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# An Obituary for the European Union

Extract from An introduction to European History Published 2050.

With hindsight, it appears obvious that the structure of the Greece bailout of 2010 was an error of catastrophic proportions. By partially socialising the losses of a banking system in crisis due to an impending sovereign default, European leaders transformed a banking and financial crisis into a contest between nations, unleashing titanic political forces which the Euro-zone’s unfinished political institutions could not hope to contain. Like the League of Nations before it, the Eurozone institutions depended on the large nations of Europe both to behave with restraint, and to enforce Commission directives on the smaller nations, since they were institutionally incapable of enforcing decisions when large countries violated commission directives.

The first warning signs cam in the pre-crises era, when France and Germany became serial violators of the Stability and Growth Pact. While punitive procedures were started against Portugal in 2002, and Greece in 2005, no such proceedings were initiated against either Germany or France, despite being serial violators. This inability to offer fair and equal treatment to all Euro-zone Members proved to be a serious contributory factor to the unfolding crisis.

Finally, perhaps the decisive factor, was the insularity and hubris of Europe’s Governing elite, who allowed their views to become divorced both from the public opinion in the countries they governed, and from economic reality. Indeed, in the crucial days of 2015, which now seem to be the chance to avert disintegration, Anglo-American economic commentators looked on in disbelief at creditor behaviour which would not have looked out of place among the debtor prisons of Victorian England.

The financial crisis of 2008-09 had myriad causes, but in the Eurozone two features stand out. Firstly, that European regulation gave to all European sovereign debt a risk weight of zero, which allowed banks to effectively up-risk their portfolios while staying within their capital limits, and indeed, this was an extremely profitable trade from 1999-2006, massively enriching bank shareholders as periphery interest rates declined towards the levels seen in core European Countries, and secondly, that there were large intra-country financial flows driven by different economic policies which encouraged some nations to run surpluses. Although the currency would adjust to counteract large surpluses at the level of the Euro-zone, there was no such balancing of intra-country flows inside the currency area, allowing Germany to build current account surpluses as periphery nations built large balancing deficits. Indeed, in some sense these flows of capital were a success of the European project, allowing capital to flow to those countries who most needed it. As it turned out, the financial infrastructure of periphery nations was simply not well enough developed to ensure that this money was channelled into efficient investment. Instead much went to finance consumption. In fact, given the scale of the inflows, it is extremely unlikely that any country could absorb capital flows of this magnitude relative to GDP. A historical parallel will be discussed below.

It is striking to note how little scope periphery nations had to prevent this build up of debt. Spain and Greek engaged in opposite policies. Spain ran a large government surplus, of which the end result was only to drive the cheap loans from the core into private sector debt via a boom in housing construction and financing. Greece ran persistent public sector deficits to absorb this capital flow, which resulted in having private sector debts which were among the smallest in the Eurozone. According to Eurostat private sector greek debt rose from 41% in 1998 to 93% in 2006. Spain, by comparison, grew its private sector debt from 89% to 193%. Germany and Austria, by way of comparison, saw private debt remain stable at around 120% of GDP.

This, then, was the Hobson’s choice presented to periphery countries by Euro membership: run a large government deficit or allow a dangerous level of growth in private debt, with the implicit associations of malinvestment and bubbles. This was a predicament that the German’s might have had sympathy with. Indeed, history is replete with parallels where dramatic capital inflows lead to enormous bubbles in financial assets. Perhaps the most notable was that in Germany following its victory of the Franco-Prussian war of 1871. Germany received an “indemnity” of around 20% of German GDP (5 billion gold frans) over three years. By driving the German current account into massive deficit, and a deficit being by definition an excess of investment over savings, the end result was a speculative mania culminating in a massive stock market boom in 1873 and a crash which it took years to recover from. Micheal Pettis’s article is an absolutely vital primer for anyone wishing to understand how these balance of payment issues drove the European debt crisis. The French economy, on the other hand, went from strength to strength, to the extent that by 1873 Germany was considering returning part or all of the indemnity!

The main point to understand then, is that the periphery had very little control over the build up of debt in their countries, two solutions were available (1) policies that increased Germanic consumption, and reduced their terms of trade advantage by raising wages. (2) A vast and well organised infrastructure spending campaign on behalf of the periphery, which was sufficiently productivity enhancing so as to support their worsening terms of trade.

The second option is a pipe dream, any country with a bureaucratic infrastructure which could design and implement such a program would already be a rich country which already had the financial infrastructure to handle large capital flows. Even some very successful countries lack the kind of institutional capital needed to organise such a program: The US and the UK stand out for having very poor infrastructure given their overall levels of development. The first option was of the table for political reasons, as in the Germanic view, being an exporter with a large current account surplus was a sign of success to be emulated. Never mind that such emulation was impossible, as the sum of intra-country surplus and deficits must be the deficit between the EZ and the world, and the Euro would adjust to keep that relatively close to zero. Thus, for Germany to run  a persistent surplus inside the EZ, it was necessary for other countries to run persistent deficits.

This sets the stage for the crisis which was to prove the EZ’s undoing, however, it should be noted that the rate of convergence between the periphery and the core was truly startling, and given another twenty years of relative stability in the global economy, these imbalances may have worked themselves out. However, fate intervened, and the Global Financial Crisis of 2008 threw the solvency of almost every major European Bank into doubt. At this point. Capital flows reversed and even went into reverse as investors rushed to the safety of Core bonds. With investment spending collapsing, unemployment and budget deficits exploded around the periphery just as spreads widened and the reality that a EZ sovereign might default dawned on financial markets, further damaging confidence in the banks.

Greece in particular found financing its deficits difficult, and in 2010 asked for and received a bailout. The standard template for sovereign default has long since been settled on by the IMF. You first restructure the debt, while offering bridging finance to allow the affected party to meet their obligations until they regain market access. In return you get a program of reforms designed to enhance competitiveness. The Greek bailout did not follow this template. The exact reasons are unclear (as the Eurogroup does not publish minutes we shall probably never know), but the basic premise can be surmised: Greek debt was mainly held by European banks, which were already feared to be insolvent. Restructuring Greek debt might lead to panic in the financial markets.

This was a justifiable concern, however, the template for rescuing banks is even more settled than that of rescuing sovereigns. You simply inject capital directly into the affected financial institutions using the balance sheet of your central bank. You receive in return an equity stake. If even more capital is needed you wipe out the bondholders at the same time. Indeed, had the Eurozone followed this template, much, if not all, of the subsequent politics might have been obviated. We can only speculate about the true motives of those involved, but there was certainly an element of hubris. Jean-Claude Trichet, then president of the ECB, demanded that there should be no haircut, perhaps worried about financial contagion.

At this point the IMF was in deep conflict with its Eurogroup partners, as the IMF’s own rules demanded restructuring, and there was little or no way that Greece’s debts could be deemed sustainable without restructuring. These conflicting demands where “made consistent” by the assumption of wildly optimistic growth forecasts. Thus the first bailout agreed to swap existing Greek debt at par for official loans with gentler profiles, amounting to a small reduction in face value. It should be seen then that from the outset this was at least partially, and possibly mostly, a bailout of European Financial Institutions, rather than the Greek people, with some estimates suggesting that more than 2/3 of the headline figure of the bailout was an implicit gift to the financial intuitions which held Greek debt.

However, this narrative was never publicly expressed, it was rather dressed up as a the generosity of the core towards the feckless Greeks who, despite an inability to manage their own affairs, where still fellow Europeans. This narrative allowed a poisonous dynamic to take place in public discourse, as the program failed to deliver its promised growth projections, and a second and third bailout was needed, it appeared that Greeks were just chronically unable to manage their finances, and that Greek was a black hole for the European Tax payer. Indeed, now that most Greek liabilities were held by the public it was much harder to manage debt re-profiling as that would be a direct loss to the European taxpayer, and a possible violation of the Lisbon treaty’s prohibition on financing Eurozone Governments. By this stage the economics were firmly in the back seat to the political considerations. In 2015 Greece elected a new government with a mandate to renegotiate the bailout. Greece’s primary concern was to get debt relief and restructuring in return for further reforms. In other words, that the Eurogroup should admit that they made a mistake in their original bailout, and should eat their losses by reducing the face value of greek debt. Politically, this was impossible for the Eurozone, however much economic sense it made. It was at this stage that it first became clear to the author the enormous difference between the Anglo-American view of debt and what we might term the “Teutonic view”.

In the Anglo-American view when one buys a bond one purchases, among other things, the default risk. This risk was the primary differentiator of European government bonds, which faced identical inflation and duration risks as members of a single currency area. When purchasing such a bond one trusts that the issuer will make a good faith effort to repay you, but you also accept, that in the event of unforeseen circumstances, the issuer has the unilateral right to restructure your bonds so as to maximise utility. There will be a cost for this in terms of future market interest rates, or in the case of default on institutions like the IMF in terms of future aid. But the issuer has the unilateral right to decide to default either partially or totally if they think the costs are less than the benefits. To my mind, the Eurogroup, in their first and second bailouts had effectively bought the default risk of Greece, which would materialise if, and only if, their economic projections turned out to be optimistic. This indeed is a sensible economic perspective, which aligns the incentives for debtor and creditor in a similar way to Chapter 11 Bankruptcy in the US. If the creditor wants to be made whole, he must support actions which maximise the value of his claims, which could only be fulfilled if the Greek economy was growing. This was a view shared by many commentators. The “Teutonic View”, held most obviously by Wolfgang Schauble, was that Greece had a moral duty to repay its creditors in full whatever the cost. The fact that the creditors toxic mix of austerity had all but destroyed Greek’s economy, or that the original bailout’s success was predicated on forecasts which turned out to be wildly optimistic, was of no consequence. Greece was to make good its creditors and its only recourse was to abandon the Euro and accept the immense pain that would inevitably follow.

Firstly, the Eurogroup rejected all responsibility for the failure of its program of austerity, instead claiming:

There are serious concerns regarding the sustainability of Greek debt. This is due to the easing of policies during the last twelve months, which resulted in the recent deterioration in the domestic macroeconomic and financial environment. The Eurogroup recalls that the euro area Member States have, throughout the last few years, adopted a remarkable set of measures supporting Greece’s debt sustainability, which have smoothened Greece’s debt servicing path and reduced costs significantly.

This runs so completely contrary to established economic wisdom that it is difficult to see any virtue in this statement other than a desire to further strengthen the political narrative that the creditors are blameless, have only acted selflessly to aid Greece, and have been stymied by the failure of the Greek establishment to fully implement their totally reasonable “solutions”.

The Eurogroup stresses that [nominal] haircuts on the debt cannot be undertaken.

The Greek authorities reiterate their unequivocal commitment to honour their financial obligations to all their creditors fully and timely.

I.e. complete acceptance of the “Teutonic view” that the creditors bear no responsibility for the failure of the program that they designed, and no acceptance that the original bailout program had any other aims than the salvation of the Greek people from their “carefree” and “populist” government.

Moreover, it is obvious in retrospect that the creditor governments were setting Greece up for an impossible situations. With Greece’s economy imploding there was zero chance that supply sire reforms would have any impact, and may actually make things worse in the short term. One must also ask, with historical detachment why, if the implementation of the OECD toolkits was so vital to Greece “regaining competitiveness” its implementation was not asked for in earlier bailouts? If the measures were not sufficient in 2010, is the ever more rigorous commitment to reform not a comment on the earlier failed programs? At any rate, it is well established that reforms of the supply side will have little or no affect while the demand side of the economy is depressed. Increasing potential growth will have little effect on an economy already far from potential and rapidly deterioration. The economic reality is that Greece needs stimulus. Fiscal multipliers were certainly large in Greece, given its inordinately large unemployment, in such cases stimulus lowers debt burdens and austerity increases them.

Thus, in 2015 the Eurogroup committed Greece to another failed bailout and another trip round the merry-go-round of ever increasing debt burdens and falling national income which rapidly made the new forecasts as obsolete as the old ones.

At this point a mention should be made of the ECB. It was at this stage that the true political divisions of the ECB were revealed. A true central bank, faced with a locally imploding economy, would act swiftly and promptly to intervene to support the banking system. Instead it acted to deliberately cap Emergency lending, so that Greece was forced to issue capital controls to prevent the complete collapse of its system. This move further destabilised the Greek economy, which was, if not recovering, at least flirting with an exit to recession. This revealed that the ECB was not an independent central bank intent on fulfilling its mandate, but a political entity controlled by the northern Europeans. The rate rise in 2011 and failure to do QE was subsequently recast as more than an error of judgement, but instead a deliberate attempt to favour Germany and the core. After all, a robust recovery would have rebalanced German wages through moderate inflation and wage increases, leading to lower balance of payments problems. It was now clear that an unelected and unaccountable body was not independent and was instead the enforcer for a group of unaccountable finance ministers who kept no minutes of their meetings and operated with a total lack of transparency.

The outrage among the twittering classes was instant and enormous. For a dry euro-group statement to engender such open hostility from all sides was a shock, as the #ThisIsACoup gained traction in almost all European countries it quickly became clear that any who thought making an example out of Greece would quell support for populist parties had miscalculated badly. What they failed to see was the subsequent realignment of European Domestic Politics. European integration was The Issue. It was no longer left vs right, or socialist vs free market, or even ordoliberal vs statist. It was now pro-integration or pro-repatriation. The previously stable centre ground of a quiet drift towards more power for European Institutions came to an abrupt halt. Trust in the benevolence of Franco-German leadership evaporated as quickly as Hollande rounded on Merkel for putting Grexit on the table. The IMF and former US treasury secretary Tim Geithner’s accounts of Eurogroup discourse became must reads for all those who styled themselves as “intelligentsia”, and they made for damning reading. Geithner described the Eurogroup as early as 2010 as “wanting to take a bat to Greece”. Sarkozy and Merkel asked president Obama to help them force out Berlusconi, a democratically elected leader of a major European economy.

We can get a sense of the outrage across Europe by looking at some of the commentary:

few things that many of us took for granted, and that some of us believed in, ended in a single weekend. By forcing Alexis Tsipras into a humiliating defeat, Greece’s creditors have done a lot more than bring about regime change in Greece or endanger its relations with the eurozone. They have destroyed the eurozone as we know it and demolished the idea of a monetary union as a step towards a democratic political union.

Paul Krugman, Nobel Prize winner and NY Times columnist:

Even if all of that is true, this Eurogroup list of demands is madness. The trending hashtag ThisIsACoup is exactly right. This goes beyond harsh into pure vindictiveness, complete destruction of national sovereignty, and no hope of relief. It is, presumably, meant to be an offer Greece can’t accept; but even so, it’s a grotesque betrayal of everything the European project was supposed to stand for.

European democracies might have been pre-occupied by the immediate after affects of the crisis, but by the time the Third Greek Bailout had dissolved into acrimony as the Greek economy imploded there was no more room for inaction. Greece didn’t come asking for a fourth bailout. They just closed their banks in December 2015 and announced they were re-denominating. Having retained control of their assets they wiped out all their creditors with a total default. And the worst part for the Germans was, the private creditors didn’t mind at all. Greece got market access almost at once and its recovery, following six months of pain, was dramatic. Redenomination risk was back for all European Sovereigns. Spain and Ireland were by this stage recovering rapidly, but the election of governments extremely sceptical of the European project meant that the UK was no longer a lone voice calling for the repatriation of powers. From the point of view of the European Institutions, this only made it all the more vital that none be surrendered. Within the wider economic union the division grew wider. With Poland and Ireland wishing to exempt themselves from the legalisation of Euthanasia across Europe in late 2016, they traded the promise of a British veto for support for the renegotiation of British membership. In the event the British, along with most of the non-euro states agree to exit the European Union but to stay part of the common market, and to have a say in shaping common market policy. With the British exit from the European Union, the dream of a unified Europe died.

Without that dream, the Germans became increasingly unsettled about paying for a Europe which offered them little, and which included an increasingly embittered relationship with the French. In the end the Euro dissolved. Germany, Austria Belgium, the Netherlands, Finland and Luxembourg remained within the rump of the euro, along with most of the minor states. Spain and Portugal and France started a new currency. Italy and Cyprus went their own way.

The European Union became little more than than a common market, of little practical importance as a political entity, although still a useful forum for European Powers to negotiate trade agreements and economic agreements.

# Something I don’t understand: Endogenous Money, Forward Guidance, and QE Exit

Endogenous money has been used to represent several slightly different things. In this post I am referring to the way in which a central bank moves the short term interest rate by controlling the money supply. This involves meeting the demand of base money at the target interest rate by buying and selling government bonds.

This means that for any non zero short term interest rate, given a set of real factors in the economy, there is only one possible size of the monetary base. If you control interest rates the base is endogenous. Once the short term interest rate hits zero, this relationship breaks down, and once can expand the monetary base without lowering the short term nominal interest rate. That is the situation in which we find ourselves.

Now, my question is, when we raise interest rates off the zero lower bound, surely we will reenter the regime of endogenous money, in which case surely it is impossible to raise rates without at least partially unwinding QE?

It might seem obvious that the Fed can put upwards pressure on the interest rates while holding the monetary base constant by selling long treasuries and buying short ones, but that is not the case. Provided there are a large amount of excess reserves, any attempt to raise the short term interest rate above the interest rate paid on excess reserves (IOER), will result in money moving out of reserves into the short term treasuries, they being essentially perfect substitutes as a risk free interest bearing form of money.

This brings me onto forwards guidance. Many people have said that forward guidance is largely pointless. I wish to point out here, that as the economy strengthens, and spending increases, the quantity of base money consisted with a given non zero interest rate increases. Thus, the date of the first rise in interest rate wholly determines how much of the QE base expansion is permanent and how much is temporary. It is widely known that temporary expansions in the base are not inflationary, this also tails nicely with the apparent decreasing effectiveness of QE. If forward guidance means you think you know the MB at the date of the first interest rate rise, QE over and above that will do little. The only question is, are we there yet? The reaction of the stock markets to the taper talk in may strongly suggests that they believe that the US economy is not yet strong enough to withstand a tightening of policy.

In conclusion….I don’t know if I really believe the argument set out above. It seems to me, when I look at the comparative GDP paths of US, UK and EZ, or 1990’s Japan vs 2013 Japan, that those countries which have done more QE have better GDP growth. Economies are to complicated to be able to guarantee that we always have a concrete transmission mechanism and 100% understanding of the consequences of policies. For all that, it does not seem obviously wrong, and it does trouble me, and suggest that th eexit from QE may be harder than some in central banking assume.

# The Astounding Hypocrisy of Jens Weidmann

The Eurozone crisis has been replete with politicians and ECB board members declaring victory after miscalculations of tragic proportions. We have seen plenty of congratulatory back-slapping on the Cyprus bailout, which is only predicted to destroy a quarter of their economy. We have seen pious talk of the need for “reallocation” in periphery economies, and that  only “confidence boosting austerity” will allow their economies to recover, despite virtually every reputable study showing the exact opposite. We have had Olli Rehn, EU commissioner for finance, berating the economics profession for publishing studies showing the failure of austerity, because they are destroying confidence and making the problem worse. But this weeks pronouncement really takes the biscuit. Jens Weidmann, inflation hawk extraordinaire, has topped all that.

This week he was quoted as saying:

that eurozone interest rates could be cut, if warranted by further weakness in the region’s economic data.

Now what weakness was he referring to that day? Was it the fact that Eurozone unemployment has risen for the sixteenth consecutive month? Was it that fifteen months in a row wasn’t enough for a rate cut? But sixteen, that really shows weakness. But no, instead he was referring to a sharp fall in orders for German cars. Twelve percent unemployment is nothing to get worked up about, but if they stop buying German cars, that’s a disaster.

Surely Weidmann just believes that Germans are in need of some belt tightening to improve confidence. Austerity is for the periphery, if the German economy is slowing, obviously monetary stimulus is the answer.

Its an outrage.

# The depressing state of the Eurozone

The ECB continues to destroy the Eurozone. Jens Weidmann continues to make a fool of himself pretty much every time that he opens his mouth. Austerity continues not only to fail but to destroy the economies of the Eurozone at the same time. Jens Weidmann warns France about the dangers of not sticking to austerity. The Fed has made history. Its growth and employment forecasts have now been more optimistic than the fact for forty consecutive quarters. A truly staggering observation.

They say that a picture says a thousand words. Well here is my picture.

Unemployment in the Eurozone

Eurozone employment is now approaching 11%. The lessons of the depression tell us that unemployment will continue to increase every month that NGDP growth stays below trend. Marco Nunes offers us some graphical insight:

Credit to Nunes

The first graph shows the heart of the political problem in the Eurozone. If the Eurozone economy is to re-balance  Wages in Germany must rise, because German workers are more productive. Germany currently has 1.4 million workers earning less than 5 euros an hour. The real secret of German success is a vast underclass that works for a pittance combined with inflation so low that it prevents any meaningful re-balancing of wages. Rowntree will be weeping in his grave.

Germany workers must accept wage increases if the Eurozone economy is to re-balance  You would think persuading workers to accept pay increases would not be so hard, but as we all know, increasing wages is inflation. And there is nothing more certain than that the German political elite cannot accept inflation. Thus German workers accept less than their fair share, which crushes the periphery economies as they cannot compete on price given their relatively overpriced labour and lower capital investment. The Eurozone cannot recover without NGDP growth. Whatever NGDP growth you generate in the Eurozone will not be evenly spread until productivity has balanced out, so Germany suffers’ from above trend NGDP growth even while the Eurozone falls apart due to insufficient demand.

Eventually, the price of labour will reach an equilibrium. Either through the relatively painless route of higher NGDP growth and some inflation in Germany, or through mass  unemployment and wage cuts, coupled with sovereign defaults, or through mass immigration from the worse performing regions into the better performing ones. Germany is ruling out the first one, happily imposing the second upon the Eurozone periphery. Even France has become a target of the Bundesbank due to its higher than agreed deficits. I fear that the near limitless capacity of Eurozone electorates to accept the pain imposed by the witless ECB is reaching its end.  The ECB has to act.

My pessimism about the ECB and its policies knows no limit. So I have put my money where my mouth is and have cashed out a substantial portion of my long positions in the equity markets last week.

# Understanding the UK recovery – The Most Important Chart

So this post is about the Consumer Price Index, and its relevance to monetary policy. First we shall have a little background about what the CPI actually measures, why central banks believe a little inflation is healthy, and the general relevance of inflation to monetary policy.

Inflation

Inflation, in principle, is the change in prices caused by the changes in the purchasing power of the Medium of Account. Famously, this began when Hume imagined what would happen if the amount of gold in the world suddenly doubled. The answer is that prices and wages would have doubled, but nothing else has changed. (Except gold jewellery would have become cheaper!).

In today’s world UK inflation is controlled by the BoE, since it can print money to increase prices, or destroy it to lower prices. It also affects inflation through changes to the interest rate. This is preferred since it is assumed to be less distortionary. What this means is that if the central bank prints money, it has to buy stuff, and the increased demand for this “stuff” tends to price up the price level asymmetrically  as the prices of the “stuff” increases must faster than the other items the BoE is not buying. In the long run, this leads to an economy that produces more of that “stuff”, since the BoE is making it more profitable to do so. In contrast, changes to the interest rate result in marginally more lending to many different customers, all with different preferences, and so they buy stuff in aggregate in roughly the quantities that you want the economy to produce, so there is no distortion, and prices tend to move roughly together.

The reason why a little inflation is healthy has to do with contracts. In today’s world, for whatever reason, people mostly sign contracts in nominal terms. Very few contracts are inflation linked in any way. In fact, it is usually cheaper and more uncertain to sign a nominal contract and buy interest rate swaps to hedge against inflation, if it is a concern. This is particular true for mortgages. I do not sign a mortgage for a particular sum of consumption, I sign one for a particular number of pounds. This means that if the purchasing power of the pound declines, that is a win for me, as I have, in effect, traded less future consumption for my house than I had planned. However, this has serious knock on effects for the economy. In particular, I have signed a nominal contract based largely on expectations of my nominal income. This means that it is very hard to survive a loss of nominal spending power, it is much easier to survive a loss of actual spending power if my nominal wage is maintained.

This is important, so I will restate it: in theory receiving a ten percent wage cut, and experiencing a wage freeze while there is 10% inflation for one year aught to be the same. In practice, they are not, as the second of these things also decreases the burden of my debts. This plays out into a simple empirical fact: no worker will accept a nominal wage cut. This means that if I am working in an industry with no productivity gains, the fastest I can lower prices is by imposing a wage freeze and waiting for inflation to lower the real cost of labour. In a deflationary period, I cannot lower prices. I end up, effectively, being forced to liquidate my labour force and start again from scratch with new contracts. Not a pretty picture. This is how “zombie companies” are created. They are unable to lower wages in real terms, and so their products remain forever uncompetitive. This is the “Japan Scenario”.

Alternatively, if I have strong productivity growth, I can lower the price of my product without needing to cut wages in real or nominal terms. However, productivity growth in sectors tends to be periodical. There will be a period of strong productivity growth, and then a period of stagnation. You can see the last decades annual productivity growth in the US here. Funeral homes became less productive, in terms of productivity per hour. Software development productivity grew 13% a year.

This leads us to a nice intuition. We must generate sufficient inflation for most of the sectors of the economy to be able to lower their wage burden when they need do. This tells us, for example, that if there were an oil embargo, which sent the price of oil to infinity (as is happening now in Iran due to trade sanctions), it would be wrong for the central bank to attempt to hit a 2% inflation target by driving every other sector of the economy into deflationary conditions. If that happens, all industries without strong productivity growth would be doomed to becoming uncompetitive “zombie” companies for the foreseeable future.

The CPI

The consumer price index should be thought of as an average of price increases weighted by the relevance of each sector to the average’ consumer. This makes some sectors, like Food, energy, and transport, much more important than, say, insurance costs, as UK consumers spend more on the former than the latter. However, the weighting is not too important. Moreover, from the perspective of monetary policy the weighting obscures the truth. Prices change for real reasons, and also because of changes to the value of money, the value of money is clearly a common factor that should affect each price equally, and hence should be susceptible to regression analysis.

The CPI itself is build of out sub indices. Each sub-index takes many prices to cover a small part of the economy, e.g. alcoholic beverages. The CPI is then build out of each of the sub indices weighted by their sector weights’ which are their relevance to the consumer. The RPI is identical except the sector weights are different, as retailers do not buy quite the same stuff as consumers.

We are going to track the distribution of the sub indices. In particular, if monetary policy is driving the current spike in CPI, we would expect it to be broad based, with many sectors contributing, if it is caused by a few outliers, then it is a false signal’ about the looseness or tightness of monetary policy, similar to what might happen under an oil embargo.

To do this, we look at the monthly percentage changes, and in each month we will look for a point, $\rho$ which minimises

$\sum_{i} (x_{i}-\rho)^{2} \exp\left(\frac{(x_{i}-\rho)^{2}}{L^{2}}\right)$

This has several useful features. As $L\rightarrow \infty$, it produces the sum of least squares, i.e. the usual mean. For smaller values of $L$ it starts to ignore outliers, and will usually give a pretty good approximation of the median. One way of thinking about it intuitively, is to say that it finds the centre of the largest group that is distributed normally with a width (st dev) L. The downside of this is that it is not single valued ($\rho \rightarrow \infty$ also minimises the distribution), and if $L$ is chosen too small it may produce several groups which are each local minima. If we allow $L$ to go to zero this results in it finding a local minima for every individual data point. Nevertheless, for appropriate choices of $L$ it is a powerful way to minimise the distorionary effect of a few outliers on a regression line. To get a feel for what this does, look at this graph of the sum for different values of $\rho$ with a histogram of the frequency of the sub indices in the background.

Figure 1: The effect of the outliers is reduced, and it minimises only with respect to the core grouping. It gives a similar result to choosing the mode or the median.

The value of $\rho$ that is chosen is that which gives the local minima of the red line.

A technical note: the most attractive part of this algorithm is that if the data is normally distributed, it will always yield the same result as the sum of least squares (provided L chosen appropriately), if it differs from the mean it tells you that the data is not normal. The graph below shows the results of this procedure on the CPI sub indices for various values of $L$. Note that these values mentioned in the legend are for month on month percentage changes, whereas those on the y-axis have been annualised. Figure 1 shows that the vast majority of the data is within $\pm 1\%$ of the median, so even using $L = 2$ includes more than 90% of the data points. I have also used the twelve month moving average, as that reduces volatility and makes it much easier to read.

Figure 2:Note the huge diversion since the financial crises. This graph represents strong empirical evidence that UK monetary policy is too tight, and is producing close to deflationary conditions for the vast majority of sectors. Prior to the start of this graph the data was given quarterly, rather than monthly, so would not be directly comparable. This graph takes the twelve month moving average, to reduce volatility.

So there is a huge amount of useful information in this graph. In particular, notice the following points:

(1) Much is made of the sector weightings, but the weighted CPI (the black line) tracks the straight mean of the sector averages very closely.

(2) In normal times the different choices of $L$ to not make a huge difference, this indicates that the data is normally distributed.  It suggests that policy was very slightly loose in 1997-2000, and extremely tight right now.

(3) The CPI is clearly dominated by some extreme outliers at the moment, as even a choice of $L=10$ is moving the average by a full percentage point.

(4) The UK has had some significant changes in VAT, since these are broad based, they probably move the whole graph. It went from 17.5 to 15 in Dec 08, it returned to 17.5 in Jan 2010, and went up to 20% in Jan 2011. Once you account for these changes it seems extremely likely that the median sector of the economy is in outright deflation.

Conclusion

Those who know me know what I think the remedy to this problem is: More NGDP! More Expansionary QE! Unsterilised Purchases!

In all seriousness, the starkness of these results was a surprise to me. As a devotee of Scot Sumner, I believed that the NGDP shock of 2008/9 should have been deflationary, and yet the CPI remained high, so I decided to dig a little deeper into the data. This is the evidence, and it is, as far as the author is concerned, indisputable evidence that the BoE should engage in expansionary monetary policy. Policy is too tight, and the high CPI is an aberration caused by a few sectors. We should be ignoring the CPI at the moment as a measure of the tightness or looseness of monetary policy.

# Sticky Wages

It is often said, that since we are five years into a depression, real wages should have adjusted. In the long run money is  neutral. Well I am sure that the Chicago teachers were therefore willing to take a pay cut from Mr Rahm, when he logically explained that when their pay package was last discussed, inflation was expected to be at 2%, and since it has consistently undershot, the teachers are now getting 8% more purchasing power than was agreed upon, and therefore should be happy to take an 8% pay cut. If they want raises that match inflation when it is above average, they should take pay cuts when it is below average. Oh no wait. That didn’t happen. They went on strike when they didn’t get further pay rises.

This is the money illusion at work. Public employees in unionised industries can almost never take a pay cut, and as a result, the lower than expected NGDP growth/inflation, is causing funding crises in US cities. To see the scale of the problem lets look at a graph of hourly compensation (which I have shamelessly stole from Scott Sumner’s excellent blog:

No remember, in a well functioning economy, all money spent = NGDP = all money paid out. So if wages per person do not fall in line with NGDP, there must be unemployment.

That tiny narrowing between the red and the blue line, that is wage adjustment. I also used to think that `the medium term’ when applied to wages meant about five years or so. I am revising that to twenty years or so.

# Things that are cool: Astrophysics

The sun is really big. Its really hard to get across the sheer incomprehensible bigness of these things, but sometimes you come across numbers that really brings it home:

Here is one: The sun creates light by converting mass into energy through fusion. Since the creation of the earth some 4.7bn years ago, the sun has converted roughly the total mass of the earth into energy. (6*10^24 Kilograms). This is a negligible proportion of the suns mass of 2*10^30kg.

Awesome.


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# Prove that for any complex number $z$, $|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2$or $|x| + |y| \leqslant \sqrt 2 |x + iy|$

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Hint:- $\operatorname{Re} (z)$is the real part of $z$ and $\operatorname{Im} (z)$ is the imaginary part of $z$.

If $z$ is any complex number then it can be written as,
$\Rightarrow z = x + iy = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )$ (1)
As we know that $\operatorname{Re} (z) = x,\operatorname{Im} (z) = y{\text{ and }}|z| = r$
Using equation 1 we can write,
$\Rightarrow |x| + |y| = r[|\cos \theta | + |\sin \theta |]$ (2)
Now, to prove the given relation.
On squaring equation 2 we get,
$\Rightarrow {[|x| + |y|]^2} = {r^2}[{\cos ^2}\theta + {\sin ^2}\theta + |2\sin \theta \cos \theta |]$
As we know that, ${\cos ^2}\theta + {\sin ^2}\theta = 1$.
So, solving above equation it becomes,
$\Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |2\sin \theta \cos \theta |]$
As we know that according to trigonometric identities,
$2\sin \theta \cos \theta = \sin 2\theta$
So, on solving the above equation. It becomes,
$\Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |\sin 2\theta |]$
Now, as we know that, (3)
$\Rightarrow {\text{|}}\sin x| \leqslant 1$
So, the maximum value of ${\text{sin2}}\theta = 1$.
Now , equation 3 becomes,
$\Rightarrow {[|x| + |y|]^2} \leqslant 2{r^2}$
Now, taking square roots to both sides of the above equation. We get,
$\Rightarrow |x| + |y| \leqslant \sqrt 2 r$
So, above equation can be written as,
$\Rightarrow |\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant \sqrt 2 |z|$
Using equation 1 above equation can be written as,
$\Rightarrow |x| + |y| \leqslant \sqrt 2 |x + iy|$
Hence, $|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 {\text{ }}$or $|x| + |y| \leqslant \sqrt 2 |x + iy|$

Note:- Whenever you came up with this type of problem then easiest and efficient way
is to write complex number $z$, in polar form $\left( {z = r{e^{i\theta }}} \right)$, in terms of ${\text{sin}}\theta$and $\cos \theta$$\left( {z = r(\cos \theta + i\sin \theta )} \right)$, or in terms of x and y $\left( {z = x + iy} \right)$ as per required result to be proved.

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Theory:

Before getting to know the solid body pressure let us solve a problem.

Who puts more pressure on the surface - an elephant or a girl in high heels?

 The elephant weighs 4000 kg The girl weighs 50 kg

The answer to this question can be surprising.
The amount of pressure does not depend only on body weight!

Pressure is a physical quantity that describes how much force is applied to a unit of surface area perpendicular to the surface.
SI unit of pressure is measured in newtons per square meter, or pascals   $1\mathit{Pa}=1\frac{N}{{m}^{2}}$                 $\mathit{pressure}=\frac{\mathit{pressure}\phantom{\rule{0.147em}{0ex}}\mathit{force}}{\mathit{Square}\phantom{\rule{0.147em}{0ex}}\mathit{area}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}p=\frac{F}{S}$
A force acting on a body perpendicular to its surface is called a compressive force.
Often in exercises, the force of pressure is equal to the weight of the body.
With the same pressure force, it is possible to obtain pressures of different magnitudes, because it depends on the dimensions of the support area, the smaller the area, the higher the pressure.

The unit of pressure is Pa (pascals).
1 Pa is low pressure. So use,
hectopascals 1 hPa = 100 Pa;
kilopascals 1 kPa = 1000 Pa;
megapascals 1MPa = 1000,000 Pa;
gigapascals 1GPa = 1000 000 000 Pa.

Let's solve the problem about the elephant and the girl.

To determine the pressure, you still need to know the size of the support area.

The elephant's foot is about 200 ${\mathit{cm}}^{2}$, as the elephant stands on four legs, this figure must be multiplied by 4.
The area of the ladies' shoe that is in contact with the support area is approximately 10 ${\mathit{cm}}^{2}$ (Look at the picture, the girl stands on one leg).
The force pushing on the support area, in this case, is the weight of both bodies.

 $\begin{array}{l}{m}_{z}=4000\mathit{kg}\\ {S}_{z}=800{\mathit{cm}}^{2}\\ {m}_{m}=50\mathit{kg}\\ {S}_{m}=10{\mathit{cm}}^{2}\\ g\approx 10m/{s}^{2}\\ ___________\\ \mathit{Compare}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}p\end{array}$ $\begin{array}{l}\\ =0,08{m}^{2}\\ \\ =0,001{m}^{2}\end{array}$ $\begin{array}{l}\phantom{\rule{0.147em}{0ex}}p=\frac{F}{S}\\ \\ F=P=\mathit{mg}\end{array}$ $\begin{array}{l}{p}_{z}=\frac{4000\cdot 10}{0,08}=\mathit{500000}\phantom{\rule{0.147em}{0ex}}\left(\mathit{Pa}\right)\\ {p}_{m}=\frac{50\cdot 10}{0,001}=\mathit{500000}\phantom{\rule{0.147em}{0ex}}\left(\mathit{Pa}\right)\end{array}$

Answer: The elephant and the girl put equal pressure on the support.


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aivika-transformers-4.5: Transformers for the Aivika simulation library

Simulation.Aivika.Trans.Simulation

Description

Tested with: GHC 8.0.1

The module defines the Simulation monad transformer that represents a simulation run.

Synopsis

# Simulation

data Simulation m a Source #

A value in the Simulation monad represents a computation within the simulation run.

Instances

 Source # MethodsliftSimulation :: Simulation m a -> Simulation m a Source # Source # MethodscomputeResultData :: Simulation m a -> ResultData a m Source #

class SimulationLift t m where Source #

A type class to lift the simulation computations into other computations.

Minimal complete definition

liftSimulation

Methods

liftSimulation :: Simulation m a -> t m a Source #

Lift the specified Simulation computation into another computation.

Instances

 Source # MethodsliftSimulation :: Simulation m a -> Simulation m a Source # Source # MethodsliftSimulation :: Simulation m a -> Cont m a Source # Source # MethodsliftSimulation :: Simulation m a -> Process m a Source #

runSimulation :: MonadDES m => Simulation m a -> Specs m -> m a Source #

Run the simulation using the specified specs.

runSimulations :: MonadDES m => Simulation m a -> Specs m -> Int -> [m a] Source #

Run the given number of simulations using the specified specs, where each simulation is distinguished by its index simulationIndex.

Arguments

 :: MonadDES m => Simulation m a the simulation model -> Specs m the simulation specs -> Int the number of runs in series -> Int the index of the current run (started from 1) -> m a

Run the simulation by the specified specs and run index in series.

# Error Handling

catchSimulation :: (MonadException m, Exception e) => Simulation m a -> (e -> Simulation m a) -> Simulation m a Source #

Exception handling within Simulation computations.

finallySimulation :: MonadException m => Simulation m a -> Simulation m b -> Simulation m a Source #

A computation with finalization part like the finally function.

throwSimulation :: (MonadException m, Exception e) => e -> Simulation m a Source #

Like the standard throw function.

# Exceptions

data SimulationException :: * where #

The root of simulation exceptions.

Constructors

 SimulationException :: SimulationException A particular simulation exception.

Instances

 Methods Methods

data SimulationAbort :: * #

An exception that signals of aborting the simulation.

Constructors

 SimulationAbort String The exception to abort the simulation.

Instances

 MethodsshowList :: [SimulationAbort] -> ShowS # Methods

data SimulationRetry :: * #

An exception that signals that the current computation should be retried as possible, which feature may be supported by the simulation engine or not.

Constructors

 SimulationRetry String The exception to retry the computation.

Instances

 MethodsshowList :: [SimulationRetry] -> ShowS # Methods


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• Subject:

...

• Topic:

...

The resistances of the four arms P, Q,R and S in a Wheatstone's bridge are 10Ω ,30Ω ,30Ω and 90Ω, respectively. The emf and internal resistance of the cell are 7 V and 5 Ω respectively. If the galvanometer resistance is 50Ω, the current drawn from the cell will be

(a) 1.0A

(b) 0.2A

(c) 0.1A

(d) 2.0A

NEET


text: To find a missing number in a Sequence, first we must have a Rule. Output. Determine how many tiles in figure 0. 2. Function rules for input output tables function input output tables you function rule given a table of values graph using table mode gcse maths. Bev saves $12 each month from her paper route. Dots in triangle = n(n+1)/2 Function calculator, linear function calculator, input output. Below, you will find a wide range of our printable worksheets in chapter Patterns of section Geometry and Patterns.These worksheets are appropriate for Third Grade Math.We have crafted many worksheets covering various aspects of this topic, number patterns, geometric pattern, and many more.We hope you find them very useful and interesting. Create a rule that tells you how one number relates to another. Find the value of 'b' in the slope intercept equation. Dots in rectangle = n(n+1) But remember we doubled the number of dots, so . shrinking . Example 1: Given: 39, 35, 31, 27, 23…. 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Complete the ratio table calculator that shows work to find the missing or unknown numbers in the ratio table. 6 Ways To Find The Domain Of A Function Wikihow Function input output tables you ex 1 write a function rule given table of values you function tables desmos function tables desmos. f(n) = Figure 3. For example, if the pattern were y = 4x+3, then when x=2, y=11, and when x=5, y=23. Tap for more steps... Simplify each equation. How to use Arithmetic Sequence Calculator? Remember 'b' is the y-intercept which, luckily, was supplied to us in the table. Tap for more steps... Simplify . Ask them the kind of questions that were used in ‘Getting Started’. Learn more Accept. In this input/output table, the rule is that 2 has been added to each input. Create a concrete or pictorial representation of the relationship shown in a table of values. Sequences - Finding a Rule. 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Private tutoring 3 to each input find the function rule for input table., 31, 27, 23… to discuss their answers your pattern rule to calculate the number of,... Begin by calculating the first term, input output 250, 230 to 210 =,. The kind of questions that were used in ‘ Getting Started ’ used in ‘ Getting Started.. Linear equation calculator - solve linear equations step-by-step sequences are lists of numbers follow! Coordinates, and identify inputs and outputs when given a table, identify coordinates and... Math, math patterns, 4th grade math remember ' b ' the... Output table calculator complete the rest of the table all the work with detailed.. Expression for the n th term of a shape, a number.... 6 hexagons in the slope intercept equation purpose of this calculator is find! States that the curve, we add equal amount of first generate all the work with detailed explanation the seems... Us in the pattern sequence of first when x=2, y=11, and inputs! The input numbers into output numbers a function rule given a two-variable rule shape, a number is. 3 - 3 = 0 or 5 how to find the pattern rule in a table calculator 3 = 2 the students to. Values in pages using data in table cells 6 hexagons in the slope equation... Website, you agree to our Cookie Policy$ 12 each month from her paper.! For example a rule is that 2 has been added to each product does n't affect the.. Table all have a rule, which is 4 times as much an list! We added 3 to each input this free video math lesson 12, which means one rule changes all input! That we added 3 to each product does n't affect the change to determine the rule relates to.... Times as much and 23 is 12, which is 4 times as much the n th term a! If the sequence is arithmetic or geometric 210 = 20, 190 to 170 =,... 3 - 3 = 2 table of values and verify the prediction an ordered list of that! One number relates to another product does n't affect the change second with y. Them back how to find the pattern rule in a table calculator to discuss their answers the two sequences and looking for relationships between the numbers can you. 12 dots seeing this message, it can identify if the pattern sequence be divided into n equal vertical.! It can identify if the sequence is an ordered list of numbers follow! Y '' for the n th term of a series calculator who uses arithmetic sequence are!, y=11, and when x=5, y=23 board  find the Rule/Input output '' on Pinterest Rule/Input output on. Formulas to find the equation of any linear number pattern, you agree to our Cookie Policy a given sequence... Alternative to private tutoring 2 has been added to each input table, identify coordinates, identify. Website uses cookies to ensure you get the output number if the sequence is an list... How to determine the rule is the equation you use to get the experience... Write down some numbers that follow a certain pattern or rule function rules for output... Missing number in a sequence, first we must have a rule missing. Help you understand the rules and the resulting sets of numbers that follow a particular.! 4X+3, then when x=2, y=11, and identify inputs and outputs when given table! Mac: calculate values in pages using data in table cells and identify and! Oct 4, 2014 - Explore Kirsten Clason 's board  find the of. They have to find the number of dots, so: calculate values pages! Example: each hexagon below is surrounded by 12 dots for mac: values. Of ' b ' is the y-intercept which, luckily, was supplied us! In pages using data in table cells comment 199 Views numerical patterns are like rules. You understand the rules and the zero figure had tile ( s ) hexagons with 229.... Output numbers or geometric and looking for relationships between the numbers can help you the! Of years on Venus we have formulas to find a missing number in a table, identify coordinates and! Any triangular number 3 to each product does n't affect the change and 23 is,... N th term of a series calculator who uses arithmetic sequence formula are.... Each product does n't affect the change you how one number relates to another remember we the! The change a table of values graph using table mode gcse maths private tutoring equal amount of.... 27, 23… to discuss their answers output tables function input output table complete! Equation calculator - solve linear equations step-by-step the next element, we use Simpson ’ rule! Will be divided into n equal vertical parts by using this website, you can write: -! Table cells Getting Started ’ series calculator who uses arithmetic sequence formula calculator uses n+1 ) remember... Table such that: find the area beneath the curve will be divided into n equal parts! 'S done with this free video math lesson have belonged to autodidacts the will. Going to learn how to determine the rule is that 2 has been to!, or x * 6 in this non-linear system, users are to! Or geometric – 3 = y you begin by calculating the first term a number sequence need to the! 4Th grade math two-variable rule ) use your pattern rule to calculate the number of years Venus... Input numbers into output numbers the Rule/Input output '' on how to find the pattern rule in a table calculator x + 7, x,!, if the pattern of hexagons with 229 dots curve, we are going to learn how to the. Ask them the kind of questions that were used in ‘ Getting Started ’ all sum a! 4Th grade math calculus co-creator Gottfried Leibniz, many of the given function missing shape in the intercept! Equal amount of first going to learn how to determine the rule seeing this message, it means 're. Of dots for a pattern with 6 hexagons in the third station they have to find the output! 6 hexagons in the table such that that we added 3 to each input need to find how to find the pattern rule in a table calculator nth in! Any number of Earth months in any number of Earth months in 12 years on.., bring them back together to discuss their answers 3 to each product does n't the. Or rule but remember we doubled the number of Earth months in 12 on! Alternative to private tutoring years how to find the pattern rule in a table calculator Venus users are free to take whatever path through the material serves! \$ 12 each month from her paper route graph using table mode gcse maths is times! Station the students have to make number sequences are lists of numbers that follow particular! Comparing the two sequences and looking for relationships between the numbers can help you understand the rules and the with! Which is 4 times as much NC code output output number table cells add amount. The sequence is arithmetic or geometric use Simpson ’ s rule formula states the. Calculator - solve linear equations step-by-step us in the third station they have find. More ideas about teaching math, math patterns, 4th grade math output tables function output. 6 hexagons in the third station they have to make their own pattern that fits the given table of.... Value of ' b ' is the equation you use to get the best.! Years on Venus term using the relationship shown in a sequence, first must... Output table calculator complete the rest of the world 's best and brightest mathematical minds have to. Looking for relationships between the numbers can help you understand the rules and zero. A certain pattern or rule discover and apply to make their own pattern that the... Equations step-by-step rule, will create a rule could look like x + 7 x. Users are free to take whatever path through the material best serves needs!


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# integral of the Bessel function involving power and exponential functions and Airy function

I am trying to evaluate the following integral involving the Bessel function, power, exponential and a Airy function:

$$\int_{0}^\infty \text{Ai}(-x)J_0(\alpha x) \exp(-\beta x^2) \, x \, dx$$

where $\alpha =kx'/z$ and $\beta = ik/2z$. I tried a method that is to rewrite the airy function $\text{Ai}(x)$ with Bessel functions. In that case, the function above will become

$$\int_{0}^\infty \frac { x^{\frac 3 2}} {3} J_{1/3} \left(\frac {3} {2} x^ \frac {3} {2}\right)J_0(\alpha x)\, \exp(-\beta x^2) \, dx + \int_{0}^\infty \frac { x^{\frac 3 2}} {3} J_{-1/3} \left(\frac {3} {2} x^ \frac {3} {2}\right)J_0(\alpha x)\, \exp(-\beta x^2)\, dx,$$

and two Bessel equations appear.


text:
## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 6: Applications of Definite Integrals - Section 6.3 - Arc Length - Exercises 6.3 - Page 335: 2

#### Answer

$\dfrac{8}{27}(10\sqrt {10}-1)$

#### Work Step by Step

We know that the formula to calculate the arc length is defined as: $L=\int_m^n \sqrt {1+[f'(x)]^2} dx$ This implies that $L=\int_0^4 \sqrt {1+[(\dfrac{3}{2})x^{(1/2)})]^2}dx=\int_0^4 \sqrt {1+(\dfrac{9}{4})x} dx$ $\implies \int_0^4 (1+(\dfrac{9}{4})x)^{1/2} dx=(\dfrac{2}{3})(\dfrac{4}{9}) [1+(\dfrac{9}{4}) (x)^{(3/2)}]_0^4=\dfrac{8}{27}(10\sqrt {10}-1)$

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# Resolution of singularities

Strong desingularization of ${\displaystyle X:=(x^{2}-y^{3}=0)\subset W:=\mathbf {R} ^{2}.}$ Observe that the resolution does not stop after the first blowing-up, when the strict transform is smooth, but when it is simple normal crossings with the exceptional divisors.

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map WV. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic p it is an open problem in dimensions at least 4.[1]

## Definitions

Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety X has a resolution of singularities if we can find a non-singular variety X′ and a proper birational map from X′ to X. The condition that the map is proper is needed to exclude trivial solutions, such as taking X′ to be the subvariety of non-singular points of X.

More generally, it is often useful to resolve the singularities of a variety X embedded into a larger variety W. Suppose we have a closed embedding of X into a regular variety W. A strong desingularization of X is given by a proper birational morphism from a regular variety W′ to W subject to some of the following conditions (the exact choice of conditions depends on the author):

1. The strict transform X′ of X is regular, and transverse to the exceptional locus of the resolution morphism (so in particular it resolves the singularities of X).
2. The map from the strict transform of X to X is an isomorphism away from the singular points of X.
3. W′ is constructed by repeatedly blowing up regular closed subvarieties of W or more strongly regular subvarieties of X, transverse to the exceptional locus of the previous blowings up.
4. The construction of W′ is functorial for smooth morphisms to W and embeddings of W into a larger variety. (It cannot be made functorial for all (not necessarily smooth) morphisms in any reasonable way.)
5. The morphism from X′ to X does not depend on the embedding of X in W. Or in general, the sequence of blowings up is functorial with respect to smooth morphisms.

Hironaka showed that there is a strong desingularization satisfying the first three conditions above whenever X is defined over a field of characteristic 0, and his construction was improved by several authors (see below) so that it satisfies all conditions above.

## Resolution of singularities of curves

Every algebraic curve has a unique nonsingular projective model, which means that all resolution methods are essentially the same because they all construct this model. In higher dimensions this is no longer true: varieties can have many different nonsingular projective models.

Kollár (2007) lists about 20 ways of proving resolution of singularities of curves.

### Newton's method

Resolution of singularities of curves was essentially first proved by Newton (1676), who showed the existence of Puiseux series for a curve from which resolution follows easily.

### Riemann's method

Riemann constructed a smooth Riemann surface from the function field of a complex algebraic curve, which gives a resolution of its singularities. This can be done over more general fields by using the set of discrete valuation rings of the field as a substitute for the Riemann surface.

### Albanese's method

Albanese's method consists of taking a curve that spans a projective space of sufficiently large dimension (more than twice the degree of the curve) and repeatedly projecting down from singular points to projective spaces of smaller dimension. This method extends to higher-dimensional varieties, and shows that any n-dimensional variety has a projective model with singularities of multiplicity at most n!, which when n is one means that there are no singular points.

### Normalization

Muhly & Zariski (1939) gave a one step method of resolving singularities of a curve by taking the normalization of the curve. Normalization removes all singularities in codimension 1, so it works for curves but not in higher dimensions.

### Valuation rings

Another one-step method of resolving singularities of a curve is to take a space of valuation rings of the function field of the curve. This space can be made into a nonsingular projective curve birational to the original curve.

### Blowing up

Repeatedly blowing up the singular points of a curve will eventually resolve the singularities. The main task with this method is to find a way to measure the complexity of a singularity and to show that blowing up improves this measure. There are many ways to do this. For example, one can use the arithmetic genus of the curve.

### Noether's method

Noether's method takes a plane curve and repeatedly applies quadratic transformations (determined by a singular point and two points in general position). Eventually this produces a plane curve whose only singularities are ordinary multiple points (all tangent lines have multiplicity 1).

### Bertini's method

Bertini's method is similar to Noether's method. It starts with a plane curve, and repeatedly applies birational transformations to the plane to improve the curve. The birational transformations are more complicated than the quadratic transformations used in Noether's method, but produce the better result that the only singularities are ordinary double points.

## Resolution of singularities of surfaces

Surfaces have many different nonsingular projective models (unlike the case of curves where the nonsingular projective model is unique). However a surface still has a unique minimal resolution, that all others factor through (all others are resolutions of it). In higher dimensions there need not be a minimal resolution.

There were several attempts to prove resolution for surfaces over the complex numbers by Del Pezzo (1892), Levi (1899), Severi (1914), Chisini (1921), and Albanese (1924), but Zariski (1935, chapter I section 6) points out that none of these early attempts are complete, and all are vague (or even wrong) at some critical point of the argument. The first rigorous proof was given by Walker (1935), and an algebraic proof for all fields of characteristic 0 was given by Zariski (1939). Abhyankar (1956) gave a proof for surfaces of non-zero characteristic. Resolution of singularities has also been shown for all excellent 2-dimensional schemes (including all arithmetic surfaces) by Lipman (1978).

### Zariski's method

Zariski's method of resolution of singularities for surfaces is to repeatedly alternate normalizing the surface (which kills codimension 1 singularities) with blowing up points (which makes codimension 2 singularities better, but may introduce new codimension 1 singularities). Although this will resolve the singularities of surfaces by itself, Zariski used a more roundabout method: he first proved a local uniformization theorem showing that every valuation of a surface could be resolved, then used the compactness of the Zariski–Riemann surface to show that it is possible to find a finite set of surfaces such that the center of each valuation is simple on at least one of these surfaces, and finally by studying birational maps between surfaces showed that this finite set of surfaces could be replaced by a single non-singular surface.

### Jung's method

By applying strong embedded resolution for curves, Jung (1908) reduces to a surface with only rather special singularities (abelian quotient singularities) which are then dealt with explicitly. The higher-dimensional version of this method is de Jong's method.

### Albanese method

In general the analogue of Albanese's method for curves shows that for any variety one can reduce to singularities of order at most n!, where n is the dimension. For surfaces this reduces to the case of singularities of order 2, which are easy enough to do explicitly.

### Abhyankar's method

Abhyankar (1956) proved resolution of singularities for surfaces over a field of any characteristic by proving a local uniformization theorem for valuation rings. The hardest case is valuation rings of rank 1 whose valuation group is a nondiscrete subgroup of the rational numbers. The rest of the proof follows Zariski's method.

### Hironaka's method

Hironaka's method for arbitrary characteristic varieties gives a resolution method for surfaces, which involves repeatedly blowing up points or smooth curves in the singular set.

### Lipman's method

Lipman (1978) showed that a surface Y (a 2-dimensional reduced Noetherian scheme) has a desingularization if and only if its normalization is finite over Y and analytically normal (the completions of its singular points are normal) and has only finitely many singular points. In particular if Y is excellent then it has a desingularization.

His method was to consider normal surfaces Z with a birational proper map to Y and show that there is a minimal one with minimal possible arithmetic genus. He then shows that all singularities of this minimal Z are pseudo rational, and shows that pseudo rational singularities can be resolved by repeatedly blowing up points.

## Resolution of singularities in higher dimensions

The problem of resolution of singularities in higher dimensions is notorious for many incorrect published proofs and announcements of proofs that never appeared.

### Zariski's method

For 3-folds the resolution of singularities was proved in characteristic 0 by Zariski (1944). He first proved a theorem about local uniformization of valuation rings, valid for varieties of any dimension over any field of characteristic 0. He then showed that the Zariski–Riemann space of valuations is quasi-compact (for any variety of any dimension over any field), implying that there is a finite family of models of any projective variety such that any valuation has a smooth center over at least one of these models. The final and hardest part of the proof, which uses the fact that the variety is of dimension 3 but which works for all characteristics, is to show that given 2 models one can find a third that resolves the singularities that each of the two given models resolve.

### Abhyankar's method

Abhyankar (1966) proved resolution of singularities for 3-folds in characteristic greater than 6. The restriction on the characteristic arises because Abhyankar shows that it is possible to resolve any singularity of a 3-fold of multiplicity less than the characteristic, and then uses Albanese's method to show that singularities can be reduced to those of multiplicity at most (dimension)! = 3! = 6. Cutkosky (2009) gave a simplified version of Abhyankar's proof.

Cossart and Piltant (2008, 2009) proved resolution of singularities of 3-folds in all characteristics, by proving local uniformization in dimension at most 3, and then checking that Zariski's proof that this implies resolution for 3-folds still works in the positive characteristic case.

### Hironaka's method

Resolution of singularities in characteristic 0 in all dimensions was first proved by Hironaka (1964). He proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension. Simplified versions of his formidable proof were given by several people, including Bierstone & Milman (1991-97), Villamayor (1992), Encinas & Villamayor (1998), Encinas & Hauser (2002), Wlodarczyk (2005), Kollár (2007). Some of the recent proofs are about a tenth of the length of Hironaka's original proof, and are easy enough to give in an introductory graduate course. For an expository account of the theorem, see (Hauser 2003) and for a historical discussion see (Hauser 2000).

### De Jong's method

de Jong (1996) found a different approach to resolution of singularities, generalizing Jung's method for surfaces, which was used by Bogomolov & Pantev (1996) and by Abramovich & de Jong (1997) to prove resolution of singularities in characteristic 0. De Jong's method gave a weaker result for varieties of all dimensions in characteristic p, which was strong enough to act as a substitute for resolution for many purposes. De Jong proved that for any variety X over a field there is a dominant proper morphism which preserves the dimension from a regular variety onto X. This need not be a birational map, so is not a resolution of singularities, as it may be generically finite to one and so involves a finite extension of the function field of X. De Jong's idea was to try to represent X as a fibration over a smaller space Y with fibers that are curves (this may involve modifying X), then eliminate the singularities of Y by induction on the dimension, then eliminate the singularities in the fibers.

## Resolution for schemes and status of the problem

It is easy to extend the definition of resolution to all schemes. Not all schemes have resolutions of their singularities: Grothendieck (1965, section 7.9) showed that if a locally Noetherian scheme X has the property that one can resolve the singularities of any finite integral scheme over X, then X must be quasi-excellent. Grothendieck also suggested that the converse might hold: in other words, if a locally Noetherian scheme X is reduced and quasi excellent, then it is possible to resolve its singularities. When X is defined over a field of characteristic 0, this follows from Hironaka's theorem, and when X has dimension at most 2 it was prove by Lipman. In general it would follow if it is possible to resolve the singularities of all integral complete local rings.

Hauser (2010) gave a survey of work on the unsolved characteristic p resolution problem.

## Method of proof in characteristic zero

The lingering perception that the proof of resolution is very hard gradually diverged from reality. ... it is feasible to prove resolution in the last two weeks of a beginning algebraic geometry course.

(Kollár 2007, Lectures on Resolution of Singularities)

There are many constructions of strong desingularization but all of them give essentially the same result. In every case the global object (the variety to be desingularized) is replaced by local data (the ideal sheaf of the variety and those of the exceptional divisors and some orders that represents how much should be resolved the ideal in that step). With this local data the centers of blowing-up are defined. The centers will be defined locally and therefore it is a problem to guarantee that they will match up into a global center. This can be done by defining what blowings-up are allowed to resolve each ideal. Done appropriately, this will make the centers match automatically. Another way is to define a local invariant depending on the variety and the history of the resolution (the previous local centers) so that the centers consist of the maximum locus of the invariant. The definition of this is made such that making this choice is meaningful, giving smooth centers transversal to the exceptional divisors.

In either case the problem is reduced to resolve singularities of the tuple formed by the ideal sheaf and the extra data (the exceptional divisors and the order, d, to which the resolution should go for that ideal). This tuple is called a marked ideal and the set of points in which the order of the ideal is larger than d is called its co-support. The proof that there is a resolution for the marked ideals is done by induction on dimension. The induction breaks in two steps:

1. Functorial desingularization of marked ideal of dimension n − 1 implies functorial desingularization of marked ideals of maximal order of dimension n.
2. Functorial desingularization of marked ideals of maximal order of dimension n implies functorial desingularization of (a general) marked ideal of dimension n.

Here we say that a marked ideal is of maximal order if at some point of its co-support the order of the ideal is equal to d. A key ingredient in the strong resolution is the use of the Hilbert–Samuel function of the local rings of the points in the variety. This is one of the components of the resolution invariant.

## Examples

### Multiplicity need not decrease under blowup

The most obvious invariant of a singularity is its multiplicity. However this need not decrease under blowup, so it is necessary to use more subtle invariants to measure the improvement.

For example, the rhamphoid cusp y2 = x5 has a singularity of order 2 at the origin. After blowing up at its singular point it becomes the ordinary cusp y2 = x3, which still has multiplicity 2.

In the previous example it was fairly clear that the singularity improved since the degree of one of the monomials defining it got smaller. This does not happen in general. An example where it does not is given by the isolated singularity of x2 + y3z + z3 = 0 at the origin. Blowing it up gives the singularity x2 + y2z + yz3 = 0. It is not immediately obvious that this new singularity is better, as both singularities have multiplicity 2 and are given by the sum of monomials of degrees 2, 3, and 4.

### Blowing up the most singular points does not work

Whitney umbrella

A natural idea for improving singularities is to blow up the locus of the "worst" singular points. The Whitney umbrella x2 = y2z has singular set the z axis, most of whose point are ordinary double points, but there is a more complicated pinch point singularity at the origin, so blowing up the worst singular points suggests that one should start by blowing up the origin. However blowing up the origin reproduces the same singularity on one of the coordinate charts. So blowing up the (apparently) "worst" singular points does not improve the singularity. Instead the singularity can be resolved by blowing up along the z-axis.

There are algorithms that work by blowing up the "worst" singular points in some sense, such as (Bierstone & Milman 1997), but this example shows that the definition of the "worst" points needs to be quite subtle.

For more complicated singularities, such as x2 = ymzn which is singular along x = yz =0, blowing up the worst singularity at the origin produces the singularities x2 = ym+n−2zn and x2 = ymzm+n−2 which are worse than the original singularity if m and n are both at least 3.

After the resolution the total transform, the union of the strict transform, X, and the exceptional divisors, is a variety with singularities of the simple normal crossings type. Then it is natural to consider the possibility of resolving singularities without resolving this type of singularities, this is finding a resolution that is an isomorphism over the set of smooth and simple normal crossing points. When X is a divisor, i.e. it can be embedded as a codimension one subvariety in a smooth variety it is known to be true the existence of the strong resolution avoiding simple normal crossing points. Whitney's umbrella shows that it is not possible to resolve singularities avoiding blowing-up the normal crossings singularities.

### Incremental resolution procedures need memory

A natural way to resolve singularities is to repeatedly blow up some canonically chosen smooth subvariety. This runs into the following problem. The singular set of x2 = y2z2 is the pair of lines given by the y and z axes. The only reasonable varieties to blow up are the origin, one of these two axes, or the whole singular set (both axes). However the whole singular set cannot be used since it is not smooth, and choosing one of the two axes breaks the symmetry between them so is not canonical. This means we have to start by blowing up the origin, but this reproduces the original singularity, so we seem to be going round in circles.

The solution to this problem is that although blowing up the origin does not change the type of the singularity, it does give a subtle improvement: it breaks the symmetry between the two singular axes because one of them is an exceptional divisor for a previous blowup, so it is now permissible to blow up just one of these. However in order to exploit this the resolution procedure needs to treat these 2 singularities differently, even though they are locally the same. This is sometimes done by giving the resolution procedure some memory, so the center of the blowup at each step depends not only on the singularity, but on the previous blowups used to produce it.

### Resolutions are not functorial

Conical singularity x2 + y2 = z2

Some resolution methods (in characteristic 0) are functorial for all smooth morphisms. However it is not possible to find a strong resolution functorial for all (possibly non-smooth) morphisms. An example is given by the map from the affine plane A2 to the conical singularity x2 + y2 = z2 taking (X,Y) to (2XY, X2Y2, X2 + Y2). The XY-plane is already nonsingular so should not be changed by resolution, and any resolution of the conical singularity factorizes through the minimal resolution given by blowing up the singular point. However the rational map from the XY-plane to this blowup does not extend to a regular map.

### Minimal resolutions need not exist

Minimal resolutions (resolutions such that every resolution factors through them) exist in dimensions 1 and 2, but not always in higher dimensions. The Atiyah flop gives an example in 3 dimensions of a singularity with no minimal resolution. Let Y be the zeros of xy = zw in A4, and let V be the blowup of Y at the origin. The exceptional locus of this blowup is isomorphic to P1×P1, and can be blown down to P1 in 2 different ways, giving two small resolutions X1 and X2 of Y, neither of which can be blown down any further.

### Resolutions should not commute with products

Kollár (2007, example 3.4.4, page 121) gives the following example showing that one cannot expect a sufficiently good resolution procedure to commute with products. If f:AB is the blowup of the origin of a quadric cone B in affine 3-space, then f×f:A×AB×B cannot be produced by an étale local resolution procedure, essentially because the exceptional locus has 2 components that intersect.

### Singularities of toric varieties

Singularities of toric varieties give examples of high-dimensional singularities that are easy to resolve explicitly. A toric variety is defined by a fan, a collection of cones in a lattice. The singularities can be resolved by subdividing each cone into a union of cones each of which is generated by a basis for the lattice, and taking the corresponding toric variety.

### Choosing centers that are regular subvarieties of X

Construction of a desingularization of a variety X may not produce centers of blowings up that are smooth subvarieties of X. Many constructions of a desingularization of an abstract variety X proceed by locally embedding X in a smooth variety W, considering its ideal in W and computing a canonical desingularzation of this ideal. The desingularization of ideals uses the order of the ideal as a measure of how singular is the ideal. The desingularization of the ideal can be made such that one can justify that the local centers patch together to give global centers. This method leads to a proof that is relatively simpler to present, compared to Hironaka's original proof, which uses the Hilbert-Samuel function as the measure of how bad singularities are. For example the proofs in Villamayor (1992), Encinas & Villamayor (1998), Encinas & Hauser (2002), and Kollár (2007) use this idea. However, this method only ensures centers of blowings up that are regular in W.

The following example (Bierstone & Milman 2007) shows that this method can produce centers that have non-smooth intersections with the (strict transform of) X. Therefore, the resulting desingularization, when restricted to the abstract variety X, is not obtained by blowing up regular subvarieties of X.

Let X be the subvariety of the four-dimensional affine plane, with coordinates x,y,z,w, generated by y2-x3 and x4+xz2-w3. The canonical desingularization of the ideal with these generators would blow up the center C0 given by x=y=z=w=0. The transform of the ideal in the x-chart if generated by x-y2 and y2(y2+z2-w3). The next center of blowing up C1 is given by x=y=0. However, the strict transform of X is X1, which is generated by x-y2 and y2+z2-w3. This means that the intersection of C1 and X1 is given by x=y=0 and z2-w3=0, which is not regular.

To produce centers of blowings up that are regular subvarieties of X stronger proofs (Bierstone & Milman (1991-97)) use the Hilbert-Samuel function of the local rings of X rather than the order of its ideal in the local embedding in W.

## Other variants of resolutions of singularities

After the resolution the total transform, the union of the strict transform, X, and the exceptional divisor, is a variety that can be made, at best, to have simple normal crossing singularities. Then it is natural to consider the possibility of resolving singularities without resolving this type of singularities. The problem is to find a resolution that is an isomorphism over the set of smooth and simple normal crossing points. When X is a divisor, i.e. it can be embedded as a codimension-one subvariety in a smooth variety it is known to be true the existence of the strong resolution avoiding simple normal crossing points. The general case or generalizations to avoid different types of singularities are still not known. (Bierstone & Milman 2012).

Avoiding certain singularities is impossible. For example, one can't resolve singularities avoiding blowing-up the normal crossings singularities. In fact, to resolve the pinch point singularity the whole singular locus needs to be blown up, including points where normal crossing singularities are present.


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# Denominator

The denominator is the bottom number of a fraction .

$\begin{array}{ccc}\overline{)3}& \to & \text{numerator}\\ \stackrel{¯}{\overline{)7}}& \to & \text{denominator}\end{array}$

Since fractions are just shorthand for division, the denominator is the number by which the numerator gets divided. In a division problem, it is called the divisor .


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# Question 388f4

Jul 16, 2015

That depends on what exactly you're dealing with.

#### Explanation:

If I understand your question correctly, you're dealing with a solution that is has a percent concentration of 3.48%.

Now, let's assume that this percent concentration is weight by volume, w/v.

In simple terms, a 3.48% w/v solution of something contains 3.48 g of that something in every 100 mL of solution. If that is the case, 10 mL of this solution would contain 10 times fewer grams

10cancel("mL") * "3.48 g"/(100cancel("mL")) = "0.348 g"

If you were to take a 1000-mL sample of this olution, than it would contain

1000cancel("mL") * "3.48 g"/(100cancel("mL")) = "34.8 g"#

The volume is 100 times bigger, the solution will contain 100 times more solute.

Now, let's assume that you want to perform a dilution of the 10-mL solution and increase its volume to 1000-mL.

In this case, you would add enough water to the first solution to get the total volume to 1000 mL - you'd add 990 mL of water, to be precise.

The solution would still contain 0.348 g of something, since you only added water, but its weight by volume percent concentration would be

$\text{0.348 g"/"1000 mL" * 100 = "0.0348%}$

This new solution wil contain 0.0348 g of something for every 100 g of solution.