text:
# Poverty and Economic Inequalities in Peru during the Boom in Growth: 2004–14

Open Access

In the first decade of the twenty-first century, Latin America experienced strong growth that was primarily attributable to high export prices and growing demand from China. Moreover, democratic transitions in the region brought to power governments with highly contrasting economic policies and different visions of the sectors that were driving growth. These governments also differed in terms of the social policies they implemented to combat poverty and inequality. Countries with ‘heterodox’ policies (Brazil, Ecuador, Bolivia, Argentina and Venezuela) that promoted efforts to better distribute the fruits of growth increased social expenditure and encouraged, to a greater or a lesser degree, productive diversification, particularly in the internal market. Countries with ‘orthodox’ policies (Chile, Colombia and Peru) promoted foreign investment in the primary export sector (mining, oil, fishing, soybean cultivation, etc.), which was considered the main driver of growth, and implemented conservative fiscal and monetary policies that created a climate of confidence for investors and led to stable exchange rates and prices.

This chapter attempts to assess the events of the last decade in terms of distributive aspects by comparing the cases of countries that applied heterodox policies with those that implemented orthodox policies. The study focuses primarily on Peru, where governments combined a ‘leftist’ ideology—which brought them to power—with economic policies that were close to the ‘Washington Consensus’. The author examines the results of this phase of rapid growth in terms of poverty reduction and assesses to what extent these results have been accompanied by (and possibly have been achieved thanks to) a drop in inequality and the growth of the middle class. This phenomenon is seen by some as a guarantor of political stability and by others, as a cauldron of conflict. Finally, the degree to which social expenditure and taxation can play distributive roles in this new phase of slower growth is explored.

## 1 Introduction

During the first decade of the new millennium, Latin America posted strong growth. This was due primarily to higher prices for exports coupled with growing demand from China. This led to surpluses in external accounts but also generated a significant reduction in the fiscal deficit and allowed public expenditure to be increased. Countries with ‘heterodox’ policies (Brazil, Argentina, Ecuador, Bolivia and Venezuela) promoted efforts to better distribute the fruits of growth by increasing social expenditure and encouraging, to a greater or lesser degree, productive diversification, particularly in the internal market. Countries with ‘orthodox’ policies (Chile, Colombia and Peru) promoted foreign investment in the primary export sector (mining, oil, fishing, soybean, etc.), which was considered the main driver of growth, and implemented conservative fiscal and monetary policies. This created a climate of confidence for investors and led to stable exchange rates and prices. A deterioration in the terms of trade and a downturn in exports to China, which began in 2010, has blocked expansion in the region, and growth levels are close to zero in some cases and negative in others.

After having experienced rapid growth throughout the expansion cycle that accompanied the boom in commodity prices, Latin American economies, and in particular those that are the most dependent on raw material exports (mainly oil, metals and soybean) to China, have suffered a brutal drop in their growth rates. After three years of weak growth, in 2015 growth in Latin America turned negative (−0.9 per cent). Some countries, including Brazil, Venezuela and Ecuador, have reported negative growth rates, while the downturn has—with a few exceptions—spread to the other countries of the region (World Bank, 2016). The perspectives for growth are not encouraging according to the forecasts published by both the World Bank (World Bank, 2016) and the International Monetary Fund (imf) (imf, 2016), and indicate that a growth phase characterised by relatively moderate growth is on the horizon.

During the decade of sustained growth, various countries implemented active policies to combat poverty (conditional cash transfers) and increased social expenditure, which widened access to education and health. In this period, Latin America was the region that posted the most significant reductions in poverty (Inchauste et al., 2014). Nevertheless, and despite macroeconomic performance (or perhaps thanks to it) and a drop in poverty, the highest rates of inequality in the region have fallen only moderately and continue to be among the highest in the world (de Ferranti et al. 2004; Gasparini et al., 2009).1

The drop in exports has led the currency position to fall, causing depreciation in the exchange rate and inflationary pressures. These inflationary pressures have, in turn, eaten away at household purchasing power, which has attenuated the pace of poverty reduction. Private investments have fallen and some countries have adopted restrictive fiscal policies, which has further exacerbated the economic downturn. In this new context, it is important to examine the impact that the downturn/recession has had on inequalities (de la Torre et al., 2014).

Although social expenditure targeting the poor also grew during the expansive phase, which to a certain extent reconciled growth with redistribution, during the current period of lower growth it is not feasible to expect that expansion alone will reduce poverty and inequality. Nevertheless, there is still a wide margin for redistributive policies given that social expenditure remains very low and progressive direct taxes, unlike in developed countries, play a much less important if not an insignificant role in reducing inequalities. This requires a great deal of political willingness, and a social consensus must be in place to back these policies.

The different governments that have set up shop in Peru since 1990 have chosen to implement neo-liberal policies, which include suppressing or reducing the role of state enterprise while making the job market more flexible. The primary-exporter model has not been questioned and on the contrary moves have been made to promote foreign investment in the mining and oil sectors, where profitability has risen due to high international prices. Weak productive diversification has focused on exporting non-traditional products, most of which are agricultural and generated by large, high-technology farming interests.

This chapter is divided into thirteen sections. Section 2 examines the contrasting evolutions of poverty, inequality and polarisation in countries that apply heterodox policies versus those that implement orthodox approaches by breaking down the main microeconomic factors that drive these evolutions in each context. In the third section, and in those that follow, we will examine the case of Peru, where macro orthodox policies are combined with micro ‘heterodox’ policies. Based on a first-hand analysis of household surveys for the last 12 years, we study the behaviour of poverty during the different stages of growth (Section 3) and answer the question of the extent to which growth was pro-poor by highlighting the contrasts between the period of rapid growth and the period of deceleration (Section 4). In Section 5 we examine how the national macroeconomic situation (linked to exchange rate depreciation) and the situation on the international front (boom in raw material prices; strong demand) made their mark through price increases, for food in particular.

The reduction in poverty has lifted a large number of households from the throes of hardship, but the question is to what extent has this population become part of the ‘new middle class’ or has this group merely increased the numbers of vulnerable individuals who are on the verge of falling back into poverty? This point will be addressed from a dynamic perspective (poverty transitions) in Section 6, and from a static perspective through indicators of inequality and polarisation (Sections 8 and 9). In Section 7, we will examine the (direct) role played by public transfer policies in reducing poverty. In Section 10, we will examine the sources of income that account for changes in income inequality. The perception of inequality and the role of the state in redistribution leads to the formation of political coalitions that favour more state intervention through redistributive policies to reduce inequalities. This topic, which is particularly relevant in the current situation of a change of administration, will be discussed in Section 11 through an analysis of the opinion surveys from Latinobarómetro and the World Values Survey. Finally, in the last section we will explore the extent to which fiscal policy, through progressive direct taxes, can serve as a redistributive instrument to reduce income inequalities while generating resources for social policies that favour the poor.

## 2 Evolution of Poverty and Inequality in Latin America2

Judging by the noteworthy reduction in poverty levels, the last ten years have not been a ‘lost decade’ for the countries in the region. All of the countries studied, both those that apply ‘heterodox’ policies (Brazil, Argentina, Ecuador and Bolivia) and those that implement policies inspired by the Washington Consensus, saw their poverty levels fall by nearly half (Figure 8.1), with the notable exceptions of Chile and Argentina, which posted reductions of 67 per cent and 70 per cent, respectively, despite having followed very different orientations with regard to economic policy. Brazil and Peru, despite differences in policy orientations, have posted a very similar evolution in their incidences of poverty. Regardless of the fact that we still do not have harmonised data from the last two years, the figures published by the national institutes of statistics, although not comparable in terms of poverty levels, lead us to believe that poverty levels basically stopped falling in 2013, although we have yet to see a reversal of the downward trend.3 Looking beyond economic policies and the international context, the initial conditions of inequality affect the extent to which growth can reduce inequality. As such, social policies that combine poverty reduction and inequality reduction can be more successful than policies that seek simply to promote growth by creating favourable conditions for private investment.

Latin America, rightly, has the reputation for being the continent with the highest income inequality in the world (de Ferranti et al., 2004). Thanks to a decade of strong growth, different governments in the region have had the opportunity to implement redistributive policies to reduce inequalities (Lopez-Calva and Lustig, 2010). Reduction of the fiscal deficit, the availability of currency, an expansion in the supply of education, and—in the case of countries involved in mining or oil—the distribution of royalties to local governments have undoubtedly had differentiated effects.

A decrease in the Gini coefficient (which expresses total inequality when equal to 1 and perfect equality when close to 0) for total income has been widespread (Figure 8.2). In Brazil, Chile and Colombia inequality levels remain high and the decline has been less pronounced. Inversely, Argentina and Peru have posted ongoing decreases in income inequality throughout the period (Lustig et al., 2013a). Beyond these nuances, the convergence of trends means that growth, coupled with redistributive policies, has had a strong redistributive impact in both ‘heterodox’ and ‘orthodox’ countries. This fact encourages research to discover more about the implementation of universal social protection policies in various countries in the region.

In the context of the poverty and inequality reduction that transpired over the last decade, academics and international bodies have focused their attention on the expanding middle class, which is seen as platform for social cohesion and the consolidation of democracy. Growing internal markets, political stability, and demand for education, among other factors have allowed some emerging economies to escape the ‘middle income trap’. Nevertheless, little is known about the real dimension of this expansion. In order to make an initial approximation, we present (Figure 8.3) the evolution of the polarisation posited by Foster and Wolfson (2010).

In general, the polarisation indexes in all of the countries diminish and converge to values between 0.40 and 0.45. The exception is Colombia, which showed no improvements in inequality or in the polarisation index. Brazil and Bolivia, which reported the highest levels of inequality and polarisation at the beginning of the decade of 2000, are also the countries that posted an increase in the population in the intermediate segments of distribution, which made them less polarised.

## 3 The Drop in Poverty in the Most Recent Decade of Growth

Peru has experienced one of the most significant drops in poverty in the region during the last decade (Figure 8.4). Between 2004 and 2014, during a period of strong and sustained macroeconomic growth, the incidence of poverty in the country fell 36 percentage points (a reduction of 61 per cent). The average poverty gap (the population mean of the relative distance of spending by the poor from the poverty line itself) also fell considerably, going from 22.1 per cent in 2004 to 5.8 per cent in 2014. Along the same lines, poverty severity, which measures heterogeneity in spending among the poor, also dropped significantly during the same period.

Figure 8.5 shows that the decrease in macroeconomic growth rates has been accompanied by a less significant decrease in poverty. Moreover, it is evident that the ‘poverty elasticity’ of growth in the period 2009–13 fell with respect to 2005–08 (the ratio of the relative variation in poverty to the relative variation of the gross domestic product (gdp) dropped, on average, from 1.33 to 1.23). It is indeed surprising that the drop in the poverty rate during the short period of the recession of 2009 and during the downturn of 2013 was proportionally more significant that the variation in gdp. This, to a large extent, may be due to the fact that the drop and downturn in gdp was concentrated in the ‘export and investment component’, which affected the ‘consumer’ component in the mid and long terms rather than in the short run. Another possible explanation could lie in the growing importance of social programmes and of monetary transfers to the poor in particular. We should put to rest the widely held belief that a decrease in the pace of poverty reduction can be explained as follows: as poverty levels fall, it becomes more difficult for the remaining poor to exit poverty given that those who remain poor represent the hard core of the poorest among the poor. The poverty gap (mean solely among the poor) fell from 37.7 per cent in 2004 to 25.6 per cent in 2014, although it is also true that little variation was reported in the last five years (it fell only 3.5 percentage points). An examination of the urban/rural divide, where poverty gaps have been historically high, shows that the relative risk of poverty (both total and extreme poverty) increases more in rural households than in their urban counterparts.4

## 4 How Pro-poor Has Growth Been?

One of the arguments in favour of maintaining the primacy of the primary-export model is that growth will eventually, like a tide that lifts all boats at the same pace, equally benefit the poorest households without the need for redistributive ‘welfare’ policies. Growth could be described as pro-poor when it favours the poor more than the non-poor, while neutral growth generates no bias either for or against the poor (homothetic growth of expenditure or income). On the other hand, growth can be termed anti-poor when household income or expenditure in the wealthiest segments grows more than those in the lower range of income distribution. Pro-poor growth reduces the gap between poor households and other households and can (although it does not necessarily) lead to a reduction in inequality as measured by the Gini coefficient.

Below (Box 8.1), we analyse the extent to which growth has been, or has not been, pro-poor by decomposing the variation in poverty into two parts: (1) variation due to ‘pure’ growth and (2) variation due to redistribution. It is important to note that redistribution can occur due to growth that is biased in favour of sectors that are intensive in terms of unskilled labour or if it is associated with a relative decline in the performance of higher education, transfers between households and public donations, among other factors.

The greater the gaps and the severity of poverty, the more difficult it will be to reduce poverty through neutral economic growth. In this way, inequality and poverty are intimately linked.

Decomposition of poverty variations into the effects of growth and redistribution

In the decomposition of poverty variations in terms of the growth effect and the redistribution effect, three distributions are compared: the distribution observed in t-1, the simulated distribution that applies the average growth rate to all households, and the distribution observed in period t. The growth effect is the difference in the incidence of poverty divided by the simulated distribution and the distribution in t-1 (using expenditures at constant prices and the poverty line in t). The distribution effect is the difference of poverty incidences between the distribution in t and the simulated distribution in t. (See Figure A.8.1 in the Annex for details of the non-parametric decomposition.)

Figure 8.6 presents, year by year, the growth and redistribution components of annual poverty changes and annual percentage point poverty variation. With the exception of the first year, the years that posted strong growth were relatively neutral in terms of distribution and on one occasion the result was slightly anti-poor (2012). The last three years of moderate growth were proportionally pro-poor. In 2015, the reduction in poverty was due, primarily and for the second consecutive year, to the redistribution effect, which outpaced the growth effect.

In the years 2004 to 2015, the total variation in the incidence of poverty was 36.9 percentage points; the growth effect was 25.3 points and the distribution effect 11.6 points. Consequently, less than a third of the significant decrease in poverty was due to the redistributive component of growth, which, as we have already mentioned, includes not only the effect of social policy but also structural and situational changes in the economy. Growth driven by exports from the primary sector that is not accompanied by comparable development in production for the internal market, coupled with still moderate social expenditure, foreshadows a situation in which the decrease in poverty during the low growth period will be relatively modest. Additionally, the redistributive component of growth posted for the period 2011–15 accounted for only 25.9 per cent of the reduction in poverty (Table 8.1).

## 5 Distributive Effects of Price Increases

One of the channels through which macroeconomic factors affect the distribution of income is by way of the differentiated effects generated by absolute and relative price variations. Price increases primarily affect those who are not able to index their income, while changes in relative prices can lead to a larger loss of buying power in the poorest homes if said variation is stronger in terms of food than non-food items. In the case of Peru and other developing countries, the effects also depend on whether or not households are net producers or consumers, in this case, of food.

In 2004–15, the general price level (measured through the Consumer Price Index—CPI0) increased by an average annual rate of 3.9 per cent with a peak of 7.6 per cent during the 2008 crisis (meaning an accumulated increase of 43.1 per cent between 2004 and 2015). Total poverty and extreme poverty lines increased in similar proportions (from 42.9 per cent and 58.5 per cent, respectively). The food price index increased by an average rate of 5.3 per cent. To what extent did these variations affected inequality and poverty?

In the calculation of these effects, we must consider, on one hand, the fact that consumption structures differ according to spending levels and, on the other, the spatial difference in price levels. As such, the deflation of nominal expenditure considers the specific weighting of the different groups of expenditure per household. In terms of the disparities of price levels, we deflated expenditure with the spatial deflator of multilateral prices calculated by the Peruvian National Institute of Statistics (Instituto Nacional de Estadistica e Informatica: inei). Figure 8.7 shows the differences between the evolution of nominal expenditure and of real expenditure for each of the expenditure deciles between 2004 and 2015. It is evident that losses due to inflation were higher for the poorest deciles while the richest deciles suffered no losses due to inflation.

This differentiated effect of inflation by decile is also correlated with poverty reduction. The incidence of poverty is found by comparing the nominal expenditure of the current year with the poverty line of the previous year considering the current year’s prices. The direct effect of the increase in prices is, as such, the difference between the incidence using the line at last year’s prices and the nominal expenditures of the current period with regard to the variation of the incidence with lines at prices that have been indexed to inflation (Figure 8.8). If we look at the period 2010–15, poverty fell by 9 points but if no inflation had been in play, the reduction would have been 17.7 points. The net effect is that inflation has offset the drop in poverty by 8.7 points (or 11.6 points if we consider the period 2004–15).

## 6 Vulnerability of Households

The decrease in poverty observed over the last decade, measured through static indicators (of incidence, gap and severity), does not reveal the dynamic of poverty, particularly in terms of the incoming and outgoing flows of poverty. The same poverty rate is compatible with very different sizes of flows of population that enter and exit poverty—which would, in this case, give the same net balance. It is important to note that economic insecurity and vulnerability to poverty (measured as a risk of falling into poverty) should be considered dimensions of well-being.

To approximate the measurement of vulnerability, we consider the individual trajectories of households using longitudinal data (household panel) and identify, for two successive periods, the proportion of households that remain in poverty, those that fall into poverty, those that exit poverty and those that have never been poor.

Figure 8.9 shows that the reduction in poverty measured was due primarily to a decrease in the number of households that have always been poor while the proportion of vulnerable households remains practically constant, at around 10 per cent of total households.

Seen from another angle, we can consider the population that has not been poor every year, but which experienced at least one episode of poverty of at least one year during the period, as part of the vulnerable population. The Peruvian National Household Survey (Encuesta Nacional de Hogares, enaho) uses a subsample panel that follows the same households every year. Table 8.2 shows the results of the household panel (2007–11). Slightly more than one household out of ten (12.4 per cent) was poor in each and every one of the five years considered while a little less than half (47.2 per cent) experienced no episodes of poverty in the period 2007–11. The most surprising result is the very high proportion of the population living in households that are vulnerable to poverty (transitory poor), given that four out of ten experienced at least one episode of poverty. This proportion contrasts strongly with the proportion of poor in 2011, the last year of the period. In other words, in 2011, the poverty ‘halo’ comprised 66.3 per cent of the population and not only the 25.9 per cent that lived in poverty in that year.

## 7 Social Programmes and Poverty Reduction

We have seen that part of the reduction in poverty is due to growth that is moderately pro-poor. In this section, we will examine the extent to which targeted public transfers explain this pro-poor bias in growth. To accomplish this, we will examine different components of real expenditure based on forms of acquisition. Total expenditure includes both purchases and goods and services acquired through self-consumption, self-supply, in-kind payments, transfers from other households and public transfers. We will focus primarily on the last component, public transfers.

Public transfers include different transfer programmes (for example, the programmes Juntos, Pension 65, Beca 18, etc.) that have a direct impact (through the transfer amount) and an indirect impact (through induced behaviours) on the well-being of households.

In 2014, public transfers represented 3.4 per cent of average total household expenditure; this percentage does not differ significantly from one year to the next (minimum variations, limited within the range of 3.1 per cent to 3.6 per cent). The ‘efforts’ made by the various governments in the fight against poverty has not led them to allot larger transfer amounts to households in proportion to average household expenditure. Given that these are targeted transfer programmes, we should examine the relative importance of these public transfers in different segments of expenditure distribution. Figure 8.10 also presents these proportions for the five poorest deciles and for the national average. For decile 1, the poorest, public transfers reached a peak in 2006 (11.8 per cent of total expenditure) and fell during three successive years to 7.2 per cent in 2009. Transfers then rebounded in 2010 only to fall again in 2012; growth reported in the final two years brought the transfer amount back to levels comparable to those seen in 2007 (8.4 per cent of total expenditure in 2014). The profile of the evolution observed for the second and third poorest deciles is very similar to that of the five poorest deciles but the proportion of public transfers represents an average of 2.2 and 3.6 percentage points less, respectively. In the period 2004–14, public transfers represented, on average, 8.5 per cent of the expenditure of the extremely poor. These proportions remained relatively stable over that decade.

What would the poverty level have been without public transfers? If we consider only the direct impact of such transfers, in Figure 8.11 we see that the impact would have increased over time. In 2004, the absence of transfers would have increased poverty by 1.9 supplementary points while in 2014 poverty would have increased by 3.9 points. Two sub-periods can be identified: the first comprises the years 2004–08, during which the absence of public transfers would have meant a 2.6 point increase in the incidence of poverty, which is lower than the amount found in the second period, which encompasses the years 2009–14 (1.3 points of supplementary poverty.)

What amount would the non-poor have to transfer, as a proportion of their total household expenditure, to allow poor households to cover the cost of a basic food basket? This would give us an idea of the dimensions that policies with regard to taxation and to transfers to households must contemplate in order to eradicate poverty.

It is important to note that the strong reduction in poverty was accompanied by a drop in the amount associated with the redistributive effort over the course of the period examined. Although ‘eliminating’ total poverty would have involved, in 2004, mobilising the equivalent of about 24 per cent of the expenditure of non-poor households, in 2015 this percentage fell to only 3 per cent (Figure 8.12). If the objective were more modest and consisted solely of eliminating extreme poverty rather than total poverty (extreme and non-extreme), then the redistributive effort would have represented about 10 per cent of the expenditure of the non-poor in 2004 and only 1 per cent in 2015. In other words, given fiscal reserves and the fact that fiscal pressure is low in Peru (around 16 per cent) in comparison to countries with comparable gdp levels, suppressing poverty through transfers to poor households is an attainable goal.

## 8 The Evolution of Inequality

The levels of income inequality, measured with the Gini coefficient, are relatively high and show a slight downward trend beginning in 2008 (Figure 8.13). The same trend is observed in the case of real per capita expenditure inequality. The income Gini coefficients fluctuate between 0.51 and 0.44 and those relative to expenditure, between 0.41 and 0.35 (Annex, Figure A.8.2).

Despite the important gaps in income and expenditures between households in urban and rural areas, the levels of inequality within each of these ambits are high and very similar.

Another way of examining changes in inequality is by comparing the relative shares of total expenditure of each of the quintiles of distribution for two different periods (Figure 8.14). If we consider total expenditure and its distribution in quintiles, in 2004 and 2015 we can see that the decrease in inequality led to the fifth richest quintile losing 5 percentage points, redistributing one percentage point to each of the remaining quintiles with the exception of the second poorest quintile, which received two percentage points of expenditure.

The ratio of average expenditures for the richest quintile to those of the poorest quintile reached a peak in 2007 and declined progressively and regularly until 2011. From that point on, gap reduction is basically imperceptible. There are no significant changes in other portions of the distribution if we compare the ratios of the 75th percentile with those of the 25th and 50th percentiles given that these percentiles remain basically constant throughout 2004–15.

## 9 The Polarisation of the Distribution of Income

The level and evolution of inequality, as measured by the Gini coefficient, tell us little about expansion or the lack thereof in the middle segments’ share of total income (often considered as equivalent to the expansion of the middle class) or if, on the contrary, we are facing a growing polarisation in income. Polarisation can lead to a loss of social cohesion and to higher levels of social conflict whereas the expansion of the middle class drives more citizen participation to monitor the political class and public policies.

To measure polarisation, we will use the index proposed by Wolfson, which is directly related to the Gini index. The Gini index measures the area between the diagonal and the curve of Lorenz in relation to the lower area of the diagonal; the polarisation index examines the relative importance of both extremes of the distribution (Figure 8.15).

Our estimates for the polarisation index for both income and real per capita expenditures show a downward trend, which is more pronounced in the case of income than in that of expenditures (Figure 8.16). We cannot discard the possibility that there is an underestimation of the degree of inequality that stems from the fact that it is difficult for surveys to capture certain sources of income in the richest households. Rejection rates (unwillingness to participate in surveys), under-declaring and sampling problems also impede an acceptable estimate of very high income. Comparing income aggregates estimated by national accounts and by household surveys has led several authors to attempt to correct the Gini coefficients. Nevertheless, very often the hypotheses made concerning the distribution of discrepancies are, by direct consequence, responsible for the observed results although there is no way of corroborating these hypotheses. Combining the information from household surveys with tax records is a promising but still pending task in Peru.

## 10 The Contributions of the Job Market and Social Policies to Variations in Poverty and Inequality: A Shapley Decomposition

A reduction in poverty or in inequality may be associated with the behaviour of labour income, remittances or private transfers, extraordinary income or even public transfers through different social programmes. As seen earlier in this chapter, despite differences in political orientations, the countries in the region experienced a strong reduction in poverty and inequality. It is valid to question if this reduction was achieved through the same channels: the market, for those with policies inspired by the Washington Consensus, or public transfers, in the case of countries with heterodox policies.

In an attempt to answer this question, we will use the decomposition method proposed initially by Barros et al. (2006) and subsequently applied by Kakwani et al. (2009) in Brazil and the approach used to analyse Chile, Brazil and Mexico proposed by Zepeda et al. (2009), and later extended by Azevedo et al. (2013a). This method is based on the additive nature of different components of income and on the construction of a counterfactual distribution that removes one component of income at a time and then recalculates the indicators to identify their contribution to changes in poverty and inequality. The algorithm developed by Azevedo et al. (2012) resolves issues regarding the sensitivity of results to the order in which each component of income is considered. To avoid this problem, all possible combinations are calculated and the average of their respective contributions is used as a measure of its contribution (Azevedo et al., 2012).

In Figure 8.17, each quadrant represents the relative contribution of each income component (grouped into five main categories) to variations in poverty and inequality between 2004 and 2015. The importance of public transfers, which mainly correspond to diverse social programmes, is particularly noteworthy in that they account for 16 per cent and 26 per cent of the reduction in the incidence of poverty and of the poverty gap, respectively. These transfers have also played a significant role in the observed reduction in inequalities, contributing 40 per cent to the observed drop in the Gini coefficient.

The contrast between the 2004–10 period and that of 2010–15 reveals an important shift in the sources of changes in poverty and inequality. On the one hand, the contribution of labour income to poverty reduction fell (from 80 per cent to 52 per cent). This was, without a doubt, linked to the downturn and in some cases to a recession in productive activities for the internal market, activities that absorb the majority of qualified workers. On the other hand, this significantly increased the contribution of public transfers to both poverty reduction and inequality reduction (see Table A-1 in the Annex). In the period 2004–10, transfers accounted for only one-third of the reduction in inequality (Gini), while in 2010–15 this same component represented half of the reduction in inequality.

A similar exercise, conducted by Azevedo et al. (2013b) for 14 Latin American countries for the period 2000–10, allows us to compare the case of countries with orthodox policies with that of those countries that opted not to change the patterns of growth based on primary exports and prudent approaches to fiscal and monetary policy management. This also matches the analyses by country presented by Lopez-Calva and Lustig (2010). Nevertheless, the relative contribution of pensions, non-labour income, and changes in the workforce in households compared to changes in inequality differs among countries. In countries with more highly developed institutions and better living conditions (Argentina, Brazil and Chile and to a lesser extent Colombia), income from pensions and transfers plays a more important role than it does in other countries. Peru and Ecuador share similar characteristics in terms of the preponderance of the contribution of labour income. Peru is unique in that it is the only country in the study that registers a negative contribution (which increases inequality) for non-labour income (−14 per cent). This result matches our estimates (a negative contribution of −11 per cent to Gini and −19 per cent to the Theil index, Table A-1). Our estimates for the most recent period (2010–15) show that this contribution ceases to be significant in the case of Gini but remains negative in the Theil case.

## 11 The Perception of Inequality and the Role of the State in Redistribution

The consequences of higher or lower degrees of economic inequality and social pressure that seeks to drive redistributive policies depend on the population’s perception of distributive justice, a preference for more income equality, the factors that have an impact on disparities in quality of life, and what role is devoted to the state with regard to reducing inequalities. To explore these issues, we analyse the Latinobarómetro opinion surveys and the World Values Surveys, which help us situate Peru’s case in the Latin American context (Figure 8.18).

A persistent and surprising fact is that a high percentage of the population of Peru perceives the inequality level to be very unjust. In fact, this figure exceeds those posted by countries whose inequality levels (measured by the Gini index) are actually higher. Panama, Ecuador, Bolivia and the Dominican Republic have levels of inequality that are higher than those in Peru. Nevertheless, a smaller proportion of the population of these countries is of the opinion that such a distribution is unfair or very unfair. In Peru, the origin of inequalities is considered, to a certain extent, less ‘legitimate’ than in some of the other countries in the region that have comparable or higher levels of inequality.

An analysis of the nine latest Latinobarómetro surveys sheds light on how perceptions of distributive justice evolved over time during the period of economic growth (Figure 8.19). In the period of crisis (1997–2001), the percentage of the population that considered distribution fair or very fair was just 5.2 per cent. In the initial phase of expansion, this proportion fluctuated between 8.4 per cent and 12.8 per cent. Tolerance of inequality seems to have increased moderately during the last five-year period, when the proportion of the population that believed that distributive justice was fair or very fair was equivalent to 16.5 per cent of the total.

The perception that the distribution is unfair or very unfair does not mean, in a mechanical sense, that the population believes that incomes should be levelled. The World Values Survey did not reveal that the population adhered consistently over time to a meritocratic conception of distribution that in general posits that he or she who tries harder should be paid more. Distributive injustice could be more related to the obstacles (discrimination, inequality of opportunities and inequality of wealth) that generate a scenario in which the rules of the game are not the same for all. A plausible explanation of the preference for a meritocratic conception of distribution is linked to the very high percentage of independent, informal workers in the country, who—as their own ‘bosses’—have an income that is not part of a salary mediation scheme and instead comes directly from the market, which rewards greater or lesser effort. This may lead some to lose sight of the fact that these individuals compete with other informal workers to obtain a ‘piece of the pie’, since the demand for their products/services does not change with alterations in the number of informal workers. In sectors with low entry costs, it will be necessary to spend more hours working in order to maintain income levels. The absence of a potentially conflictual capital/work relationship means that the relations of subordination in which independent workers are embedded—independent workers who profess that they are in favour of meritocratic rules for income generation—remain invisible.

A comparison of countries in the region that have a lower proportion of independent workers than Peru demonstrates even more clearly the specificity of meritocratic and non-redistributive preferences for income equality. This meritocratic conception was found in less than half of the opinions registered in Argentina, Brazil, Chile, Colombia and Uruguay, while in Peru approximately 8 out of 10 adhere to this belief (Table 8.3).

The preponderant idea that wealth is expandable, and can increase in a way that is sufficient for all, runs counter to the idea that wealth is fixed and only a few will enrich themselves at the expense of others, and is consistent with the meritocratic conception of income distribution. At the same time, not seeing wealth as fixed attenuates potential distributive conflicts, particularly when this vision is shared by the upper class and the middle and working classes. If we compare (Table 8.4) the year 2008, which posted a peak in growth, with the year 1996, which marked the end of a deep economic crisis, we see a progression in the proportion of individuals whose perception leans towards the idea that wealth is expandable and does not give rise to distributive conflicts. This progression is situated at around 10 percentage points for the middle class, the working class and the lower class, as well as for the upper and upper-middle classes.

The opinion that an individual’s income should be aligned with his or her efforts, coupled with the conception that wealth is expandable and the belief that wealth can be obtained by some without compromising the wealth of others, is consistent with the majority opinion that the government should have less of a stake in regulating income obtainment, which is best left to individuals. Here, nonetheless, we observe some differences between the middle and working classes and the upper and upper-middle classes (Table 8.5). After having assigned a more important role to the government with regard to the distribution of income in 1996 and 2001 (a period of a drop in income), in 2008 a lower proportion of the lower classes assigned a preponderant role to the government with regard to efforts to distribute income.

## 12 Fiscal Policy as a Redistributive Instrument

The downturn in Chinese growth, which had been one of the main drivers of growth in various countries in the region, has also implied the end of the cycle of high prices for raw materials and as such a significant drop in fiscal resources that will inevitably have repercussions that may lead to a contraction in public expenditure. It is in this new context of slower growth that it is likely that questions will be raised about the possible ways through which poverty and inequality could be reduced.

Latin America has often been characterised as the region with the world’s highest level of income inequality. Nevertheless, it is important to note that the distribution of primary income or market income in the member countries of the Organisation for Co-operation and Economic Development (oecd) is as—if not more—unequal than that seen in many Latin American countries. It is also important to remember that fiscal policies and social expenditure are the mechanisms that offset, to a large extent, market disparities. In the case of Latin American countries, fiscal policy plays a very marginal role in attenuating income inequalities given that indirect taxes (on added value) predominate and are regressive by nature (Lustig et al., 2013b). Social expenditure has a redistributive impact on poor households but, as we have seen, its reach is very limited in Peru’s case (Figure 8.20).

If we compare the situation in Peru with that of other countries in the region, it is evident that income taxes play a less important role than social expenditure, and than education and health expenditure in particular (Figure 8.21). Cash transfers are more important in Chile, Mexico, Uruguay and Brazil.

The margin for implementing policies via which to fulfil the dual objectives of reducing poverty and inequalities is very broad (Goñi et al., 2011; Blofield, 2011). Nevertheless, this option will hit a wall when it comes to generating social consensus with regard to its timeliness and appropriateness. The findings regarding the perception of distributive justice and the role assigned to the market and that assigned to the state allows us to predict that redistributive fiscal policies have a long way to go before they can be institutionalised as policies that combat poverty and inequality.

## 13 Conclusion

In the new context of low growth, poverty reduction must be accompanied by policies that aim to reduce inequality—policies that must, on one hand, drive growth in productive activities for the internal market and, on the other, propitiate a demand shock through an increase in social expenditure. This will offer the additional benefit of contributing to reducing gaps in access to quality public goods, which will, in turn, reduce inequality of opportunities. The viability of an alternative strategy for development in this new context requires a different vision of tax policies, which—unlike the frameworks in place in neighbouring countries—have not been used as a policy instrument for achieving more redistributive justice. The amounts needed to close the poverty gap represented barely 2.9 per cent of the expenditure of non-poor households in 2015. It is commonly believed that after the spectacular reduction in poverty posted in the last decade, it may be more difficult to reduce poverty and inequality further given that the households that have yet to exit poverty fall very far below the poverty line. Nevertheless, this idea does not hold up if we test the data: the aforementioned reduction in poverty was accompanied by a reduction in poverty gaps. The low level of indebtedness and the reserves accumulated during high-growth phases (in terms both of currencies and of fiscal reserves) provide a wide margin for applying distribution policies via more social expenditure, improving quality coverage for education and health and universalising social programmes (Pension 65 and Cuna Mas, among others). The redistributive effect of both types of interventions in the countries of the region falls significantly short of that observed in oecd countries.

The Peruvian case combines a strong dependency on raw materials with direct distributive policies (conditional transfer programmes) and indirect distributive policies (distribution of the canon5 and mining and oil royalties to local authorities). The vulnerability of this social model will become manifest in the new phase of weak or moderate growth and will simultaneously affect fiscal income and its distributive capacity as well as household incomes. In this context, where the tax rate and the fiscal base are low and social expenditures barely increase, the fragility of the growth model will not be remediated without a social pact in favour of better distribution of income, which entails increasing fiscal pressure on income to finance social expenditure in favour of the poorest homes. The current political context in Peru, and the population’s preferences, are unlikely to favour this option. Nevertheless, problems of public safety, which are without a doubt associated with an unequal distribution of the fruits of growth, have become a top priority among the population and may pressure the Peruvian state into practicing more redistribution or more repression. The new government will have to resolve this dilemma.

## Annex

### Non-parametric Decomposition of the Variations of Poverty: Growth and Redistribution*

This method is based on the link that exists between the Lorenz curve and the cumulative density function (cdf). As pointed out by Maasoumi and Mahmoudi (2013), the cdf function can be used to highlight the redistribution effect given that the Lorenz curve (normalised for the average value) is nothing more than the inverse of the distribution function:

L(p) = 1/μcdf −1(π) dπ
.

Given that it is postulated that cdf1 and cdf2 have the same average value, Maasoumi and Mahmoudi (2013) show that

The portion of the change in the incidence of poverty that can be attributed to growth is summarised, according to the authors cited, with the following expression:

The variation in the poverty rate is equal to $ΔP=PF2;z−PF1z.$

The intersection between the vertical line (of poverty) and the F1 curve gives us (on the axis of the coordinates) the incidence of poverty (H1).

A new curve is constructed (F1*) of accumulated frequency (cdf), multiplying the expenditures of all of the households (F1) by the average rate of expenditure growth.

## References

• AzevedoJ.-P.V.Sanfelice and M.C.Nguyen (2012) Shapley Decomposition by Components of a Welfare Measure (Washington, D.C.: World Bank) mimeo.

• AzevedoJ.-P.V.Sanfelice and G.Inchauste (2013a) Decomposing the Recent Inequality Decline in Latin America Policy Research Working Paper 6715 (Washington, D.C.: World Bank) http://documents.worldbank.org/curated/en/597661468054543060/Decomposing-the-recent-inequality-decline-in-Latin-America (accessed on 14 September 2016).

• AzevedoJ.-P.G.InchausteS.OlivieriJ.Saavedra and H.Winkler (2013b) Is Labor Income Responsible for Poverty Reduction? A Decomposition Approach Policy Research Working Paper 6414 (Washington, D.C.: World Bank) http://documents.worldbank.org/curated/en/882051468340821632/Is-labor-income-responsible-for-poverty-reduction-a-decomposition-approach (accessed on 14 September 2016).

• BarrosR.M.de CarvalhoS.Franco and R.Mendonça (2006) ‘Uma análise das principais causas da queda recente na renda brasileiraEconômica8(1) pp. 117147.

• BlofieldM. (ed.) (2011) The Great Gap: Inequality and the Politics of Redistribution in Latin America (Pennsylviana: Pennsylviana State University Press).

• BourguignonF. (2015) ‘Appraising income inequality databases in Latin AmericaJournal of Economic Inequality13(4) pp. 557578 DOI: 10.1007/s10888-015-9304-4.

• de FerrantiD.G.PerryF.Ferreira and M.Walton (2004) Inequality in Latin America: Breaking With History? World Bank Latin American and Caribbean Studies (Washington, D.C.: World Bank) https://openknowledge.worldbank.org/handle/10986/15009 (accessed on 14 September 2016).

• de la TorreA.E.Levy YeyatiG.BeylisT.DidierC.Rodriguez Castelan and S.Schmukler (2014) Inequality in a Lower Growth Latin America LAC Semiannual Report October (Washington, D.C.: World Bank) https://openknowledge.worldbank.org/handle/10986/20413 (accessed on 14 September 2016).

• ECLAC IEF (2014) Los efectos de la política fiscal sobre la redistribución en América Latina y la Unión Europea Colección de Estudios N°8 (Madrid: Programa EUROsociAL) http://sia.eurosocial-ii.eu/files/docs/1412088027-Estudio_8_def_final.pdf (accessed on 1 July 2017).

• FosterJ.-E. and M.Wolfson (2010) ‘Polarization and the decline of the middle class: Canada and the U.S.Journal of Economic Inequality8 pp. 247273http://hdl.handle.net/10.1007/s10888-009-9122-7.

• GaspariniL.G.CrucesL.Tornarolli and M.Marchionni (2009) A Turning Point? Recent Developments on Inequality in Latin America and the Caribbean Working Paper 81 (La Plata: CEDLAS) http://cedlas.econo.unlp.edu.ar/archivos_upload/doc_cedlas81.pdf (accessed on 14 September 2016).

• GaspariniL. and N.Lustig (2011) The rise and fall of income inequality in Latin America Working Paper 118 (La Plata: CEDLAS) http://cedlas.econo.unlp.edu.ar/download.php?file=archivos_upload/doc_cedlas118.pdf (accessed on 14 September 2016).

• GoñiE.H.López and L.Servén (2011) ‘Fiscal Redistribution and Income Inequality in Latin AmericaWorld Development39(9) pp. 15581569 DOI: 10.1016/j.worlddev.2011.04.025.

• IMF (International Monetary Fund) (2016) World Economic Outlook. Too Slow for Too Long. (Washington, D.C.: IMF).

• InchausteG.J.-P.AzevedoB.Essama-NssahS.OlivieriT.Van NguyenJ.Saavedra-Chanduvi and H.Winkler (2014) Understanding Changes in Poverty. Directions in Development—Poverty (Washington, D.C.: World Bank) https://openknowledge.worldbank.org/handle/10986/19445 (accessed on 14 September 2016).

• KakwaniN.M.Neri and H.H.Son (2009) Linkages between pro-poor growth social programs and labour market: the recent Brazilian experience Research Paper 2009.26 (Helsinki: UNU-WIDER) http://hdl.handle.net/10419/45140.

• López-CalvaL.F. and N.Lustig (2010) Declining Inequality in Latin America: A Decade of Progress? (Washington, D.C.: Brookings Institution Press).

• LustigN.L.-F.López-Calva and E.Ortiz-Juárez (2013a) ‘Declining Inequality in Latin America in the 2000s: The Cases of Argentina, Brazil, and MexicoWorld Development44 pp. 129141 DOI: 10.1016/j.worlddev.2012.09.013.

• LustigN.C.Pessino and J.Scott (2013b) The Impact of Taxes and Social Spending on Inequality and Poverty in Argentina Bolivia Brazil Mexico Peru and Uruguay: An Overview CEQ Working Paper 13http://econ.tulane.edu/RePEc/pdf/tul1313.pdf (accessed on 15 September 2016).

• MaasoumiE. and V.Mahmoudi (2013) ‘Robust growth-equity decomposition of change in poverty: The case of Iran (2000–2009)The Quarterly Review of Economics and Finance53 pp. 268276.

• RavallionM. (2016) The Economics of Poverty: History Measurement and Policy (New York and Oxford: Oxford University Press).

• SzékelyM. and P.Mendoza (2015) ‘Is the Decline in Inequality in Latin America Here to Stay?Journal of Human Development and Capabilities16 pp. 397419. DOI: 10.1080/19452829.2015.1050320.

• World Bank (2016) Global Economic Prospects (Washington, D.C.: World Bank).

• YamadaG.J.-F.Castro and J.-L.Bacigalupo (2012) ‘Desigualdad monetaria en un contexto de rápido crecimiento económico: El caso reciente del PerúRevista Estudios Económicos24 (Lima: Banco Central de Reserva del Perú) pp. 6577http://www.bcrp.gob.pe/docs/Publicaciones/Revista-Estudios-Economicos/24/ree-24-yamada-castro-bacigalupo.pdf (accessed on 15 September 2016).

• ZepedaE.D.AlarcónF.Veras and R.Osorio (2009) Changes in earnings in Brazil Chile and Mexico: disentangling the forces behind pro-poor change in labour markets Working Paper 51 (Brasilia: International Policy Centre for Inclusive Growth) http://www.ipc-undp.org/pub/IPCWorkingPaper51.pdf (accessed on 15 September 2016).

Diagnosing the evolution of inequality in Latin America in the last decade is subject to several methodological limitations linked to problems with household surveys in terms of estimating non-labour income without bias and impediments to capturing information for the highest income percentiles (Bourguignon, 2015; Ravallion, 2016; Yamada et al., 2012).

In this section, we use poverty and inequality indicators from the sedlac database. The primary sources of data are as follows: Bolivia, 2003 is in fact 2002; Argentina, ephc 2003–12 is the average of the first and second half years; Brazil, Nueva pnad 2003, 2004–13 including the northern rural region; Peru, enaho; Colombia, ech 2004–05, geih 2008–13; Ecuador, enemdu. This database was homogenised in the construction of the aggregate incomes as well as the values of the poverty lines (usd 4 in purchasing power parity (ppp) from 2005). We decided not to use eclac’s indicators given the existence of a number of methodological problems and the lack of detailed information regarding imputations (Bourguignon, 2015).

For an analysis of growth and inequality reduction in the first decade of the twenty-first century, see Gasparini and Lustig, 2011; Székely and Mendoza, 2015. For national official poverty rates (2003–15) see Annex, Figure A.8.3.

The odds ratio of rural households went from 1.42 to 2.03 and from 2.53 to 3.42 for total poverty and extreme poverty, respectively.

The government transfers half of the taxes collected from the mining industry to the regions and municipalities where the minerals are extracted. This is what is called the ‘canon’.

If the inline PDF is not rendering correctly, you can download the PDF file here.

# Alternative Pathways to Sustainable Development: Lessons from Latin America

## Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 705 705 7


text:
4.9

# Worded questions and working with three or more fractions

Sometimes a question written in words can appear more difficult. Written questions require the student to extract the important information, create a mathematical model and identify the type of calculation they are needing to perform.

## Example

Steve and Sheila buy a digital TV recorder. Steve fills $\frac{2}{5}$ of the recorder with his programmes and Sheila fills $\frac{1}{3}$ of the recorder with her programmes.

a) What fraction of the recorder is full?
b) What fraction of the recorder is empty?

Steve deletes one of his recordings which takes up $\frac{1}{10}$ of the recorder.

c) What fraction of the recorder do Steve’s recordings fill now?

From the question we have identified that there are two fractions representing two parts of the same whole, the digital recorder: $\frac{2}{5}$ and $\frac{1}{3}$.

To answer part a) we need to make the denominators the same:

The denominators of our two fractions are 3 and 5, both prime numbers, so we use the lowest common multiple, 15, as our denominator.

To get our equivalent fractions of fifteenths, we multiply the numerator and denominator of each fraction as shown above. $\frac{2}{5}$, multiplying top and bottom by 3, becomes $\frac{6}{15}$ and $\frac{1}{3}$, multiplying top and bottom by 5, becomes $\frac{5}{15}$.

We can now add the two fractions together as they have the same denominator:

This means, $\frac{11}{15}$of the recorder is full.

b) To find how much of the recorder is empty, we know that the ‘whole’ recorder can be represented as ‘1’, and this can be represented as $\frac{15}{15}$ (using the denominator of our previous answer). It’s then straightforward to take away $\frac{11}{15}$ from $\frac{15}{15}$, to get $\frac{4}{15}$. Alternatively we could say that the difference between $\frac{15}{15}$ and $\frac{11}{15}$ is $\frac{4}{15}$, so $\frac{4}{15}$ of the disc is empty.

c) The final part of the question asks us to take $\frac{1}{10}$ of $\frac{2}{5}$ away from $\frac{2}{5}$.

First we find the equivalent fraction of $\frac{2}{5}$ with 10 as the denominator. This gives $\frac{4}{10}$.

Now we take $\frac{1}{10}$ away from $\frac{4}{10}$ to get $\frac{3}{10}$.

Hence, Steve’s recordings now take up $\frac{3}{10}$ of the recorder.

## Discuss

Work through this question again, where are the points in this question where students might make a mistake? Can you come up with any general advice for tackling worded questions? Share in the comments below.

## Problem worksheet

Now complete questions 9, 10 and 11 from this week’s worksheet.
As a reminder, the worksheet can be found in the first step of this week.

format: plain


text:
# NCERT Solution for Linear equation Chapter 4 Exercise 4.4

In this page we have NCERT book Solutions for Class 9th Maths Linear equations in two variables for EXERCISE 4.4 on page 77. Hope you like them and do not forget to like , social share and comment at the end of the page.
Question 1
Give the geometric representations of y = 3 as an equation
(i) in one variable
(ii) in two variables
Solution
(i) The representation of the y=3 on the number line is shown in below figure where y=3 is treated as an equation in one variable

(ii) This can be treated as equation in two variable as
$0 \times x + y =3$
Drawing the graph by putting x=0 and x=1

So ,it is a line parallel to X-axis

Question 2
Give the geometric representations of 2x + 9 = 0 as an equation
(i) in one variable
(ii) in two variables
Solution
(i) The representation of the $2x+9=0$ on the number line is shown in below figure where x=-9/2 =4.5 is treated as an equation in one variable

(ii) This can be treated as equation in two variable as
$2x + 0 \times y +9=0$
Drawing the graph by putting y=0 and y=2

So ,it is a line parallel to Y-axis


text:
# Register pressure in Compute Shader

I'm in the process of writing a Ray Tracer using DirectCompute / HLSL. First, eye rays are generated (one per pixel). Then, rays are traced, shaded and reflected in a loop. Also, shadow rays for each light source are generated and tested for occlusion. As a scene structure, I'm using a KD Tree with Ropes$^1$.

I have analyzed the shader with Joshua Barczak's Pyramid and it seems that I have a major problem with register pressure, mainly vectors. On a Fiji GCN architecture, the shader uses 85 SGPRs and 81 VGPRs, limiting the number of simultaneous wavefronts per SIMD to 3 (VPGR).

Furthermore, I have implemented a simple counter in my shader, that atomically increases a value when a thread starts, decreases it when a thread ends and keeps track of the maximum number of shaders that were concurrently running. I've managed to get rid of some data and bring it up from 8192 to 9216, resulting in a correlated speedup of ~%13. When I use an empty dummy shader, I get 16384 concurrent threads.

I've tried to get rid of excessive variables, especially vectors, by keeping their lifetime as short as possible, and also tried storing some variables in groupshared memory and reading / writing directly from there, all without any change in register count whatsoever.

Are there any practical tips on how to take pressure of the registers? Is this even as much of a problem as I think it is?

I have also thought about splitting my Ray Tracer into various kernels. This should, the way I understand it, take pressure of the registers significantly, but also takes quite some coding effort on my part. Does this sound like an idea worth trying?


text:
May 1, 2012
By

(This article was first published on O Beautiful Code - R, and kindly contributed to R-bloggers)

Avoid R’s warning feature. This is particularly important if you use R in production; when you regularly run R scripts as part of your business process. This is also important if you author R packages. Don’t issue warnings in your own code and treat warnings in 3rd party code as errors that are explicitly suppressed. I’ll discuss a strategy to implement this in a little bit, but first lets discuss the warning feature and the justification for this advice.

## The Warning

A warning cautions users without halting the execution of a function. It basically says “although I can and will give you an answer, there might be a problem with your inputs. Thus, the computation could be flawed.” For example, the correlation function issues a warning when an input vector has a standard deviation of 0. Rather than raising an error and halting execution, the function issues a warning and returns NA.

> cor( c( 1 , 1 ), c( 2 , 3 ) )
[1] NA
Warning message:
In cor(c(1, 1), c(2, 3)) : the standard deviation is zero

To issue a warning simply call the warning() function

Division = function( x , y )
{
if ( y == 0 ) { warning( "Divide by 0" ) }
return( x / y )
}
> Division( 1 , 0 )
[1] Inf
Warning message:
In Division(1, 0) : Divide by 0

Multiple warnings can be issued during a function call. They can originate from the function itself or from child functions. R stores all warnings and prints them out after the function returns.

ParentFunction = function( )
{
warning( "warning from parent" )
ChildFunction1()
ChildFunction2()
warning( "second warning from parent" )
return( TRUE )
}

ChildFunction1 = function() {  warning( "warning from child" ) }
ChildFunction2 = function() {  warning( "second warning from child" ) }

> ParentFunction()
[1] TRUE
Warning messages:
1: In ParentFunction() : warning from parent
2: In ChildFunction1() : warning from child
3: In ChildFunction2() : second warning from child
4: In ParentFunction() : second warning from parent

## Avoid Warning

You should avoid warnings (as a producer and a consumer) for three reasons:

Functions should either succeed or fail and it should be as simple as that. Warnings introduce a third state. A function author must consider when to issue a warning, how to message it, and how the function will proceed in the warning/caution state. Its difficult enough to write a good function without this burden. On the flip side, a function consumer must decide how to handle a warning. Do you continue or halt? Does it depend on the warning? If so, do you switch on specific warning messages or keywords within the message? What if the warning message changes with an updated package? It’s far less complex when a function just fails; there’s less for both parties to do.

2. Most warnings are actually errors.
This is anecdotal, but I’d bet it generalizes well: 80% of the time I see warnings when there’s a bug in the inputs to a function call.

3. You will miss critical errors.
In an automated/production environment dealing with warnings is a pain. You might miss warnings outright because no human will see it. Per #2 above, this could mean bad things. Otherwise, you will have to pipe a script’s output to a file, parse it to detect the warning and notify someone. Lets say you pull that off. You are still betting that someone will heed the warnings in a timely manner. Also, you’re left with a time-consuming and ambiguous task of deciding which warnings to addressed. If, however, your script just failed, I’m guessing you will find out pretty quick and you will have to pay attention to the issue.

## Step1: Stop Warning

Instead of issuing a warning, do one of the following:

1. Call stop("error message here"). This will generate an error and halt execution.
2. Add a function parameter that specifies the action to take when the inputs are malformed. This is mostly likely a boolean (logical) parameter that indicates whether the function should fail or gives the function permission to proceed. Use a descriptive parameter. Some good prefixes are ignore and fail. For example, I’ve added parameters failSd0 and ignoreSd0 below.
Cor( x , y , failSd0 )
{
if ( failSd0 && equals( sd( x ) , 0 ) ) { stop( "the standard deviation of x is 0" ) }
if ( failSd0 && equals( sd( y ) , 0 ) ) { stop( "the standard deviation of y is 0" ) }
...
}

Cor( x , y , ignoreSd0  )
{
if (  ( !ignoreSd0 ) && equals( sd( x ) , 0 ) ) { stop( "the standard deviation of x is 0" ) }
if (  ( !ignoreSd0 ) && equals( sd( y ) , 0 ) ) { stop( "the standard deviation of y is 0" ) }
...
}

Just deciding between approach #1 and approach #2 is a useful exercise. It challenges you to judge the utility of a function in the face of specific categories of input (i.e. correlation when an input vector has standard deviation of 0). You might admit that a function’s outputs have little to no value when the inputs have a particularly quality. In that case, you would use approch #1 and just stop() the execution. This is the least complex thing to do.

Although solution #2 is a bit more work, it provides a wonderful benefit to your users. It gives them information about how not call your function. In other words, it helps them minimize failure. This information is nicely embedded into the parameter list. When a user sees the parameter failSd0 it triggers the most obvious question: “Is it possible that my x or y has a standard deviation of 0? Maybe I should check the code that populates x and y just in case.”

## Step2: Strategically Maneuver 3rd Party Warnings

To consume 3rd party code (i.e. CRAN packages) that issues warnings, do the following:

1. Treat all warnings are errors.
Take the complexity out of the equation by first assuming that all warnings are in fact errors. Simply call options( warn = 2 ). R will convert the first warning it encounters into an error and halt execution. I recommend putting this into your Rprofile.site file so that this treatment is always applied.

2. Selectively suppress warnings.
There are times when a function issues a warning and you are certain that your inputs are correct and you are happy with the results. In this case, wrap the function call in suppressWarnings().

> options( warn = 2 )
> ParentFunction()
Error in ParentFunction() : (converted from warning) warning from parent
> suppressWarnings( ParentFunction() )
[1] TRUE

This is a case-by-case opt-in. Don’t wrap everything in suppressWarnings()! Be sure to add a clear justification in the comments:

# FictitiousFunction warns when myValue is 0.  myValue will always be 0.
# This only affects result\$ABC output which is fixed at NA.  We don't use this output.
# All other elements of the result list are as expected.
suppressWarnings( FictitiousFunction( myValue ) )

Going-forward, your code could find itself in a state that triggers a 3rd party warning that you’ve never seen before. Congrats! This is a quality problem. Your code will raise an error. You can investigate the issue and in turn learn more about the package you depend on and about the behavior of your own code. And you can always suppressWarnings(). Your code will become more robust and more resilient.

## Practice What You Preach

I have many thousands of lines of R code in production and I employ the strategy described above to avoid warnings. It’s a beautiful thing - I have never seen a single raw warning from R. My code simply passes or fails; there’s no third-state to deal with. Also, I sleep well knowing that my nightly processes don’t mask errors that are expressed as warnings.


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Permutation Problems

## Using the nPr function found in the Math menu under PRB  on the TI calculator

0%
Progress
Practice Permutation Problems
Progress
0%
Evaluate Permutations Using Permutation Notation

Do you know how to calculate a permutation in permutation notation? Take a look at this dilemma.

Find \begin{align*} _{8}P _{2}\end{align*}

To figure out this permutation, you will need to understand permutation notation. Pay attention and you will learn all that you need to know in this Concept.

### Guidance

Order is important in some situations and not important in others. For example, in following a cake recipe, the order in which the events take place is important. You need to crack the eggs before you mix them with the flour. Similarly, you put the icing on the cake only after it has baked.

In buying the ingredients to make a cake, on the other hand, order is not important. Does it matter if you buy the flour before the eggs or the milk before the icing? It doesn’t, so you would say that order is NOT important in buying cake ingredients.

For solving many problems in which order is important, you can use permutations. A permutation is a selection of items in which order is important. To use permutations to solve problems, you need to be able to identify the problems in which order, or the arrangement of items, matters.

We can find the number of permutations in a group if you include all members of that group. For example, suppose there are 3 cabs in front of a hotel, Acme, Bluebird, and Checker. If all 3 line up to wait for the next customer, the number of different lineups, or permutations, of 3 items taken 3 at a time is:

Again, this is the permutation for three cabs lined up three at a time. We could also say that this is three objects taken three at a time.

The most efficient way to calculate permutations uses numbers called factorials.

Factorials are special numbers that represent the product of a series of descending numbers.

The symbol for a factorial is an exclamation sign. Take a look.

To compute the values of factorials, simply multiply the series of numbers.

We can use factorials to calculate permutations.

Suppose you have 6 items and you want to know how many arrangements you can make with 4 of the items.

Order matters in this problem, so you need to find the number of permutations there are in 6 items taken 4 at a time.In permutation notation you write the following.

\begin{align*}{\color{red}_{6}}P {\color{blue}_{4}} \ \Longleftarrow \end{align*} 6 items taken 4 at a time

In general, permutations are written as:

\begin{align*}{\color{red}_{n}}P {\color{blue}_{r}} \ \Longleftarrow {\color{red}n}\end{align*} items taken \begin{align*}{\color{blue}r}\end{align*} at a time

To compute \begin{align*}{_n}P{_r}\end{align*} you write:

To compute \begin{align*}_{6}P _{4}\end{align*} just fill in the numbers:

\begin{align*}\frac{720}{2}\end{align*}

\begin{align*}360\end{align*}

Notice that it is the product of the values in descending order that tells us how many permutations are possible.

You can use this method to solve any number of permutations.

Find each permutation.

#### Example A

Find \begin{align*} _{4}P _{3}\end{align*}

Solution: \begin{align*}24\end{align*} options

#### Example B

Find \begin{align*} _{12}P _{2}\end{align*}

Solution: \begin{align*}132\end{align*} options

#### Example C

Find \begin{align*} _{8}P _{6}\end{align*}

Solution: \begin{align*}20,160\end{align*} options

Now let's go back to the dilemma from the beginning of the Concept.

Find \begin{align*} _{8}P _{2}\end{align*}

This is the same as saying 8 options taken 2 at a time.

We can multiply to solve the permutation.

\begin{align*}8 \times 7 = 56\end{align*}

There are 56 options.

### Vocabulary

Permutation
a selection of items in which order is important.
Factorial
a special number that represents the product of a series of descending values.

### Guided Practice

Here is one for you to try on your own.

Find \begin{align*} _{7}P _{3}\end{align*}

Solution

Step 1: Understand what \begin{align*} _{7}P _{3}\end{align*} means.

\begin{align*} {\color{red}_{7}}P {\color{blue}_{3}} \ \Longleftarrow \end{align*} 7 items taken 3 at a time

Step 2: Set up the problem.

Step 3: Fill in the numbers and simplify.

There are 210 possible permutations.

### Practice

DirectionsFind each permutation.

1. Find \begin{align*} _{7}P _{2}\end{align*}
2. Find \begin{align*} _{6}P _{3}\end{align*}
3. Find \begin{align*} _{5}P _{4}\end{align*}
4. Find \begin{align*} _{5}P _{5}\end{align*}
5. Find \begin{align*} _{9}P _{3}\end{align*}
6. Find \begin{align*} _{9}P _{7}\end{align*}
7. Find \begin{align*} _{11}P _{3}\end{align*}
8. Find \begin{align*} _{12}P _{3}\end{align*}
9. Find \begin{align*} _{6}P _{2}\end{align*}
10. Find \begin{align*} _{14}P _{3}\end{align*}
11. Find \begin{align*} _{15}P _{3}\end{align*}
12. Find \begin{align*} _{11}P _{4}\end{align*}
13. Find \begin{align*} _{16}P _{2}\end{align*}

Directions: Use permutations to solve each problem.

1. Mia has 7 charms for her charm bracelet – a heart, a moon, a turtle, a cube, a bird, a hoop, and a car. Into how many different orders can she arrange the 7 charms?
2. One of the charms in Mia’s bracelet in problem 6 above fell off. How many fewer arrangements are there now?

### Vocabulary Language: English

combination

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
factorial

factorial

The factorial of a whole number $n$ is the product of the positive integers from 1 to $n$. The symbol "!" denotes factorial. $n! = 1 \cdot 2 \cdot 3 \cdot 4...\cdot (n-1) \cdot n$.
n value

n value

When calculating permutations with the TI calculator, the n value is the number of objects from which you are choosing.
Permutation

Permutation

A permutation is an arrangement of objects where order is important.
r value

r value

When calculating permutations with the TI calculator, the r value is the number of objects chosen.


text:
# How do you take the derivative of tan(x - y) = y/(1 + x^2) ?

Sep 16, 2015

Have a look, but check my maths.

#### Explanation:

In this case you'll need to derive implicitly considering that here $y$ represents a function of $x$ (a little bit...difficult to work out!) and we derive also $y$:
so if you want, say, to derive $y$ you get $1 \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$

so we get (using the Chain Rule on the left and the Quotient Rule on the right):

$\frac{1}{\cos} ^ 2 \left(x - y\right) \cdot \left[1 - 1 \frac{\mathrm{dy}}{\mathrm{dx}}\right] = \frac{\frac{\mathrm{dy}}{\mathrm{dx}} \left(1 + {x}^{2}\right) - 2 x y}{1 + {x}^{2}} ^ 2$

rearrange to isolate $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\frac{{\left(1 + {x}^{2}\right)}^{2}}{\cos} ^ 2 \left(x - y\right) - \frac{{\left(1 + {x}^{2}\right)}^{2}}{\cos} ^ 2 \left(x - y\right) \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dx}} \left(1 + {x}^{2}\right) - 2 x y$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left[\left(1 + {x}^{2}\right) + \frac{{\left(1 + {x}^{2}\right)}^{2}}{\cos} ^ 2 \left(x - y\right)\right] = 2 x y + \frac{{\left(1 + {x}^{2}\right)}^{2}}{\cos} ^ 2 \left(x - y\right)$

Multiply the non-fractions by ${\cos}^{2} \frac{x - y}{\cos} ^ 2 \left(x - y\right)$ to get common denominators, cancel out those denominators, then divide by the resultant left-side bracketed contents:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x y {\cos}^{2} \left(x - y\right) + {\left(1 + {x}^{2}\right)}^{2}}{\left(1 + {x}^{2}\right) {\cos}^{2} \left(x - y\right) + {\left(1 + {x}^{2}\right)}^{2}}$


text:
# lab01 : Picobot

lab01 true Picobot Tue 08/08 08:30AM Thu 08/10 05:00PM

If you find typos or problems with the lab instructions, please report them on Piazza

# Learning Goals

• Reason about the behavior of a simulated robot in a simple state-machine language
• Methodically trace the execution of a simple state-machine program
• Find and fix errors in a simple state-machine program
• Write a program to control a simulated robot so that it navigates through both an empty world and a maze environment
• Save and submit your files for feedback

The main goals for this part of the lab are (and remember, these are minimal goals, eager for extensions):

• Write a program that directs your robot, Picobot, to completely “cover” (i.e., visit all the non-wall spaces) an empty room
• Write a program to do the same for a maze

## First steps

• Create a new github repo under the “ucsd-cse-spis-2017” organization. Your github repo should be called spis17-lab01-Name1_Name2 (where Name1 and Name2 are the pair names).

• Create three text files in your git repo: answers.txt, emptyroom.txt and maze.txt. The file answers.txt should contain your answers to all the questions in the following section. emptyroom.txt and maze.txt should contain your Picobot code for the empty room and maze exercises. See additional instructions in the subsequent sections.

## Picobot Simulator: Feel free to jump right in…

Once you have logged on to your account on a terminal in the computer lab and managed to open a web browser (which we assume you have done if you are reading this), you are ready to dive into Picobot. You may already feel comfortable with Picobot from the exercises we did in the interactive portion of this lab. If that’s the case and you’re ready to start writing your code, you can jump right to the Picobot simulator and start writing programs:

Again, your goals are to write a program to cover the empty room and the maze. Feel free to play around for a while–playing around is often the best way to get going. But many people eventually find this frustrating and crave a deeper understanding of what’s going on. Furthermore, it is an important skill not only to solve the problems, but to understand exactly why your solution works. So once you’ve gotten your feet wet playing around, please move on to the next section.

Here are some steps we suggest to help you get the most out of your Picobot experience. Even if you’ve solved the two main goals, you should at least skim the section below to make sure your understanding is as deep as it can be. Also there are some fun challenge problems in step 5.

Read Chapter 1.2 of the book CS for All. You are welcome to read the first part of the book as well, but it won’t help you that much with Picobot, at least not directly.

As you read, make sure you can answer the following questions. Discuss the answers with your partner. Ask a tutor if you’re stuck or want to check your answers. Put your answers to the following questions in the answers.txt file in your git repo. Commit your file to submit your answers.

1. Describe the following rules in English:

 0 N*x* --> W 0
1 x**x --> N 2

2. What is the difference between the “*” and the “x” in the description of Picobot’s surroundings?

3. Write a Picobot rule that would cause Picobot to crash into a wall (Picobot will report an error if you try this)

4. In the Picobot simulator, what will happen if we replace the “X” in the fourth (i.e., last) starting rule with a W. First try to predict, and then run the simulation to see what happens.

5. In the Picobot simulator, modify the starting rules so that Picobot moves from East to West instead of from North to South.

## Program Picobot to solve the empty room

Write a Picobot program to make Picobot visit every cell in the empty room, no matter where it starts. Hints to get you started: As you saw if you answered question 4 above, a simple change gets you a good part of the way there, covering half the room. But you’ll probably notice picobot gets stuck when it reaches the NW or SW corner, because it doesn’t have a rule that applies to this surrounding. You’ll need to add one. However, before you do, you should think carefully about your algorithm. What is it that you want Picobot to do when it reaches one of those corners? One option is to immediately head to the east wall, and then start moving N, S and W as it was doing before. Another option is to begin a N, S, and E sweep back the other way. Either way you are going to need at least one new state. We encourage you to think carefully about what this state or states will mean before you start implementing rules. Once you are finished with this problem, save your code for the empty room solution in the file emptyroom.txt in your git repo, before moving on to the next problem.

## Program Picobot to solve the maze

Finally, once you have successfully covered the empty room, use the MAP “–>” button to find the maze, and implement the maze covering program using the “right hand rule” as described in the book in Section 1.2.8. Again, we provide some hints to get you started:

Think carefully about your states. You will likely want to use 4 states where each state represents a different direction that you are currently moving.

Consider carefully which wall your hand will be on if you are heading in each direction, and this will help you figure out which direction to move, and when to change states. For example, if you are heading north, your hand is on the east wall. So as long as there is a wall to your east, and no wall to the north, you keep going north, dragging your hand on the wall. If the wall to the east suddenly disappears, then you turn in that direction to follow the wall. If you suddenly hit a wall to the north, then your hand sweeps around the corner between the eastern and northern walls and you turn in the process so that you are facing west and your hand is now on the northern wall. Make sure you understand ALL of the possible turns and transitions before you start writing rules. If anything is unclear, ask a tutor. We repeat: do not write rules until you have a firm understanding of what each state means, and how different environments affect your direction and forward progress. It’s all about keeping your right hand on the wall… Before continuing, save your code for the maze solution in the file maze.txt in your github repo. You can use the same file to solve the maze for multiple rooms.

## SUPER-challenge problems

When you’ve mastered the maze and you’re ready for more, try some optional SUPER-challenge problems. You don’t need to submit these. For example, you could challenge yourself with the following problems:

• for the empty room, see if you can use only 6 rules

• for the empty room, can you build a machine that covers each cell in the room exactly once?

• for the maze, see if you can use only 8 rules

Scroll through the different maps and see if you can write Picobot programs to solve these other rooms. Can you write a single program that works for ALL provided mazes? Continuing with the Roomba vacuum cleaner analogy, you might know that the edges of a room get dirtier than the middle of the room (dust kicked up accumulates near the walls). Can you program Picobot to cover the cells that are adjacent to walls at least twice and cover interior cells exactly once? Can you formulate other criteria you might want to solve? Are there unsolvable requirements?


text:
Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 Title Keywords Abstract Author All
Search Results: 1 - 10 of 100 matches for " "
 Page 1 /100 Display every page 5 10 20 Item
 Kurt Johansson Mathematics , 2005, Abstract: The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution $\exp(-\exp(-x))$, the Gumbel distribution and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density $\exp(-x)$ and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $M=M_0+\sqrt{2S} V$, with $M_0$ diagobal with independent elements and $V$ from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.
 Paul-Olivier Dehaye Mathematics , 2006, Abstract: A classical question for a Toeplitz matrix with given symbol is to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis (Toeplitz minors, J. Combin. Theory Ser. A, 97 (2002), pp. 252--271) obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom (On the limit of some Toeplitz-like determinants, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 1194--1196) used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy- Widom identity is a differentiated version of the classical Jacobi-Trudi identity.
 A. J. McCafferty Mathematics , 2007, Abstract: In this note, we find sufficient conditions for an operator with kernel of the form $A(x)B(y)-A(x)B(y)/(x-y)$ (which we call a Tracy-Widom type operator) to be the square of a Hankel operator. We consider two contexts: infinite matrices on $\ell^2$, and integral operators on the Hardy space $H^2(\mathbb{T})$. The results can be applied to the discrete Bessel kernel, which is significant in random matrix theory.
 Physics , 2008, DOI: 10.1103/PhysRevE.79.031117 Abstract: In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists in removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristc of extreme values of an uncorrelated sequence, is obtained.
 Physics , 2015, Abstract: While originally discovered in the context of the Gaussian Unitary Ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the Tracy-Widom distribution plays an important role. In our contribution we study one-dimensional systems with domain wall initial conditions. For an appropriate choice of parameters the profile develops a rarefaction wave, while maintaining the initial equilibrium states far to the left and right, which thus serve as infinitely extended thermal reservoirs. For two distinct model systems we will demonstrate that the properly projected time-integrated current has a deterministic contribution, linear in time $t$, and fluctuations of size $t^{1/3}$ with a Tracy-Widom distributed random amplitude.
 Statistics , 2013, DOI: 10.1007/s10955-014-1058-z Abstract: We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature $\beta$ tends to $0$. We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy-Widom $\beta$ law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index $k$. Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in [L. Dumaz and B. Vir\'ag. Ann. Inst. H. Poincar\'e Probab. Statist. 49, 4, 915-933, (2013)]. We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when $\beta\to 0$. As an application, we investigate the maximal eigenvalues statistics of $\beta_N$-ensembles when the repulsion parameter $\beta_N\to 0$ when $N\to +\infty$. We study the double scaling limit $N\to +\infty, \beta_N \to 0$ and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of [A. Edelman and B. D. Sutton. J. Stat. Phys. 127, 6, 1121-1165 (2007)] and [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944 (2011)] from our later study of the stochastic Airy operator.
 Bálint Vet？ Mathematics , 2014, Abstract: We consider the q-Hahn TASEP which is a three-parameter family of discrete time interacting particle systems. The particles jump to the right independently according to a certain q-Binomial distribution with parallel updates. It is a generalization of the discrete time q-TASEP which is the q-deformed totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1). For step initial condition, we prove that the current fluctuation of q-Hahn TASEP at time t is of order $t^{1/3}$ and asymptotically distributed as the GUE Tracy-Widom distribution. We verify the KPZ scaling theory conjecture for the q-Hahn TASEP.
 Mathematics , 2012, DOI: 10.1088/1751-8113/46/1/015202 Abstract: We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.
 Mathematics , 2013, Abstract: We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.
 Igor Rumanov Physics , 2014, Abstract: In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $\beta$. Using this general result, the case $\beta=6$ is further considered here. This is the smallest even $\beta$, when the corresponding Lax pair and its relation to Painlev\'e II (PII) have not been known before, unlike cases $\beta=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $\beta=6$ involving the PII function in the coefficients, is found, which allows one to compute asymptotics for the distribution function. The ODE is a consequence of a linear system of three ODEs for which the local Painlev\'e analysis yields series solutions with exponents in the set $4/3$, $1/3$ and $-2/3$.
 Page 1 /100 Display every page 5 10 20 Item


text:
# $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.

Denote the (Banach) space (in fact an algebra under convolution), $l^1(G)=\{\sum_{g\in G}\lambda_g g| \sum_{g\in G}|\lambda_g|<\infty, \lambda_g\in \mathbb{C}\}$.

Note that $\mathbb{Z}G\subset l^1(G)$ and we can consider $l^1(G)$ as a right $\mathbb{Z}G$-module, my question is:

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

If it is true, any reference (maybe for related topics)? Otherwise, any counterexample? Especially, is it true for the Heisenberg group?

RK: I have asked a general one in MSE but no answer appeared, so I asked here a more specific one.

-
Just a comment - any group $G$ is canonically isomorphic to the opposite group $G^{op}$, where multiplication is reversed (i.e. $g\cdot h := hg$). So $\mathbb{Z} G$ is left-Noetherian iff it is right-Noetherian. But I don't know how the $l^1$ condition interacts with standard homological algebra. Can you prove what you want for $G = \mathbb Z$? –  Peter Samuelson Jul 21 '13 at 0:51
I am not sure it has been proved when $G=\mathbb{Z}$, but perhaps you may find it helpful to read theorem 3.1(4) and corollary 3.5 in the following paper:arxiv.org/abs/1103.1567 –  Jiang Jul 21 '13 at 2:36
And "how the $l^1$ condition interacts with standard homological algebra" is more or less what I want to know. –  Jiang Jul 21 '13 at 2:37

Here are a few observations that are too long for a comment but don't provide a full answer by any means.

As Peter Samuelson observed left-right questions aren't significant in this setting since $\mathbb{Z}G$ is isomorphic to $\mathbb{Z}G^{op}$.

Also, note that $\mathbb{C}G$ is a flat $\mathbb{Z}G$-module since $\mathbb{C}$ is a flat $\mathbb{Z}$ module. Thus, by standard base change arguments, to prove that $l^1(G)$ is flat over $\mathbb{Z}G$ it would suffice to prove that it is flat over $\mathbb{C}G$. At least in the case $G$ is polycyclic-by-finite, $\mathbb{C}G$ will also be (left and right) Noetherian.

Now I'm going to restrict to the case $G=\mathbb{Z}$, so $\mathbb{C}G\cong \mathbb{C}[x,x^{-1}]$, the ring of Laurent polynomials in $x$. This ring is a principal ideal domain.

Now I refer to Weibel's Introduction to homological algebra (CUP) Proposition 3.2.4. A left $R:=\mathbb{C}[x,x^{-1}]$-module $B$ is flat as such if and only if $\mathrm{Tor}_1^R(R/I,B)=0$ for every right ideal $I$ of $R$.

But we've already observed that every (right) ideal $I$ is of the form $fR$ in this case, and (for $f\neq 0$), we can compute $\mathrm{Tor}_1^R(R/fR,B)$ explicitly as $[f]B :=\{ b\in B\mid fb=0\}$.

In other words, to answer your question positively for $G=\mathbb{Z}$ it suffices to prove that $l^1(G)$ has no elements killed by $f$ for any non-zero $f\in \mathbb{C}G$. I think it should be possible to prove this by elementary arguments although I haven't even begun to check.

-
It occurs to me that in the final paragraph one might as well assume that $f$ is irreducible in $R$. Since (by the fundamental theorem of algebra) such an $f$ is a unit times $(x-\lambda)$ for $\lambda\in \mathbb{C}$ non-zero it should be very easy to complete the case $G=\mathbb{Z}$ by hand. –  Simon Wadsley Sep 17 '13 at 10:54
in this paper ams.org/journals/proc/1998-126-03/S0002-9939-98-04025-8/… the author mentioned at the last line of page 721 that $0\neq f\in \mathbb{C}(\mathbb{Z})$ is a uniform nonzero divisor. –  Jiang Sep 19 '13 at 2:01
How to modify your reduction to the case such that $G$ is a non-commutative polycyclic-by-finite group, say the discrete Heisenberg group? In particular, what does a general right ideal $I$ look like in $\mathbb{C}G$? –  Jiang Sep 19 '13 at 2:06
I'm not even sure that it is easy to extend my strategy to $G=\mathbb{Z}^n$ for $n>1$, although it may be easier than I think. Classifying right ideals in $\mathbb{C}G$ for $G$ a discrete Heisenberg group is probably hard. To get an idea why, see arxiv.org/abs/math/0102190. The first sentence of section 7 of your reference suggests that $Tor^1_{\mathbb{C}G}(\mathbb{C}G/f\mathbb{C}G,l^1(G))=0$ for all non-zero $f\in \mathbb{C}G$ whenever $G$ is torsionfree polycyclic. However, it isn't clear to me how hard this result was to prove already in that generality. –  Simon Wadsley Sep 19 '13 at 13:27