Rúa administration to implement austerity measures. Distress escalated to full-scale crisis, climaxing with the largest debt default in history in December 2001. Following the default, Argentina faced exclusion from international capital markets and 41% inflation in 2002. Legal battles with a subset of the bondholders continue to this day.

With this history, it is not surprising that both inflation and IMF relations are sensitive issues for Argentina. Bevacqua says that Secretary Moreno "said that if we didn’t aim for zero inflation, we were unpatriotic.” The  irony is that the deceptive data, intended to improve appearances, fooled no one, and made the country look worse rather than better. Consumers in Argentina are well aware that inflation is several times the official rate. A consumer survey run by Torcuato di Tella University has measured inflation expectations near 30% for several years.

 Source: Torcuato di Tella University. Yellow bars indicate median expectation and gray line indicates mean expectation.
MIT economist Alberto Cavallo published an investigation into Argentina's official price index in the widely-read Journal of Monetary Economics in 2012. Here is the abstract:
Prices collected from online retailers can be used to construct daily price indexes that complement ofﬁcial statistics. This paper studies their ability to match ofﬁcial inﬂation estimates in ﬁve Latin American countries, with a focus on Argentina, where ofﬁcial statistics have been heavily criticized in recent years. The data were collected between October 2007 and March 2011 from the largest supermarket in each country. In Brazil, Chile, Colombia, and Venezuela, online price indexes approximate both the level and main dynamics of ofﬁcial inﬂation. By contrast, Argentina’s online inﬂation rate is nearly three times higher than the ofﬁcial estimate.
Cavallo suggests that "the way the data is being altered is far simpler than commonly assumed. INDEC is a large organization, with many employees involved with the data collection and construction of the price indexes. Instead of changing the prices at the item level, it is probably easier for the government to change the aggregate numbers, which are seen by just a handful of people at the end of the CPI calculation process."

He adds, in conclusion, that "There is no obvious reason for why the government continues to manipulate the ofﬁcial price indexes. Some economists point to lower interest payments for inﬂation-linked bonds, while others highlight the fact that, by using artiﬁcially low inﬂation estimates in the budget, the government can avoid distributing any excess tax income to the provinces. However, these short-term resources are negligible next to the negative effects and uncertainty the manipulation has introduced in the economy."

Regarding Argentina's inflation-linked bonds, doubts about data accuracy began to wipe away their market in 2007. Argentina originally issued inflation-linked bonds in 1973, and the "linkers" came to play a large role in public debt management. Even as late as 2009, a third of Argentine debt was linked to inflation. But in September 2009, Argentina launched a new bond, the Bonar 2015, as part of a debt swap that replaced much of the inflation-indexed paper.

The collapse of Argentina's inflation-indexed bond market, though destructive, pales in comparison to the harms that have been inflicted on workers and consumers. A government-imposed price freeze implemented in February, and another in June, appear unsurprisingly ineffective. The mismatch between official data and generally perceived cost-of-living increases makes satisfactory wage negotiations between trade unions and the Ministry of Labor impossible. Erosion of purchasing power is likely much worse in the informal sector, which accounts for an estimated 40% of Argentine workers. In addition, strict capital controls fuel a large currency black market.

Lagarde describes the IMF censure of Argentina as a "yellow card." When Lagarde makes her report to the IMF Executive Committee later this month, if the data is not suitable, she will give a "red card." It is not clear how harmful any potential IMF sanctions could be. Argentina is already the only Group of 20 country that refuses to allow the IMF to conduct an annual review of the economy known as the Article IV consultation, and may not be overly concerned with further deterioration of relations with the IMF. A likely, if unsatisfactory, outcome is some knuckle-rapping and maintenance of the status quo for all practical purposes. The IMF could suspend Argentina's voting and related rights and begin the process of compulsory withdrawal from the fund. A top international priority should be the minimization of contagion to other emerging markets-- which should be possible, since the situation is sufficiently Argentina-specific. The way out of the mess in Argentina is difficult to foresee. The way into the mess is outlined in the timeline below.

### Timeline

1970s: Argentina borrows heavily to fund industrialization efforts.

1973: Argentina issues inflation-indexed bonds.

1980s: Argentine inflation heats up in the midst of Latin American debt crisis.

March 1990: Argentina's hyperinflation reaches peak.

1990s: Argentina implements IMF-directed structural reforms.

1998-2002: Argentina suffers severe recession.

June 31, 2001: Argentina swaps $29.5 billion of debt and defers$7.8 billion in interest payments.

December 5, 2001: IMF announces it will not release $1.3 billion in aid to Argentina because austerity measures are insufficient. December 23, 2001: Argentina defaults on$100 billion in debt.

May 25, 2003: Néstor Kirchner becomes President, succeeding Carlos Menem.

2007: Kirchner replaces several statisticians at INDEC with political appointees.

December 10, 2007: Christina Fernandez de Kirchner becomes President.

September 2, 2009: Argentina swaps 16.7 billion pesos ($4.34 billion) of inflation-linked bonds for newly-issued 2014 and 2015 bonds to extend maturities and reduce financing costs. January 2010: Standoff between Central Bank President Martín Redrado and President Kirchner over use of central bank reserves to pay debt. Redrado is fired, then reinstated by courts. January 30, 2010: Central Bank President Redrado resigns. Redrado accuses government of data manipulation. February 3, 2010: Mercedes Marcó del Pont appointed President of Central Bank. February 1, 2012: IMF Executive Board gives Argentina 180 days to implement specific measures to address quality of inflation data. September 17, 2012: IMF Executive Board determines Argentina has made insufficient progress in implementing remedial measures. February 1, 2013: IMF Executive Board censures Argentina. February 4, 2013: Argentina announces two-month price freeze. September 29, 2013: Deadline for Argentina to address concerns with inflation and GDP data. November 13, 2013: Deadline for Christine Lagarde's report to IMF Executive Board on Argentina's progress. #### 9 comments: 1. Add to your list of events the arbitrary bureaucratic blocking of imports from Brazil since the beginning of 3003 1. BREAKING NEWS: USD Currency Collapse China Causes The USD To Fall? USD Dollar Drops 50% In Value Overnight? Hey there I don’t know you have heard this yet… But the USD Dollar is on the verge of collapse. In fact since 1973 it’s been on a downward trend… Dollar-Index-Past-Forty-Years USD national debt is at an all time high and now financial experts are saying that China is starting to sell their debt holdings to the secondary market. This means it won’t be too long until the USD crumbles in value! >>Watch This Video To Learn More<< I don’t want to be the bringer of bad news, but this is seriously not good for the US since we already have domestic problems of our own, such as immigration and unemployment. Our economy can not withstand another hit. This time the USD might actually collapse. >>Watch This Video To Learn More<< Make sure you watch it before it’s taken offline. Speak soon. [Mr Mark Fidelman] 2. Correction: ...beginning of 2013 3. Good post. One minor, but important detail: I think it is misleading to say that “In the 1970s, like several other Latin American countries, Argentina borrowed heavily to fund its industrialization efforts.” Argentina borrowed heavily under military rule after the coup in 1976. Economic liberalization was the main objective of the economic policy of that period, and the opening up of domestic markets was intended to shut down inefficient local industries that couldn’t adapt to international competition, and thus produced a considerable deindustrialization. Private sector did most of the borrowing, benefiting from financial deregulation and the appreciation of the peso (In the 70s Argentina and Chile differed from the other Latin American countries, where the government did most of the borrowing). Government’s mostly spent their foreign loans on huge infrastructure projects and to pay for the fiscal deficit. The big industrialization efforts in Argentina (1930-1975) preceded the huge increase in indebtedness (1975-1982). I would also like to ask you why “Erosion of purchasing power is likely much worse in the informal sector, which accounts for an estimated 40% of Argentine workers.”? Is it just beacause things generally tend to be worse in the informal sector? Or are you considering any specific mechanism? Tor O. 4. Enough paradigm maintenance please! Reality check: Argentina has the fastest growing economy in the Americas. It is far more industrialized than the USA. (25% of GDP is manufacturing vs. 11%) I'd be willing to bet 150,000 RMB that Americans will begin emigrating in large numbers to Argentina within five years. Argentina is going to be wealthier than the USA for most of this century. Always bet on countries with an industrial sector and willingness to defend it. 5. No mention of how the Currency board failed? While it did lower inflation initially, it amplified recessions and it had to be abandoned. Maybe Argentina should try an independent Central Bank with inflation targeting. 6. The problem seems to have been caused by the IMF's insane "austerity" prescription in the beginning. I can't see why Argentina would give a damn about the IMF. At this point, the correct trade policy for most countries -- those which are large enough -- is autarky. Argentina's having some trouble pulling it off but it's the right move. 7. Hello Everybody, My name is Mrs Sharon Sim. I live in Singapore and i am a happy woman today? and i told my self that any lender that rescue my family from our poor situation, i will refer any person that is looking for loan to him, he gave me happiness to me and my family, i was in need of a loan of S$250,000.00 to start my life all over as i am a single mother with 3 kids I met this honest and GOD fearing man loan lender that help me with a loan of S\$250,000.00 SG. Dollar, he is a GOD fearing man, if you are in need of loan and you will pay back the loan please contact him tell him that is Mrs Sharon, that refer you to him. contact Dr Purva Pius,via email:(urgentloan22@gmail.com) Thank you.

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## Search Result

### Search Conditions

Years
All Years
for journal 'PTP'
author 'S.* Matsumoto' : 13
total : 13

### Search Results : 13 articles were found.

1. Progress of Theoretical Physics Vol. 77 No. 5 (1987) pp. 1232-1239 : (5)
Quantization Formula for the Schrödinger Equation with Repulsive Potential
Takeshi Yamashiro, Shuichi Matsumoto and Kiyotaka Kakazu
2. Progress of Theoretical Physics Vol. 78 No. 1 (1987) pp. 166-175 : (5)
Einstein-Gravity in Fibre Bundle Formalism
Kiyotaka Kakazu and Schuichi Matsumoto
3. Progress of Theoretical Physics Vol. 78 No. 4 (1987) pp. 932-950 : (5)
Supergravity on Principal Fibre Bundle
Kiyotaka Kakazu and Schuichi Matsumoto
4. Progress of Theoretical Physics Vol. 79 No. 5 (1988) pp. 1022-1027 : (5)
Chaos on the Super Riemann Surface
Shuji Matsumoto and Yukinori Yasui
5. Progress of Theoretical Physics Vol. 79 No. 6 (1988) pp. 1431-1450 : (5)
Construction of $N = 2$ Supergravity on Principal Fibre Bundle
Kiyotaka Kakazu and Schuichi Matsumoto
6. Progress of Theoretical Physics Vol. 82 No. 2 (1989) pp. 246-253 : (1)
Physical Interpretations on Reduction of Wave Packet in the Machida-Namiki Theory
Shoju Kudaka, Shuichi Matsumoto and Kiyotaka Kakazu
7. Progress of Theoretical Physics Vol. 82 No. 4 (1989) pp. 665-681 : (1)
Generalized Coherent State Approach to Quantum Measurement
Shoju Kudaka, Shuichi Matsumoto, and Kiyotaka Kakazu
8. Progress of Theoretical Physics Vol. 84 No. 1 (1990) pp. 1-6 : (5)
Canonical Formulation of Supermechanics
Shuji Matsumoto
9. Progress of Theoretical Physics Vol. 85 No. 3 (1991) pp. 631-642 : (5)
Precise Test of the Standard Model in $e^{-}e^{+}$ Annihilation at TRISTAN Energy $\sqrt{s} = 60$ GeV
Seiji Matsumoto, Kenji Tominaga, Osamu Terazawa and Minoru Biyajima
10. Progress of Theoretical Physics Vol. 90 No. 1 (1993) pp. 35-42 : (1)
A Reexamination of the Wigner and Araki-Yanase Theorem
Shuichi Matsumoto
11. Progress of Theoretical Physics Vol. 98 No. 1 (1997) pp. 9-28 : (1)
Quantum Dissipation in Open Harmonic Systems
Izumi Joichi, Shigeki Matsumoto and Motohiko Yoshimura
12. Progress of Theoretical Physics Vol. 120 No. 1 (2008) pp. 77-98 : (b)
Effective Potential of Higgs Field in Warped Gauge-Higgs Unification
Naoyuki Haba, Shigeki Matsumoto, Nobuchika Okada and Toshifumi Yamashita
13. Progress of Theoretical Physics Vol. 125 No. 6 (2011) pp. 1311-1316 : (b)
Baryogenesis from Dark Sector
Naoyuki Haba and Sigeki Matsumoto


text:
# How large would a chunk of ice have to be to do lasting harm if it were 'thrown' at Earth?

I'm working on a story involving life on Europa being upturned dramatically by contact from a probe sent from Earth. I know the chances of life, let alone intelligent life, are pretty slim on Europa, but I'm ignoring that for the sake of the story. The driving force of the story is that a probe that's come from Earth crashes/malfunctions after melting through the ice.The probe sinks to the bottom of a trench and sets off some sort of ecological disaster/collapse, that then has cascading effects on the civilizations living there. In retaliation, a fairly advanced society that has adapted to living in the ice (most of society is less advanced and lives on the sea floor), makes plans to retaliate against the Earth by breaking off a chunk of the crust and hurling it at Earth.

How big of a piece of ice would be needed to cause considerable damage to Earth? I'm thinking even if it burns up entirely on impact, if it adds enough water to our atmosphere it could really do some damage, but I'm not really sure what sort of size would make sense? I'm thinking it would need to be as little as possible while still big enough to ensure it does real harm to Earth. It would probably not be going very fast since they don't have engines attached or anything like that, and are purely working with controlling water pressure to push it the way they want (my idea is that they are basically creating their own plumes as initial propulsion out of orbit, and then just riding that momentum or possibly sling-shotting around other celestial bodies to compensate). I'm also just going to assume they have the capacity to calculate the trajectory correctly. So they are reasonably sure it will hit Earth.

I don't know much about physics, so I'm not sure how much Jupiter's gravity would have an impact, or whether they would be able to use that somehow to slingshot it around at their target and maybe increase it's speed that way, or if it would most likely just never be able to break free of Jupiter's orbit at all without a more powerful propulsion.

• "It would probably not be going very fast": Anything coming from outer space will come with at least 11.2 km/s (40,320 km/h, 25,054 mph). (Because mechanics is fully reversible; the minimum speed with which something can come to Earth from outer space is equal to the minimum speed needed to lauch it from Earth to reach outer space.) Jul 30 at 4:22
• Is that like terminal velocity? Like, once it comes to Earth, that's the min speed it would hit? Or does it have to be coming at that speed before it gets to Earth if it wants to make it? Jul 30 at 4:41
• That's the speed with which is will be approaching Earth, assuming is started with zero speed at infinity. ("Terminal velocity" is something else entirely.) For an example of what happens when a chunk of ice enters the atmosphere at cosmic speeds, see the Tunguska event. Jul 30 at 4:42
• @AlexP Good point, but wouldn't the minimum be Earth's escape velocity + Eruopa's + Jupiter's combined escape velocity at the orbit of Europa? And Brooke, what do you mean by "lasting harm" and "considerable damage?" Would the crater and odd genetic disorders resulting from the Tunguska Event be considered "lasting harm?" (No practical harm was done.) Or are we thinking Chicxulub crater kind of damage? (Think "dead dinosaurs.") Or something else? Can't judge the size of the chunk without knowing what you expect for damage (please be specific).
– JBH
Jul 30 at 6:07
• @JBH I think since they don't have the power to control where it hits on Earth, it would have to be more than the Tunguska Event. I don't think it's necessary to be extinction-level (though the one's responsible wouldn't be sad if it was). They are retaliating for an event that upended their whole society and caused massive ecological devastation. So they'd want something that would hurt humans regardless of where it hit the planet. I was thinking, since it's primarily ice, how big would it have to be to cause water levels to rise a la melting glaciers, but that might be too big? Jul 30 at 12:48

20 Kilometer Ice Cube

From Wikipedia, the asteroid that killed the dinosaurs is estimated at 10-15 km in diameter and hit the Earth at about 20km/s.

The mass and speed together give the amount of energy that was suddenly absorbed into the Atmosphere. Exactly what the projectile was made of does not matter.

D'Monlord says 20km/s is about the same escape velocity you need to escape from Jupiter's gravity well and the projectile will, regardless of what happens in between, hit Earth at least 11km/s. So let's split the difference and say it hits Earth at 20km/s.

Asteroids can be up to 5 times denser than ice. That means an ice asteroid of the same mass has $$\sqrt[3]5 \simeq 1.71$$ times the diameter.

So we get 10-15 km times 1.71 which is about 20km in diameter.

"how big would it have to be to cause water levels to rise a la melting glaciers"

According to Wikipedia Earth's ocean surface area is 361132000 km^2. To raise level by 1 meter you would need 361132 km^3 of water or ~392534 km^3 of ice which would be a cube with side of ~73 km.

Escape velocities were mentioned in comments. 11.186 km/s for Earth, 2.025 km/s for Europa herself, ~19.2 km/s for Jupiter from Europa's surface.

Ice cube with 73 km side would not melt in the atmosphere. Atmosphere is thin, at this speed and right angle it would penetrate the atmosphere in seconds and hit Earth harder than what killed the dinosaurs. Kinetic energy of this magnitude doesn't just dissipate in atmosphere.

But most of this energy has to be first supplied to this chunk of ice by Europeans. And if they can do that launching ice cube looks like a bad idea. Accelerating giant piece of ice to escape velocity using water is terribly difficult. It melts and breaks. I can't think of any plausible way of achieving it.

I would consider something else. E.g. Europeans are not carbon based but extremely proficient in building nano-machines. So they make a machine capable to replicate itself using any protein and send about a cup of it to Earth.

• I do have to point out that, contrary to what your intro suggests, melting glaciers do not raise water levels. Because of the very mechanics of buoyancy, a partially floating glacier displaces exactly the same amount of water than it does once it turns its full mass to water. Global warming makes water levels raise because of thermal expansion, not because of melted ice. It is a misconception that unfortunately hurts the credibility of many people that have the otherwise good intention of warning against the dangers of global warming. Melting glaciers is an ecological disaster on its own. Aug 1 at 6:29


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Quantifying abundances of transcripts using kallisto
1
0
Entering edit mode
11 months ago
Kumar ▴ 120

Hi, I am analyzing RNA-Seq data. I have 190 PE samples (control: male, female, mutant: male, female) and generated featureCount matrix. For the differential gene expression, I used deseq2 with adjusted 0.05 p-value.

On the other hand, I used kallisto for quantifying abundances of transcripts with the same input data.

My question is that I am getting very less number of transcripts in differential expression after kallisto analysis in comparison to gene analysis results. However, the number of DE should be more in abundances of transcripts rather than a gene-based method.

Here are the results:

Results: Deseq2 gene expression using STAR aligner

adjusted p-value < 0.05
LFC > 0 (up)       : 606, 1.2%
LFC < 0 (down)     : 169, 0.34%
outliers [1]       : 7, 0.014%
low counts [2]     : 40617, 81%

(mean count < 6)

Deseq2 result with kallisto:

out of 138 with nonzero total read count
LFC > 0 (up)       : 124, 90%
LFC < 0 (down)     : 14, 10%
outliers [1]       : 0, 0%
low counts [2]     : 0, 0%
(mean count < 0)

rna-seq kallisto next-gen • 422 views
0
Entering edit mode
11 months ago

1) It is not clear to me that DESeq is intended for finding DE transcripts. Genes, sure, but not transcripts

2) Kallisto is smarter about assigning ambiguously assigned reads, I'd be using its gene counts, not FeatureCounts. Then you don't have to run two totally separate pipelines.

0
Entering edit mode

Hi, I am new in this analysis. I ran the kallisto and got all .tsv files then I merged these all files to make a single matrix file. It looks like this: Therefore, you mean I do not need to do differential expression analysis with this data, if so how I can use these data for RNA-seq analysis. Thank you!

#

                      270   515 574
ENST00000415118.1   0   0   0
ENST00000434970.2   0   0   0
ENST00000448914.1   0   0   0
ENST00000604642.1   0   0   0
ENST00000603326.1   0   0   0

0
Entering edit mode

You should use tximport first (or tximeta)

0
Entering edit mode

Yes! I am using tximport.

Here is my script: However, the issue here is that my samples number are not equal so I am not sure how to improve the script instead of using "each=93". Please suggest.

names(files) <- paste0("sample", 1:187)
txi.kallisto <- tximport(files, type = "kallisto", txOut = TRUE)
head(txi.kallisto$counts) sampleTable <- data.frame(condition = factor(rep(c("mutant", "control"), each=93))) rownames(sampleTable) <- colnames(txi.kallisto$counts)
dds <- DESeqDataSetFromTximport(txi.kallisto, sampleTable, ~condition)

dds <- DESeq(dds)

Since I was not sure about the data structure, so I prepared a single .csv matrix file of all samples' count files using cut and paste commands and I am using the following script: Please suggest if this way is correct.

dat <- read.table("/outfile-kallisto.txt", header=TRUE)
dat <- as.matrix(dat)
dds <- DESeqDataSetFromMatrix(countData = dat,
design= ~ Group + Gender)

dds <- DESeq(dds)

0
Entering edit mode

You need to stop and figure out what that line with the "each=93" is doing, then you will know how to modify it for your own needs. Or, making a metadata table outside R is fine too.

Are you completely sure that txOUT = true is what you want to use with DESeq?

0
Entering edit mode

I figured out the updates. Here is my updated script: However, the genes in differential gene expression are very few numbers.

names(files) <- paste0("sample", 1:187)
txi.kallisto <- tximport(files, type = "kallisto", txOut = TRUE)
head(txi.kallisto$counts) sampleMetaData <- data.frame(Group=c(rep(c("mutant"), 94), rep(c("control"), 93)), Gender=rep(c(rep(c("F"), 130), rep(c("M"), 57)))) rownames(sampleMetaData) <- colnames(txi.kallisto$counts)
dds <- DESeqDataSetFromTximport(txi.kallisto, sampleMetaData, design= ~ Group + Gender)
res <- results(dds)
dds <- DESeq(dds)

results

out of 171018 with nonzero total read count
LFC > 0 (up)       : 975, 0.57%
LFC < 0 (down)     : 12, 0.007%
outliers [1]       : 0, 0%
low counts [2]     : 73970, 43%
(mean count < 2)


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# 6: The Four Fundamental Forces

The fundamental forces are normally divided in four groups, of the four so-called “fundamental” forces.

• 6.1: Prelude to the Fundamental Forces
The fundamental forces are normally divided in four groups, of the four so-called “fundamental” forces.
• 6.2: Gravity
The theory of gravity can be looked at in three ways: (1) The old fashioned Newtonian gravity,  (2) Einstein’s theory of gravity and (3)quantum theory. Latter is still under development.
• 6.3: Electromagnetism
• 6.4: Weak Nuclear Force
The weak nuclear force is manifests itself through nuclear β decay.
• 6.5: Strong Nuclear Force
The strong force is what keeps nuclei together. It is described by a theory called quantum chromodynamics (QCD), which described the forces between fermions called quarks that make up the hadrons. These forces are mediated by spin-1 bosons called gluons.

Thumbnail: The Feynman diagram for the weak decay of a neutron.

This page titled 6: The Four Fundamental Forces is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


text:
# zbMATH — the first resource for mathematics

Results on the propositional $$\mu$$-calculus. (English) Zbl 0553.03007
In this paper we define and study a propositional $$\mu$$-calculus $$L\mu$$, which consists essentially of propositional modal logic with a least fixpoint operator. $$L\mu$$ is syntactically simpler yet strictly more expressive than Propositional Dynamic Logic (PDL). For a restricted version we give an exponential-time decision procedure, small model property, and complete deductive system, thereby subsuming the corresponding results for PDL.

##### MSC:
 03B45 Modal logic (including the logic of norms) 68Q65 Abstract data types; algebraic specification
Full Text:
##### References:
 [1] De Bakker, J.W., Mathematical theory of program correctness, (1980), Prentice-Hall Englewood Cliffs, NJ [2] De Bakker, J.W.; De Roever, W., A calculus for recursive program schemes, (), 167-196 · Zbl 0238.68006 [3] Chandra, A.; Kozen, D.; Stockmeyer, L., Alternation, J. assoc. comput. Mach., 28, 1, 114-133, (1981) · Zbl 0473.68043 [4] Emerson, E.A.; Clarke, E.M., Characterizing correctness properties of parallel programs using fixpoints, (), 169-181 · Zbl 0456.68016 [5] Emerson, E.A.; Clarke, E.M., Design and synthesis of synchronization skeletons using branching-time temporal logic, (), 52-71 [6] Fischer, M.; Ladner, R., Propositional dynamic logic of regular programs, J. comput. system sci., 18, 2, 194-211, (1979) · Zbl 0408.03014 [7] Hitchcock, P.; Park, D.M.R., Induction rules and termination proofs, (), 225-251 [8] Halpern, J.; Reif, J., The propositional dynamic logic of deterministic, well-structured programs (extended abstract), Proc. 22nd IEEE symp. on foundations of computer science, 322-334, (1981) [9] Kozen, D., A representation theorem for models of ∗-free PDL, (), 352-362 [10] Kozen, D., On the duality of dynamic algebras and Kripke models, (), 1-11 [11] Kozen, D., On induction vs. ∗-continuity, (), 167-176 [12] D. Kozen, On the expressiveness of μ, Unpublished manuscript. [13] D. Kozen, Small models for the propositional μ-calculus, Unpublished manuscript. [14] Kozen, D., Results on the propositional μ-calculus, Proc. 9th internat. colloq. on automata, languages, and programming, 348-359, (1982) [15] Kozen, D.; Parikh, R., An elementary proof of the completeness of PDL, Theoret. comput. sci., 14, 113-118, (1981) · Zbl 0451.03006 [16] Kozen, D.; Parikh, R., A decision procedure for the propositional μ-calculus, () · Zbl 0564.03012 [17] Park, D.M.R, Fixpoint induction and proof of program semantics, (), 59-78 · Zbl 0219.68007 [18] Pratt, V.R., A near optimal method for reasoning about action, J. comput. systems sci., 20, 231-254, (1980) · Zbl 0424.03010 [19] Pratt, V.R., A decidable μ-calculus, Proc. 22nd IEEE symp. on foundations of computer science, 421-427, (1981), (Preliminary Rept.) [20] De Roever, W.P., Recursive program schemes: semantics and proof theory, () · Zbl 0344.68001 [21] Streett, R., Propositional dynamic logic of looping and converse, Proc. 13th ACM symp. on theory of computing, 375-383, (1981) [22] Scott, D.; De Bakker, J.W., A theory of programs, (1969), IBM Vienna, Unpublished manuscript
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
If a text index has multiple fields in Mongo, how can I do a search in only one field since the weight is applied to the index and not to the query?

1 Answer

I assume you are using MongoDB version 2.4 to 3.x

You have no other choices than using MongoDB Text search.

below two approaches may helpful.

1. Indexing Multiple Fields (Compound Indexing)

db.yourcollection.createIndex({"textfield1":"text","textfield2":"text"})

Compound index is combination of both indices, works only if we search on both fields with given values.

2. Indexing the Entire Document (Wildcard Indexing)

db.yourcollection.createIndex({"$**":"text"}) Though the option two is not recommended. But if you use more text fields those are searchable every time, saves you time. One index fits for all. reference: Full-Text Search in MongoDB (Just to take examples from it) 建立完索引后, 可以用如下的查询： db.collectionName.find({$text: {\$search: "keyword"}})


text:
# APPENDIX C - Graphs of Second-Degree Equations - C Exercises: 17

Ellipse

#### Work Step by Step

Given: $4x^{2}+y^{2}=1$ $\frac{x^{2}}{(\frac{1}{2})^{2}}-\frac{y^{2}}{1^{2}}=1$ which is the standard form of the equation of an ellipse.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# New Streaming Algorithms for High Dimensional EMD and MST

We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an n-point set X ⊂{1,2,…,Δ}^d, and computing the Earth Mover Distance (EMD) between two multi-sets A,B ⊂{1,2,…,Δ}^d of size n. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of Õ(log n) and using polylog(n,d,Δ)-space only. Furthermore, our algorithm for EMD can be compressed to a single pass with a small additive error. Previously, the best known sublinear-space streaming algorithms for either problem achieved an approximation of O(min{log n , log (Δ d)}log n) [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20]. For MST, we also prove that any constant space streaming algorithm can only achieve an approximation of Ω(log n), analogous to the Ω(log n) lower bound for EMD of [Andoni-Indyk-Krauthgamer '08]. Our algorithms are based on an improved analysis of a recursive space partitioning method known generically as the Quadtree. Specifically, we show that the Quadtree achieves an Õ(log n) approximation for both EMD and MST, improving on the O(min{log n , log (Δ d)}log n) approximation of [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20].

## Authors

• 195 publications
• 17 publications
• 9 publications
• 11 publications
09/25/2019

### Streaming PTAS for Binary ℓ_0-Low Rank Approximation

We give a 3-pass, polylog-space streaming PTAS for the constrained binar...
09/30/2018

### Streaming Algorithms for Planar Convex Hulls

Many classical algorithms are known for computing the convex hull of a s...
08/16/2021

### A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

We prove a lower bound on the space complexity of two-pass semi-streamin...
02/20/2018

### Sublinear Algorithms for MAXCUT and Correlation Clustering

We study sublinear algorithms for two fundamental graph problems, MAXCUT...
07/23/2020

### Lower Bounds and Hardness Magnification for Sublinear-Time Shrinking Cellular Automata

The minimum circuit size problem (MCSP) is a string compression problem ...
01/21/2020

### Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently ...
07/12/2020

### Streaming Algorithms for Online Selection Problems

The model of streaming algorithms is motivated by the increasingly commo...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.


text:
Synopsis: Better transistors through quantum mechanics

More compact transistors may be possible by harnessing the quantum properties of semiconductor heterostructures.

As chip designers seek to pack more transistors onto every square millimeter of silicon real estate, they run up against a fundamental problem: small devices become overwhelmed by quantum effects. However, some researchers are seeking to turn the tables by pressing quantum mechanics into service as part of the transistor design. In a paper in Physical Review B, Adam Sciambi of Stanford University, California, and his collaborators report their proof-of-principle experiments on a new kind of quantum transistor.

The team has created a two-layer $\text{GaAs/AlGaAs}$ quantum well heterostructure, in which the wave function of one layer extends into the second to modulate the tunneling current between the layers. In this design, a voltage on the first quantum well causes that layer to be depleted of carriers, which changes the subband energy level in the well. As the subband energy approaches the top of the quantum well potential, the wave function extends further and further out toward the second layer. When the wave function overlaps the second layer, the tunneling current can increase as much as two orders of magnitude, a substantial degree of gating leverage.

Although the reported design only works at cryogenic temperatures, a different choice of materials, for example, graphene, may allow operation at more technologically relevant temperatures. – David Voss

More Features »

Announcements

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Subject Areas

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Synopsis: Flip-Flopping the Bands

A pair of semiconductor quantum wells with an inverted band structure hosts electrons whose spins are almost all in the same quantum state.   Read More »

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Focus: Negative Resistance with a Single Atom

Current flowing through a single silicon atom can be made to decrease with increasing voltage, potentially allowing the integration of a new type of component into microelectronic circuits. Read More »

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Synopsis: Valley of the Dichalcogenides

A magnetic field can be used to change the “valley” states that emerge in certain semiconductors. Read More »


text:
Subsections

# Partial Derivatives and their Geometric Interpretation

## Purpose

The purpose of this lab is to acquaint you with using Maple to compute partial derivatives.

## Getting Started

To assist you, there is a worksheet associated with this lab that contains examples and even solutions to some of the exercises. You can copy that worksheet to your home directory by going to your computer's Start menu and choose run. In the run field type:

\\filer\calclab

when you hit enter, you can then choose MA1024 and then choose the worksheet

Pardiff_start_B11.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.

## Background

For a function of a single real variable, the derivative gives information on whether the graph of is increasing or decreasing. Finding where the derivative is zero was important in finding extreme values. For a function of two (or more) variables, the situation is more complicated.

### Partial derivatives

A differentiable function, , of two variables has two partial derivatives: and . As you have learned in class, computing partial derivatives is very much like computing regular derivatives. The main difference is that when you are computing , you must treat the variable as if it was a constant and vice-versa when computing .

The Maple commands for computing partial derivatives are D and diff. The Getting Started worksheet has examples of how to use these commands to compute partial derivatives.

## Exercises

1. Compute the three distinct second order partial derivatives of

at the point using the diff command and then again using the D command.

2. Given the function

a)
Plot the function and the plane on the same graph. Use intervals , , .
b)
Find the derivative of in the plane.
c)
Graph the two-dimensional intersection of the plane and .
d)
Does your two-dimensional graph look like the intersection from your three-dimensional graph? Be sure to use the same ranges to properly compare and rotate the 3-D graph.

3. Find the equation of the plane tangent to the surface at the point and find the equation of the plane tangent to the graph at . Plot both tangent planes on the same graph as the surface over the intervals and . Be sure to rotate the graph to see that the planes are tangent to the surface.


text:
Article | Open

# Accessing valley degree of freedom in bulk Tin(II) sulfide at room temperature

• Nature Communicationsvolume 9, Article number: 1455 (2018)
• doi:10.1038/s41467-018-03897-3
Accepted:
Published:

## Abstract

The field of valleytronics has promised greater control of electronic and spintronic systems with an additional valley degree of freedom. However, conventional and two-dimensional valleytronic systems pose practical challenges in the utilization of this valley degree of freedom. Here we show experimental evidences of the valley effect in a bulk, ambient, and bias-free model system of Tin(II) sulfide. We elucidate the direct access and identification of different sets of valleys, based primarily on the selectivity in absorption and emission of linearly polarized light by optical reflection/transmission and photoluminescence measurements, and demonstrate strong optical dichroic anisotropy of up to 600% and nominal polarization degrees of up to 96% for the two valleys with band-gap values 1.28 and 1.48 eV, respectively; the ease of valley selection further manifested in their non-degenerate nature. Such discovery enables a new platform for better access and control of valley polarization.

## Introduction

The development of valleytronics offers great potential in providing an additional degree of freedom that may open a new field beyond that of electronics and spintronics1,2. Over the past four decades, the presence of valleytronics in cubic systems such as Si3,4, AlAs5,6, and even recently, diamond7 has been established, where strong biases have been utilized to distinguish three pairs of degenerate conduction band minima. More recently, developments in two-dimensional (2D) materials have sparked the revival of valleytronics with enlightening research in graphene and various transition metal dichalcogenides (TMDCs)8,9,10,11,12. By probing odd layers8,9 of such low dimensional materials, and/or the application of certain experimental conditions, such as cryogenic temperatures10,11 and/or strong electric12 or magnetic field13, valleytronic behavior can be detected. However, such requirements inevitably pose a plethora of practical challenges that create a high barrier in advancing the technology towards practical applications.

We report a model system in Tin(II) sulfide (SnS) that may present plausible solutions to the above-mentioned issues and provide a completely new platform with easy access to the valley degree of freedom. SnS has a Pmcn crystal structure with an orthorhombic unit cell comprising puckered layers analogous to black phosphorus14,15. This puckered-layered structure, gives rise to the high anisotropy between the armchair (y) and zigzag (x) directions, which then gives rise to directional behavior manifested in, for example, the Raman response16, in-plane electronic mobility17, and photoactivity18 of SnS. Such anisotropy can also be correlated to the reciprocal space. It has been demonstrated theoretically that, in monolayer SnS, there exist a pair of non-degenerate band gaps along the in-plane orthogonal high symmetry axes, which correspond to the y and x axes in real space19. These band gaps correspond to two sets of distinct valleys that can be directly accessed using light with different linear polarizations.

Here, we demonstrate the presence of two non-degenerate band gaps in bulk SnS that are, analogous to the monolayer case, effectively valleys with high polarization degrees. We follow the convention of labeling high symmetry k-points as previously reported in ref. 20, where the ΓY and ΓX axes in the reciprocal unit cell correspond to the y and x directions in real space. Density-functional theory (DFT) calculations yield a band structure with two local band gaps along the ΓY and ΓX axes, denoted by Eg,ΓY and Eg,ΓX, respectively. These valleys along the ΓY and ΓX axes have unique advantages over previously reported conventional and 2D valleytronic materials: (i) we can clearly differentiate two sets of valleys in multiple ways without the application of external bias fields as the valleys are inherently non-degenerate, enabling direct access to the valley degree of freedom, (ii) the valleys can be effectively polarized under ambient conditions without low temperature, and (iii) bulk SnS was utilized and can be prepared with relative ease, as it is not limited by the requirement of centrosymmetry breaking as in certain 2D materials.

## Results

### Material preparation and characterization

The crystal structure (left and right panels) and the corresponding band structure (middle panel) of SnS is presented in Fig. 1a, with the high armchair–zigzag anisotropy further emphasized in Fig. 1b. We synthesized bulk SnS microplates via physical vapor deposition (PVD) and observed their flat morphology and Sn:S ratio of 1:1 via SEM-EDX (Fig. 1c). Raman spectra (Fig. 1d) show four characteristic peaks of SnS (three Ag and one B3g). We utilized prior understanding of the polarization-dependent Raman responses16,21 of orthorhombic IV–VI monochalcogenides to determine the armchair (y) and zigzag (x) directions of the sample (Fig. 1d, Supplementary Fig. 1 and Supplementary Note 1). (The directions can also be directly observed from the optical microscopy image (Fig. 1c) due to the preferred growth facets in bulk crystals as demonstrated using surface energy arguments21.)

### Valley selectivity via absorption measurements

Figure 2 shows the optical setup configuration used for the reflection (R), transmission (T), and photoluminescence (PL) measurements, where the sample is rotated with respect to the polarization axis of the linearly polarized excitation light, giving an angle θ. (For simplicity, the armchair (y) direction, as previously determined, is set as θ = 0°.)

We observe clear valley selectivity in the R and T spectra of SnS, shown in Fig. 3a, Supplementary Fig. 2, and Supplementary Note 2. The reflectivity of our sample remains stable at short wavelengths until the onset of fluctuations due to Fabry–Pérot interference oscillations22, which denotes the region where the absorption of SnS decreases rapidly. This agrees with the observation that the percentage of transmitted light is very small at wavelengths shorter than the oscillations. Distinct onsets are observed when the polarization of light is aligned to the armchair (zigzag) directions, respectively. From R and T, we obtained the corresponding wavelength-dependent absorption coefficients23,24, and hence the Tauc plots25 of SnS when aligned to the two axes directions. The Tauc plots (Fig. 3c) give band-gap values of 1.48 and 1.28 eV, clearly distinguishing the two sets of valleys.

The selectivity of the absorption to polarized light can be quantified by the degree of optical dichroic anisotropy, which is given by the ratio αzigzag/αarmchair in our system (Fig. 3c). We see that SnS is able to maintain a αzigzag/αarmchair > 200% for a range of more than 0.2 eV and also possesses a maximum αzigzag/αarmchair > 600%. This wide range and large anisotropy may potentially relax the strict requirements usually present in valleytronic systems.

Because the valley behavior in SnS is due to polarization-dependent band-gap transitions, we further investigated the temperature dependence of the absorption coefficients along the two axes directions. From the plots of α (Fig. 3d), we see that the x-axis intercept increases with decreasing temperatures, which agrees with the typical behavior of band gaps. The variations in band-gap values with temperature are then fitted with the Varshni equation26 (Fig. 3e), which shows that the valley behavior extends across the whole investigated temperature range.

### Selection rules for valley selectivity

Our observation of polarization dependence of the valley peaks in bulk crystals of SnS has also been theoretically verified. Using an approach similar to the analysis of monolayer SnS19,27,28,29, we demonstrate, via DFT results, the application of light polarized along the y-axis (x-axis) in real space will predominantly allow excitation of electrons from the valence band to the conduction band of the band gap along ΓY (ΓX) in reciprocal space.

As shown in Fig. 3b, the band structure of bulk SnS has two nearly direct transitions T y and T x . There are no transitions with energies near band gap along the ΓZ and ZU directions.

In the electric dipole approximation, the transition rate between two states due to the electromagnetic perturbation is proportional to $i ^ ∙ ⟨ i ∣ r ∣ f ⟩ 2$, where |i,f〉 denote the initial and final states, $i ^$ is the direction of the polarization of the electromagnetic wave, and r is the position operator. Depending on the composition of the |i〉 and |f〉, the matrix element 〈i|r|f〉 can have non-vanishing components only in certain directions. If the component in the direction i vanishes, the transition is not allowed. Therefore, by calculating the dipole matrix elements 〈i|r|f〉, one can predict which light polarization is required for the state transition. The transition is allowed if the direct product of the irreducible representations that correspond to i, f, and r gives the fully symmetric representation A1.

With this in mind, we first turn to the transition T x . The wavevector (k) points along the ΓX direction (except for Γ and X) belong to the C2v point group, with the twofold rotation symmetry axis C2 along x. Looking at the composition of the valence and the conduction bands, we see that they are composed of orbitals, which are even under the reflection across the xy plane, but odd with respect to the xz plane, and therefore correspond to the B2 representation of the C2v point group. The character table for C2v (adapted from ref. 30) is presented in Table 1. The position vector x possesses an A1 representation. Thus, for light polarized along x, 〈i|x|f〉 transforms as B2A1B2 = A1 and the transition is allowed. However, for light polarized along y, the transition is forbidden because 〈i|y|f 〉 transforms as B2B1B2 = B1.

Similarly, the k points along the ΓY direction (except for Γ and Y) belong to the C2v point group but in this case with C2 along the y direction. By analyzing the signs of the orbitals that make up both the conduction and valence bands under various symmetry operations of the C2v point group, both bands were determined to belong to the A1 representation. Therefore, the transition is allowed for light polarized along y (A1 representation), as 〈i|y|f〉 now transforms as A1A1A1 = A1 instead. The transition is now forbidden for light polarized along x (B2 representation) because 〈i|x|f〉 now transforms as A1B2A1 = B2.

### Valley selectivity via photoluminescence

An important figure of merit for valleytronic systems is the polarization degrees of the valleys, P11, which can be obtained by comparing the PL intensities at each valley. For previously demonstrated 2D systems, P2D has been defined as11:

$P 2 D = I σ - - I ( σ + ) I σ - + I ( σ + ) ,$

where $I σ - ∕ +$ represents the PL intensities that are left and right circularly polarized, respectively. As the transitions at each valley in SnS is purely symmetry-dependent, PL at each valley will be linearly polarized along the same direction as the incident excitation-allowed light; PL conducted under parallel polarization will thus demonstrate valley selectivity for both absorption and emission and can be used to calculate the polarization degrees between the non-degenerate valleys. We define these as the intervalley polarization degrees:

$P intervalley , Γ X = I Γ X ∥ θ = 9 0 ∘ - I Γ Y ∥ θ = 9 0 ∘ I Γ X ∥ θ = 9 0 ∘ + I Γ Y ∥ θ = 9 0 ∘ and P intervalley , Γ Y = I Γ Y ∥ θ = 0 ∘ - I Γ X ∥ θ = 0 ∘ I Γ Y ∥ θ = 0 ∘ + I Γ X ∥ θ = 0 ∘$

where Pintervalley,i and Ii||(θ) stands for the polarization degree and PL intensity under parallel polarization for the i valley, respectively. Pintervalley,i are lower-bound estimations of the polarization degrees at both sets of SnS valleys due to certain logical simplifications made because of the non-degeneracy of the valleys. Such estimations include (i) comparing the intensities at the two valleys at 0° and 90° only under parallel polarization, (ii) taking the intensity values from the fitted polar plots, and (iii) using normalized intensity at each valley to eliminate issues of differing emission and detection efficiencies at the two valleys.

For PL measurements, linearly polarized lasers at 532, 633, and 785 nm were used to excite the SnS plate, giving similar emission peaks, which confirm the PL nature of the peaks. A polarizer was placed before the detector parallel to the excitation polarization (Fig. 2) and the sample was rotated to give different θ values.

We observed two PL peaks at 817 and 995 nm (Fig. 4). The polarization dependence of both PL peaks is clearly observed. The 817 nm peak maximizes at the polarization that minimizes the 995 nm peak, and vice versa, demonstrating a 90° phase shift between the peaks and hence the orthogonality in polarization selectivity of the two sets of valleys responsible for such PL emissions. (Note the existence of a higher energy, broad peak centered at 760 nm that has similar polarization dependence as the 817 nm peak. We attribute the presence of the two PL peaks to the presence of two peaks in the valence band along the ΓY direction, which can potentially allow two transitions of different energy values from the conduction band to the valence band. As both peaks lie on the ΓY axis, they will obey the same selection rules. This observation may also be caused by other higher order transitions which is beyond the scope of discussion in this work.) For the peaks at 817 nm (ΓY) and 995 nm (ΓX), we calculated Pintervalley,ΓY = 92% and Pintervalley,ΓY = 62%, respectively.

The uniqueness of SnS as a valleytronic system lies in the non-degeneracy of the valleys, which, unlike previous 2D materials, can be viewed as not only having an additional valley degree of freedom, but one that can be accessed via two routes; between valleys and within valleys, thereby allowing an additional valley selection pathway for applications. For anisotropy within each valley, we define the intravalley polarization degrees:

$P intravalley , Γ X = I Γ X ∥ θ = 9 0 ∘ - I Γ X ⊥ θ = 9 0 ∘ I Γ X ∥ θ = 9 0 ∘ + I Γ X ⊥ θ = 9 0 ∘ and P intravalley , Γ Y = I Γ Y ∥ θ = 0 ∘ - I Γ Y ⊥ θ = 0 ∘ I Γ Y ∥ θ = 0 ∘ + I Γ Y ⊥ θ = 0 ∘$

We identified the maximum PL intensities for both valleys (θ = 90° for ΓX and θ = 0° for ΓY) and conducted PL measurements under cross polarization (second polarizer placed perpendicular to excitation polarization (Fig. 2)), obtaining near-zero Ii(θ) for both valleys (Fig. 4b, c), giving Pintravalley,ΓY = 95% and Pintravalley,ΓX = 96%, respectively. The large difference between Ii(θ) and Ii||(θ) at each valley further reinforces the notion of strong valley emission anisotropy at both valleys. Our P values are amongst the highest polarization degrees reported, even rivaling those of monolayer dichalcogenides obtained from experiments conducted at cryogenic temperatures8,11,31.

Our PL results are complemented by the absorption data, which provides a definite proof of the selectivity towards the absorption of polarized light, specifically backing the claim that the valleys indeed possess selectively for both absorption and emission of light.

## Discussion

In summary, we have conducted reflection/ transmission and PL measurements that show the existence of two band gaps at 1.28 and 1.48 eV in bulk SnS under ambient conditions and without additional biases. These band gaps reside along the ΓY and ΓX axes in reciprocal space, are strongly and solely excited by y- and x-polarized light and also mostly emits y- and x-polarized light, respectively, effectively serving as non-degenerate valleys. Our system also has, nominally, the advantage of superior optical dichroic anisotropy of up to 600%, intravalley polarization degrees of 96 and 95%, respectively, and intervalley polarization degrees of 62% and 92%, respectively. Such attributes make SnS a model system and allow future work to be conducted with more controllability and potential for practical applications.

## Methods

### Sample preparation and characterization

SnS microplates were either mechanically exfoliated or synthesized using physical vapor deposition (PVD) on fused silica substrate (for PL measurements) and on Si/SiO2 substrates (for reflection measurements). 99.5% −4 mesh SnS powder from Alfa Aesar is used for both exfoliation and PVD synthesis. The PVD method used is adapted from ref. 16.

SEM was conducted using a FEI Quanta 3D FEG system.

### Optical measurements

PL measurements were conducted using a Horiba Jobin Yvon LabRAM ARAMIS automated scanning confocal Raman microscope under 532, 633, or 785 nm excitation.

Reflection and transmission measurements were carried out using our in-house setup coupled to a Horiba iHR320 spectrometer and Horiba Synapse CCD detector. We used the reflection spectrum of a clean gold substrate (transmission spectrum of a clean fused SiO2 substrate) to normalize the reflection (transmission) spectra by considering the theoretical reflection (transmission) of gold (fused SiO2) obtained from reported literature values32,33. Absorption coefficients, α, were obtained from the normalized reflection, R, and transmission, T data via $α=- 1 d ln T 1 - R$23,24 and plotted, disregarding the thickness, d, in a self-consistent manner without a loss of generality. Band-gap values were then obtained by determining the x-intercept of the corresponding direct Tauc plots25, plotted as (αhν)2 against , where E =  is the incident photon energy.

Reflection, transmission and PL samples were rotated at 10° intervals in a counter-clockwise manner with respect to the excitation light to obtain an angle of θ between the y-axis and the axis of polarization.

### DFT calculations

First-principles calculations based on density-functional theory were used to compute the electronic structure. The calculations were performed using the Quantum ESPRESSO code34.The core electrons were treated using a Troulier–Martins pseudopotential35. The Kohn–Sham orbitals were expanded in a plane-wave basis with a cutoff energy of 50 Ry. The exchange and correlation interaction were described using the PBE functional34. Dipole matrix elements were calculated with the epsilon code, which is part of the Quantum ESPRESSO package36.

### Data availability

The data that support the findings of this study are available from the corresponding author upon request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

This work is supported by Bakar Fellowship. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

## Author information

### Affiliations

1. #### Department of Materials Science and Engineering, University of California, Berkeley, CA, 94720, USA

• Shuren Lin
• , Roger Li
• , Sujung Kim
• , Xi Wang
•  & Jie Yao
2. #### The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

• Shuren Lin
•  & Emory M. Chan
3. #### Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, 6 Science Drive 2, Singapore, 117546, Singapore

• Alexandra Carvalho
• , Aleksandr Rodin
• , Lídia Carvalho
•  & Antonio H. Castro Neto
4. #### School of Geography and Biological Information, Nanjing University of Posts and Telecommunications, 210023, Nanjing, People’s Republic of China

• Shancheng Yan

• Jie Yao

### Contributions

J.Y. coordinated the work. S.L. and J.Y. designed the experiments. S.L. conducted the optical measurements and analyzed the data. S.Y., E.M.C, and X.W. conducted the optical measurements. A.C., A.R., L.C., and A.H.C.N carried out the DFT calculations and electric dipole analysis. S.L., R.L., and S.K. synthesized and exfoliated the materials. S.L. and J.Y. wrote the manuscript with the help from all authors.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Jie Yao.


text:
# Todd Trimble classical well-foundedness

Recall that an object $X$ of a topos $\mathbf{E}$ is inhabited if the unique map $!: X \to 1$ is an epimorphism. Recall also one of the formulations of “well-foundedness” in set theory: given a binary relation $\prec_X \hookrightarrow X \times X$, we say that $\prec_X$ is classically well-founded if every subset $P$ of $X$ with an element possesses a $\prec_X$-minimal element. Formally, this is the assertion

$\forall (P: \Omega^X) \;\; \exists (x: X)\; P(x) \Rightarrow \exists (x_0: X) \; P(x_0) \wedge \neg \exists (y: X)\; P(y) \wedge (y \prec_X x_0)$

which can be interpreted in an arbitrary topos $\mathbf{E}$.

###### Proposition

Suppose that in a topos $\mathbf{E}$, there is an object $X$ that carries an inhabited classically well-founded binary relation $\prec_X$. Then the law of excluded middle holds in $E$.

###### Proof

We must show that for any proposition $Q: 1 \to \Omega$, we have $Q \vee \neg Q = true: 1 \to \Omega$.

First observe that if $j: J \hookrightarrow X$ is any subobject, then the induced relation $\prec_J$ defined by the pullback

$\array{ \prec_J & \hookrightarrow & J \times J \\ \downarrow & & \downarrow _\mathrlap{j \times j} \\ \prec_X & \hookrightarrow & X \times X }$

is also classically well-founded, as any inhabited subobject $R$ of $J$ may be regarded as an inhabited subobject $j R$ of $X$, and any $\prec_X$-minimal (generalized) element of $j R$ can be viewed as a $\prec_J$-minimal element of $R$.

The subobject $A$ of $X$ defined by the formula

$(x_1: X) \exists (y: X) y \prec_X x_1$

is inhabited since $\prec_X$ is. By classical well-foundedness, the object $I_0$ of $\prec_X$-minimal elements of $X$ is inhabited; notice this is precisely $\neg A$. As $A$ itself is inhabited, classical well-foundedness implies that the object $I_1$ consisting of $\prec_X$- (or $\prec_A$-)minimal elements of $A$ is also inhabited. Obviously $I_0$ and $I_1$ are disjoint subobjects of $X$, and so their coproduct is their union in $X$, a subobject $I = I_0 + I_1 \hookrightarrow X$. As we already observed, the induced relation $\prec_I$ on $I$ is classically well-founded.

Let $P: I \to \Omega$ be the predicate

$I_0 + I_1 \stackrel{! + !}{\to} 1 + 1 \stackrel{(Q, true)}{\to} \Omega.$

As $P$ is $true$ on $I_1$ and $I_1$ is an inhabited object, we see $\exists (x: I) P(x)$ is $true$. It follows from classical well-foundedness that

$true \;\; = \;\; \exists (x_0: I) P(x_0) \wedge \neg \exists (y: I) P(y) \wedge (y \prec_I x_0).$

But as $I = I_0 + I_1$, this proposition is equivalent to a join of two propositions

(1)$\exists (x_0: I_0)\; P(x_0) \wedge \neg \exists (y: I) P(y) \wedge (y \prec_I x_0)$

and

(2)$\exists (x_1: I_1)\; P(x_1) \wedge \neg \exists (y: I) P(y) \wedge (y \prec_I x_1).$

For the first (1), we have

$\exists (x_0: I_0)\; P(x_0) \wedge \neg \exists (y: I) P(y) \wedge (y \prec_I x_0) \;\; = \;\; Q$

as the left side reduces to $\exists (x_0: I_0) P(x_0)$ (since $\neg \exists (y: I) P(y) \wedge (y \prec_I x_0)$ is $true$ over the context $(x_0: I_0)$ of minimal elements), and $\exists (x_0: I_0) P(x_0) = Q$ since $P$ restricted to $I_0$ is $Q !: I_0 \to \Omega$.

For the second (2), this reduces to $\exists (x_1: I_1) \neg \exists (y: I) P(y) \wedge (y \prec_I x_1)$ because $P(x_1) = true$ over the context $(x_1: I_1)$ (i.e., $P$ restricted to $I_1$ is $true !: I_1 \to \Omega$). Furthermore, over the context $(x_1: I_1)$ we have $y \prec_I x_1 \vdash y \in I_0$, and so

$\array{ \exists (x_1: I_1)\; \neg \exists (y: I) P(y) \wedge (y \prec_I x_1) & \vdash & \exists (x_1: I_1)\; \neg \exists (y: I) P(y) \wedge y \in I_0 \\ & = & \exists (x_1: I_1)\; \neg Q \\ & = & \neg Q }$

Thus

$true \;\; = \;\; \exists (x_0: I)\; P(x_0) \vee \neg \exists (y: I) P(y) \wedge (y \prec_I x_0) \;\; \vdash \;\; Q \vee \neg Q$

as was to be shown.

Revised on May 31, 2014 at 08:58:58 by Todd Trimble


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 Trudy MIAN: Year: Volume: Issue: Page: Find

 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 315–335 (Mi tm3498)

Random $A$-permutations and Brownian motion

A. L. Yakymiv

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: We consider a random permutation $\tau _n$ uniformly distributed over the set of all degree $n$ permutations whose cycle lengths belong to a fixed set $A$ (the so-called $A$-permutations). Let $X_n(t)$ be the number of cycles of the random permutation $\tau _n$ whose lengths are not greater than $n^t$, $t\in[0,1]$, and $l(t)=\sum_{i\leq t,i\in A}1/i$, $t>0$. In this paper, we show that the finite-dimensional distributions of the random process $\{Y_n(t)=(X_n(t)-l(n^t))/\sqrt{\varrho\ln n}$, $t\in[0,1]\}$ converge weakly as $n\to\infty$ to the finite-dimensional distributions of the standard Brownian motion $\{W(t),t\in[0,1]\}$ in a certain class of sets $A$ of positive asymptotic density $\varrho$.

DOI: https://doi.org/10.1134/S0371968513030217

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 298–318

Bibliographic databases:

UDC: 519.212.2+519.218.1

Citation: A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 315–335; Proc. Steklov Inst. Math., 282 (2013), 298–318

Citation in format AMSBIB
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