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# What Are Interest Rates And Can They Be Artificially Low Or High?

Many economic commentators believe that interest rates in America and around the world are “artificially low”. Indeed, I too have used the term in the past to refer to the condition in Europe that saw interest rates across the member states converge to a uniformly low level at the introduction of the Euro, only to diverge and soar in the periphery during the ongoing crisis.

So what is an interest rate? An interest rate is the cost of money now. As Eugen von Böhm-Bawerk noted, interest rates result from people valuing money in the present more highly than money in the future. If a business is starting out, and has insufficient capital to carry out its plans it will seek investment, either through selling equity in the ownership of the business, or through credit from lenders. For a lender, an interest rate is their profit for giving up the spending power of their capital to another who desires it now, attached to the risk that the borrower will default.

In monetary economies, money tends to be distributed relatively scarcely. In a commodity-based monetary system, the level of scarcity is determined by the physical limits of how much of a commodity can be pulled out of the ground. In a fiat-based monetary system, there is no such natural scarcity, but money’s relative scarcity is controlled by the banking system and central bank that lends it into the economy. If money was distributed infinitely widely and freely, there would be no such thing as an interest rate as there would be no cost to obtaining money now, just as there is no cost to obtaining a widely-distributed and freely-available commodity like air (at least on the face of the Earth!). Without scarcity money would lose its usability as a currency, as there is no incentive to trade for a substance which is uniformly and effectively infinitely available to everyone. So an interest rate is not only the cost of money, but also a symptom of its scarcity (and, as Keynes pointed out, a key mechanism through which rentiers profit).

So, where does the idea that interest rates can be made artificially low or artificially high arise from?

The notion of an artificially low or high interest rate implies the existence of a natural interest rate, from which the market rate diverges. It is a widely-held notion, and indeed, Ron Paul made reference to the notion of a natural rate of interest in his debate with Paul Krugman last year. A widely-used definition of the “natural rate of interest” appears in Wicksell (1898):

There is a certain rate of interest on loans which is neutral in respect to commodity prices, and tends neither to raise nor to lower them.

This is easy to define and hard to calculate. It is whatever interest rate yields a zero-percent inflationary level. Because interest rates have a nonlinear relationship with inflation, it is difficult to say precisely what the natural interest rate is at any given time, but Wicksell’s definition specifies that a positive inflation rate means the market rate is above the natural rate, and a negative inflation rate means the market is below the natural rate. (Interestingly, it should be noted that the historical Federal Funds Rate comes pretty close to loosely approximating the historical difference between 0 and the CPI rate, despite questions of whether the CPI really reflects the true price level due to not including housing and equity markets which often record much greater gains or greater losses than consumer prices).

The notion of a natural rate of interest is interesting and helpful — certainly, high levels of inflation can be challenged through decreasing interest rates (or more generally increasing credit-availability), and deflation can be challenged by decreasing interest rates (or more generally increasing credit availability). If the goal of monetary policy is price stability, then the notion of a “natural interest rate” as a guide for monetary policy is useful.

But policies of macrostabilisation have been strongly questioned by the work of Hyman Minsky, which posited the idea that stability is itself destabilising, because it leads to overconfidence which itself results in malinvestment and credit and price bubbles.

Austrian Business Cycle Theory (ABCT) developed by Ludwig von Mises and Friedrich Hayek, most influentially in Mises’ 1912 work The Theory of Money and Credit, theorises that the business cycle is caused by credit expansion (often fuelled by excessively low interest rates) which pours into unsustainable projects. The end of this credit expansion (as a result of a collapse resulting from excessive leverage, or from the failure of unsustainable projects, or from general overproduction, or for some other reason) results in a panic and bust. According to ABCT, the underlying issue is that the banking system made money cheaply available, and the market rate of interest falls beneath the natural rate of interest, manifesting as price inflation.

I do not dispute the idea that bubbles tend to coincide with credit expansion and easy lending. But it is tough to say whether credit expansion is a consequence or a cause of the bubble. What is the necessary precursor of an unsustainable credit expansion? Overconfidence, and the idea that prices will just keep going up when sooner or later the credit expansion will run out steam. This could be the overconfidence of central bankers, who believe that macrostabilisation policies have produced a “Great Moderation”, or the overconfidence of traders who hope to get rich quick, or the overconfidence of homeowners who see rising home prices as an easy opportunity to remortgage and consume more, or the overconfidence of private banks who hope to make bumper gains on loans or loan-related securities (Carl Menger noted that fractional reserve banking and credit-fuelled bubbles originated in economies with no central bank, in contradiction of those ABCT-advocates who go so far as to say that without central banking there would be no business cycle at all).

And is price stability really “natural”? Wicksell (and other advocates of a “natural rate of interest” like RBCT and certain Austrians) seem to imply so. But why should it be the norm that prices are stable? In competitive markets — like modern day high-tech markets — the tendency may be toward deflation rather than stability, as improving technology lowers manufacturing costs, and firms lower prices to stay competitive with each other. Or in markets for scarce goods — like commodities of which there exists a limited quantity — the tendency may be toward inflation, as producers may have to spend more to extract difficult-to-extract resources form the ground. Ultimately, human action in market activity is unpredictable and determined by the subjective preferences of all market participants, and this applies as much to the market for money as it does for any market. There is no reason to believe that prices tend toward stability, and the empirical record shows a significant level of variation in price levels under both the gold standard and the modern fiat system.

Ultimately, if interest rates are the cost of money, and in a fiat monetary system the quantity and availability of money is determined by lending institutions and the central bank, how can any interest rate not be artificial (i.e. an expression of the subjective opinions, forecasts and plans of those involved in determining the availability of credit and money including governments and central bankers)? Even under a commodity-money system, the availability of money is still determined by the lending system, as well as the miners who pull the monetary commodity or commodities out of the ground (and any legal tender laws that define money, for example monetising gold and demonetising silver).

And if all interest rates in contemporary markets are to some degree artificial this raises some difficult questions, because it means that the availability of capital, and thus the profitability (or unprofitability) of rentiers are effectively policy choices of the state (or the central bank).

# Are Markets Informationally Efficient?

A key assumption in many mainstream macroeconomic models (both formal and informal) is the Efficient Market Hypothesis. Very simply, this is the belief that markets are informationally efficient — that they reflect information with little (or no) delay, leaving few (or no) arbitrage opportunities.

So the real question here is what information do markets (and by markets, I mean free markets where market participants are free to pay and receive any negotiated value for an asset) really reflect? When we see the price of an asset or asset class fluctuating, what does this movement signify? Is it fluctuation in the fundamental value of the asset? Is it just a fluctuation in the market’s perception of an asset? Is it some combination of these two factors? Or is it just random noise? Fundamentally, markets are composed of a series of transactions, each between a bidder and a seller. Each transaction in itself reflects a discrete set of information — specifically, what the bidder and the seller are willing to pay for, and take for that specific asset. This in turn is typically (although not always!) influenced by a some or all of the following: what others are willing to pay and accept for an asset presently, the use-value of an asset, notions of fundamental value (price-over-earnings, stock-to-flow, EBITDA, cashflow, etc), notions of momentum and what others may be willing to pay and accept for an asset in the future (trendlines, gut feelings, “hot stock tips”, etc).

An intriguing addendum to this is that the automation of trading (high-frequency trading) has created bidders and sellers who are acting on the instructions of algorithms. As these instructions are programmed by humans — usually automating some form of technical analysis — the only real difference is that of (extreme) speed. The beliefs reflected in high-frequency trading reflect the underlying algorithmic instructions programmed by the humans who created the algorithm.

Ultimately, whenever we purchase an asset for the purpose of speculation or investment (and even use-value — prices can change, and the price we paid last week or last year could end up looking very expensive, or very cheap) we are taking a guess as to whether the current bid or sell value is worth it. Each agent makes their guess based on a different set of data and expectations. What the prices in markets signify is the operation of this mechanism — different agents evaluating information and making guesses about the future.

Let’s consider the example of Bitcoin, the price of which is currently soaring. Some choose to buy Bitcoins based on momentum, or their liking of the cryptography, or Bitcoin’s inherent deflationary bias or some other positive belief. This is a speculation that the price may continue to climb. Some may choose to buy bitcoins based on their use-value, as an anonymous, decentralised currency that can be used to buy a wide array of things. Holders of Bitcoins may be motivated to sell by the fact that the price has risen since they bought or mined their coins, or by the belief that bitcoin is “in a bubble”, or some other negative belief.

What the market reflects is the net weight of different opinions and resultant human actions. If those who are motivated to buy outweigh those who are motivated to sell, the price  rises and vice versa. This means that the beliefs of big players in a particular market can have strange and disproportionate effects. Consider the effect of the Hunt Brothers’ attempts to coin the silver market in 1980. The price of silver rose from $11 an ounce in September 1979 to almost$50 an ounce in January 1980, as the Hunt Brothers bought more and more. The market was very efficient at reflecting the fact that the Hunt brothers were willing to buy more than the market could supply at lower prices. And once the Hunt Brothers faced margin calls, the market quickly adjusted to reflect the fact that they were now selling instead of buying, and prices fell.

That’s what (transparent) markets are guaranteed to reflect — bidders and sellers, supply and demand. This information is still useful to firms trying to gauge what, and how much to produce.  Everything else — the information that bidders and sellers are acting upon — is not necessarily reflected in market activity. Very often, bidders and sellers are brought to the market by new information regarding a large number of things — price changes, earnings, business decisions, technologies and inventions, macroeconomic data, etc — but there is no systematic or reliable way to predict what humans will respond to, or how they will respond. Human psychology and human action in this sense is totally unstable and nonlinear — consider the recent contrast in market reaction to earnings data from Apple and Google. This instability is an alternative explanation for why consistently beating the market is indeed very difficult, as the Efficient Market Hypothesis implies.

And prices do not even reflect an aggregation of sentiment toward an asset or asset class — they only reflect the sentiment of those who are involved in the market, in proportion to their level of buying and selling activity. This means that the opinions of big players who buy or sell a lot, are reflected many times more than those of small players who buy or sell a little. And irrationality can create a feedback loop — if stock prices are rising, and macroeconomic fundamentals are weak, many market participants may initially be sceptical. Yet as more participants pile into the stock market purely for reasons of sustained upward momentum, more and more participants may begin to suspend their disbelief, if only to not miss out on a profit opportunity. This is one mechanism (of infinitely many) through which price bubbles can form.

Yet accurately reflecting supply and demand is not the same thing as informational efficiency. Empirical data show that arbitrage opportunities are widely exploitable and exploited even in modern marketsOne of the largest forms of high frequency trading is of course statistical arbitrage. This reality should probably be a final nail in the coffin of the idea that markets reflect anything more than the actions of bidders and sellers. Unfortunately, very many models rest on the assumption of informational efficiency in markets, meaning that this approach is very unlikely to die out any time soon.

# What is Profit?

In neoclassical macroeconomic models that assume perfect competition, there can in the long run be no such thing as profit — defined as revenue left over after all costs have been subtracted.

Clearly, in the real world where many businesses have lived and died profitably there is no such thing as perfect competition, and therefore the neoclassical models that treat profit as a short-run anomaly are working from an unrealistic assumption.

My definition of profit is that profit is what happens when a business’s input transactions are priced less than its output transactions. That is, the sum of the cost of a business’s inputs from those it buys is transacted for a lower price than the sum of those it sells its goods and services to. Because transactions are assumed to be voluntary — and when they are not voluntary, any residual gain is theft, not profit — and businesses are assumed to try to negotiate the best price in both inputs and outputs, any profit is due to those who purchase the business’s output valuing the output higher than those who sold the business its inputs. That an output or input transactor would accept a profit-creating price could be for any number of perceived reasons: convenience, or expertise, or prestige, or necessity, or even outright trickery. Their decision to accept the price is subject to their own subjective valuation, and it is the difference between prices that creates the profit.

Marx and Lenin represented this idea as surplus value; that businesses make a profit by extracting uncompensated labour value out of their workers. But why not the other transactors? In my view, profit is derived from the sum of the business’s transactions with all of its transactors: consumers, supplies, labour (etc). Workers (etc) cannot extract a greater share of the firm’s revenue than they can negotiate, and at various points in history (including the present day) the working class seems to have had little real leverage for negotiation.

In my view, any model that attempts to represent real world markets should begin from the historical fact of profit and loss, and the historical fact of a disequilibrium between input transactions and output transactions.

# A Critique of the Methodology of Mises & Rothbard

I find myself in the middle of a huge blowup between Max Keiser and Tom Woods over Mises, Menger and Austrian economics and feel that this is an opportune moment to express some doubts I have regarding contemporary Austrian methodology.

I am to some extent an Austrian, on three counts.

First, I subscribe to the notion that value is subjective; that goods’ and services’ values differ according to different individuals because they serve various uses to various users, and that value is entirely in the eye of the beholder.

Second, I subscribe to the notion that free markets succeed because of the sensitive price feedback mechanism that allocates resources according to the real underlying shape of supply and demand and conversely the successful long-term allocation of labour, capital and resources by a central planner is impossible (or extremely unlikely), because of the lack of a market feedback mechanism.

Third, I subscribe to the notion that human thought is neither linear nor rational, and the sphere of human behaviour is complicated and multi-dimensional, and that attempts to model it using linear, mechanistic methods will in the long run tend to fail.

It is not, then, the overall drift of Misesean-Rothbardian economics that I find problematic — indeed, I often find myself drawing similar conclusions by different means — but rather the methodology.

I reached my views — some of which new evidence will eventually wash away — through a lot of theorising mixed with much careful observation and consideration of case studies, historical examples and all sorts of real world data. I love data; and one of the things that attracted me toward thinking and writing about economics is the beautiful superabundant growth of new data opened up to the world by computers and the internet. No, it is not universal or complete, and therefore building a perfect predictive model is not possible, but that is not the point. If I want to know how the corn price in the USA moved during the first half of the twentieth century, the data is accessible. If I want to know the rate of GDP growth in Ghana in 2009, the data is accessible. If I want to know the crime rate in France, the data is accessible.

Miseseans choose to reach their conclusions not from data, but instead from praxeology; pure deduction and logic.

This is quite unlike the early Austrians like Menger who mainly used a mixture of deductionism and data.

Praxeology rests on the fundamental axiom that individual human beings act, that is, on the primordial fact that individuals engage in conscious actions toward chosen goals. This concept of action contrasts to purely reflexive, or knee-jerk, behavior, which is not directed toward goals. The praxeological method spins out by verbal deduction the logical implications of that primordial fact. In short, praxeological economics is the structure of logical implications of the fact that individuals act.

Our statements and propositions are not derived from experience. They are not subject to verification or falsification on the ground of experience and facts.

This is completely wrongheaded. All human thought and action is derived from experience; Mises’ ideas were filtered from his life, filtered from his experience. That is an empirical fact for Mises lived, Mises breathed, Mises experienced, Mises thought. Nothing Mises or his fellow praxeologists have written can be independent of that — it was all ultimately derived from human experience. And considering the Austrian focus on subjectivity it is bizarre that Mises and his followers’ economic paradigm is wrapped around the elimination of experience and subjectivity from economic thought.

If, as I often do, I produce a deductive hypothesis — for instance, that the end of Bretton Woods might produce soaring income inequality — it is essential that I refer to data to show whether or not my hypothesis is accurate. If I make a deductive prediction about the future, it is essential that I refer to data to determine whether or not my prediction has been correct.

Exposing a hypothesis to the light of evidence augments its strong parts and washes away its weaker ones. When the evidence changes, I change my opinion irrespective of what my deductions led me to believe or what axioms those deductions were based upon. Why reach the conclusion that central planning can induce civilisational failure through pure logic when the historical examples of Mao’s China and Stalin’s Russia and Diocletian’s Rome illustrate this in gory detail?

This is elementary stuff. Deduction is important — indeed, it is a critical part of forming a hypothesis — but deductions are confirmed and denied not by logic, but by the shape of the evidence. In rejecting modelling — which has produced fallacious work like DSGE and RBCTbut also some relatively successful models like those of Minsky and Keen — praxeologists have made the mistake of rejecting empiricism entirely. This has confined their methods to a grainier simulation; that of their own verbal logic.

It is not necessary to define a framework through mathematical models in order to practice empirical economics. Keynes was cited by Rothbard in support of the notion that economics should not be fixated on mathematical models:

It is a great fault of symbolic pseudo-mathematical methods of formalizing a system of economic analysis, that they expressly assume strict independence between the factors involved and lose
all their cogency and authority if this hypothesis is disallowed: whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep “at the back of our heads” the necessary reserves and qualifications and the adjustments which we have to make later on, in a way in which we cannot keep complicated partial differentials “at the back” of several pages of algebra which assume that they all vanish. Too large a proportion of recent “mathematical” economics are mere concoctions, as
imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.

And I agree. But nowhere did any of the figures cited by Rothbard; not Keynes, nor Wild, nor Frola, nor Menger endorse a wholly deductionist framework. All of these theorists wanted to work with reality, not play with logic. Create a theory; test; refine; test; refine; etc.

Praxeologists claim that praxeology does not make predictions about the future, and that any predictions made by praxeologists are not praxeological predictions, but instead are being made in a praxeologist’s capacity as an economic historian. But this is a moot point; all predictions about the future are deductive. Unless predictions are being made using an alien framework (e.g. a neoclassical or Keynesian model) what else is the praxeologist using but the verbal and deductive methodology of praxeology?

It has been the predictive success of contemporary Austrian economists — at least in identifying general trends often ignored by the mainstream — that has drawn young minds toward Misesean-Rothbardian economics.

Of those economists who predicted the 2008 crisis, a significant number were Austrians:

Yet Miseseans including Peter Schiff damaged their hard-earned credibility with a series of failed predictions of imminent interest rate spikes and hyperinflation of the dollar by 2010.

That is not to say that interest rate spikes and high inflation cannot emerge further down the line. But these predictive failures were symptomatic of deduction-oriented reasoning; Miseseans who forewarned of imminent hyperinflation over-focused on their deduction that a tripling of the monetary base would produce huge inflation, while ignoring the empirical reality of Japan, where a huge post-housing-bubble expansion of the monetary base produced no such huge inflation. Reality is often far, far, far more complex than either mathematical models or verbal logic anticipates.

Like all sciences, economics should be driven by data. For if we are not driven by data than we are just daydreaming.

As Menger — the Father of Austrianism, who favoured a mixture of deductive and empirical methods — noted:

The merits of a theory always depends on the extent to which it succeeds in determining the true factors (those that correspond to real life) constituting the economic phenomena and the laws according to which the complex phenomena of political economy result from the simple elements.

Austrianism would do well to return to its root — Menger, not Mises.

# The Origin of Money

Markets are true democracies. The allocation of resources, capital and labour is achieved through the mechanism of spending, and so based on spending preferences. As money flows through the economy the popular grows and the unpopular shrinks.  Producers receive a signal to produce more or less based on spending preferences. Markets distribute power according to demand and productivity; the more you earn, the more power you accumulate to allocate resources, capital and labour. As the power to allocate resources (i.e. money) is widely desired, markets encourage the development of skills, talents and ideas.

Planned economies have a track record of failure, in my view because they do not have this democratic dimension. The state may claim to be “scientific”, but as Hayek conclusively illustrated, the lack of any real feedback mechanism has always led planned economies into hideous misallocations of resources, the most egregious example being the collectivisation of agriculture in both Maoist China and Soviet Russia that led to mass starvation and millions of deaths. The market’s resource allocation system is a complex, multi-dimensional process that blends together the skills, knowledge, and ideas of society, and for which there is no substitute. Socialism might claim to represent the wider interests of society, but in adopting a system based on economic planning, the wider interests and desires of society and the democratic market process are ignored.

This complex process begins with the designation of money, which is why the choice of the monetary medium is critical.

Like all democracies, markets can be corrupted.

Whoever creates the money holds a position of great power — the choice of how to allocate resources is in their hands. They choose who gets the money, and for what, and when. And they do this again and again and again.

Who should create the monetary medium? Today, money is designated in the ivory towers of central banks and allocated through the banking system. Historically, in the days of commodity-money, money was initially allocated by digging it up out of the ground. Anyone with a shovel or a gold pan could create money. In the days of barter, a monetary medium was created even more simply, through producing things others were happy to swap or credit.

While central banks might claim that they have the nation’s best democratic interests at heart, evidence shows that since the world exited the gold exchange standard in 1971 (thus giving banks a monopoly over the allocation of money and credit), bank assets as a percentage of GDP have exploded (this data is from the United Kingdom, but there is a similar pattern around the world).

Clearly, some pigs are more equal than others:

Giving banks a monopoly over the allocation of capital has dramatically enriched banking interests. It is also correlated with a dramatic fall in total factor productivity, and a dramatic increase in income inequality.

Very simply, I believe that the present system is inherently undemocratic. Giving banks a monopoly over the initial allocation of credit and money enriches the banks at the expense of society. Banks and bankers — who produce nothing — allocate resources to their interests. The rest of society — including all the productive sectors — get crumbs from the table. The market mechanism is perverted, and bent in favour of the financial system. The financial system can subsidise incompetence and ineptitude through bailouts and helicopter drops.

Such a system is unsustainable. The subsidisation of incompetence breeds more incompetence, and weakens the system, whether it is government handing off corporate welfare to inept corporations, or whether it is the central bank bailing out inept financial institutions. The financial system never learned the lessons of 2008; MF Global and the London Whale illustrate that. Printing money to save broken systems just makes these systems more fragile and prone to collapse. Ignoring the market mechanism, and the interests of the wider society to subsidise the financial sector and well-connected corporations just makes society angry and disaffected.

Our monopoly will eventually discredit itself through the subsidisation of graft and incompetence. It is just a matter of time.

# Bernanke vs Greenspan?

Submitted by Andrew Fruth of AcceptanceTake

Bernanke and Greenspan appear to have differing opinions on whether the Fed will monetize the debt.

Bernanke, on behalf of the Federal Reserve, said in 2009 at a House Financial Services Committee that “we’re not going to monetize the debt.

Greenspan, meanwhile, on Meet the Press in 2011 that “there is zero probability of default“ because the U.S. can always print more money.

But they can’t both be true…

There is only 0% probability of formal default if the Fed monetizes the debt. If they refuse, and creditors refuse to buy bonds when current bonds rollover, then the U.S. would default. But Ben said the Fed will never monetize the debt back on June 3, 2009. That’s curious, because in November 2010 in what has been termed “QE2″ the Fed announced it would buy $600 billion in long-term Treasuries and buy an additional$250-$300 of Treasuries in which the$250-$300 billion was from previous investments. Is that monetization? I would say yes, but it’s sort of tricky to define. For example, when the Fed conducts its open market operations it buys Treasuries to influence interest rates which has been going on for a long time — way before the current U.S. debt crisis. So then what determines whether the Fed has conducted this egregious form of Treasury buying we call “monetization of the debt?” The only two factors that can possibly differentiate monetization from open market operations is 1) the size of the purchase and 2) the intent behind the purchase. This is how the size of Treasury purchases have changed since 2009: Since new data has come out, the whole year of 2011 monetary authority purchases is$642 billion – not quite as high as in the graph, but still very high.

Clearly you can see the difference in the size of the purchases even though determining what size is considered monetization is rather arbitrary.

Then there’s the intent behind the purchase. That’s what I think Bernanke is talking about when he says he will not monetize the debt. In Bernanke’s mind the intent (at least the public lip service intent) is to avoid deflation and to boost the economy – not to bail the United States out of its debt crisis by printing money. Bernanke still contends that he has an exit policy and that he will wind down the monetary base when the time is appropriate.

So In Bernanke’s mind, he may not consider buying Treasuries — even at QE2 levels — “monetizing the debt.”

The most likely stealth monetization tactics Bernanke can use — while still keeping a straight face — while saying he will not monetize the debt, will be an extreme difference between the Fed Funds Rate and the theoretical rate it would be without money printing, and loosening loan requirements/adopting policies that will get the banks to multiply out their massive amounts of excess reserves.

If, for example, the natural Fed Funds rate — the rate without Fed intervention — is 19% and the Fed is keeping the rate at 0%, then the amount of Treasuries the Fed would have to buy to keep that rate down would be huge — yet Bernanke could say he’s just conducting normal open market operations.

On the other hand, if the banks create money out of nothing via the fractional reserve lending system and a certain percentage of that new money goes into Treasuries, Bernanke can just say there is strong private demand for Treasuries even if his policies were the reason behind excessive credit growth that allowed for the increased purchase of Treasuries.

Maybe Bernanke means he will not monetize a particular part of the debt that was being referred to in the video. Again, though, he could simply hide it under an open market operations 0% policy or encourage the banking system to expand the money supply.

Whatever the case, if you ever hear Bernanke say “the Federal Reserve will not monetize the debt” again, feel free to ignore him. When he says that, it doesn’t necessarily mean he won’t buy a large quantity of Treasuries with new money created out of nothing.

Remember, Greenspan says there’s “zero probability of default” because the U.S. can always print more money. Does Greenspan know something here? There’s only zero probability if the Fed commits to monetizing the debt as needed. If Greenspan knows something there will be monetization of the debt, even if Bernanke wants to call it something else.

# Out of the Liquidity Trap?

Professor Krugman has produced an interesting graph that — according to his calculation — suggests that while we’re not quite out of the liquidity trap, we are getting closer:

It’s a useful contribution, that shows just what the Federal Reserve does in terms of trying to match interest rates to the broader inflationary outlook. The liquidity trap at the zero bound is clearly visible — the Fed cannot cut rates below the zero bound, which renders traditionally monetary policy essentially useless. (Austrians will of course interject here that traditional monetary policy is worse than useless, but that is another story for another day).

If the liquidity trap is ended, we should eventually see higher demand (Krugman’s point is broadly that stimulus would fight off the problem of the liquidity trap and solve the problem sooner). The Krugmanites think that demand is the only problem, and higher demand (even if that is down the line and later than Krugman would like it) will cure our economic woes.

I completely disagree and believe that depressed demand is not the main problem, but merely a symptom. I believe that the credit contraction that occurred in 2008 was a direct product of various non-monetary challenges that America faces, almost none of which have been solved, or will be solved by an end to the liquidity trap:

The three main problems are a lack of confidence stemming from high systemic residual debt, deindustrialisation in the name of globalisation (& its corollary, financialisation and that sprawling web of debt and counter-party risk), and fragility and side-effects (e.g. lost internal productivity due to role as world policeman) coming from America’s petroleum addiction.

In the months and years to come we will see who is right.

If everyone tries to save more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population because of the decrease in consumption and economic growth. The paradox is, narrowly speaking, that total savings may fall even when individual savings attempt to rise, and, broadly speaking, that increase in savings may be harmful to an economy.

It is in many ways a microcosm of Keynesian economics — mostly right, but in a few details spectacularly and outlandishly wrong

(A small detour: In many cases, these few details matter — big time. Austrian economics, of course, is the inverse of this — mostly wrong, but in a few details spectacularly and outlandishly right. I remain unsure as to which set of ideas gives more insight; although I am deeply impressed by some insights of Austrian economics, I generally think the only sensible economic outlook is to scavenge for whatever explanations fit the facts.)

If many people in the economy cease economic activity and hoard their capital, of course the economy will slow down — both in nominal (GDP) and generally also in real terms. Satisfying supply and demand requires a constant and continuous flow of productivity, and goods, and services and money.

Of course there is no paradox there: lowered economic activity is almost always deleterious (the exception being the rare cases when a type of economic activity is itself deleterious to the health of the economy.)

But here is where the paradox of thrift really falls down: saving is not necessarily the same thing as hoarding. Hoarding is saving without investment. And there’s a good chunk of evidence to suggest that more saving with investment is correlated with stronger economic health.

Facts reinforce the idea that spending is no cure-all for what ails America. Most countries experiencing full employment and rapid growth do so while saving at very high rates. China is growing like crazy with a saving rate of more than 30 percent. Japan also saved at very high rates when it was booming. Since then, both rates have plummeted.

The U.S. has also done better when saving was high. In the 1950s and 1960s, saving averaged 14 percent of national income, which grew at 4.4 percent a year in real terms. In the 1990s and 2000s, saving averaged 5.1 percent and national income increased only 2.4 percent.

The connection between saving and growth runs through domestic investment. Countries that save invest not only by building inventory for tomorrow; they also invest in physical capital that makes workers more productive.

U.S. saving is highly correlated with domestic investment; when we save, we primarily invest here at home in starting businesses, buying equipment, and building factories.

In reality, Keynes (and his antecedents) identified an important economic eventuality — a slowdown resulting from hoarding. However, associating this with saving — and subsequently tilting policy toward consumption — is really a misidentification. Societies need investment to grow, and investment requires saving.

# Stiglitz vs Krugman

A very interesting front is opening up regarding the current state of America.

Some economists believe that the main problem in America is a lack of demand, defined as the desire to buy, the willingness to buy, and the ability to pay for it

From Paul Krugman:

There is nothing — nothing — in what we see suggesting that this current depression is more than a problem of inadequate demand. This could be turned around in months with the right policies. Our problem isn’t, ultimately, economic; it’s political, brought on by an elite that would rather cling to its prejudices than turn the nation around.

The implication here is that people just don’t have the money in their pockets to spend at the levels they were five years ago, and the solution is (through whatever means) giving them that money.

As well as the obvious (and accurate) Austrian retort that demand in 2006 was being pushed skyward as part of a ridiculous and entirely artificial debt-financed bubble, other economists believe that a lack of demand is just a symptom of other underlying symptoms. I myself believe that the three main problems are a lack of confidence stemming from high systemic residual debt, deindustrialisation in the name of globalisation (& its corollary, financialisation and that sprawling web of debt and counter-party risk), and fragility and side-effects (e.g. lost internal productivity due to role as world policeman) coming from America’s petroleum addiction.

Now Joe Stiglitz has weighed in in a lengthy and essential Vanity Fair piece:

The trauma we’re experiencing right now resembles the trauma we experienced 80 years ago, during the Great Depression, and it has been brought on by an analogous set of circumstances. Then, as now, we faced a breakdown of the banking system. But then, as now, the breakdown of the banking system was in part a consequence of deeper problems. Even if we correctly respond to the trauma—the failures of the financial sector—it will take a decade or more to achieve full recovery. Under the best of conditions, we will endure a Long Slump. If we respond incorrectly, as we have been, the Long Slump will last even longer, and the parallel with the Depression will take on a tragic new dimension.

Many have argued that the Depression was caused primarily by excessive tightening of the money supply on the part of the Federal Reserve Board. Ben Bernanke, a scholar of the Depression, has stated publicly that this was the lesson he took away, and the reason he opened the monetary spigots. He opened them very wide. Beginning in 2008, the balance sheet of the Fed doubled and then rose to three times its earlier level. Today it is \$2.8 trillion. While the Fed, by doing this, may have succeeded in saving the banks, it didn’t succeed in saving the economy.

Reality has not only discredited the Fed but also raised questions about one of the conventional interpretations of the origins of the Depression. The argument has been made that the Fed caused the Depression by tightening money, and if only the Fed back then had increased the money supply—in other words, had done what the Fed has done today—a full-blown Depression would likely have been averted. In economics, it’s difficult to test hypotheses with controlled experiments of the kind the hard sciences can conduct. But the inability of the monetary expansion to counteract this current recession should forever lay to rest the idea that monetary policy was the prime culprit in the 1930s. The problem today, as it was then, is something else. The problem today is the so-called real economy. It’s a problem rooted in the kinds of jobs we have, the kind we need, and the kind we’re losing, and rooted as well in the kind of workers we want and the kind we don’t know what to do with. The real economy has been in a state of wrenching transition for decades, and its dislocations have never been squarely faced. A crisis of the real economy lies behind the Long Slump, just as it lay behind the Great Depression.

At the beginning of the Depression, more than a fifth of all Americans worked on farms. Between 1929 and 1932, these people saw their incomes cut by somewhere between one-third and two-thirds, compounding problems that farmers had faced for years. Agriculture had been a victim of its own success. In 1900, it took a large portion of the U.S. population to produce enough food for the country as a whole. Then came a revolution in agriculture that would gain pace throughout the century—better seeds, better fertilizer, better farming practices, along with widespread mechanization. Today, 2 percent of Americans produce more food than we can consume.

What this transition meant, however, is that jobs and livelihoods on the farm were being destroyed. Because of accelerating productivity, output was increasing faster than demand, and prices fell sharply. It was this, more than anything else, that led to rapidly declining incomes. Farmers then (like workers now) borrowed heavily to sustain living standards and production. Because neither the farmers nor their bankers anticipated the steepness of the price declines, a credit crunch quickly ensued. Farmers simply couldn’t pay back what they owed. The financial sector was swept into the vortex of declining farm incomes.

The cities weren’t spared—far from it. As rural incomes fell, farmers had less and less money to buy goods produced in factories. Manufacturers had to lay off workers, which further diminished demand for agricultural produce, driving down prices even more. Before long, this vicious circle affected the entire national economy.

The parallels between the story of the origin of the Great Depression and that of our Long Slump are strong. Back then we were moving from agriculture to manufacturing. Today we are moving from manufacturing to a service economy. The decline in manufacturing jobs has been dramatic—from about a third of the workforce 60 years ago to less than a tenth of it today. The pace has quickened markedly during the past decade. There are two reasons for the decline. One is greater productivity —t he same dynamic that revolutionized agriculture and forced a majority of American farmers to look for work elsewhere. The other is globalization, which has sent millions of jobs overseas, to low-wage countries or those that have been investing more in infrastructure or technology. (As Greenwald has pointed out, most of the job loss in the 1990s was related to productivity increases, not to globalization.) Whatever the specific cause, the inevitable result is precisely the same as it was 80 years ago: a decline in income and jobs. The millions of jobless former factory workers once employed in cities such as Youngstown and Birmingham and Gary and Detroit are the modern-day equivalent of the Depression’s doomed farmers.

The consequences for consumer spending, and for the fundamental health of the economy—not to mention the appalling human cost—are obvious, though we were able to ignore them for a while. For a time, the bubbles in the housing and lending markets concealed the problem by creating artificial demand, which in turn created jobs in the financial sector and in construction and elsewhere. The bubble even made workers forget that their incomes were declining. They savored the possibility of wealth beyond their dreams, as the value of their houses soared and the value of their pensions, invested in the stock market, seemed to be doing likewise. But the jobs were temporary, fueled on vapor.

So far, so excellent. Stiglitz first shovels shit over the view of Fisherian debt-deflation as the main cause of the slump in demand — debt-deflation is a symptom, and a very nasty one, but not really a cause. Second, Stiglitz also correctly notes that today’s ailments are the result of social, infrastructural and productive upheaval in the real economy. He correctly identifies the leading trend here — manufacturing (and, it should be added, primary industry) has been ripped out of America by the forces of globalisation, and the powerful pull of cheaper wages. This is a strong explanation of why Krugman’s view — that the only thing missing is demand, and that government can fix that in an instant — is nonsense.

As I wrote earlier this month:

The point here is that economic health — and real industrial output, measured in joules, or in “needs met” — and money circulation are in reality almost totally decoupled. Getting out of a depression requires debt erasure, and new organic activity, and there is absolutely no guarantee that monetary easing will do the trick on either count. Most often, depressions and liquidity traps are a reflection of underlying structural and sociological problems, and broken economic and trade systems. Easing kicks the can down the road a little, and gives some time and breathing room for those problems to be fixed, but very often that just doesn’t happen. Ultimately, societies only take the steps necessary (e.g. a debt jubilee) when their very existence seems threatened.

The simple expansionary recipe for getting out of depressions is a sad smile, a false promise of an easy route out of complex and multi-dimensional problems. It gives succour to the simplistic, and those enamoured with linear thinking.

Stiglitz continues:

What we need to do instead is embark on a massive investment program—as we did, virtually by accident, 80 years ago—that will increase our productivity for years to come, and will also increase employment now. This public investment, and the resultant restoration in G.D.P., increases the returns to private investment. Public investments could be directed at improving the quality of life and real productivity—unlike the private-sector investments in financial innovations, which turned out to be more akin to financial weapons of mass destruction.

Now, I don’t really have a problem with the idea that government can do some good. If people in a democracy choose to solve problems via public spending, well, that’s part of the bargain in a democratic state. Even Adam Smith noted that government should fund “certain great institutions” beyond the reach of private enterprise.

But here we reach the great problem with Stiglitz’s view:

The private sector by itself won’t, and can’t, undertake structural transformation of the magnitude needed—even if the Fed were to keep interest rates at zero for years to come. The only way it will happen is through a government stimulus designed not to preserve the old economy but to focus instead on creating a new one. We have to transition out of manufacturing and into services that people want—into productive activities that increase living standards, not those that increase risk and inequality.

And there we have the fundamental problem with statism: central planners — particularly in a nuanced and huge area like the economy — are prone to invest in all of the wrong things, mostly because they do not grasp what the problems are, or how to solve them. (Or, more nefariously, corruption).

The United States spent the last decade (arguably longer) and trillions of dollars embroiled in wars aimed at keeping oil cheap, and maintaining the flow of global goods precisely because America is dependent upon those things. America does not play global policeman out of nicety or vanity — she does it out of economic necessity. That is precisely because America let globalisation take away all of her industry, making her dependent not only on the continued value of her paper dollar, but on the flow of global trade in energy and goods.

Investing more money in services will leave America dependent on these contingencies. And dependency is fragility — and the more fragile America becomes, the more aggressive she becomes in maintaining and controlling the flow of global goods.

Any stimulus package ought to instead be focussed on making America energy independent, and encouraging innovative new forms of manufacturing (e.g. 3-D printing) that can undercut Chinese labour.

So while Stiglitz must be commended for seeing through the Bernankean haze, it is extremely disappointing that his alternative to that quagmire is spending more money to allow America to become even more dependent on foreign goods and oil.

Government has the spending power to fix many of the structural problems, but just won’t for whatever reasons (corruption, myopia, paralysis).

And while America is dependent on foreign goods and energy, she is prone to not only waste huge amounts of productive capital on war and weapons, but she also risks serious economic damage from events such as oil shocks, geostrategic shocks, regional wars, and — well — anything that might slow down or endanger global trade. Her need to police the world makes her even hungrier for oil, which means she spends more money on the world, which makes her hungrier for oil.

# Artificially Low Interest Rates in Europe

My chart of the day, illustrating a pretty brutal reversion to the mean:

Of course, all interest rates in a fiat system are artificial. Interest rates are the price of money, and if a central bank is determining the level of money, then they are in effect determining the level of interest, which is one reason why sovereigns who borrow in their own currency do not tend to face a danger of rising interest rates even at high levels of borrowing.

The post-Euro low-rates euphoria was a cunning trick: the single monetary policy disguised each state’s true fiscal picture. Fiscal union might have prevented this blowup, but introducing it now seems unlikely given Germany’s severe aversion to such a thing.

If AIG is considered ‘too big to fail’ what does that make the Eurozone given the very high levels of integration across the global economy today? (I don’t have the answer, but I think we can all guess).


text:
# zbMATH — the first resource for mathematics

Geodesic balls and Chern numbers of Kähler manifolds. (English) Zbl 0788.53065
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 127-135 (1992).
The author gives a summary of his results about the volume of geodesic balls which appeared in [Nagoya Math. J. 116, 181-189 (1989; Zbl 0683.53061)] and which are closely related to the volume conjectures of A. Gray and the reviewer [Acta Math. 142, 157-198 (1979; Zbl 0428.53017)]. He focuses on Kähler manifolds $$M$$ and considers the conjectures in relation with conditions on the generalized Chern numbers or Chern classes, in particular when $$\dim M = 4$$.
For the entire collection see [Zbl 0764.00002].
##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds
##### Keywords:
volume of geodesic balls; Kähler manifolds; Chern numbers

format: plain


text:
# zbMATH — the first resource for mathematics

Spectral topologies of dually residuated lattice-ordered monoids. (English) Zbl 1080.06023
Summary: Dually residuated lattice-ordered monoids ($$DR\ell$$-monoids for short) generalize lattice-ordered groups and include for instance also GMV-algebras (pseudo MV-algebras), a non-commutative extension of MV-algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.
##### MSC:
 06F05 Ordered semigroups and monoids 06D35 MV-algebras
##### Keywords:
$$DR\ell$$-monoid; prime ideal; spectrum
Full Text:


text:
My Math Forum > Math Two-stage decoding based on Euclidean division

 Math General Math Forum - For general math related discussion and news

 June 8th, 2018, 02:43 PM #1 Newbie   Joined: Jul 2016 From: Ames, IA Posts: 3 Thanks: 0 Two-stage decoding based on Euclidean division Define the following quantity: $$X=abs^{2\ell}+ce+Fs^{\ell}$$ where $$F=ae+cb$$ Assumptions: $a,b,c,e$ are unknown integers $s,\ell$ are known positive integers $X$ is known My goal is to find $F$. The general idea is simple. Since we know $X$, we first find the remainder of the division $X/s^{2l}$. That would be $ce+Fs^{\ell}$. Then from this quantity we subtract the remainder of the division $(ce+Fs^{\ell})/s^\ell$ which yields $Fs^{\ell}$. Then just divide by $s^\ell$. Now, if all variables were positive integers under the constraint $F0$). Definition 1 $$x=qd+r$$ in which case the $0\leq r0$, then $R=r$ and $Q=q$ - If $x<0$, then $r=R+d$ and $Q=q+1$ Matlab has an implementation for both definitions. For my initial problem if I use Definition 1 the result for $F$ is wrong if $X<0$ (it differs from the correct one by $s^\ell$ which is expected) and correct if $X>0$. If I use Definition 2, the result for $F$ is correct if $X>0$ and for $X<0$ it's sometimes correct and sometimes wrong. So my strategy is flawed but I am not sure what is wrong. Also, some assumptions need to hold a priori for the general case where some variables can be negative, to have a solution and I think that those should be $|a|,|b|,|c|,|e|,|F|<0.5(s^{\ell}-1)$.

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# Finding minimum distance of points along a 2D grid

This topic is 2502 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Here's an interesting question for everyone.

Say you have a 2D grid with length X and width Y. You have N points in this grid (existing in some cell of the grid). It is possible for more than one point to be in the same grid position. Moving a point to a neighboring spot in the grid (horizontal and vertical, no diagonal movement) takes 1 unit. Find the spot on the grid that allows you to move all points to one grid cell with the least amount of travel cost.

+ - + - P - + - +
| | | | |
P - + - + - + - +
| | | | |
+ - P - + - + - +

(the 5 |'s are suppose to be spaced out but I guess the editor doesn't like that)

+'s are grid cells, - are horizontal movements | are vertical movements, P is a point in a grid cell. The answer for this example is position (2, 2) requiring 4 moves, 2 from the top middle point, 1 each of the other two points.

My brute force solution to this was to check the required distance traveled for each node in the grid. The one (or more than one) with the minimum traveled distance is the solution. This requires O(XY) comparisons.

I know there is a better solution to this and I believe that this problem is very similar to finding the center of many points, just forced to fit along a grid. I thought of taking the average of all points but that doesn't work for a lot of cases.

Anyone have suggestions on how to solve this with a better time complexity than the brute force method?

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I think what you want to look for is the center of the sample points. That is find the center in the given random sample points such that the distance between all points to that center is minimal. Google center star algorithm. I think this problem is in NP and thus you might have to settle for an approximation algorithm. Note, I maybe wrong about my method but the above is my initial thought.

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Use a code block if you want it to look right:+ - + - P - + - + | | | | | P - + - + - + - + | | | | | + - P - + - + - +
Your brute force algorithm is actually O(NXY) because for each cell it has to calculate the distance required to get there from each of the N points.
That's not terribly bad, but if the grid is 5000 * 5000 and there are 2000 points then it could be slow.

I have an idea that is O(NX + NY) ...
Basically you calculate the cost for the horizontal and vertical movements of all points separately, then find the minimum sum of the combinations of horizontal moves plus vertical moves.

E.g. for the above board you calculate the cost of moving all points to each column
Column => Cost
1 => 3
2 => 2
3 => 3
4 => 6
5 => 9

Row => Cost
1 => 3
2 => 2
3 => 3

Now you iterate over the rows and columns and pick the minimum sum from each.
2 is smallest from the columns and 2 is smallest from the rows = total of 4, at position (2, 2)
All this does is avoid repeatedly calculating the same horizontal costs for each vertical position and vice versa.

It might be possible to do better, but I'd probably just do this.

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Couldn't you do a flood fill of distances from each point? The pseudocode for my suggestion (you need to keep track of the distances from each point to each location):
1. Fill in the locations with a distance 1 higher then the distance you last looked at (so if you looked at locations 1 distant from each point you would look at locations 2 distant from each points) from each point
2. Examine all the points that you filled in step 1 to see if any of them have distances to all points, and rank them on the sum of the distances
3. Either you stop because you know that you have the smallest such sum (admittedly I do not know how to do this at the moment) or go back to step 1

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This problem can be solved in linear time to the number of the input points.

As iMalc pointed, the problem is separable to two applications of 1D scans through the input points, to first solve for X, and then for Y. This is the case since you don't allow diagonal movements. Taxicab distance makes things simpler!

To solve the problem in 1D:

- You are given an array of input points N_1, N_2, ..., N_k, integers, and you need to find an integer K such that

a) Max | N_i - K | is minimized, or

b) Sum | N_i - K | is minimized.

I am unsure which one you were after, so for both:

In the case A, take the average of the minimum and maximum value, round to integer.

In the case B, first sort the points into ascending order (can be done in linear time with radix or bucket sort), then perform a linear pass through the array. At each iteration, you can compute the "penalty" caused by leaving the past points behind, and the gain received by moving the center point forward. At some point, these two balance, and you know you have the optimal center point. (In fact, I'm thinking it could perhaps be converted into log-time binary search, but I'd have to implement it to see if that's the case).

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Thanks guys. Both are good algorithms that perform better than brute force! Appreciate the help.

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Actually, I think the problem is as simple as computing the median of the X coordinates and the median of the Y coordinates. Indeed, this can be solved in linear time.

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# A signal sampling related doubt

1. Mar 3, 2013

### dexterdev

Hi PF,
I have a signal which is sum of 2 sinusoids having frequencies 6KHz and 12KHz. Now if I sample it at a rate of 16KHz and pass it through a ideal Low pass filter having 16KHz cutoff frequency, what is the signal I obtain. What frequency contents does it have. Please help me visualize in frequency domain using comb.

-Devanand T

2. Mar 4, 2013

### milesyoung

A magnitude plot of the spectrum of your signal will show sinusoids at +-6 kHz and +-12 kHz (usually pictured as vertical lines).

Passing this signal through the ideal sampler will replicate its spectrum at integer multiples of your sampling frequency - maybe try to make a quick sketch of this. The 6 kHz and 12 kHz sinusoids will thus fold down and produce aliases at +-10 kHz and +-4 kHz, respectively, and your brick-wall low pass filter will cut off anything above/below +-16 kHz.

The resulting signal will thus have components at +- 4, 6, 10 and 12 kHz.

3. Mar 4, 2013

### AlephZero

Either this is a "trick" question about the Nyquist frequency and aliasing, or there are some typos in your question.

4. Mar 4, 2013

### milesyoung

I don't see the confusion. Using a brick-wall low-pass filter at 16 kHz for reconstructing the signal from its samples is perfectly valid.

5. Mar 4, 2013

### dexterdev

thanks milesyoung

6. Mar 4, 2013

### dexterdev

this is just a competitive exam question from India.

7. Mar 4, 2013

### skeptic2

This is an interesting problem especially if we change a few parameters. First let's specify the two signals are in phase such the the maxima of the 12 kHz signal coincide with the maxima and minima of the 6 kHz signal and that the sample time is 12 kHz and coincides with the maxima of the 12 kHz signal. Now with a Fourier Transform we can reconstruct both signals even though the sampling frequency is below the Nyquist limit of the higher frequency.

8. Mar 4, 2013

### AlephZero

What (if anything) do you think a low pass filter with a cut-off frequency of 16 KHz will do, after you have already sampled the data with a Nyquist frequency of 8 kHz??

The specification of the filter makes no sense, unless you are supposed to guess that you convert the signal back to analog or re-sample it at a higher rate before you filter it.

9. Mar 4, 2013

### marcusl

This makes no sense. If you specify the signals as you did, then they are already known exactly and you don't need an FT. If you don't know what they are, then you are lost: The 12 kHz signal aliases to 0 (DC). The 6 kHz signal, because it is at exactly 1/2 the sampling frequency, is ambiguous--you can't tell if it is at 6 kHz or 0. There's no chance of correctly identifying what you have.

10. Mar 4, 2013

### milesyoung

Are you somehow under the impression that the low-pass filter is there as an anti-aliasing filter? In that case, of course it's absurd to place it _after_ sampling with a cutoff frequency that high.

The low-pass filter is there as an ideal signal reconstruction device. A zero-order hold usually takes its place in practical systems.

11. Mar 5, 2013

### rbj

the question does strictly speaking make sense. it's just that the sampling setup does not. there is no difference between sampling the 12 kHz sinusoid at $f_s$ = 16 kHz is the same as sampling a 4 kHz sinusoid. that's why they call it an "alias".

so, if you filter with cutoff of 16 kHz, what you will see are components at 4 kHz and 6 kHz as well as 10 kHz and 12 kHz (and at their negatives).

12. Mar 5, 2013

### rbj

it could be an anti-imaging filter, but the sampling rate is still 1/2 of what it should be.

oh, there certainly could be an additional LPF after the ZOH, to smooth it out a little. BTW, the ZOH is inherent to the D/A converter. you don't really replace anything with a device called a "ZOH". it's just there.

13. Mar 5, 2013

### milesyoung

The OP's question is a classic example that asks you to consider the effect of sampling a signal below its Nyquist rate. It makes perfect sense and there's no ambiguity.

Yes, it's a anti-imaging filter (a reconstruction filter). The sampling frequency doesn't have to be any value in particular - the OP isn't asking what sampling frequency to choose in order to avoid aliasing.

Did I say there couldn't?

I'm well aware of that, but I haven't said anything with regards to the actual implementation of the filters - I have only included descriptions of mathematical models in my posts, i.e. the DAC is usually modeled as a ZOH.

14. Mar 5, 2013

### sophiecentaur

If you sample a 12kHz signal at 16kHz, no amount of filtering will remove the aliasing (except a notch filter at 4kHz).

15. Mar 6, 2013

### milesyoung

Again, the OP isn't asking how to avoid aliasing, i.e. the purpose is not to reconstruct the original signal. If he/she wants to bandlimit the signal to 16 kHz then that's perfectly valid.

16. Mar 6, 2013

### AlephZero

But in the real world, "wanting to bandlinit a signal to 16kHz" when it is already bandlinited to 8KHz is nonsensical. It should also be nonsensical in the world of "competetive exam questions", IMO.

There's nothing nonsensical about the first part of the question (i.e. answering it requires an understanding of aliasing), but the filtering part is meaningless.

17. Mar 6, 2013

### milesyoung

The sampled signal is certainly not bandlimited - the star transform of the original signal replicates its spectrum at integer multiples of the sampling frequency.

18. Mar 6, 2013

### sophiecentaur

If this thread is really the result of a test question then the setter should be shot. You can set up 'nonsense' scenarios, informally in class and make sure they are resolved properly before people go away but if this question was set merely to show that oversampling is dangerous then it was worded wrongly and can only confuse and worry the poor students.
If it had been worded "Why wouldn't the following system work? etc. etc." then no one would be looking for a reason for it to work and then they'd stand a chance..

19. Mar 6, 2013

### milesyoung

I guess I just don't see what it is about this question that rubs people the wrong way. There's no denying that this is a terrible setup if you want to avoid aliasing, but I don't see that as part of the question.

As a test question, I don't think its uncommon to make the student include only the first few aliases. If the low-pass filter wasn't included after sampling, the signal wouldn't be bandlimited.

20. Mar 6, 2013

### rbj

i thought it was a clear question of what may not have been intended. but it had a clear answer. it had 4, 6, 10, and 12 kHz in the output. might not have been what the original questioner expected.


text:
Any suggestions/improvements for the following custom thread-pool code?

import threading
from Queue import Queue

def __init__(self, function, in_queue, out_queue):
self.function = function
self.in_queue, self.out_queue = in_queue, out_queue
super(Worker, self).__init__()

def run(self):
while True:
if self.in_queue.empty(): break
data = in_queue.get()
result = self.function(*data)
self.out_queue.put(result)

def process(data, function, num_workers=1):
in_queue = Queue()
for item in data:
in_queue.put(item)

out_queue = Queue(maxsize=in_queue.qsize())
workers = [Worker(function, in_queue, out_queue) for i in xrange(num_workers)]

for worker in workers:
worker.start()

in_queue.join()

while not out_queue.empty():
yield out_queue.get()

• I recommend looking into the multiprocessing.pool.ThreadPool object, also explained here. – Theuni Dec 7 '12 at 21:03

1. The function concurrent.futures.ThreadPoolExecutor.map is built into Python 3 and does almost the same thing as the code in the post. If you're still using Python 2, then there's a backport of the concurrent.futures package on PyPI.

2. But considered as an exercise, the code here seems basically fine (apart from the lack of documentation), and the remaining comments are minor issues.

3. Worker.run could be simplified slightly by writing the loop condition like this:

while not self.in_queue.empty():
# ...

4. There's no need to pass a maxsize argument to the Queue constructor—the default is that the queue is unbounded, which is fine.

5. The only use of the workers list is to start the workers. But it would be easier to start each one as you create it:

for _ in range(num_workers):
Worker(function, in_queue, out_queue).start()

6. The results come out in random(-ish) order because threads are nondeterministic. But for many use cases you would like to know which input corresponds to which output, and so you'd like the outputs to be in the same order as the inputs (as they are in the case of concurrent.futures.ThreadPoolExecutor.map).


text:
# Trajectories of particles with spin in Einstein-Cartan theory

The Einstein-Cartan theory is a generalisation of General Relativity insofar as the condition that the metric affine connection is torsion-free is dropped. In other words, the space time is a Riemannian manifold together with the datum of a metric affine connection (which may differ from the Levi-Civita connection by a suitable contorsion tensor).

In this case, geodesics (paths that locally extremalise the length, and which are given by a variational principle) generally differ from auto-parallels. As far as I know, the trajectories of spin-less particles in Einstein-Cartan theory are usually assumed to be geodesics (rather than autoparallels) so they don't feel the difference between the given connection and the Levi-Civita connection. (By the way, is there a good reference for this statement?)

My question is, how classical particles with spin are supposed to behave? Will they also travel along geodesics with the only difference that their spin direction will evolve according to the contorsion tensor (viewed as a so(1,3)-valued one-form)?

-

No. Particles with spin will feel the torsion not only through their spin precession, since the equations of motion for them (Mathisson-Papapetrou equations) will contain the asymmetric part of the connection.

One source for the question is the review

Hehl, F. W., Von der Heyde, P., Kerlick, G. D., & Nester, J. M. (1976). General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys., 48(3), 393. (it has an online version).

From there we learn:

We have already indicated that photon and spinless test particles sense no torsion. A test particle in $U_4$ theory, one which could sense torsion, is a particle with dynamical spin like the electron. Its equation of motion can be obtained by integrating the conservation law of energy-momentum (3.12). In so doing, we obtain directly the Mathisson-Papapetrou type equation${}^{20}$ for the motion of a spinning test particle (Hehl, 1971; Trautman, 1972c)${}^{21}$ Adamowicz and Trautman (1975) have studied the precession of such a test particle in a torsion background. All these considerations seem to be of only academic interest, however, since torsion only arises inside matter. There, the very notion of a spinning test particle becomes obscure (H. Gollisch, 1974, unpublished). Only neutrinos, whose spin self interaction vanishes, seem to be possible candidates for $U_4$ test particles.

(Hehl, 1971) reference here is apparently original result on the motion of test particle with spin:

Hehl, F. W. (1971). How does one measure torsion of space-time?. Phys. Lett. A, 36(3), 225-226. (http://dx.doi.org/10.1016/0375-9601(71)90433-6)

For a more modern notation (tetrad formalism) for the mentioned Mathisson-Papapetrou equations you can use the thesis:

Laskoś-Grabowski, P. (2009). The Einstein–Cartan theory: the meaning and consequences of torsion. Master's thesis pp. 17-19

The references there should provide all further information.

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text:
### Clinton Boys

I am an Australian data scientist and mathematician, living in Tel Aviv.

# Seeing the functional light

I learnt to program in Python in high school. I didn’t take computer science as a high school subject, but rather took a once-a-week extracurricular course with a mild-mannered teacher called Mr Huxley and a few other kids. I learnt the basics of control structures, solved basic programming problems, and entertained myself by “polluting the namespace” with variables whose names were offensive jibes against my teammates:

while peter.is_an_asshole():
print 'Peter is an asshole!'

Over the next ten years I came to depend on Python as my only programming language, using it for scripting, data munging, scraping, and more sophisticated data analysis and machine learning through a bunch of great libraries.

Recently I’ve been learning about functional programming. I have a very mathematical background, so I was instantly struck by how similar many formalisms in functional programming are to mathematics. You can see mathematical definitions and descriptions of sets, proofs by induction, formal constructions of objects, all translated very neatly into a programming paradigm.

An axiomatic approach

For example, introductory courses on functional programming usually contain some sort of exercise which walks the student through constructing the integers, or the rational numbers. This is an important mathematical idea that is somewhat obscured by programming procedurally.

In an exercise I did in the early stages of learning functional programming in Scala, I had to define sets in a functional way.

object FunSets {
type Set = Int => Boolean
def contains(s: Set, elem: Int): Boolean = s(elem)
def singletonSet(elem: Int): Set = (x: Int) => if(x == elem) true else false

def union(s: Set, t: Set): Set = (x: Int) => s(x) || t(x)

def intersect(s: Set, t: Set): Set = (x: Int) => s(x) && t(x)

def diff(s: Set, t: Set): Set = (x: Int) => s(x) && !t(x)

def filter(s: Set, p: Int => Boolean): Set = (x: Int) => if (p(x)) s(x) else false
}

The above code will seem very strange for someone not used to functional program, but it perfectly encodes the mathematical idea of a set of integers as a map with domain the integers and a boolean range (i.e. the “set membership function” for a set S maps an integer to True if the integer is a member of S, and to False otherwise). This allows for very elegant descriptions of common set operations like union, intersection and difference by simple logical operators on the function values.

Another moment of clarification when learning functional programming comes the first time you are able to distill a complicated “pipeline” formula into a single elegant line of code. Programming functionally teaches you to break all actions into their smallest possible pieces which build upon each other to produce more complicated actions and procedures mean the ultimate expressions of what your code is trying to do become very succinct and usually extremely readable.

    def decode(tree: CodeTree, bits: List[Bit]): List[Char] = {

def iterate(remaining: CodeTree, bits: List[Bit]): List[Char] = remaining match {
case Leaf(c, _) if bits.isEmpty => List(c)
case Leaf(c, _) => c :: iterate(tree, bits)
case Fork(left, right, _, _) if bits.head == 0 => iterate(left, bits.tail)
case Fork(left, right, _, _) => iterate(right, bits.tail)
}
iterate(tree, bits)
}

The above code uses Huffman coding to decode the bit sequence bits using the code tree tree and returns the sequence of actual characters encoded by the bit sequence. Although good programmers will do this anyway, functional programming teaches the modularity evident in this example: in order to define decode we must first define the classes CodeTree, Leaf and Fork, as well as the function iterate, and the methods isEmpty, head and tail on a List[Bit] object. Changes made to any of these smaller pieces then propograte throughout the code.

Typing

The strangest thing about functional languages for someone who is learning about them for the first time is that they are usually strongly typed, which represents a very alien departure from Python. In Python variables rarely need to have their type declared. This is a great thing for quick scripting, as it can make the translation from English to code much quicker and easier.

What is difficult to understand (and I didn’t believe was possible at first) is that nearly every single bug in any program that causes it to fail at runtime is caused by an issue that could be caught by a strong typing system at compile time.

Conclusion

Of course there is no “winning” paradigm here. People argue about the benefits of procedural versus functional programming, static versus strong typing, all the time, attempting to demonstrably show which is the better style. I think it’s great to know both. First of all, without knowing both paradigms it is impossible to be aware of the weaknesses and limitations of the one you use exclusively. And most importantly, being able to program fluently in both paradigms gives you the flexibility to choose the right tools for the task at hand.


text:
Computes a nonlinearity statistic based on Teräsvirta's nonlinearity test of a time series. The statistic is $$10X^2/T$$ where $$X^2$$ is the Chi-squared statistic from Teräsvirta's test, and T is the length of the time series. This takes large values when the series is nonlinear, and values around 0 when the series is linear.

nonlinearity(x)

## Arguments

x a univariate time series

A numeric value.

## Author

Yanfei Kang and Rob J Hyndman

## Examples

nonlinearity(lynx)
#> nonlinearity
#>    0.8959046


text:
### Calculate the average atomic weight when given isotopic weights and abundancesFifteen Examples

To do these problems you need some information. To wit:

(a) the exact atomic weight for each naturally-occuring stable isotope
(b) the percent abundance for each isotope

These values can be looked up in a standard reference book such as the "Handbook of Chemistry and Physics." The values can also be looked up via many online sources. The ChemTeam prefers to use Wikipedia to look up values.

The unit associated with the answers to the problems below can be either amu or g/mol, depending on the context of the question. If it is not clear from the context that g/mol is the desired answer, go with amu (which means atomic mass unit).

By the way, the most correct symbol for the atomic mass unit is u. The older symbol (which the ChemTeam grew up with) is amu (sometimes seen as a.m.u.) The unit amu is still in use, but you will see u used more often.

This problem can also be reversed, as in having to calculate the isotopic abundances when given the atomic weight and isotopic weights. Study the tutorial below and then look at the tutorial linked just above.

Example #1: Calculate the average atomic weight for carbon.

 mass number isotopic weight percent abundance 12 12.000000 98.93 13 13.003355 1.07

Solution:

To calculate the average atomic weight, each isotopic atomic weight is multiplied by its percent abundance (expressed as a decimal). Then, add the results together and round off to an appropriate number of significant figures.
(12.000000) (0.9893) + (13.003355) (0.0107) = 12.0107 amu

This is commonly rounded to 12.011 or sometimes 12.01.

The answers to problems like this tend to not follow strict significant figure rules. Consult a periodic table to see what manner of answers are considered acceptable.

Example #2: Nitrogen

 mass number isotopic weight percent abundance 14 14.003074 99.636 15 15.000108 0.364

Solution:

 (14.003074) (0.9963) + (15.000108) (0.0037) = 14.007 amu (or 14.007 u) (isotopic weight) (abundance) + (isotopic weight) (abundance) = average atomic weight

A point about the term 'atomic weight:' When discussing the atomic weight of an element, this value is an average. When discussing the atomic weight of an isotope, this value is a value that has been measured experimentally, not an average.

Example #3: Silicon

 mass number isotopic weight percent abundance 28 27.976927 92.23 29 28.976495 4.67 30 29.973770 3.10

Solution:

(27.976927) (92.23) + (28.976495) (4.67) + (29.973770) (3.10) = 2808.55 u

There is a problem with the answer!!

The true value is 28.086 u. Our answer is too large by a factor of 100.

This is because I used percentages (92.23, 4.67, 3.10) and not the decimal equivalents (0.9223, 0.0467, 0.0310).

To obtain the correct answer, we must divide by 100.

 Example #4: How to Calculate an Average Atomic Weight Solution: Two points: (1) notice I wrote the same number of decimal places in the answer as were in the isotopic weights (the 184.953 and the 186.956). This is common. (2) I forgot to put a unit on the answer, so 186.207 u would be the most correct answer.

Example #5: In a sample of 400 lithium atoms, it is found that 30 atoms are lithium-6 (6.015 g/mol) and 370 atoms are lithium-7 (7.016 g/mol). Calculate the average atomic mass of lithium.

Solution:

1) Calculate the percent abundance for each isotope:

Li-6: 30/400 = 0.075
Li-7: 370/400 = 0.925

2) Calculate the average atomic weight:

x = (6.015) (0.075) + (7.016) (0.925)

x = 6.94 g/mol

I put g/mol for the unit because that what was used in the problem statement.

Example #6: A sample of element X contains 100 atoms with a mass of 12.00 and 10 atoms with a mass of 14.00. Calculate the average atomic mass (in amu) of element X.

Solution:

1) Calculate the percent abundance for each isotope:

X-12: 100/110 = 0.909
X-14: 10/110 = 0.091

2) Calculate the average atomic weight:

x = (12.00) (0.909) + (14.00) (0.091)

x = 12.18 amu (to four sig figs)

3) Here's another way:

100 atoms with mass 12 = total atom mass of 1200

10 atoms with mass 14 = total atom mass of 140

1200 + 140 = 1340 (total mass of all atoms)

Total number of atoms = 100 + 10 = 110

1340/110 = 12.18 amu

4) The first way is the standard technique for solving this type of problem. That's because we do not generally know the specific number of atoms in a given sample. More commonly, we know the percent abundances, which is different from the specific number of atoms in a sample.

Example #7: Boron has an atomic mass of 10.81 u according to the periodic table. However, no single atom of boron has a mass of 10.81 u. How can you explain this difference?

Solution:

10.81 amu is an average, specifically a weighted average. It turns out that there are two stable isotopes of boron: boron-10 and boron-11.

Neither isotope weighs 10.81 u, but you can arrive at 10.81 u like this:

x = (10.013) (0.199) + (11.009) (0.801)

x = 1.99 + 8.82 = 10.81 u

It's like the old joke: consider a centipede and a snake. What's the average number of legs? Answer: 50. Of course, neither one has 50.

Example #8: Copper occurs naturally as Cu-63 and Cu-65. Which isotope is more abundant?

Solution:

Look up the atomic weight of copper: 63.546 amu

Since our average value is closer to 63 than to 65, we concude that Cu-63 is the more abundant isotope.

Example #9: Copper has two naturally occuring isotopes. Cu-63 has an atomic mass of 62.9296 amu and an abundance of 69.15%. What is the atomic mass of the second isotope? What is its nuclear symbol?

Solution:

1) Look up the atomic weight of copper:

63.546 amu

2) Set up the following and solve:

(62.9296) (0.6915) + (x) (0.3085) = 63.546

43.5158 + 0.3085x = 63.546

0.3085x = 20.0302

x = 64.9277 amu

3) The nuclear symbol is:

${\text{}}_{29}^{65}\text{Cu}$

4) You might see this

29-Cu-65

This is used in situations, such as the Internet, where the subscript/superscript notation cannot be reproduced. You might also see this:

65/29Cu

Example #10: Naturally occurring iodine has an atomic mass of 126.9045. A 12.3849 g sample of iodine is accidentally contaminated with 1.0007 g of I-129, a synthetic radioisotope of iodine used in the treatment of certain diseases of the thyroid gland. The mass of I-129 is 128.9050 amu. Find the apparent "atomic mass" of the contaminated iodine.

Solution:

1) Calculate mass of contaminated sample:

12.3849 g + 1.0007g = 13.3856 g

2) Calculate percent abundances of (a) natural iodine and (b) I-129 in the contaminated sample:

(a) 12.3849 g / 13.3856 g = 0.92524
(b) 1.0007 g / 13.3856 g = 0.07476

3) Calculate "atomic mass" of contaminated sample:

(126.9045) (0.92524) + (128.9050) (0.07476) = x

x = 127.0540 amu

Example #11: Neon has two major isotopes, Neon-20 and Neon-22. Out of every 250 neon atoms, 225 will be Neon-20 (19.992 g/mol), and 25 will be Neon-22 (21.991 g/mol). What is the average atomic mass of neon?

Solution:

1) Determine the percent abundances (but leave as a decimal):

Ne-20 ---> 225 / 250 = 0.90
Ne-22 ---> 25 / 250 = 0.10

The last value can also be done by subtraction, in this case 1 - 0.9 = 0.1

2) Calculate the average atomic weight:

(19.992) (0.90) + (21.991) (0.10) = 20.19

Example #12: Calculate the average atomic weight for magnesium:

 mass number exact weight percent abundance 24 23.985042 78.99 25 24.985837 10.00 26 25.982593 11.01

The answer? Find magnesium on the periodic table:

Remember that the above is the method by which the average atomic weight for the element is computed. No one single atom of the element has the given atomic weight because the atomic weight of the element is an average, specifically called a "weighted" average.

See Example #7 and the example just below to see how this "no individual atom has the average weight" can be exploited.

Example #13: Silver has an atomic mass of 107.868 amu. Does any atom of any isotope of silver have a mass of 107.868 amu? Explain why or why not.

Solution:

The specific question is about silver, but it could be any element. The answer, of course, is no. The atomic weight of silver is a weighted average. Silver is not composed of atoms each of which weighs 107.868.

Example #14: Given that the average atomic mass of hydrogen in nature is 1.0079, what does that tell you about the percent composition of H-1 and H-2 in nature?

Solution:

It tells you that the proportion of H-1 is much much greater than the proportion of H-2 in nature.

Example #15: The relative atomic mass of neon is 20.18 It consists of three isotopes with the masses of 20, 21and 22. It consists of 90.5% of Ne-20. Determine the percent abundances of the other two isotopes.

Solution:

1) Let y% be the relative abundance of Ne-21.

2) Then, the relative abundance of Ne-22 is:

(100 − 90.5 − y)% = (9.5 − y)%

3) Relative atomic mass of Ne (note use of decimal abundances, not percent abundances):

(20) (0.905) + (21) (y) + (22) (0.095 − y) = 20.18

18.10 + 21y + 2.09 - 22y = 20.18

y = 0.010

Relative abundance of (note use of percents):

Ne-21 = 1.0%

Ne-22 = (9.5 − 1)% = 8.5%

Bonus Example #1: There are only two naturally-occurring isotopes of bromine in equal abundance. An atom of one isotope has 44 neutrons. How many neutrons are in an atom of the other isotope?

(a) 44     (b) 9     (c) 46     (d) 36     (e) 35

Solution:

Choice (a):

Isotopes have the same number of protons in each atom, but a different number of neutrons. The correct answer to this question is not the same number of neutrons that are in the other isotope.

Choice (b):

The number of neutrons in the various stable isotopes of a given element are almost always within a few neutrons of each other. Tin's ten stable isotopes span 12 neutrons; this is the largest span of stable isotopes the ChemTeam can think of without looking things up. Bismuth isotopes span 36 neutrons, but none of them are naturally-occurring (i.e., stable).

Also, the number of neutrons in a given atom is always fairly close to how many protons there are. There are no cases of the number of neutrons being 26 less (or more) than the atomic number.

Choice (c):

This is the correct answer. It is different from 44 and it's only 2 away from 44.

Choice (d) and (e):

The span in numbers of neutrons of naturally-occurring isotopes is not 10 or 11. It is much less, usually a span of one, two, or three neutrons. While it is true that tin isotopes span 12 neutrons, there are 8 isotopes in between the lightest and the heaviest isotopes. The span between adjacent isotopes in tin is one or two neutrons.

Bonus Example #2: Bromine has only two naturally-occurring isotopes of equal abundance. An atom of one isotope has 35 protons. How many protons are in an atom of the other isotope?

(a) 44     (b) 36     (c) 46     (d) 37     (e) 35

Solution:

This is a trick question. Both isotopes are atoms of the element bromine. All atoms of bromine, regardless of how many neutrons are present, contain the same number of protons.


text:
# Render Math Equations in iOS [duplicate]

I was wondering if there is an easy way to implement the rendering of math equations into an iOS app. It doesn't have to be interactive. All it should do is display the equation in traditional ways so when the user is dealing with complex equations it makes it easier to follow.

ex. 2^(2/(6^(1/2))) gives:

Wolfram Alpha does this along with other apps that I have seen. It doesn't have to deal with variables either.

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Have a look at this: stackoverflow.com/questions/2907045/… –  Tom Irving Apr 14 '13 at 17:59
@Tom, SirKaydian wants an easy way, and drawing formulae with Quartz2D is, as Brad answers, "not a trivial undertaking".. –  foundry Apr 14 '13 at 18:49
@RichardJ.RossIII, I wouldn't regard this as a duplicate, your linked question was asked in 2010 and self-answered in 2011, before the advent of MathML on iOS5 –  foundry Apr 14 '13 at 18:51

## marked as duplicate by Richard J. Ross III, 0x7fffffff♦, Monolo, Anoop Vaidya, Cole JohnsonApr 16 '13 at 0:03

A UIWebView can display mathML since iOS5. Your example:

<math title="2^(2/(sqrt(6))" xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle mathcolor="blue" fontfamily="sanserif" displaystyle="true">
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<msqrt>
<mrow>
<mn>6</mn>
</mrow>
</msqrt>
</mrow>
</mfrac>
</mrow>
</msup>
</mstyle>
[/itex]

This will render in a UIWebView on iOS.

There are javascript libraries that can convert for you - eg mathjax can accept ASCIIMath input such as this example 2^(2/(sqrt(6))

There is a good discussion of these issues here Tradeoff between LaTex, MathML, and XHTMLMathML in an iOS app?

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But there's a step missing here - you will now need a library to convert a 'simple' equation from 2^(2/(6^(1/2))) to it's MathML equivalent. –  Richard J. Ross III Apr 14 '13 at 18:19
@RichardJ.RossIII I only changed 6^(1/2) to sqrt(6)- the former will render "correctly" putting "0.5" into 6's superscript. That's a human authoring decision as much as anything (and maybe mathjax would render that with a sqrt symbol anyway, i didn't try)... –  foundry Apr 14 '13 at 18:22