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# Capital accumulation

Capital accumulation

Most generally, the accumulation of capital refers simply to the gathering or amassment of objects of value; the increase in wealth; or the creation of wealth. Capital can be generally defined as assets invested with the expectation that their value will increase, usually because there is the expectation of profit, rent, interest, royalties, capital gain or some other kind of return.

The definition of capital accumulation is subject to controversy and ambiguities, because it could refer to a "net addition" to existing wealth, or to a "redistribution" of wealth. If more wealth is produced than there was before, a society becomes richer; the total stock of wealth increases. But if some accumulate capital only at the expense of others, wealth is merely shifted from A to B. In principle, it is possible that a few people or organisations accumulate capital and grow richer, although the total stock of wealth of society decreases. Most often, capital accumulation involves both a net addition and a redistribution of wealth, which may raise the question of who really benefits from it most.

In economics, accounting and Marxian economics, capital accumulation is often equated with investment of profit income, especially in real capital goods. The concentration and centralisation of capital are two of the results of such accumulation (see below).

But capital accumulation can refer variously to
*working and consuming less than earned (saving or accumulating the residual)
*relying on the effects of compound interest to increase initial capital
*real investment in tangible means of production.
*financial investment in assets represented on paper.
*investment in "non-productive" physical assets such as residential real estate that appreciate in value.
*consuming less than produced by productive assets like farm land--saving or accumulating the residual
*"human capital accumulation," i.e., new education and training increasing the skills of the (potential) labour force.

Non-financial and financial capital accumulation is usually needed for economic growth, since additional production usually requires additional funds to enlarge the scale of production. Smarter and more productive organization of production can also increase production without increased capital. Capital can be created without increased investment by inventions or improved organization that increase productivity, discoveries of new assets (oil, gold, minerals, etc.), the sale of property, etc.

In modern macroeconomics and econometrics the term "capital formation" is often used in preference to "accumulation", though UNCTAD refers nowadays to "accumulation".

Harrod-Domar model

In macroeconomics, following the Harrod-Domar model, the savings ratio ($s$) and the capital coefficient ($k$) are regarded as critical factors for accumulation and growth, assuming that all saving is used to finance fixed investment. The rate of growth of the real stock of fixed capital ($K$) is:

:Delta K} over K} = {Delta K} over Y} over {K over Y = {s over k }

where $Y$ is the real national income. If the capital-output ratio or capital coefficient ($k=\left\{K over Y\right\}$) is constant, the rate of growth of $Y$ is equal to the rate of growth of $K$. This is determined by $s$ (the ratio of net fixed investment or saving to $Y$) and $k$.

A country might for example save and invest 12% of its national income, and then if the capital coefficient is 4:1 (i.e. \$4 billion must be invested to increase the national income by 1 billion) the rate of growth of the national income might be 3% annually. However, as Keynesian economics points out, savings do not automatically mean investment (as liquid funds may be hoarded for example). Investment may also not be investment in fixed capital (see above).

Assuming that the turnover of total production capital invested remains constant, the proportion of total investment which just maintains the stock of total capital, rather than enlarging it, will typically increase as the total stock increases. The growth rate of incomes and net new investments must then also increase, in order to accelerate the growth of the capital stock. Simply put, the bigger capital grows, the more capital it takes to keep it growing and the more markets must expand.

The measurement of accumulation

Accumulation can be measured as the monetary value of investments, the amount of income that is reinvested, or as the change in the value of assets owned (the increase in the value of the capital stock). Using company balance sheets, tax data and direct surveys as a basis, government statisticians estimate total investments and assets for the purpose of national accounts, Input-output tables national balance of payments and Flow of funds statistics. Usually the Reserve Banks and the Treasury provide interpretations and analysis of this data. Standard indicators include Capital formation, Gross fixed capital formation, fixed capital, household asset wealth, and foreign direct investment.

Organisations such as the International Monetary Fund, UNCTAD, the World Bank Group, the OECD, and the Bank for International Settlements used national investment data to estimate world trends. The Bureau of Economic Analysis, Eurostat and the Japan Statistical Office provide data on the USA, Europe and Japan respectively.

Other useful sources of investment information are business magazines such as Fortune, Forbes, The Economist, Business Week etc. as well as various corporate "watchdog" organisations and NGO publications. A reputable scientific journal is the "Review of Income & Wealth". In the case of the USA, the "Analytical Perspectives" document (an annex to the yearly budget) provides useful wealth and capital estimates applying to the whole country.

Psychology, sociology and ethics of capital accumulation

There have been numerous psychological and sociological studies of the motivations of investment behaviour by individuals. Most of these suggest that the propensity to accumulate capital is associated with qualities such as an intelligent understanding of property ownership, a positive attitude towards money, the ability to seize a money-making opportunity, and a desire to acquire more wealth. These are not innate or genetic qualities, but learned through social experience.

However, even if a strong motivation for enrichment or social improvement exists, the business, government, legal, climate, local culture or social instability may prevent this motivation from being realised. Hernando de Soto for example argues that the reason why poor countries are poor is mainly because of the absence of a legal-cultural infrastructure of "asset management" and of formalised and enforced private property rights. Many systems seemed designed to keep a small minority in power so they can consume more. This power minority exist as a parasite on the common people--consuming much more than they produce. One popular argument in this respect remains the vicious cycle of poverty: the poor are poor because they are poor. Critics of this argument object it is an uninformative and unhelpful tautology.

Greed and desire can play a very important role in capital accumulation, but are not a necessary requirement. Indeed according to Max Weber's study of capitalism and the Protestant ethic, frugality, sobriety, deferred consumption and saving were among the key values of the rising bourgeoisie in the age of the Reformation.

Some economic historians (e.g. David Landes, Gregory Clark (economist)) refer to national psychology and argue that some nations or cultures (e.g. Europe) are inherently better equipped for capital accumulation, due to cultural habits, customs and values.

Other economic historians (e.g. Paul A. Baran) have argued that psychological factors explain very little, because a nation which previously had a low level of accumulation can suddenly "take off". In that case, the causes must be sought in the prevailing social relations.

Controversies about the ethics of accumulation have occurred ever since commercial trade began. If informal and formal prostitution is regarded as the oldest profession, the first ethical debate about accumulation must have occurred tens of thousands of years ago at the very least. The problem is that trade or market forces do not create any particular morality of their own, beyond the requirement to meet contractual obligations that settle transactions. Some forms of trade may be accepted, others rejected, but there exists no general moral principle for this which can be derived from the trade itself.

A good contemporary illustration of this problem is the gigantic increase in total reported crime and the grey economy or shadow economy after the deregulation of world markets from the 1980s, and the marketisation of the USSR and China. But ancient philosophers and theologians already knew about the problem, which is why they were intensely preoccupied with the politics of the “rule of law” and its enforcement.

The main ethical questions concern which "routes to wealth" are morally justifiable, and what entitles individuals and groups to appropriate amounts of wealth, in particular wealth which they have not themselves created. The medieval economists invented theories of a just price and the moral debate surfaces again these days e.g. in the controversies about fair trade, imperialism and Islamic banking. Neo-liberal theory emphasises that a "good" person is one who creates new wealth by deferring consumption or improving production, while socialist theory says a "good" person should be forced to share their wealth however accumulated. The most popular moral theories are similar to that of John Rawls.

Karl Marx illustrated his analysis with sarcastic comments about “Christian accumulation”; some forms of accumulation were believed to be compatible with Jesus Christ, while other were not; some forms of accumulation were forgiven by God afterwards, others were not. Martin Luther for example raged against usury and extortion.

Marxism-Leninism is hostile to all private property and market activity. It must be kept in mind that the "private property" that Marx refers to is the ownership of the means of production by a generally small elite of wealthy entrepreneurs. The proletariat, or laborer, is inferior to all aspects of production--including labor, the products or services made, and revenue; and therefore the division of labor and its products must be equally redistributed to avoid the control and degradation of an unknown bourgeoisie.

But because capital accumulation does not presuppose any particular or specific "moral system", accumulation can also continue regardless of any particular morality advocated by popes, presidents, queens, journalists, pop stars, business tycoons or anybody else. All that is required is (1) the ability to own assets and trade in them and (2) sufficient income beyond subsistence and (3) the will to defer consumption to be able to accumulate capital.

Marxian concept of capital accumulation

In Karl Marx's critique of political economy, capital accumulation refers to the operation whereby a sum of money is transformed into a larger sum of money (capitalism is this money-making activity, although Marx often equates capitalism with the capitalist mode of production). Here, capital is defined essentially as economic or commercial asset value in search of additional value or surplus-value. This requires property relations which enable objects of value to be appropriated and owned.

According to Marx, capital accumulation has a double origin, namely in trade and in expropriation, both of a legal or illegal kind. The reason is that a stock of capital can be increased through a process of exchange or "trading up" but also through directly taking an asset or resource from someone else, without compensation. David Harvey calls this accumulation by dispossession. Marx does not discuss gifts and grants as a source of capital accumulation, nor does he analyze taxation in detail. Nowadays the tax take is so large (i.e. 25-40% of GDP) that some authors refer to state capitalism.

The continuation and progress of capital accumulation depends on the removal of obstacles to the expansion of trade, and this has historically often been a violent process. As markets expand, more and more new opportunities develop for accumulating capital, because more and more types of goods and services can be traded in. But capital accumulation may also confront resistance, when people refuse to sell, or refuse to buy (for example a strike by investors or workers, or consumer resistance). What spurs accumulation is competition; in business, if you don't go forward, you go backward, and unless the law prevents it, the strong will exploit the weak.

In general, Marx's critique of capital accumulation is that the human chase after wealth and self-enrichment leads to inhuman consequences. The enrichment of some is at the expense of the immiseration of others, and competition becomes brutal. The basis of it all is the exploitation of the labour effort of others. When the "economic cake" expands, this may be obscured because all can gain from trade. But when the "economic cake" shrinks, then capital accumulation can only occur by taking income or assets from other people, other social classes, or other nations. The point is that to exist, capital must always grow, and to ensure that it will grow, people are prepared to do almost anything.

Concentration and centralization

According to Marx, capital has the tendency for concentration and centralization the hands of richest capitalists. Marx explains:

"It is concentration of capitals already formed, destruction of their individual independence, expropriation of capitalist by capitalist, transformation of many small into few large capitals ... Capital grows in one place to a huge mass in a single hand, because it has in another place been lost by many ... The battle of competition is fought by cheapening of commodities. The cheapness of commodities demands, caeteris paribus, on the productiveness of labour, and this again on the scale of production. Therefore, the larger capitals beat the smaller. It will further be remembered that, with the development of the capitalist mode of production, there is an increase in the minimum amount of individual capital necessary to carry on a business under its normal conditions. The smaller capitals, therefore, crowd into spheres of production which Modern Industry has only sporadically or incompletely got hold of. Here competition rages ... It always ends in the ruin of many small capitalists, whose capitals partly pass into the hands of their conquerors, partly vanish." ("Das Kapital", vol.1, ch. 25)

The rate of accumulation

In Marxian economics, the "rate of accumulation" is defined as (1) the value of the real net increase in the stock of capital in an accounting period, (2) the proportion of realised surplus-value or profit-income which is reinvested, rather than consumed. This rate can be expressed by means of various ratios between the original capital outlay, the realised turnover, surplus-value or profit and reinvestments (see e.g. the writings of the economist Michal Kalecki).

Other things being equal, the greater the amount of profit-income that is disbursed as personal earnings and used for consumptive purposes, the lower the savings rate and the lower the rate of accumulation is likely to be. However, earnings spent on consumption can also stimulate market demand and higher investment. This is the cause of endless controversies in economic theory about "how much to spend, and how much to save".

In a boom period of capitalism, the growth of investments is cumulative, i.e. one investment leads to another, leading to a constantly expanding market, an expanding labor force, and an increase in the standard of living for the majority of the people.

In a stagnating, decadent capitalism, the accumulation process is increasingly oriented towards investment on military and security forces, real estate, financial speculation, and luxury consumption. In that case, income from value-adding production will decline in favour of interest, rent and tax income, with as a corollary an increase in the level of permanent unemployment.

As a rule, the larger the total sum of capital invested, the higher the return on investment will be. The more capital one owns, the more capital one can also borrow and reinvest at a higher rate of profit or interest. The inverse is also true, and this is one factor in the widening gap between the rich and the poor.

Ernest Mandel emphasized that the rhythm of capital accumulation and growth depended critically on (1) the division of a society's social product between "necessary product" and "surplus product", and (2) the division of the surplus product between investment and consumption. In turn, this allocation pattern reflected the outcome of competition among capitalists, competition between capitalists and workers, and competition between workers. The pattern of capital accumulation can therefore never be simply explained by commercial factors, it also involved social factors and power relationships.

The origin of capital accumulation in trade

In the simplest circuit of commercial trade, a sum of money $M$ is loaned and returned with interest as the larger sum $M\text{'}$. Or, as a variation, $M$ is traded for another currency, which rises in value. In counter-trade (a form of barter in which money may be used only to value goods and services), a commodity $C$ exchanges for another commodity $C\text{'}$, which may also result in a larger sum of value. Marx calls the additional value surplus-value.

In a slightly more complex trading circuit, a sum of money $M$ buys a commodity $C$ which upon sale yields a larger sum of money $M\text{'}$, which can be reinvested. Alternatively, the circuit $C-M-C\text{'}$ could substitute for $M-C-M\text{'}$ but in this case the enlarged value consists of commodities rather than of money. These circuits are basic to merchant trade.

In the more developed trading circuit of capitalism, however, $M$ buys inputs $C$ (means of production and labour-power) which through new production creates outputs $C\text{'}$ and upon sale yield a larger sum of money $M\text{'}$. In this case, we are no longer dealing with merchant capitalism, but with capitalist industry (the capitalist mode of production: all or most of the inputs and outputs of production are available as marketed commodities, and the costs & benefits of total production are rationally calculated in price terms.

In modern capitalism, the circuits of finance, commerce and production have become exceedingly complex, often lack transparency and may involve multilateral exchanges or a lot of fictitious capital. The daily trading volume in the world's foreign exchange markets was estimated at \$1.88 trillion in 2004, as against \$590 billion in 1989 (current dollars) (Der Spiegel, special edition 4/2005, p. 107). By comparison, the New York Stock Exchange daily volume is said to be around \$25 billion a day, and the international futures markets are said to trade about \$35 billion worth of contracts a day. Speculative trading makes up the bulk of the daily trading volumes. Most rich people do not want to bother with the financial management of most of their wealth, and know little about it. Investment specialists make their money from investing the money of the rich using their superior market knowledge, contacts, networks and commercial skills.

The circuit of capital accumulation from production

Strictly speaking, capital has accumulated only when realised profit income has been "reinvested" in capital assets. But the process of capital accumulation in production has, as suggested in the first volume of Marx's Das Kapital, at least 7 distinct but linked moments:

*The initial investment of capital (which could be borrowed capital) in means of production and labor power.
*The command over surplus-labour and its appropriation.
*The valorisation (increase in value) of capital through production of new outputs.
*The appropriation of the new output produced by employees, containing the added value.
*The realisation of surplus-value through output sales.
*The appropriation of realised surplus-value as (profit) income after deduction of costs.
*The reinvestment of profit income in production.

All of these moments do not refer simply to an "economic" or commercial process. Rather, they assume the existence of legal, social, cultural and economic power conditions, without which creation, distribution and circulation of the new wealth could not occur. This becomes especially clear when the attempt is made to create a market where none exists, or where people refuse to trade.

In fact Marx suggests that the original or primitive accumulation of capital often occurs through violence, plunder, slavery, robbery, extortion and theft. He argues that thecapitalist mode of production requires that people must be forced to work in value-adding production for someone else, and for this purpose, they must be cut off from sources of income other than selling their labor power.

imple and expanded reproduction

In volume 2 of Das Kapital, Marx continues the story and shows that, with the aid of bank credit, capital in search of growth can more or less smoothly mutate from one form to another, alternately taking the form of money capital (liquid deposits, securities, etc.), commodity capital (tradeable products, real estate etc.), or production capital (means of production and labor power).

His discussion of the simple and expanded reproduction of the conditions of production offers a more sophisticated model of the parameters of the accumulation process as a whole. At simple reproduction, a sufficient amount is produced to sustain society at the given living standard; the stock of capital stays constant. At expanded reproduction, "more" product-value is produced than is necessary to sustain society at a given living standard (a surplus product; the additional product-value is available for investments which enlarge the scale and variety of production.

Yet there is no economic law according to which capital is necessarily re-invested in the expansion of production; that depends on anticipated profitability, market expectations and perceptions of investment risk. All that Marx proves is that in capitalism production of output is "conditional" on capital accumulation, i.e. at least in the longer term, if production is not profitable, it will close down.

Ernest Mandel introduced the additional concept of "contracted economic reproduction", i.e. reduced accumulation where business operating at a loss outnumbers growing business, or economic reproduction on a decreasing scale, for example due to wars, natural disasters or devalorisation.

Balanced economic growth requires that different factors in the accumulation process expand in appropriate proportions. But markets themselves cannot spontaneously create that balance, in fact what drives business activity is precisely the imbalances between supply and demand: inequality is the motor of growth. This partly explains why the worldwide pattern of economic growth is very uneven and unequal, even although markets have existed almost everwhere for a very long time. It also explains government regulation of market trade and protectionism.

Capital accumulation as social relation

"Accumulation of capital" sometimes also refers in Marxist writings to the reproduction of capitalist social relations (institutions) on a larger scale over time, i.e., the expansion of the size of the proletariat and of the wealth owned by the bourgeoisie.

This interpretation emphasizes that capital ownership, predicated on command over labor, is a social relation: the growth of capital implies the growth of the working class (a "law of accumulation"). In the first volume of Das Kapital Marx had illustrated this idea with reference to Edward Gibbon Wakefield's theory of colonisation:

In the third volume of Das Kapital, Marx refers to the "fetishism of capital" reaching its highest point with "interest-bearing capital", because now capital seems to grow of its own accord without anybody doing anything. In this case,

Different forms of capital accumulation

Essentially, in capitalism the production of output "depends" on the accumulation of capital. The propensity to invest in production therefore depends a lot on expectations of profitability and sales volume, and on perceptions of market risk. If production stops being profitable, or if sales drop sharply, or if there is social instability, capital will exit more and more from the sphere of production. Or if it cannot or does not, rationalisation investments will be undertaken, to amalgamate unprofitable enterprises into profitable units.

As a corollary, capital accumulation may be the accumulation of "production capital" (industrial assets), or the accumulation of "money capital" (financial assets), or the accumulation of "commodity capital" (products, real estate etc. which can be traded).

But irrespective of whether the additional capital value (or surplus-value happens to take the form of profit, interest, rent, or some kind of tax impost or royalty income, what drives the accumulation process is the perpetual search for more surplus-value, for added value as such.

This requires a constant supply of a labor force which can conserve and add value to inputs and capital assets, and thus create a higher value. Normally, the socio-economic compulsion to work for a living in capitalist society is legally enforced and regulated by the state, for example through workfare and strict conditions for receiving an unemployment benefit.

Although capital accumulation does not necessarily require production, ultimately the basis for it is value-adding production which makes net additions to the stock of wealth. Capital can accumulate by shifting the ownership of assets from one place to another, but ultimately the total stock of assets must increase. Other things being equal, if production fails to grow sufficiently, the level of debt will increase, ultimately causing a breakdown of the accumulation process when debtors cannot pay creditors.

Capital "accumulation" does not necessarily require trade either, although capital presupposes trade, and the ability to exchange goods for money. The reason is that wealth can be amassed through illegal or legalised expropriation (robbery, plunder, theft, piracy, slavery, embezzlement, fraud and so on). However, a continuous and cumulative accumulation process always presupposes that capital ownership is secure. Consequently, military and police forces have typically been necessary for capital accumulation on a larger scale, to protect property.

In medieval society, typically the bourgeoisie could not protect its capital assets permanently from attacks, which meant that the accumulation process was interrupted, and remained limited in scope. Today however, capitalists can own billions of dollars worth of assets which are well-protected against crime (see the annual Merrill-Lynch survey of the world's wealthy). With the aid of private banking it is easier to obscure or hide the wealth that one owns.

Regime of accumulation

Both the Regulation School of French Marxist economists, inspired by the original writings of Michel Aglietta and developed by Robert S. Boyer, as well as the American social structure of accumulation school founded by the economists Samuel Bowles and David Gordon have emphasized that the processes of capital accumulation occur within a social regime of accumulation.

In other words, a specific political and socio-economic environment is required that enables sustained investment and economic growth. This environment is created partly by state policy, but partly by also by technological innovations, changes in popular culture, commercial developments, the media, and so on. An example of such a regime often cited here is that of Fordism, named after the enterprise of Henry Ford. As the pattern of accumulation changes, the regime of accumulation also changes.

Similar ideas also surface in institutional economics. The main insight here is that market trade cannot flourish without regulation by a legal system plus the enforcement of basic moral conduct and private property by the state. But the regime of accumulation responds to the total experience of living in capitalist society, not just market trade.

Environmental criticism of capital accumulation

The environmental criticism of capital accumulation focuses on four main ideas.

Firstly, there is the problem of externalities. This means that public or privately owned industry incurs costs, including environmental and health costs, which are not charged or priced. This happens for example when effluents are discharged on land, water or in the air, which can cause pollution or despoilation of terrains. In recognition of this, environmental taxes are sometimes imposed.

Secondly, commercial activities which may be rational from the point of view of a narrow public or private enterprise may not be rational from the point of view of the larger society, or from the point of view of the biosphere, especially when they involve the destruction of natural habitats of flora and fauna, pollution and entropy.

Because a natural resource happens to be a freely available good (for example fish in the open sea), it may be over utilized be either public or private enterprises. Or, a lot of energy may be wasted producing and transporting a good to the consumer. Or, the disturbance of subsistence economics by commerce may cause overpopulation by not controlling population by starvation.

Thirdly, goods and services may be produced for public or private profit in ways which are directly or indirectly harmful to human life, either because of the nature of the use-value involved, or because of the techniques used to produce them, or because they encourage consumer habits with harmful effects.

Finally, business and cultural ethics may often not be reconcilable with some human ethics or good environmental ethics. This means for example that the imputation of a price to an environmental cost, or imposing an environment tax may be insufficient as a policy, because some things which have value simply have no price.

Nowadays environmental concerns are an essential part of so-called socially responsible business and corporate governance. However, opinion is divided about whether a capitalist market economy can be ecologically sustainable. Some argue that the experience of wide spread environmental destruction in the Soviet Union and China proves that state socialism or command economy can be ecologically worse than capitalism. Today [2005] some environmentalists consider capitalism, or the "free market system" as it is usually called, incapable of complying with the basic requisites of a sustainable and respectful habitation of planet earth. A major problem, inherent in some free market production dynamics, is the constant desire to constantly expand production. In this particular regard, critics point to the penchant to plan in short-term cycles, and with a narrow concern about the fortunes of only a single country, firm or business entity, thereby ignoring the cumulative effect brought to bear on the biosphere by the entire production system.

In the 1970s, some environmentalists argued for a policy of "zero economic growth" in "affluent" Western societies. However, when a long recession began in that decade, halving economic growth rates, most people became more concerned about mass unemployment. Thus, the proponents of zero growth lost popularity. Nowadays, the popular concept is sustainable economic development or growth. But interpretations of what that means can differ wildly. One difficulty is that predictions of future resource scarcity are usually based on extrapolation from the past, "assuming present trends will continue", but they may not.

Capital accumulation and risk

Most capital accumulation involves risk, because capital is committed to an investment without perfect certainty about future earnings. A capital asset could gain value, but it could also lose value in the future. Owners of capital (investors) therefore typically diversify their investment portfolio, and try to minimise the risks involved in investments by every possible means.

In the course of two centuries of capital accumulation based on industrialisation the intensive economising and exploitation of human labour, and technological innovation,

*the value of the assets that are invested in has become very large
*the markets traded in extend around the globe
*the deregulation of markets has increased the level of market uncertainty
*the volume of speculative capital has grown enormously
*the banking industry dominates the ownership of capital assets.

This has led to an enormous expansion of the insurance industry and of the profession of risk management. As a corollary, this powerfully stimulates the construction of mathematical models which aim to assess how probable it is that particular "risky events" will occur. Some sociologists such as Frank Furedi claim that an exaggerated and unhealthy preoccupation or anxiety about risks has infiltrated the whole of modern society.

Speculation - making money from price differentials or price fluctuations - is "justified" as follows: "The roles of speculators in a market economy are to absorb risk and to add liquidity to the marketplace by risking their own capital for the chance of monetary reward." However, speculation often also occurs with borrowed capital. In this case, capital is borrowed at a low rate of interest, and reinvested at a higher return.

Capital accumulation and military wars

Wars typically causes the diversion, destruction and creation of capital assets as capital assets are both destroyed or consumed and diverted to types of production needed to fight the war. Many assets are wasted and in some few cases created specifically to fight a war. War driven demands may be a powerful stimulus for the accumulation of capital and production capability in limited areas and market expansion outside the immediate theatre of war. Often this has induced laws against perceived and real war profiteering.

War destruction can be illustrated by looking at World War 2. Industrial war damage was heaviest in Japan, where 1/4 of factory buildings and 1/3 of plant & equipment were destroyed; 1/7 of electric power-generating capacity was destroyed and 6/7 of oil refining capacity. The Japanese merchant fleet lost 80% of their ships. In Germany in 1944, when air attacks were heaviest, 6.5% of machine tools were damaged or destroyed, but around 90% were later repaired. About 10% of steel production capacity was lost. In Europe, the United States and the Soviet Union enormous resources were accumulated and ultimately dissipated as planes, ships tanks, etc. were built and then lost or destroyed.

Germany's total war damage was estimated at about 17.5% of the pre-war total capital stock by value, i.e. about 1/6. In the Berlin area alone, there were 8 million refugees lacking basic necessities. In 1945, less than 10% of the railways were still operating. 2395 rail bridges were destroyed and a total of 7500 bridges, 10,000 locomotives and more than 100,000 goods wagons were destroyed. Less than 40% of the remaining locomotives were operational.

However, by the first quarter of 1946 European rail traffic, which was given assistance and preferences (by western appointed military governors) for resources and material as an essential asset, regained its prewar operational level. At the end of the year, 90% of Germany's railway lines were operating again. In retrospect, the rapidity of infrastructure reconstruction appears astonishing.

Initially, in May 1945, newly installed President Harry S. Truman's directive had been that no steps would be taken towards economic rehabilitation of Germany. In fact, the initial industry plan of 1946 prohibited production in excess of half of the 1938 level; the iron and steel industry was allowed to produce only less than a third of pre-war output. These plans were rapidly revised and better plans were instituted. In 1946, over 10% of Germany's physical capital stock (plant & equipment) was also dismantled and confiscated, most of it going to the USSR. By 1947, industrial production in Germany was at 1/3 of the 1938 level, and industrial investment at about 1/2 the 1938 level.

The first big strike wave in the Ruhr occurred in early 1947 - it was about food rations and housing, but soon there were demands for nationalisation. The US appointed military Governor (Newman) however stated at the time that he had to power to break strikes by withholding food rations. The clear message was: "no work, no eat". As the military controls in Western Germany were nearly all relinquished and the Germans were allowed to rebuild their own economy with Marshal Plan aid things rapidly improved. By 1951, German industrial production had overtaken the prewar level. The Marshall Aid dollars were important, but, after the currency reform (which permitted German capitalists to revalue their assets) and the establishment of a new political system, much more important was the commitment of the USA to "rebuilding" German capitalism and establishing a free market economy and government, rather than keeping Germany in a weak position. Initially, average real wages remained low, lower even than in 1938, until the early 1950s, while profitability was unusually high. So the total investment fund, aided by credits, was also high, resulting in a high rate of capital accumulation which was nearly all reinvested in new construction or new tools. This was called the German Economic Miracle or "Wirtschaftswunder"("Source": Armstrong, Glyn & Harrison 1984).

In the United States in World War II the large investments in industrial plant necessitated by the war brought some advantages; but the costs of dead, waste and debt would have never been under taken by any rational government for the slight advantages.

In modern times, it has often been possible to rebuild physical capital assets destroyed in wars completely within the space of about 10 years, except in cases of severe pollution by chemical warfare or other kinds of irreparable devastation. However, damage to human capital has been much more devastating, in terms of fatalities (in the case of world war 2, about 55 million deaths), permanent physical disability, enduring ethnic hostility and psychological injuries which have effects for at least several generations.

New developments in capital accumulation

New trends in capital accumulation include:

*financialisation (the extraordinarily strong growth of the international financial markets. This is trade in financial claims to current and future income. As a corollary, the proportion of national income which consists of interest income and rentier income increases. The International Swaps and Derivatives Association reported in September 2006 that the outstanding nominal value of swaps and derivatives at the end of June 2006 was \$283 trillion - nearly ten times the combined GDP of the US, Canada, the EU, Japan, and China; or ten times the value of total US home equity (each being valued at about \$34 trillion). According to Standard & Poor's, world stock market capitalization is about \$41 trillion. Of total swaps and derivatives, some \$26 trillion was in the fastest growing area, credit default swaps.

*Modern information technology makes it possible to engage in very complex investment projects and shift funds extremely quickly from one placement to another in space and time. This increases the rotation speed of capital and raises the profit rate, but can also increase potential financial risks.

*the growing controversies about intellectual property rights and the protection (or security) of ideas which can make money for the owner. Increasingly, the basic conditions necessary for a good, service or idea to become a tradeable commodity are theoretically defined.

*ongoing privatisation of assets which were previously under public ownership. The IMF estimates suggest that in two decades since 1985 more than \$2 trillion US dollars (in 2005 values) worth of state assets were privatised worldwide. Typically, these assets also rise sharply in value within a few years, because they involve enterprises occupying monopoly positions (e.g. utilities) which thus provide guaranteed profits. If profits dry up in the private sector, capitalists plunder public assets paid for by all citizens, with the argument that if they run them, supply will be more efficient.

*The enormous increase in capital gains from rising property values in the richer countries, especially in the housing market. US tax data for fiscal 2000 showed that "realised" capital gains in the USA peaked at an estimated "\$644.3 billion worth of income" while US GDP in 2000 was at US\$9,817.0 billion, in other words "realised" capital gains assessed for tax purposes were equal to 6.5% of GDP at that point (total capital gains would be larger). Yet GDP, being a measure of value added in production, does not even include this "hidden" personal and business income.

*A growing proportion of capital assets which is not productively invested (overcapitalisation), together with an increase in the amount of consumer debt and liabilities. Some observers see the cause as being an increase in the gap between rich and poor, which causes only sluggish demand growth. "Debt management" has become a distinct and profitable business.

*The crisis of numerous pension funds providing a large amount of investment capital, which are alleged to be badly managed.

*An international "competition of currency values" strongly influenced by speculative capital, which has a big effect on the pattern of international trade. The magnitudes involved can be gauged e.g. from the currency conversion ratios used to establish purchasing power parity. For example, India's GDP valued at "ppp" becomes five times larger. This tends to stimulate counter-trade.

*The acceleration of the concentration and centralisation of capital internationally in very large corporations. The Fortune Magazine "Global 500" largest corporations in 2004 employed more people than the whole workforce of Germany. The after-tax profit volume of the Fortune Global 500 was said to be \$731 billion, the combined asset value was \$60.8 trillion, gross income (revenues) \$14.8 trillion, and stockholders equity \$6.8 trillion. For comparison, world GDP in 2004 was valued at \$40.9 trillion (World Bank).

*The Merrill lynch/CapGemini World Wealth Report 2005 covering High Net Worth Individuals (HNWI) claims the fortunes of the world's millionaires and billionaires grew strongly in 2004, increasing by 8.2% to US\$30.8 trillion in one year. Driven by North America & Asia–Pacific, this represents "the highest growth of HNWI wealth in more than three years".

*Dollarisation - more US currency now circulates outside the US than inside it, and some countries such as Ecuador and El Salvador have adopted the US dollar as national currency. "Dollar hegemony" is maintained by large Asian, Arab and European investments in the United States.

*the tendency for corporate investment to orient towards activities which secure good short-term returns for shareholders. This is called "value-based management". Most corporate executive officers (CEO's) cite profitability as their prime concern.

*an increasing preoccupation with the conditions for extending credit, and with all sorts of risk factors. World markets are increasingly sensitive to events and disturbances which might cause social instability or panics.

*the declining overall significance of business start-ups, in the sense of enterprises creating new products and services, rather than being just tax-shelters or secondary employment (whether this is a permanent trend remains to be seen).

*the growth of criminal (or illegal) accumulation as measured by crime reports, including business crime and corruption such as fraud, embezzlement, money laundering, insider trading, smurfing and theft, but also prostitution, forced labour, slavery, war plunder etc. The volume of illegal international transactions is now said to be around \$1 trillion a year, equal to the GDP of Spain or Canada. National Geographic has reported there are about 27 million slaves in the world. ILO estimates of forced labor are a little over a dozen million. There are possibly 70 million people involved around the world in prostitution of one form or another. But there are many more, employed or unemployed, in "intermediate" positions. Traditional sociological categories may not describe their situation accurately, but a growing "underclass" (which may not be an accurate label) is a policy concern for many governments.

*the most ignored aspect is the changing structure of the international workforce in its totality, specifically the number employed by specific employment status and by income, in different sectors. But just as Marx's Law of Accumulation predicted, the working class has grown enormously within 2 centuries. Deon Filmer estimated that 2,474 million people participated in the worldwide non-domestic labour force in the mid-1990s. Of these around a fifth, 379 million people, worked in industry, 800 million in services, and 1,074 million in agriculture. The majority of workers in industry and services were wage & salary earners - 58 percent of the industrial workforce and 65 percent of the services workforce. But a big portion were self-employed or involved in family labour. Filmer suggests the total of employees worldwide in the 1990s was about 880 million, compared with around a billion working on own account on the land (mainly peasants), and some 480 million working on own account in industry and services.

A few references to works of theory

*Michel Aglietta, A Theory of Capitalist Regulation.
*Elmar Altvater, Gesellschaftliche Produktion und ökonomische Rationalität; Externe Effekte und zentrale Planung im Wirtschaftssystem des Sozialismus.
*Samir Amin, Accumulation on a world scale.
*Philip Armstrong, Andrew Glyn and John Harrison, Capitalism since World War II.
*Paul A. Baran, The Political Economy of Growth.
*Gregory Clark (economist) A Farewell to Alms: A Brief Economic History of the World.
*Milton Friedman & Rose Friedman, Free to choose.
*P. Groenewegen (ed.), Economics and ethics.London: Routledge 1996.
*Henryk Grossman, The Law of Accumulation and Collapse of the Capitalist System.
*Andre Gunder Frank, World accumulation, 1492 - 1789. New York 1978
*Rudolf Hilferding, Finance Capital.
*Rosa Luxemburg, The Accumulation of Capital.
*Ernest Mandel, Marxist Economic Theory.
*Karl Marx, Das Kapital Vol. 1, Part 7 and Vol. 2, Part 3.
*Seymour Melman, Profits without production.
*Michael Perelman, Steal this Idea: the Corporate Confiscation of Creativity.
*Joan Robinson, Essays in the Theory of Economic Growth.
*Harry Rothman, Murderous providence; A study of pollution in industrial societies.
*Vaclav Smil, China's Environmental Crisis: An Inquiry into the Limits of National Development. Armonk: M.E. Sharpe, 1992.
*Hernando de Soto, The Mystery of Capital: Why Capitalism Triumphs in the West and Fails Everywhere Else.

*Manual Velázquez, Business Ethics: Concepts and Cases.
*William J. Bernstein, The Birth of Plenty: How the Modern World of Prosperity was Launched.
*Deon Filmer, Estimating the World at Work, a background report for World Bank's World Development Report 1995 (Washington DC, 1995).
*Willem van Schendel and Itty Abraham (eds), Illicit Flows and Criminal Things. States, Borders, and the Other Side of Globalization. Bloomington, Indiana University Press, 2005; ISBN 0-253-34669-X
*Joshua S. Goldstein, War and economic History [http://www.joshuagoldstein.com/jgeconhi.htm]

ee also

*Accumulation by dispossession
*Capital formation
*Capital
*Capitalism
*Capitalist mode of production
*Commodity fetishism
*Culture of capitalism
*Das Kapital
*DEMOLOGOS
*Fixed capital
*Gross fixed capital formation
*History of theory of capitalism
*Investment
*Investment-specific technological progress
*Law of accumulation
*Prices of production
*Primitive accumulation of capital
*Productive and unproductive labour
*Proletarianization
*Relations of production
*Return on capital
*Simple commodity production
*Surplus value
*Unequal exchange
*Value investing

* [http://www.diva-portal.org/diva/getDocument?urn_nbn_se_su_diva-378-1__fulltext.pdf Growth, Accumulation, Crisis: With New Macroeconomic Data for Sweden 1800-2000 by Rodney Edvinsson]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• capital accumulation — ˌcapital accumuˈlation noun [uncountable] ECONOMICS the process of saving money rather than spending it: • There should be a move from taxes on capital accumulation to taxes on consumption. * * * capital accumulation UK US noun [U] (also capital… …   Financial and business terms

• Capital Accumulation — This refers to profits that a company uses to increase its capital base. Capital accumulation involves acquiring more assets that can be used to create more wealth or that will appreciate in value. Alternatively, capital accumulation can also… …   Investment dictionary

• capital accumulation — Within Marxism , this is the central dynamic of capitalist development and refers to the process whereby capital is expanded through the production, appropriation, and realization of surplus value . It cannot be understood apart from capitalist… …   Dictionary of sociology

• Capital intensity — is the term in economics for the amount of fixed or real capital present in relation to other factors of production, especially labor. At the level of either a production process or the aggregate economy, it may be estimated by the capital/labor… …   Wikipedia

• Accumulation by dispossession — is a concept presented by the Marxist academic David Harvey, which defines the neoliberal changes in many western nations, from the 1970s and to the present day, as being guided mainly by four practices. These are privatization, financialization …   Wikipedia

• capital and interest — ▪ economics Introduction       in economics, a stock of resources that may be employed in the production of goods and services and the price paid for the use of credit or money, respectively.       Capital in economics is a word of many meanings …   Universalium

• Capital formation — Gross capital formation in % of gross domestic product in world economy Capital formation is a concept used in macroeconomics, national accounts and financial economics. Occasionally it is also used in corporate accounts. It can be defined… …   Wikipedia

• capital — According to modern economics, one of four factors of production the others being land, labour, and enterprise. That capital is not, first and foremost, a sum of money, was the great discovery of early political economy , which stressed that… …   Dictionary of sociology

• Capital (economics) — In economics, capital or capital goods or real capital refers to items of extensive value. The term can also be applied to the amount of wealth a person controls or is capable of controlling.Capital goods may be acquired with money or financial… …   Wikipedia

• capital formation — expansion of capital or capital goods through savings, which leads to economic growth. Bloomberg Financial Dictionary * * * capital formation ˌcapital forˈmation noun [uncountable] ECONOMICS the process of collecting capital for use in investment …   Financial and business terms


text:
# Collatz Conjecture (c++)

I am new to C++ and this is my first program. The objective of this program is to fully complete the sequence called the Collatz conjecture.

It concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. Otherwise, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

I am not familiar with the language just yet and would appreciate some (constructive) feedback. The program works as intended, however, I have a feeling that some of my code is not as elegant as it could/should be.

#include <iostream>
#include <ctime>

using namespace std;

bool isValid(int num) {
if(num <= 0) {
return false;
} else {
return true;
}
}

bool isEven(int num) {
if (num % 2 == 0) {
return true;
} else {
return false;
}
}

void calculate(int num) {
if (num == 1) {
cout << endl << "Finished!" << endl;
return;
} else if(isEven(num)) {
num = num / 2;
cout << num << " ";
calculate(num);
} else {
num = (num * 3) + 1;
cout << num << " ";
calculate(num);
}
}

int main() {

int input;
cout << "Please enter an integer greater than zero." << endl;
cin >> input;

if(isValid(input)) {
calculate(input);
} else {
cout << "Not a valid input." << endl;
return 0;
}

return 0;
}

• Has anyone tried running this with 715827883 or higher? :) – Brian Sutherland Sep 12 '17 at 22:48
• 113383 is also apparently a very interesting number to run this program with – Brian Sutherland Sep 12 '17 at 22:51

Good job checking the input for validity and breaking up the logic into simple functions! However, some improvements can be made.

## Avoid using namespace std

This can cause name collisions because it adds every name in the std namespace to the global namespace. For a small program like this one it's unlikely that you'll run into any problems (then again, maybe not) but it's best to get into the habit of using the std:: prefix on names in the std namespace.

Alternatively, you can introduce using declarations like using std::cout; to add specific names to the global namespace.

## Avoid std::endl in favor of \n

std::endl flushes the stream, which can cause a loss in performance. In some cases you can also reduce calls to operator<< (e.g. cout << endl << "Finished!" << endl; becomes cout << "\nFinished!\n";).

You can remove some of your explicit return true and return false statements by returning the expression in the if condition. For example, you can shorten isEven() like so:

bool isEven(int num) {
return num % 2 == 0;
}

Similarly, you can shorten isValid().

## Don't mix logic and writing to cout in calculate()

Currently calculate() is not very reusable since it writes to cout. What if you wanted to write to a different stream, or if you just wanted to calculate the next term in the sequence without writing to any stream? I suggest redefining calculate() to simply return the next term in the sequence:

int next(int prev) {
if (isEven(prev))
return prev / 2;
else
return prev * 3 + 1;
}

Then in main() you can output each number returned by next() to cout, as well as check for the condition in which the term equals 1. You would replace the calculate(input); statement in main() with something like this:

while (input != 1) {
input = next(input);
std::cout << input << " ";
}

std::cout << "\nFinished!\n"

• I fixed the next function to not rely on a global variable, and the loop to terminate. I hope it matches your style. – Roland Illig Sep 12 '17 at 6:47
• Can't say I agree with the part about std::endl. If you want to see your output or logs as they happen, its essential to use endl. It can be very confusing to debug a program that buffers its output. It's good to know the implications of flushing the buffer, but if you're at the point where you're using cout to see your program flow, you need it to be as responsive as possible. – JPhi1618 Sep 12 '17 at 14:37
• However, I do agree that the endl before "Finished!" is unnecessary and should be a \n. There's nothing to flush at that point. – JPhi1618 Sep 12 '17 at 14:49
• @JPhi1618 The OP says the program works as intended, so I take that to mean he isn't debugging. And I don't see much point in flushing the stream for every term in the sequence just to see the output as it's generated. There's a time and a place for using std::endl to see the output as it is generated (I do it myself in certain situations) but I don't think this is one of them. – Null Sep 12 '17 at 14:57
• Actually, I just noticed that the OP only flushes the stream at the end of the sequence and at a few other times (right before receiving the input and right before returning from main). It seems even more superfluous in this case since it would be flushed anyway. – Null Sep 12 '17 at 15:01

Your error handling is missing an important point. When you declare the variable input, it already has a value, though that value is typically unpredictable. Reading this value from the variable invokes undefined behavior (that's an official term), which would make your program useless.

The operator std::cin >> input is not guaranteed to write a value to the variable. It only does so if the input is valid. So when I type asdf instead of an integer, the old value is preserved (or in general, partly overwritten, but for an int, that's not possible).

After that, your code reads that undefined value. It must not do this.

To solve this problem, write all code like this:

int input;
if (std::cin >> input && input > 0) {
std::cout << "The number is " << input << ".\n";
} else {
std::cerr << "error: wrong input format\n";
return 1;   // Returning 1 from the main function means an error.
}

The >> operator does not read the value of the variable, it only writes to it in the successful case. Therefore this program never reads the undefined value from the variable.

• Shouldn't you be initializing the input variable in your example? – Frank Sep 12 '17 at 7:03
• No. This usage is perfect since I'm only reading the variable after it has been definitely assigned a value. The >> operator also is a write-only operation on the variable. – Roland Illig Sep 12 '17 at 7:04
• Yes, nervermind, I misread your code for some reason. – Frank Sep 12 '17 at 7:05

Very nice use (intentional or not) of tail recursion.

Use unsigned int when you want to represent numbers greater or equal to zero

unsigned int can represent twice as many numbers as int. It also gives the compiler more information about what your program does, which lets it do its job better.

Do not include headers you are not using

You are not using <ctime> for anything, so you should not include it.

Return something else than 0 in main when the program fails

The convention is that a program returns 0 on success, and something else on failure. It's generally just 1, but some programs encode the nature of the failure in that number.

else {
cout << "Not a valid input." << endl;
return 1;
}

If you are simply testing a boolean expression and returning true or false based on the result, you could just return the result of the boolean expression.

Rather than

if (num % 2 == 0) {
return true;
} else {
return false;
}

You could simply have

return num % 2 == 0;

since the result of num % 2 ==0 already is a bool.

To do this with your isValid function, you would have to either negate the boolean expression, or change the way you are comparing the numbers.

The expression num <= 0 would become !(num <= 0), which might be a little more confusing, or simply change it to num > 0.

format: plain


text:
### Teddy Town

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

### Nine Colours

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

# Conway's Chequerboard Army

##### Stage: 3 Challenge Level:

Duncan from Warden Park School, Zach from Rosemellin School and Stephen all got the minimal method for achieving both targets $C$. Stephen explains his method for $C$:

Position five counters in the row under $A$ (row $1$), with the middle counter directly under $A$. Then place three counters in the row below that (row $2$), with the left most directly under $A$.
Two under $A$ jumps one under $A$, so there is a counter on $A$. Then the counter two to the right of $A$ jumps to be directly under $A$ in row $1$. Then this counter jumps the one on $A$ so it sites on $B$.
The counter in row $1$ two to the left of $A$ jumps to directly under $A$, and the counter in row $2$ two the the right of $A$ jumps to be directly under $A$ in row $2$. Then this counter jumps the one above to land on $A$ then jumps the next to land on $C$.

Although all of you also found a method for finding $d$, you did not find the minimal method, so using the least possible counters and moves. Andy from the Garden International School explains the minimal way of achieving the target $D$.

$6c -> 4c$
$5e -> 5c -> 3c$
$5a -> 5c$
$6a -> 6c -> 4c -> 2c$
$6e -> 6c$
$7c -> 5c$
$7a -> 7c$
$8c -> 6c -> 4c$
$7e -> 7c$
$8e -> 8c -> 6c$
$8a -> 8c$
$9c -> 7c -> 5c -> 3c -> 1c$

Andy goes on to explain why you cannot achieve further than $D$ on the grid provided.

With the counters above needed to reach $D$, then we are left with:

Which you cannot reach beyond $D$ as you would need to get counters on $D$, $B$, and below $A$. Though there is a counter on $D$ which isn't in my picture. This cannot be made.

Congratulations to Andy, can you think of another way of reaching $D$?


text:
Why doesn't ether = spacetime

1. Jan 24, 2013

Pjpic

EM radiation seems like a wave traveling through spacetime so, why isn't spacetime the same as ether?

2. Jan 24, 2013

Staff: Mentor

Because it doesn't have a definable state of motion.


text: # Latex Table Width


text:
# Where is complex Log continuous

Write down the biggest subset $D$ of $\mathbb C$ on which ${\rm Log(z)}$ is a continuous function. Explain why ${\rm Log(z)}$ is not continuous at points outside $D.$

Anyone know the answer to this?

-
Related... –  Guess who it is. Apr 18 '12 at 14:35
My answer here: math.stackexchange.com/questions/88341/… might help... –  Isaac Solomon Apr 18 '12 at 14:38

## 1 Answer

I'll tell you one possible answer, and you tell me why (I'm assuming the principal branch):

Answer: $D$ is the complex plane minus the non-positive reals.

Hint: Follow things in a semi-circle and observe the "jump" in angles.

-


text:
# Timing Rack Middlewares with metaprogramming, recursive monkey-patching, and a sprinkle of statistics

Analysing performance of your Rails or Sinatra is easy enough with New Relic, but figuring out whether the soft outer shell of your stack is under-performing is more of a challenge. We’ve written rack-timer to figure things out.

• time spent in before/after_filters (if using Rails).


text:
# More on parity matrices

Here’s a generator matrix,

$G = \left( {\begin{array} {rrrr} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 \end{array} } \right)$

A parity check matrix for this, $H$ is one where $GH^T = 0$ (hence the product of $H^T$ with a codeword is also 0, though an error word generates a non-zero output).

Two candidates for this present themselves (are there others? I can’t see them):

$H^{\prime}= \left( {\begin{array} {rrrr} 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 \end{array} } \right)$

$H^{\prime\prime}= \left( {\begin{array} {rrrr} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} } \right)$

Taking $H^{\prime}$, there are $2^k$ distinct messages (codewords), where $k = 2$, but he generator can create $2^n = 2^4$ different outputs. So we have $2^k - 1$ non-zero code words and $2^n - 1$ possible outputs, the number of detectable errors is:

$2^n - 1 - (2^k - 1) = 2^n - 2^k = 12$

The most likely error is the one with the lowest Hamming weight:

$\left( {\begin{array} {lrrrr} codewords & 0000 & 0101 & 1100 & 1001 \\e_1 & 0001 & 0100 & 1101 & 1000\\e_2 & 0010 & 0111 & 1110 & 1011 \\e_3 & 0011 & 0110 & 1111 & 1010 \end{array} } \right)$

1. prubin73

H’ works as a parity check matrix, but I think H” does not: H” times (1,1,1,1) yields (0,0), but (1,1,1,1) is not generated by G.

The maths is modulo 2, so surely each row of $GH^{\prime \prime T}$ is zero?


text:
summaryrefslogtreecommitdiffstats log msg author committer range
path: root/manual/plugins/mosaic.tex
blob: d5fdfefe6430a57573b6c99df0bf607d9bf88c20 (plain)
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  % $Id$ % % % \subsection{Mosaique} \screenshot{plugins/images/ss-mosaic}{Mosaique}{img:mosaique} This simple graphics demo draws a mosaic picture on the screen of the \dap. \begin{btnmap} \PluginUp & Change the gap between the drawing lines.\\ \PluginSelect & Restart the drawing process.\\ \PluginCancel{} or \PluginExit & Exits Mosaique demo\\ \end{btnmap}


text:
# Teacher Notes

### Why use this resource?

This puts an exponential graph alongside some real-world (and some extra-terrestrial) objects. Students can start to tackle the problem using their knowledge of powers of $2$. Before long, a calculator is beneficial as the numbers start to get large, but the entire problem can be tackled using trial and improvement with powers of $2$. Students who have been introduced to logarithms will find them a useful tool for solving equations more directly, but the resource can equally be used as a pre-logarithm introduction. The problem also involves some interesting unit conversions once we start looking at astronomical data, and just a little bit of Physics.

It could be followed by the review question Can we find bounds for $\log_2 3$?

### Preparation

A printed copy of the data could be useful for each group so that the board could be used for other things.

Access to the internet might be helpful for students to research heights and distances of other things.

If possible some A2 or even A3 paper with 1 cm grid lines would be quite helpful to help less confident students.

White boards with grid lines could be useful for starting with.

If display work is wanted as a result some name cards for the different objects could be printed out in advance and an $x$-axis on a display board ready for it.

### Possible approach

Display the problem image and draw students’ attention to the scaling of $1$cm per unit on each axis. Begin with mini white boards (portrait orientation) ask students to mark along the top edge how far along it the graph of $y=2^x$ would exit the top of the page and show boards.

Discuss what that $x$ value is and how they know. Show the further questions (these could be printed out) and discuss as a group what they might already know, need to conjecture or research. Possibly show some of the data.

Students could work in small groups making conjectures and doing calculations to confirm. They could come to mark points along an $x$-axis displayed on the board, or better still a more permanent display.

### Key questions

• How much further along will an object twice as tall be?
• Can you predict what the $x$-values for other distances would be (length of the Nile, distance from Land’s End to John O’Groats, length of the Trans-Siberian Railway)?
• What do you notice?
• What difference would it make to change the $2$ to a $3$?

### Possible support

Students could be encouraged to write a table of values, say as far as $x=10$.

Grid paper could be useful so that students could plot a graph to start to get a better feel.

Copies of the data from the data tabs could be given early to students who are unable to make size conjectures early on if whole class discussion has not been used.

### Possible extension

Students could consider how much difference it would make to change to powers of another base, say $3$ or $10$.

The Solution section also contains a nice extension idea: “Suppose instead of the graph of $y=2^x$ we had used $y=x^{10}$, or some other power of $x$. How different would this diagram be?”

A version of this resource has been featured on the NRICH website. You might like to look at some students’ solutions that have been submitted there.


text:
## Geometry: Common Core (15th Edition)

$\frac{5}{4}$
By looking at the graph, you can determine EC is 8 units. You can also tell that AC is 6 units long. To figure out AE's length, calculate using the Pythagorean theorem. $AC^{2}$ + $EC^{2}$ = $AE^{2}$ $6^{2}$ + $8^{2}$ = $AE^{2}$ 36+64=$AE^{2}$ 100=$AE^{2}$ 10=AE Write the ratio of the sides. $\frac{AE}{EC}$=$\frac{10}{8}$ Reduce by dividing the numerator and denominator by 2. $\frac{5}{4}$


text:
# Proving a Theorem about Perfect Numbers

I am trying to prove the following theorem:

Theorem. A number is perfect iff the sum of the reciprocal of its divisors, excluding $1$, is $1$.

Thus far, this is the proof that I have managed to sew:

Proof. Let $n$ be perfect. Then $2n=1+a_1+a_2+\cdots+a_m+n$, where each $a_j$ is a divisor of $n$. Moreover, let $1<a_j<a_{j+1}<n$. It follows that $2=\frac{1}{n}+\frac{a_1}{n}+\frac{a_2}{n}+\cdots+\frac{a_m}{n}+1\Longrightarrow$ $1=\frac{1}{n}+\frac{1}{a_m}+\frac{1}{a_{m-1}}+\cdots+\frac{1}{a_1}$, as required. Conversely, let $1=\frac{1}{n}+\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_m}$. It follows that $1=\frac{1}{n}+\frac{a_m}{n}+\frac{a_{m-1}}{n}+\cdots+\frac{a_1}{n}\Longrightarrow$ $n=1+a_1+a_2+\cdots+a_m$. Therefore, $n$ is perfect. $\square$

Nevertheless, I do not feel very confident about it. What do you guys think?

-
A "Corollary" means a consequence of a more general, or a more sophisticated, or a more substantial result (often these consequences are straightforward, but sometimes are rather technical). So the question is: this is a corollary to what? –  William Feb 18 '12 at 1:30
I believe that I may have messed up the terminology. Thank you for pointing that out. What I meant to express was that I am trying to give (and prove) an alternative 'definition' to perfect numbers. –  Josué Feb 18 '12 at 1:39
As pointed out by WNY, the term "Corollary" is not appropriate here, the proof uses only the definition of perfect number. Your solution is almost fine, just small wording tweaks needed. Start with "Let $1, a_1, a_2, \dots, a_m, n$ be the divisors of $n$, listed in increasing order." The second half is correct, but kind of confusing. Why not after the "Conversely, let $\dots$" sentence, write a sentence that adds $1$ to each side, then a sentence that multiplies both side by $n$. –  André Nicolas Feb 18 '12 at 1:42
Apart from slight wording changes, it is fine. One could also write: As $d$ ranges over the positive divisors of $n$, $\frac{n}{d}$ also ranges over the positive divisors of $n$. Thus $\sigma(n)=\sum_{d|n} \frac{n}{d}$. It follows that $\sigma(n)=2n$ iff $2n=\sum_{d|n} \frac{n}{d}$. Now divide both sides by $n$. (Your proof does exactly the same thing. For concreteness, it is better to use your notation than the condensed notation of this comment.) –  André Nicolas Feb 18 '12 at 1:53

Let $a$ be a perfect number, My '$a$' has $n$ factors.

According to perfect number,

$a=a_1+a_2+a_3+...a_n$

$a_1=1$

$\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+...\frac{1}{a_n}$

Notice that

$a_1.a=a$

$a_2.a_n=a$

.

.

.

$a_n.a_2=a$

You can surely take on from here.

-
I should point out that if $a = a_1 + \dots + a_n$, then $a$ has $n+1$ positive divisors, the largest of which is $a$ itself. –  JavaMan Mar 9 '13 at 8:07


text:
» » » The speed of wind is usually measured using?

The speed of wind is usually measured using?

Question

The speed of wind is usually measured using?

A)
a wind vane
B)
a wind rose
C)
an anemometer
D)
a thermometer


text:
## Category Archives: Uncategorized

by Aris Spanos

Few statisticians will dispute that R. A. Fisher (February 17, 1890 – July 29, 1962) is the father of modern statistics; see Savage (1976), Rao (1992). Inspired by William Gosset’s (1908) paper on the Student’s t finite sampling distribution, he recast statistics into the modern model-based induction in a series of papers in the early 1920s. He put forward a theory of optimal estimation based on the method of maximum likelihood that has changed only marginally over the last century. His significance testing, spearheaded by the p-value, provided the basis for the Neyman-Pearson theory of optimal testing in the early 1930s. According to Hald (1998)

“Fisher was a genius who almost single-handedly created the foundations for modern statistical science, without detailed study of his predecessors. When young he was ignorant not only of the Continental contributions but even of contemporary publications in English.” (p. 738)

What is…

View original post 1,934 more words

## Tripling a Triangle

Seen at Tanya Khovanova’s Math Blog

Note:  The triangular numbers are given explicitly by the formula $T_n = {n+1 \choose 2}$

Tagged

## The book about flies in the news

“Million-dollar book

Algorithms may be cleverer than humans but they don’t necessarily have our sense of perspective – a failing that became evident when Amazon’s price-setting code went to war with itself earlier this year.

“The Making of a Fly” – a book about the molecular biology of a fly from egg to fully-fledged insect – may have been a riveting read but it almost certainly didn’t deserve a price tag of \$23.6m (£14.3m).

It hit that figure briefly on the site after the algorithms used by Amazon to set and update prices started outbidding each other.

It is a small taste of the chaos that can be caused when code gets smart enough to operate without human intervention, thinks Mr Slavin.

Tagged , , ,

Tagged

## The March of Progress

1980: C
`printf("%10.2f", x);`
1988: C++
`cout << setw(10) << setprecision(2) << showpoint << x;`
1996: Java
```java.text.NumberFormat formatter = java.text.NumberFormat.getNumberInstance();
formatter.setMinimumFractionDigits(2);
formatter.setMaximumFractionDigits(2);
String s = formatter.format(x);
for (int i = s.length(); i < 10; i++) System.out.print(' ');
System.out.print(s);```
2004: Java
`System.out.printf("%10.2f", x);`
2008: Scala and Groovy
`printf("%10.2f", x)`

“Note the lack of semi-colon. Improvement!”

—————————————————

Tagged , ,

Tagged


text:
# $0/0$ limit question

I have an examples book with a limit exercise that I can't understand.

The limit in question is:

$$f(x,y)=\frac{x}{x+y}$$ with $x\ne-y$; $$\lim_{(x,y)\to(0,0)} f(x,y)$$

And then to solve it, it goes:

$$\lim_{(x,y)\to(0,0)} f(x,y) = \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$

Can you help me understand that? Thanks,

UPDATE: Ok, just to make sure that I got it right. I have a very similar test exercise with $4$ different options.

The following limit $$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{x^2+y^2}$$ equals:

A. $0$

B. $- \infty$

C. Doesn't exist

D. $+\infty$

My doubt is: if I consider it normally I'd say that it doesn't exist, but if I solve it using the same approach (i.e. $y=mx$) then the limit equals $0$. Which one is the right answer?

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I think you are missing something important regarding $y$. Is $y$ a straight line massing through the origin..or anything like this has been mentioned in the book. –  anonymous Aug 18 '10 at 11:05
@Chandru1: Yes, you're right. Thanks for the correction. It says that "y is a straight line with any value for m but -1" –  Kokoloko Aug 18 '10 at 11:50

The point is that this limit doesn't exist.

For it to exist, then if you take any sequence of points $(x_n,y_n)$ converging to $(0,0)$ for which $f(x_n,y_n)$ is defined, then $$\lim_{n\to\infty}f(x_n,y_n)$$ must exist and also be independent of the sequence $(x_n,y_n)$. The book is saying in essence that if you have a sequence of points of the form $(x_n,m x_n)$ (so lying on the line with gradient $m$ though the origin) and converging to the origin, then $$\lim_{n\to\infty}f(x_n,y_n)=\frac1{1+m}\qquad\qquad\qquad(*)$$ which is not independent of the sequence of points $(x_n,y_n)$, as changing $m$ will change the limit $(*)$.

-
Your answer is very accurate and similar to the one I have in the book. The problem I have, and I probably didn't explain it well enough is with the last step, where it says that (x->0)lim(x/(x+mx))=1/(1+m). Why isn't it =(0/(0+0m)) thus 0? –  Kokoloko Aug 18 '10 at 11:58
0 + 0m = 0 so limit = 0/0, so they are using L'hospital's rule. –  deinst Aug 18 '10 at 12:06
deinst, no one is using L'Hopital's rule. Kokoloko, would you have the same "problem" with $\lim_{x\to0}x/x$? –  Robin Chapman Aug 18 '10 at 12:08
@Kokoloko: You could just cancel the x on top and bottom. –  Niel de Beaudrap Aug 18 '10 at 12:09
REMARK While it's true that L'Hospital's rule is overkill here, as I remarked before on sci.math, it's a good thing that the rule applies in such cases because in general there is no algorithm for deciding if two functions are equal (or 0), so e.g. one cannot tell in general if f = 0 or f = g in f/g, but the rule may still apply and yield the correct result (whether by human or machine computation). See this thread for further discussion: groups.google.com/group/sci.math/msg/465b5953538f4049 –  Bill Dubuque Sep 29 '10 at 16:50

For your second question: is $$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{x^2+y^2}=0?$$ imagine a sequence of points $(x_n,y_n)$ converging to the origin, and think whether $f(x_n,y_n)\to0$.

I would be tempted to write these points in polar form: $x_n=r_n\cos t_n$ and $y_n=r_n\sin t_n$ as then the denominator becomes $r_n^2$ (nice) and moreover $r_n\to0$ (why?).

-
Mm... I'm not sure if I did it right, but giving values close to (0,0) to the variables seems to get results close to 0. Does that mean that the answer is 0 then? –  Kokoloko Aug 18 '10 at 13:22
Now that I think about it, changing the Cartesian variables in the original question to polar form also makes it clear that the limit does not exist: the role played here by $\theta$ is the same role played by m in your first answer. –  Ｊ. Ｍ. Aug 18 '10 at 13:30
Kokoloko, what it means is that the answer seems to be $0$ :-) (whether it is zero is another matter). Jan, changing to polars is generally a good trick when dealing with homogeneous functions. –  Robin Chapman Aug 18 '10 at 14:30
Oh, you don't have to worry about me not appreciating polar/cylindrical/spherical transformations, I've seen them vastly simplify operations that are otherwise hard to do the Cartesian way. :) –  Ｊ. Ｍ. Aug 18 '10 at 14:40
Good: so you get $r(-\cos^3 t+3\cos t\sin^2 t)$. As $(x_n, y_n)\to0$ then $r_n\to0$. So if $r$ tends to zero, what can you say about $r(-\cos^3 t+3\cos t\sin^2 t)$? –  Robin Chapman Aug 18 '10 at 18:18

I will elaborate on Robin's answer.

There is no unique limit as (x,y) goes to (0,0) --- i.e. the limit does not exist. For instance, if we approach (0,0) on the x axis, we obtain the limit

$\lim\limits_{(x,y) \to (0,0)} \dfrac{x}{x+y} = \lim\limits_{x \to 0} \dfrac{x}{x} = 1,$

and if we approach (0,0) on the y axis, we obtain the limit

$\lim\limits_{(x,y) \to (0,0)} \dfrac{x}{x+y} = \lim\limits_{y \to 0} \dfrac{0}{y} = 0.$

However, we can still consider limits along different curves. The two above limits, along the x and y axes, are two examples: the limit you describe in your question is a similar limit, on the locus of the equation y = mx. This is something different than whether there is "a limit" of f(x,y) as we approach (0,0) --- in this case, because there are different limits depending on how we approach the origin, we may consider the question of what limit one has as one approaches the origin on a particular curve.

In this case, a "curve" is a function

$\big(x(t),y(t)\big) = c(t)$,

for some function $c: \mathbb R \to \mathbb R^2$. "Approaching" (0,0) along the curve c(t) entails specifying some domain for c in which it does not cross itself (for the example of the line y = mx one could take c(t) = (tmt), in which case the domain can be the real numbers) and for which c(t) = (0,0) for some (unique) value t = T. Then, evaluating the limit of f(x,y) along the curve (x,y) = c(t) means just taking the limit of the composite function f(c(t)) as t approaches T.

In the case of approaching (0,0) on the curve c(t) = (tmt), we just take T = 0, and evaluate the limit of f(tmt) as t approaches 0. Up to a change of variables, this is just what you have done above.

-
Your answer does a great job elaborating on Robin's answer. I think I'm getting the grasp of it. I understand the your examples of approaching the x and y axis, but I don't get the one of my book (with straight lines). If you could elaborate on that I'd really appreciate it. –  Kokoloko Aug 18 '10 at 12:26
Is there something specific that you don't understand in my lower two paragraphs? Think of x and y as positions of some particle, and then let these vary with a "time" variable. The limit along a curve (or line) is the limit of f(x(t),y(t)), along that trajectory. The X and Y axes are just two possible trajectories, in which the y and x co-ordinates (resp.) are constant with time; other lines are a more general class of the same sort of thing, and special cases of the class of all continuous trajectories (x(t), y(t)) = c(t). –  Niel de Beaudrap Aug 18 '10 at 13:17
@Kokoloko Niel's answer is correct to the problem as posted. In the comments to your question you say the limit is taken along a line through the origin. This is what permits the book to substitute mx for y. Normally (and in your second example) it is required that the function be close to the limit for all points close to the target. If you take different paths approaching the target and get different limits, the limit doesn't exist. –  Ross Millikan Oct 20 '10 at 22:27

HINT The limit becomes that of $x/f(x)$ along the path $y = f(x)-x$ where $f(x)\to 0$. But this limit can be whatever you desire, by choosing $f$ weaker, comparable, or stronger than $x$ at $0$. Explicitly: choosing $f$ to be one of $\; x^{1/2},\; x/c,\; x^2$ yields $\; x/f(x)\to 0,\; c,\; \infty\;$ respectively, as $x\to0^+$.

-
I'm sorry, but I don't understand your hint. –  Kokoloko Aug 18 '10 at 15:46
E.g. choose $x^{1/2},\; cx,\; x^2\;$ for $f(x)$ –  Bill Dubuque Aug 18 '10 at 16:00
Thanks! Let me try that ;) –  Kokoloko Aug 18 '10 at 16:16

EDITED. The version I originally posted was wrong.

For the second problem, use the fact that squares are non-negative to show that $$|f(x,y)| \le 3|x|$$

From which it follows (by using the sandwich theorem) that, $$\lim_{(x,y) \to (0,0)} f(x,y)=0$$.

-

De l'Hôpital explains the last step.
If both numerator and denominator approach zero, you can apply de l'Hôpital, provided the derivative of the functions in numerator and denominator exist for x=0, which is the case.

-
That is way more than overkill... x is not zero, so we can cancel it from the top and bottom in the last step! –  Dylan Wilson Aug 18 '10 at 12:45
L'Hopital's rule should be avoided wherever possible. Using it here is just ludicrous. –  Robin Chapman Aug 18 '10 at 13:01
@Robin: Why???? (Yes, one question mark would do, but not for stackexchange, which insists on at least 15 chars for a comment) –  stevenvh Aug 18 '10 at 13:11
Search the archives of sci.math for my collected rants about L'H's rule :-) –  Robin Chapman Aug 18 '10 at 14:31


text:
## anonymous one year ago How to simplify this?? $\frac{-\sqrt{3} - 1}{1+(-\sqrt{3})*1}$ $\frac{-\sqrt{3} - 1}{1-\sqrt{3}}$ $\frac{-\sqrt{3} - 1}{1-\sqrt{3}} * \frac{1-\sqrt{3}} {1-\sqrt{3}}$ $\frac{2}{2(2-\sqrt{3})}$ Is this correct??

1. anonymous

In step 3, you should multiply the expression by the denominator's conjugate, top and bottom.

2. anonymous

That allows you to eliminate radical expressions in the denominator.

3. anonymous

Could you show me? I don't see what you mean. I thought I did multiply the expression

4. anonymous

$\frac{ -\sqrt{3}-1 }{ 1-\sqrt{3} } \times \frac{ 1+\sqrt{3} }{ 1+\sqrt{3} }$

5. anonymous

That should be your step 3.

6. anonymous

I did that and I get 2 for the numerator and $$2(2-\sqrt{3})$$ for the denominator

7. anonymous

No, if you multiply a radical expression by its conjugate, the square roots disappear.

8. anonymous

Yeah, I used the same I see that.

9. anonymous

I use - and not the conjugate

10. anonymous

$(1-\sqrt{3})(1+\sqrt{3}) = 1-\sqrt{3} + \sqrt{3} - 3 = 2$

11. anonymous

Do you agree with my above expansion?

12. anonymous

Yes

13. anonymous

Ok, so 2 is now your denominator.

14. anonymous

Let me do the top

15. anonymous

Let's multiply the numerator now.

16. anonymous

I got -2

17. anonymous

One sec I made a boo boo

18. anonymous

Check your numerator again. It should contain some square roots.

19. anonymous

I have $-\sqrt{3} -\sqrt{9} -1 -\sqrt{3}$ is this correct?

20. anonymous

Yes, you can combine the terms to make it easier to read.

21. anonymous

$-2\sqrt{3}-4$

22. anonymous

Now that you have the numerator, don't forget that you still have a denominator, -2.

23. anonymous

From what I put up where di the 4 come from?

24. anonymous

Oh your $- \sqrt{9} - 1 = -3 - 1 = - 4$

25. anonymous

26. anonymous

Ok how about eh 2 and the sqr(3) ?? from what I put up.

27. anonymous

Oh that's from $-\sqrt{3}-\sqrt{3} = -2\sqrt{3}$

28. anonymous

$\frac{ -2\sqrt{3}-4 }{ -2 } = \sqrt{3} + 2$

29. anonymous

Yep that is it. Thank you so much.

30. anonymous

Are you a qualified helper?

31. anonymous

I just like to help :)


text:
# Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime?

Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime? While answering @pedja's question about the existence of any such representations I was surprised to discover that $32$, $128$, $512$, and $2048$ cannot be represented in this way, and I've since found that the same is true for $8192$, $32768$, $131072$, $524288$, $2097152$, and $8388608$. Is there any counterexample? If not, how can it be proved?

-

From Nicolas's argument I realized that it was sufficient to check if $2^{2n+1}-7$ was prime. A much easier task than the usual brute forcing. After a few minutes of manual testing on wolfram alpha I found that $2^{39}$ could indeed be written as the sum of a prime and a safeprime:

$$7+549755813881=549755813888=2^{39}.$$

Edit: I wrote a sage program to check all of the odd powers of two less than 2000. It seems I was a bit lucky since the next examples are 715 and 1983. I checked OEIS and numbers such that $2^{n}-7$ are prime has its own page.

-
Wow! ........... (39 of these) –  Dan Brumleve Sep 14 '11 at 6:50

The congruential argument below shows that we need only worry about the safe prime $7$.

An odd power of $2$ is congruent to $-1$ modulo $3$. Let $p$ be a safe prime. Then, except in the cases $p=5$ and $p=7$, $p$ is of the form $2(6k\pm 1)+1$. But $p$ cannot be of the form $2(6k+1)+1$, so it is of the form $2(6k-1)+1$, that is, $p$ is congruent to $-1$ modulo $3$. The safe prime $5$ is also congruent to $-1$ modulo $3$. (The only safe prime that escapes this congruential net is $7$.)

So if $2^{2n+1}=p+q$, where $p$ is a safe prime other than $7$, then $q$ must be divisible by $3$. In particular, if $q$ is prime, $q$ must be $3$.

But this is not possible, since (apart from the case $p=5$) a safe prime $p$ is congruent to $3$ modulo $4$, and hence $p+3$ is congruent to $2$ modulo $4$. The case $p=5$ accounts for the decomposition $8=5+3$.

Thus if $2^{2n+1}=p+q$, where $p$ and $q$ are primes and $p$ is safe, we must have $p=7$. So the only question that remains is to look for primes $q$ of the form $q=2^{2n+1}-7$.

There likely is some literature on primes of the form $2^{2n+1}-7$. A feeble attempt at experimentation did not locate any, but there is certainly not always a simple small divisor.

-
I have: $2^{2k+1} = q + r$, $r = 2p+1$, $4^k = p + (q+1)/2$. What am I missing? –  Dan Brumleve Sep 14 '11 at 6:12
Ahh, got it. Now I wonder if there are any other classes of exceptions that invalidate the heuristic argument I suggested in my answer to the other question? –  Dan Brumleve Sep 14 '11 at 6:16
But I guess it is still possible that $2^{2k+1} = 7+q$ for some prime $q$? This is a great answer but I've unaccepted it for the moment. –  Dan Brumleve Sep 14 '11 at 6:38
@Dan Brumleve: No problem, easy come easy go. It is not often that simple congruential conditions get us this far. Yes, $7$ is the only possible exception. The weird safe primes are $5$ and $7$, and $5$ was no trouble. –  André Nicolas Sep 14 '11 at 6:51
@Dan Brumleve: A little mulling over numbers showed that the difference between $2^{2n+1}$ and a safe prime is usually divisible by $3$. Paid no attention to $7$ since it is obviously special. So had to rule out a difference of $3$ (easy) and $7$ (not easy, indeed as it turns out not possible). –  André Nicolas Sep 14 '11 at 7:06
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text:
Post Categories:   DEFAULT

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text:
m
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F Nous contacter

0

# Documents  Zhang, Shou-Wu | enregistrements trouvés : 3

O

P Q

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## Heegner points and Rankin L-series :papers from the workshop on special values of ... held at MSRI#Dec. Darmon, Henri ; Zhang, Shou-Wu | Cambridge University Press 2004

Congrès

- 367 p.
ISBN 978-0-521-83659-3

Mathematical sciences research institute publications , 0049

Localisation : Colloque 1er étage (BERK)

courbe elliptique # fonction L # série L # formule de Gross-Zagier # multiplication complexe # forme automorphe # méthode de Rankin-Selberg # intersection arithmétique # théorie d'Iwasawa # point de Heegner

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## Congruent number problem and BSD conjecture Zhang, Shou-Wu | CIRM H

Multi angle

Research talks;Number Theory

A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has proved that at least 41.9% of $n=1,2,3$ satisfy (refined) BSD in rank $0$, and at least 55.9% of $n=5,6,7$ mod $8$ satisfy (weak) BSD in rank $1$. This implies in particular that at last 41.9% of $n=1,2,3$ mod $8$ are not congruent numbers, and 55.9% of $n=5, 6, 7$ mod $8$ are congruent numbers. I will explain the ingredients used in Smith's proof: including the classical work of Heath-Brown and Monsky on the distribution F_2 rank of Selmer group of E_n, the complex formula for central value and derivative of L-fucntions of Waldspurger and Gross-Zagier and their extension by Yuan-Zhang-Zhang, and their mod 2 version by Tian-Yuan-Zhang. A thousand years old problem is to determine when a square free integer $n$ is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves $E_n : ny^2 = x^3 - x$. In fact by BSD, all $n= 5, 6, 7$ mod $8$ should be congruent numbers, and most of $n=1, 2, 3$ mod $8$ should not be congruent numbers. Recently, Alex Smith has ...

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## The Gross-Zagier formula on Shimura curves Yuan, Xinyi ; Zhang, Shou-Wu ; Zhang, Wei | Princeton University Press 2013

Ouvrage

- viii; 256 p.
ISBN 978-0-691-15591-3

Annals of mathematics studies , 0184

Localisation : Ouvrage RdC (YUAN)

variétés des Shimura # géométrie algébrique arithmétique # forme automorphe # quaternions

#### Filtrer

##### Codes MSC

Ressources Electroniques (Depuis le CIRM)

Books & Print journals

Recherche avancée

0
Z


text:
# Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5).

So I gave him an upper bound:

$$8!~3^8\cdot24!~2^{24}/2^{12}\cdot12!~2^{12}\cdot24!/4!^6\cdot24!/4!^6,$$

did some rough approximation, and said it was around 10^79.

Then I decided I'd try to give a lower bound, and came up dry.

In the Wikipedia article, we read that:

• The orientation of the eighth vertex depends on the other seven.
• The 24 edge pieces that are next to vertices can't be flipped.
• The twelfth central edge's orientation depends on the other eleven.
• The parity of a permutation of the vertices and of the edge pieces are linked.

The second point is easy to see by imagining the route that such an edge piece takes under all the various moves. (The Wikipedia article opts to give a mechanical reason instead, which is a questionable practice.) The other points are not too hard to see either.

So we divide by $3\cdot2^{24}\cdot2\cdot2$ and get the answer, around $2.83\cdot10^{74}.$

My friend does not know group theory, and the only proof I know of the independence of the rest of the stuff uses group theory to a considerably greater extent than that which proves the dependent parts dependent. Can anyone think of a simpler proof of a reasonable lower bound? One that, say, at least puts the number greater than that for Rubik's Revenge (4x4x4), which is about $7.40\cdot10^{45}$?

-

The lower bound you get is actually the exact number of reachable different-looking configurations. I'd write it as $$\frac{8! \cdot 24! \cdot 12!}{2} \cdot 3^7\cdot 2^{11}\cdot \frac{24!}{4!^6} \cdot \frac{24!}{4!^6}$$ The first factor counts the number of reachable permutations of the non-center cubies, ignoring orientation. The two middle ones count the number of possible orientations for the corners and central edges. Finally the last ones count the number of distinguishable states for each of the two sets of 24 movable center panels.

In order to see that this many different arrangements are actually possible, we can observe:

• Of course the 8 corners are all different, as are the 12 central edges.

• The 24 non-central edge pieces are all different, when we take into account that the outer end of such a piece stays the outer end no matter how we twist the cube. So seen from that end of the edge it has a distinct left and right face, and the colors of these two faces determine which piece it is.

So the two denominators of $4!^6$ for the center panels is all that is needed to take into account states that look the same but actually have physical pieces switched around. (By ignoring the middle centers completely in the count, we're implicitly taking account of the fact that they look the same in all four orientations).

So all that is needed is to convince ourselves that we can actually get to each of the positions we have counted, starting with a solved cube.

First get the $24!$ factor done with by moving the non-middle edges around such that each of them is next to the corner piece we want them to end up next to, and on the right side of it. During this phase the corner pieces themselves make no net movement. We don't care what happens to the middle edges or center pieces in this phase, so we can treat the cube as a 4×4×4 one with an irrelevant middle layer sandwiched into each dimension. Getting the non-center edges into place can be done with the 4×4×4 combination for flipping two neighbor edges together, and appropriate adaptations of that. (Here "adaptation" means "do something that brings the two edges you want to swap next to each other, not caring about anything else, then swap them, and finally do the initial something backwards").

Next get the the corners and the middle edges into the desired positions. If we work with only the two middle ones of the four cuts in each dimension, the cube will work like a plain old 3×3×3, and the non-middle edges will stay next to the corners we've just anchored them to. So the number of choices we have in this phase is exactly the number of positions on a 3×3×3 cube, namely $\frac{8!\cdot 12!}{2}3^7 2^{11}$.

In the first two phases we haven't cared about getting the center panels scrambled, but now finally handle the two factors of $\frac{24!}{4!^6}$ one by one by arranging each group of 24 center tiles as we want them. For each such group, there's a combination that cyclically permutes three centers pieces on different sides without touching anything else on the cube. Again, appropriate adaptations of this will allow us to do anything -- officially 3-cycles are not enough to do everything, but in pinch we can make the 3-cycle look like a plain swap of two pieces by choosing two of the three participants to be positions that are already of the same color, and plain swaps are (of course) enough to reach any desired pattern.

There's some tiny amount of group theory hidden in the talk about which permutations can be built up from which pieces, but it doesn't need to be phrased in those terms because the point is just that transpositions are enough to do everything. We don't need to tell that the "adaptations" of the basic operations I describe would be called conjugations by a group theorist, and we certainly don't need to mention that, say, the basic 3-cycle of center panels is constructed as a commutator.

(Note that this would be horribly slow as an actual solution algorithm, since it's optimized for making clear in a uniform way that all the states we've counted can be produced reachable).

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Chris Hardwick has come up with a closed-form solution for the number of permutations for an $n\times n\times n$ cube: http://speedcubing.com/chris/cubecombos.html

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I imagine the solution uses some group theory. – Daniel Briggs Jan 16 '13 at 23:09