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## Main

Our ancestors shifted from a diet derived entirely from hunting and gathering to one reliant primarily on farming annual grain crops that were domesticated either directly or indirectly from wild perennial progenitors1,2. This one-sow, one-harvest system underpinned the origins of civilizations and continues to feed today’s societies3. Annual grains, domesticated independently on multiple continents from the beginning of the Neolithic revolution, are currently grown on 60–80% of global croplands, supplying 80% of global food4,5. However, modern high-yielding annual crops require wholesale removal of competing vegetation to establish and typically demand high inputs of energy, pesticides and fertilizers (and, in some cases, labour) to be productive6. The combination of chronic soil disturbance and high inputs often compromises essential ecosystem services, pushing many beyond sustainable planetary boundaries7,8,9. In addition, annual crops typically provide only intermittent coverage of the soil by plants, leaving barren soil vulnerable to heavy rainfall, which carries substantial quantities of soil and nutrients off the land, reducing its fertility and resulting in the eutrophication of downstream aquatic ecosystems10,11,12. This situation occurs especially on marginal lands, which currently support 50% of the world’s population and are at risk of further degradation under annual cropping13,14.

Perennial crops have advantages over annual crops in maintaining important ecosystem functions. Perennial crops generally have a longer photosynthetic season, which increases yearly light interception and enhances productivity15. In addition to permanent living cover, deep roots of perennial crops have been shown to increase nitrogen (N) retention11,12and soil carbon accumulation16,17 and may ultimately require lower rates of fertilizer, pesticide and labour inputs9. Perennial cropping systems can also be managed to produce both grains and forages in mixed crop–livestock systems5,18,19,20,21. By improving labour efficiency and soil quality, perennial cropping systems not only improve farmers’ livelihoods but also benefit the ecological systems required to maintain productivity over the long term20,21,22.

Targeting key grains and oilseeds, including wheat, sunflower, barley, sorghum, buckwheat and rice, international researchers from numerous countries are engaged in developing the perennial counterparts of these annual crops using two breeding strategies: de novo domestication23,24,25 and interspecific hybridization26,27. The grain Kernza® is a perennial that exemplifies de novo domestication of the wild species Thinopyrum intermedium, or intermediate wheatgrass28. The perennial rice (PR) reported here was developed by interspecific hybridization. It was targeted initially at favourable (irrigated) rice production systems, in which paddy fields are bunded and ponded water depth is carefully managed. Nevertheless, rice is cultivated across a wide range of ecosystems, from flood-prone to rainfed lowland and rainfed upland, in which submergence and drought define the crop’s adaptive characteristics29. We report on progress in breeding recently commercialized PR cultivars, including their performance over regrowth cycles and locations, and evaluate the livelihood, ecological and soil benefits of PR cultivation over ten successive seasons. We then examine potential zones for sustainable PR production and consider the prospects, limitations and impacts of research, development, adoption and commercialization of this crop, and hence of perennial grains worldwide.

## Breeding PR via interspecific hybridization

A successful interspecific hybridization between the annual domesticated Asian rice Oryza sativa ssp. indica RD23, a cultivar from Thailand (Fig. 1a and Extended Data Figs. 1 and 2a), and an accession of the undomesticated African perennial and rhizomatous O. longistaminata from Nigeria (Fig. 1b and Extended Data Fig. 2b) was achieved by embryo rescue in 199627. The F1 plant (Extended Data Fig. 2c) possessed strong rhizomes, partial pollen fertility and self-compatibility, which provided foundational material for basic research on the genetics of rhizome biogenesis and development in rice and paved the way for breeding PR cultivars27,30.

From wide crosses, it can be difficult to obtain breeding lines with target traits due to linkage drag and low frequency of desirable alleles from the undomesticated parent; therefore, we observed a low frequency of F2s that were perennial, were rhizomatous and had good agronomic characteristics. To circumvent this linkage drag, we used pedigree selection from the best single plant in each generation and screened 7,200 F2s to identify rare individuals with the best combination of traits from both parents for further breeding. In 2007, an exceptional F2 individual, coded 36–1, was selected due to its high pollen fertility and seed-setting rate (greater than 85% and 60%, respectively) and moderately strong rhizome production (Extended Data Figs. 1 and 2d).

Starting from the exceptional F2 36–1, we conducted multiple rounds of self-pollination. In each generation, only one individual with short rhizomes and good pollen fertility was selected for self-pollination, and the crown of the chosen individual was retained. Consequently, we constructed a population with individuals representing successive generations from F1 to F9 (Extended Data Fig. 3 and Supplementary Table 1) and used this population to investigate rhizome- and yield-related traits in 2018. As the generations advanced, the agronomic traits gradually changed in the direction of cultivated rice. From F1 to F9, we observed a decrease in rhizome number per plant (from 17 to 4) and rhizome length (from 9.3 cm to 3.4 cm) (Extended Data Fig. 3a,b). Pollen fertility gradually increased, and plant height, tiller number, grain number per plant, seed-setting rate, panicle length and grain size (grain length, width and weight) improved to reach values similar to those of widely planted annual rice (AR) cultivars (Extended Data Fig. 3c–i). These data collectively showed that the PR lines we developed had agronomic traits resembling domesticated AR yet retained from the perennial O. longistaminata parent the ability to regrow vigorously after harvest. From next-generation sequencing, 16.16% of the PR23 genome was O. longistaminata.

More than 20 candidate PR lines derived from self-pollinating the F2 selection 36–1 were evaluated for agronomic traits and perenniality in two multi-year field trials from 2012 to 2017 at ten locations in Yunnan, China and Laos31,32 (Experiments 1 and 2; Supplementary Tables 2 and 3). Genotype-by-environment analyses revealed that broad-sense heritability was high for most traits, including yield (0.87 to 0.94) and regrowth rate (0.88 to 0.96), indicating that the selections were stable throughout the region. An especially promising line, subsequently named PR 23 (PR23) (Fig. 1c,d), exhibited high and stable grain yields over multiple years and good performance (Fig. 1e–h) at most locations33,34. PR23 was released to farmers in China in 201835, a key milestone in the commercialization of perennial grains bred via interspecific hybridization. Through further selection, three additional PR lines were successfully bred: PR24, PR25 (released as Yunda25 in China in 202036) and PR101 (Extended Data Fig. 1). PR101 has grain quality similar to its O. sativa ssp. indica parent RD23, which has a high grain length/width ratio of 3.57, whereas PR23, PR24 and PR25 have grain quality traits similar to the typical O. sativa ssp. japonica cultivar Zhonghua 11, including a lower grain length/width ratio of 2.13. These grain qualities should be widely accepted as they are similar to current consumer preferences, except in countries such as Tanzania, Ethiopia and Laos, which have specific grain quality requirements37,38.

In a second round of breeding, our strategy was to transfer perenniality into local elite O. sativa cultivars. A perennial breeding line MP3–235 (later renamed PRB3), with strong perenniality and short rhizomes but (less desirably) a 2 m plant height, strong seed shattering and low grain density (Extended Data Fig. 2e,f), was chosen to cross with an elite indica cultivar, Dianrui 449 (DR449). Three generations of backcrossing (240 lines each) to DR449 as the recurrent parent were performed, followed by several generations of self-pollination. As expected, backcrossing to an elite parent increased the frequency of progenies with desirable agronomic traits relative to the F2 population (Supplementary Table 1). From this backcross breeding, a new PR cultivar, PR107 (DR449/MP3–235//DR449) (Extended Data Figs. 1 and 2g,h), with strong perenniality, high yield and good grain quality, was released as Yunda107 in China in 202039 and as NARORICE-1 (PR107) in Uganda in 2021. Taken together, these results illustrate that our breeding strategy, based on large population size and strong selection for perenniality and pollen fertility in recombinant inbred lines derived from an F2 or backcross population, was effective.

## Performance and adoption

We initiated an experiment in 2016 to evaluate the productivity of PR23 in farm-scale plots of 1–13 ha (Experiment 3). This took place at three locations in Yunnan Province, Mengzhe, Menglian and Xinping, where rice is typically planted from seed twice a year. The grain yield of PR23 in Mengzhe over five successive years was 10.9, 8.7, 8.2, 8.1 and 5.3 Mg ha−1 in the first season of each year, respectively, and 6.6, 6.4, 6.2, 6.0 and 3.1 Mg ha−1 in the cooler second season (Fig. 2 and Supplementary Table 4). These yields (averaging 6.9 Mg ha−1) were comparable to those of the local elite AR Diantun502 (averaging 7.0 Mg ha−1) in eight successive seasons over four years but gradually declined in the ninth season (Extended Data Fig. 4a). Similarly, Menglian PR23 produced higher grain yields (averaging 7.7 Mg ha−1) than the local elite AR Yunhui290 (averaging 7.1 Mg ha−1) in eight successive seasons over four years (Extended Data Fig. 4c). In Xinping, the regrowth crop in spring 2017 (the first regrowth season) was severely damaged by rice hoppers and rats, but grain yields of PR23 were able to recover in the two subsequent seasons (averaging 6.0 Mg ha−1) and showed performance comparable to the local elite AR Wenfu6 (averaging 6.1 Mg ha−1) in the first three seasons; in the following seasons, the yield was consistently lower but stable (Extended Data Fig. 4b). Across the experiment, therefore, PR23 sustainably produced yields averaging 6.8 Mg ha−1 in each of four consecutive years (eight seasons) without re-sowing, and these were comparable to yields from the current leading AR cultivars, which averaged 6.7 Mg ha−1 after being replanted each season (Fig. 2d). Similarly, the more recently released cultivars PR25 and PR107 produced high and stable yields for at least two years and four seasons from a single planting in 2018 at multiple locations (Experiment 4), and these were comparable to yields of PR23 (Extended Data Figs. 5 and 6 and Supplementary Table 5). The high and stable grain yields maintained over eight seasons for PR23 were associated with a high regrowth rate (above 75%) and stable agronomic traits (Extended Data Fig. 4), suggesting PR would need to be resown after about four years. Alternatively, PR could be rotated with a legume pasture or a pulse or brassica crop to provide a break from pests, diseases and weeds and to support other farm enterprises such as livestock while further improving soil health.

Removal of PR after eight harvests, pests, diseases and weeds pose potential threats to successful adoption, and these are considered in detail in Methods, together with strategies to reduce risks. Briefly, adoption of PR is not without risks to the farmer, as 1–2 more herbicide applications were required in PR ratoon cycles than in transplanted AR in our experiments (Supplementary Table 6). While not encountered in our experiments, pests and diseases adapted to high N applications could pose threats under irrigated conditions, especially if pustule loads in stubble are large or insect-transmitted viruses such as tungro become established in regenerating plant crowns. Short rhizomes were intentionally selected in PR so as not to provide substantial weed potential but to permit improved chances of survival and regrowth when harsher conditions were encountered. Under irrigation, therefore, tillage may be required to expose PR crowns to soil drying, perhaps with follow-up glyphosate spraying of young regrowth if needed before re-sowing. Further details are provided in Methods, including on our pest-, disease- and weed-control strategies.

In summary, the three recently released cultivars of PR that we bred (PR23, PR25 and PR107) can produce, from a single planting, high yields over two to four consecutive years for each of four to ten cycles of growth–harvest–regrowth on plants that closely resemble modern O. sativa cultivars (Fig. 2, Extended Data Figs. 4, 5 and 6 and Supplementary Tables 4 and 5), whereas O. sativa cultivars can typically be ratooned at most only once, and with a substantial yield drop in the ratoon crop40,41,42. Therefore, these first cultivars of PR represent a step change that makes ratooning an economically attractive option for irrigated production environments that are not limited in duration by cold weather or other adverse environmental conditions. These cultivars, along with Kernza®21, are also among the first perennial grains to be commercialized. By 2020, PR23 was being grown on over 8,400 ha. In 2020, it was cultivated commercially on 3,818 ha by more than 11,000 smallholder farms, primarily in China (Fig. 2c and Supplementary Table 7). In 2021, the planting area of PR increased to 15,533 ha, including 44,752 smallholder farms, demonstrating a rapid adoption of PR—a fourfold increase in area and number of farmers in one year.

## Livelihood benefits

The prevailing irrigated annual cropping system of rice is labour intensive and thus costly, requiring seedling nurseries, ploughing, transplanting, crop management and harvesting every season. Non-labour costs also include seed, pesticides, herbicides and fertilizers. These expenses lead to low profitability33,34 and insecure livelihoods. To compare the socioeconomics of annual and PR production, we measured the grain yields, labour requirements, inputs costs and returns associated with each. These data are summarized in Supplementary Table 8, which aggregated labour costs for seedling nurseries, ploughing, transplanting, crop management and harvesting. Note that fuel costs are included within ploughing, transplanting and harvesting; fertilizer costs are included within nurseries, ploughing, transplanting and crop management; and pesticide costs are included within ploughing and crop management. Under crop management, the additional cost of each regrowth cycle in Mengzhe and Menglian is for weed control ($74 ha−1 on average), and in Xinping the further$82 ha−1 is for pest control, as noted earlier.

In the first season of our study, costs were similar between annual and PR (Fig. 3a and Supplementary Table 8) for both the labour (US$1,926–2,012 ha−1) and non-labour (US$515–787 ha−1), which accounted for 71.3–79.5% and 20.5–28.7% of total economic cost, respectively (Extended Data Fig. 7a–f). However, in the regrowth seasons, because PR does not require seedling nurseries, ploughing or transplanting to initiate each crop, it was found to have economic savings from lower labour cost (US$1,057–1,206 ha−1) and non-labour costs (US$96–201 ha−1), saving US$1,177–1,401 ha−1 (46.8–51% of the costs of AR) per regrowth season (Fig. 3b, Extended Data Fig. 7a–f and Supplementary Table 8). Moreover, in each regrowth season, PR production could save approximately 68–77 labour days ha−1 compared with AR, dramatically reducing drudgery and making the management of the PR production system much easier (Fig. 3a). Equally important is that the reduction in human labour requirements is accomplished without substitution by fossil fuel-based equipment, an important consideration as society aims to reduce greenhouse gas emissions associated with agricultural production. During 2016–2020, the net economic benefits of PR in the regrowth seasons at Mengzhe, Xinping and Menglian averaged US$882, US$109 and US$1,165 ha−1 season−1, which were 57.3%, 17.4% and 161% more than elite AR, respectively (Extended Data Fig. 7i and Supplementary Table 8). Thus, even in the case where yields of PR were significantly lower than AR (for example, Xinping), PR always achieved a greater economic return (Fig. 3a and Extended Data Fig. 7i). This was confirmed by a sensitivity analysis of net economic gain in response to changes in grain yield, labour inputs and cost inputs. For the second crop in the cycle (re-transplanted AR versus the first regrowth cycle of PR), net economic gain was insensitive to changes in labour input or cost input within each system as the major variation in these parameters was between systems. By contrast, net economic gain was highly sensitive to changes in grain yield, with an extra 1 t ha−1 transplanted AR generating an extra $436, while an extra 1 t ha−1 in each regrowth cycle of PR generated an extra$2,308. Thus, net economic gain was five times as responsive to increase in grain yield in each regrowth cycle of PR than in re-transplanted AR. Adoption of PR would thus reduce farm labour costs, energy use and technological inputs that are required for yearly tillage in annual cropping systems, thereby providing more profits to farmers than AR.

## Soil benefits

With soil tillage required for only the initial crop in the cycle, and with less soil disturbance between cycles, PR was expected to provide soil benefits over annual cropping. In the 0–40 cm soil layer, PR cropping over four years resulted in increased soil organic carbon (SOC) and total nitrogen (TN) at rates of 0.95 and 0.11 Mg ha−1 yr−1, respectively (Fig. 3b, Extended Data Fig. 8 and Supplementary Table 9). We found that the C/N ratio declined significantly in the topsoil over the years in the perennial cropping system (Extended Data Fig. 8l), which could stimulate microbial decomposition of organic matter and result in net N mineralization in the soil43. This predicted enhancement of N cycling due to lower C/N ratios contrasts with apparent net N immobilization of 0.11 Mg ha−1 yr−1 N, which is a consequence of SOC accumulation following the cessation of tillage44,45.

The combined effects of no tillage and increases in organic matter under PR appear to have influenced several other key soil properties, including soil pH, plant-available soil-water capacity and capillary and non-capillary porosity (Extended Data Fig. 8b,c). Soil pH increased significantly by 0.3–0.4 units, with all values falling within the optimal range (5.0–6.5) in rice for nutrient uptake (including phosphorus, copper, zinc and boron) and microbial activity (Fig. 3b). This finding supports earlier work43 that showed SOC stocks of rice paddy soils from 612 sites in 51 countries to be positively correlated with increases in soil pH. Significant increases in plant-available water capacity and porosity have been measured in rice paddy systems with reduced tillage46, and our measured increases in SOC are also likely to have contributed to greater plant-available soil-water capacity47 (Extended Data Fig. 8g–i).

In addition, no tillage retains the soil structure, which can enhance the oxidative capacity of soil methane-oxidizing bacteria and thereby reduce methane emissions40,41, whereas ploughing in annual rice paddies results in an increase of methane emissions by 51.11% (ref. 42). Management of perennial rice with intermittent flooding could further reduce methane emissions by reducing the amount of time soils experience anoxic conditions48. However, there may be a trade-off in which intermittent flooding favours substantially greater pulses of nitrous oxide emissions, resulting in higher total global warming potential48.

## Optimal ecological zoning

Low temperature significantly affected the regrowth of PR. In the field experiment, the regrowth rate had a significant quadratic correlation with monthly mean temperature in January 2017 (y = −2.54x2 + 86.33x − 633.98, R2 = 0.73*, P < 0.05) (Supplementary Table 10 and Extended Data Fig. 9a). On the basis of this equation, we used a mean monthly temperature of 13.5 °C in the coldest month as the minimum likely to permit at least 75% regrowth in subsequent PR crops. In the controlled environment experiment, the regrowth percentage of PR decreased from 92.2% to 55.4% when temperatures of 4 °C lasted for seven days, while regrowth decreased to 82.8% after five days and 22.4% after seven days at 0 °C (P < 0.05) (Supplementary Table 11 and Extended Data Fig. 9b). From these results, we determined that periods with an average daily temperature lower than 4 °C must last for fewer than five days for PR regrowth to be likely to remain above 75%.

On the basis of these findings for low-temperature effects on regrowth, we assessed which of 25,049 meteorological stations from the National Centres for Environmental Information (https://ncei.noaa.gov) met the criteria. The climates in 2,659 stations were considered potentially suitable for PR cultivation on the basis of cardinal temperatures for regrowth. This suggests that PR has a broad range of potential planting regions, generally within the latitudes of 40° N and 40° S (Fig. 3c).

We recognize that high temperature affecting spikelet fertility around flowering would also impose limits on zones for PR cultivation. Jagadish et al.49 reported that spikelets in rice became sterile when exposed to 33.7 °C around anthesis, although reductions commenced from 29 °C. From meiosis to anthesis to early grain expansion, Nguyen et al.50 reported spikelet sterility commenced as spikelets were exposed to 31 °C from 12 days before anthesis to 12 days after anthesis. Consequently, we conclude that exposure to temperatures of about 31 °C during the month around flowering would be detrimental to spikelet fertility in rice, and the zoning analysis should be extended for these high-temperature limitations. Nevertheless, the estimated ranges of PR and AR are likely to be similar based on temperature alone as cardinal temperatures are considered to be species characteristics51. When other factors (precipitation, evapotranspiration, landscape) are included, however, the range of AR will decline as PR will better cope with abiotic stress (due to extensive root systems and greater assimilate reserves) and in more marginal landscapes (due to lack of soil disturbance, continued soil cover, reduced leaching and nutrient loss, and reduced run-off and soil loss, with run-off and soil loss being of major concern in the sloping rainfed uplands).

## Prospects and potential

After more than 20 years of effort, the cultivation of PR has become a reality. This represents a cropping system that simultaneously achieves grain production, labour reduction and ecological security, especially for terraced and fragile farmland. PR has demonstrated good yield potential, and agronomic traits for four years and eight cropping seasons from a single planting can enhance soil fertility and reduce requirements for inputs through ecological intensification. The simplified management, reduced labour requirements and improved livelihood made possible by PR are attractive, so this technology is becoming accepted by increasingly more farmers. This is particularly important because in most countries, rural labour forces have declined due to migration of people to cities, and the application of PR will undoubtedly relieve the work burden of rural farmers, especially women and children, who often provide the unskilled labour52.

Given the recent development of PR, numerous important research questions are just beginning to receive attention. For example, we found more herbicide applications were required in regrowth cycles than in transplanted AR, suggesting the need for improved weed-management agronomy. And while not encountered here, pests adapted to high N applications could pose threats in irrigated systems, especially if pustule loads in stubble were large or insect-transmitted viruses became established in regenerating plant crowns (Performance and adoption and Methods). Finally, understanding management practices that influence the total emissions and relative balance of methane and nitrous oxide trace gasses is critical for understanding total global warming potential of this crop.

Our PR has received certification for commercialization due to the multifaceted benefits it brings, providing a promising solution that reconciles food production with environmental security and the needs of farmers. Recent advances in plant breeding, such as low-cost DNA sequencing, molecular marker-assisted selection, genomic selection and gene editing, can be expected to further accelerate the development of PR in the future, as well as facilitate the perennialization of other crops. Having succeeded in developing a PR adapted for irrigated and favourable lowland conditions, we have now begun breeding PR cultivars suitable for rainfed lowland and importantly for rainfed upland cultivation. PR has great potential for curbing serious erosion and other forms of soil degradation that are common in such marginal lands, especially in upland regions of Southeast Asia53.

## Methods

### Rice ecosystems and cultural practices

Characteristics of rice ecosystems and their cultural practices have recently been reviewed by Fukai and Wade29. Briefly, rice ecosystems are commonly classified by water availability. The most common is irrigated lowland, in which the paddy field is shallow-flooded after the crop is established until just before harvest. Another major ecosystem is rainfed lowland, in which fields are bunded like in irrigated to maintain ponded water, but with no or limited irrigation, conditions can vary depending on rainfall. Soils may be saturated and anaerobic during the growing season, but in periods of low rainfall, ponded water vanishes, the soil becomes aerobic, and plants quickly encounter water deficit. In flood-prone ecosystem, rice may be submerged, and deep-water or floating rices may be grown. Conversely in upland ecosystem, ponded water is absent, and rice is grown in aerobic soil sometimes on sloping land.

Irrigated rice has traditionally been transplanted into puddled soil, which is cultivated wet to reduce drainage and enhance ponding of water. Transplants have a competitive advantage over newly-emerging weeds, which generally die from anaerobiosis. In addition, soil nutrient availability is improved by puddling, and the paddy favours development of complex food webs for IPM. The disadvantages include labour-intensive cultivation and transplanting, high water use and greenhouse gas emissions and difficulty in establishing post-rice crops into previously puddled soils. Direct seeding into non-puddled soil, and flush irrigation, with ponded water restricted to critical growth phases, have been employed to reduce these problems, although early nutrient availability and weed problems have increased as a consequence. Double cropping of AR is common where temperatures are favourable, and rainfall or irrigation is sufficient. Where cooler conditions are encountered at higher altitude or latitude, however, only one rice crop per year may be harvested from the first season (dry season, December to May in our locations). Conversely, where hotter and drier conditions are encountered, only one rice crop per year may be harvested from the second season (wet season, June to November). PR offers the potential to produce several crop cycles from a single planting, with performance in regrowth cycles dependent on successful survival, regrowth, spikelet fertility and grain yield in the conditions encountered29.

### Experimental approach

We conducted five experiments to explore the potential and benefits of PR across field locations in China (Supplementary Table 12). In Experiments 1 and 2, we studied genotype-by-environment interactions from multi-environment trials to examine the effects of selection during breeding, enabling us to estimate variance components and heritability to elucidate patterns of PR adaptation. In Experiment 3, large-plot trials were used to study the performance, long-term productivity and livelihood benefits of PR23, the first PR cultivar released. Trials of PR25 and PR107, subsequent PR releases, were conducted to quantify performance over years and locations (Experiment 4). Finally, PR23 was used to evaluate conditional cold tolerance of PR (Experiment 5). Experimental designs, treatments, cultural practices, data collection and analysis, mean yield and sources of further information are summarized for Experiments 1–4 in Supplementary Table 6, with the main attributes explained briefly for each experiment in the following. Soil constraints, pests, diseases and weeds may pose threats to successful perennial rice cropping, so the following section considers the risks they pose and the mitigation strategies available. Finally, we provide details for socioeconomic and soil measurements and details of the statistical analysis.

### Experiment 1

Field trials were conducted in 12 location–season–year (environment) combinations at Jinghong, Menglian and Simao in Yunnan Province of China and at Na Pok in Vientiane Province of Laos in 2012 and 2013. At each location, a randomized complete blocks design was used, comprising 22 genotypes with 2–3 replicates. The 22 genotypes comprised PR derivatives obtained from the cross between O. sativa cultivar RD23 and the wild species O. longistaminata. Specific details about the development of these breeding lines were reported by Zhang et al.54. There were two crops per year at Jinghong and Menglian, but only one crop per year at Simao (first season) and Na Pok (second season). Cultural, measurement and analysis details are summarized in Supplementary Table 6. Briefly, after each harvest, stubble was cut to 10 cm so that consistent stubble for uniform regrowth was available. Mean timings of transplanting and maturation and successive cycles of stubble cut-off and maturation were recorded to define growth duration in each cycle. Regrowth percentage, plant height, grain yield and yield components were recorded from each plot, from 0.9 m2 samples31.

### Experiment 2

The trials were conducted in 19 environment combinations at Jinghong, Puer (Simao), Hongta, Wenshan, Yuanyang, Lancang and Dehong in Yunnan Province of China in 2014–2016. The trials at Jinghong (double crop) and Puer (first season) were continued for three years while the others (first season) continued for two years only. At each location, the trial was a randomized complete blocks design comprising nine genotypes with three replications. Five PR genotypes thought to differ in adaptation31, obtained from the cross between O. sativa cultivar RD23 and the wild species O. longistaminata, were included, along with three annual O. sativa cultivar controls32. Cultural, measurement and analysis details were similar to Experiment 1, except a split dressing of 72 kg ha−1 N was applied at boot stage (Supplementary Table 6)32.

### Experiment 3

In this field experiment, larger plots of PR23 of 1–13 ha were grown to examine performance, socioeconomics and impact of PR versus AR lines for validation and official release purposes, at Mengzhe, Menglian and Xinping in southern Yunnan province of China, with data collected from 2016 to 2020. Double cropping is the dominant rice cropping system at all three locations, which have a subtropical monsoon climate with mean annual temperature of 18–19.6 °C. The annual rainfall is 869–1,341 mm, which falls mainly in the second season from May to October. The PR PR23 was compared with locally elite AR cultivars selected specifically for each location, which, for Mengzhe, Xinping and Menglian, were Diantun502, Yunhui290 and Wenfu6, respectively (Supplementary Table 6).

Both PR23 and AR were established in the seedling nursery and transplanted to the field. After the first harvest, the straw of PR was cut back to 10 cm above the ground to maintain the uniformity of new tillers arising from rhizomes and to depress tillers from developing from above-ground nodes in each season, with no till being conducted across the successive regrowth cycles. By contrast, for AR, the rice stubble was ploughed after each harvest and the rice was seeded and transplanted as it was in the first season, which is a typical commercial practice in the region. At each location, PR and AR were planted in adjacent plots within the same ecological area. The rice fields were managed by the local farmers under our guidance. During the growth season of PR and AR, the labour and non-labour costs were recorded. At each harvest, yield-related traits (grain yield, duration, regrowth rate, panicle number m−2, spikelet number panicle−1, grain weight and seed-setting rate) were measured.

### Experiment 4

Field trials were conducted in Yunnan Province, China, at seven locations: Mengzhe, Menglian, Jinghong, Xinping, Lancang, Wenshan and Yiliang. Measurements at these locations took place over the course of three years, from 2018 to 2020. The management of PR was the same as in Experiment 3. PR25 and PR107 were selected as experiment material, and PR23 as selected as control. At each harvest, yield-related traits (grain yield, duration, plant height, regrowth rate, panicle number m−2, spikelet number panicle−1, grain weight and seed-setting rate) were recorded (Supplementary Table 6).

### Experiment 5

Two sub-experiments were conducted to assess the regrowth potential of PR in response to low temperature, with an initial field experiment followed by a controlled-temperature chamber experiment. The first experiment was conducted at 16 field locations in 2016 and 2017 (Supplementary Table 10). At each location, a ratoon crop was exposed to low temperatures in the field in January during early regrowth, and mean monthly temperature and regrowth percentage were recorded. A quadratic equation was fitted to the relationship between temperature and regrowth percentage, and LSD for regrowth percentage (P < 0.05) was obtained.

The second experiment was conducted in a low-temperature plant incubator (model FH-1300, Taiwan Hipoint Co. Ltd) at Yunnan University. Individual plants in 1 cm pots with three replicates were exposed to temperatures of 0 °C or 4 °C for three, five or seven days from the three-leaf stage. Regrowth percentage was recorded for each plant, and means and LSD (P < 0.05) were obtained.

Critical temperatures identified from the two sub-experiments on low-temperature effects on regrowth were then used to assess potential ecological zones for PR. Data from 25,049 meteorological stations worldwide (National Centres for Environmental Information, https://ncei.noaa.gov) were assessed using these criteria, and stations that met the criteria were mapped using GIS to identify potential ecological zones for PR worldwide.

### Risks and mitigation

The high and stable grain yields maintained over eight seasons for PR23 were associated with a high regrowth rate (above 75%) and stable agronomic traits (Extended Data Fig. 4), suggesting PR would need to be resown after about four years. We observed that although the regrowth rate showed a decreasing trend over ten seasons, the panicle number m−2 and other grain yield traits were maintained. This indicates an ability of PR to regrow and tiller, which can compensate for the minor decrease in plant stand that may occur in earlier regrowth cycles. In the fifth year (the ninth and tenth seasons), however, grain yield decreased significantly due to a significant decrease in regrowth rate (below 75%), panicle number and spikelet number per panicle (Extended Data Fig. 4). Present cultivars of PR would need to be resown in year 5, preferably by direct seeding into minimum or zero tillage, with stubble cut to a height of 10 cm and excess straw removed if necessary for passage of machinery. Alternatively, PR could be rotated with a legume pasture or a pulse or brassica crop to provide a break from pests, diseases and weeds and to support other farm enterprises such as livestock.

While low temperatures in winter31 or high temperatures and drought in summer55 may reduce regrowth or kill the plant, perennial stubble may also need to be killed under favourable conditions to establish a new crop. Short rhizomes were selected intentionally in PR so as not to provide significant weed potential but to permit improved chances of survival when harsher conditions are encountered. Under irrigation, therefore, tillage may be required to expose the crowns of PR to soil drying, perhaps with follow-up glyphosate spraying of young regrowth if needed before re-sowing. In addition, post-rice crops on puddled soil may encounter difficulties from poor soil structure29, but this problem may be reduced after several ratoon cycles as soil structure parameters generally improved here (Supplementary Table 9). This problem would be minimized by establishing PR by direct seeding without soil puddling, which is becoming more common in rice, especially under conditions of labour scarcity56.

Pests and diseases of rice have been extensively studied. Under irrigated conditions, breakouts of brown plant hopper, green leaf hopper and leaf folder are associated with high N application but can be managed through integrated pest management with split applications of N more closely matching crop demand57. Rust, blast and other leaf and panicle diseases can be severe, but resistance is available in many rice cultivars, and crossing with O. longistaminata may provide additional resistance. These strategies should also be effective for PR. Under rainfed cultivation, pests and diseases are usually less severe although gall midge can be important if ponded water is absent during tiller elongation, and thrips can be damaging locally58. Perhaps of greater concern are insect-transmitted viruses that would remain viable in crowns and diseases with a high pustule load able to re-infect from previous stubble. Viruses of concern may include tungro59 and rice yellow mottle virus in Africa, although PR107 is resistant to the latter from its O. longistaminata parent and has been released in Uganda. Nevertheless, an additional one to two sprays were needed for weed control in regrowth cycles of PR in our experiments, relative to re-transplanted AR (Supplementary Table 6).

In perennial cropping systems, weeds are often observed to build up over cycles, but this may be of less concern in irrigated rice systems, in which timely soil flooding suppresses or kills most weeds early in each crop cycle. Some weeds adapted to anaerobic conditions may survive60, but grazing by ducklings when seedlings are small56 or after harvest by livestock can suppress weeds, together with spot spraying of remaining problems with glyphosate if needed.

### Livelihood benefits

In Experiment 3, cooperating farmers were surveyed to establish the timing and details of cultural practices, labour requirements and labour and non-labour costs, which were recorded for both PR and AR systems. The total labour number (persons ha−1) was confirmed by dividing total labour cost (US$ha−1) by labour price (US$15.63 person−1 on 4 November 2021). Total output was calculated as the product of grain yield and market price (AR, US$0.55 kg−1; PR, US$0.48 kg−1 on 4 November 2021). Net economic gain was obtained by difference, from output minus input.

### Soil benefits

Soil samples were collected with three replications after rice harvest in December each year. Core samples for soil nutrients and soil physical properties were obtained over four soil depth increments (0–10, 10–20, 20–30, 30–40 cm) under the PR and AR systems. Soil bulk density was measured from undisturbed cores of known volume, at saturation (upper limit (UL)) and at wilting point (lower limit (LL)) of rice. Undisturbed cores were weighed, dried in a convection oven at 105 °C for 24 hours and weighed again to determine volumetric soil water contents at UL and LL and plant-available water capacity (UL – LL) for each depth increment. From these data, bulk density and soil porosity were also calculated.

### Soil nutrients

For soil chemical analysis, core samples were sieved through a 0.25 mm screen. SOC was extracted using the K2CrO7–H2SO4 heat treatment, and residue was determined using the FeSO4 titration of potassium dichromate method61. Soil TN was determined using the Kjeldahl method62. The total accumulations of SOC and soil nitrogen were calculated as the product of SOC × bulk density and TN × bulk density over depth increments, respectively.

### Statistical analysis

Genotype-by-environment interactions were examined by joint application of cluster and ordination analysis using CropStat 7.2 (IRRI, http://bbi.irri.org), with further details of those analyses provided elsewhere for Experiment 131 and Experiment 232. Variance components and heritability were also obtained for all parameters from both experiments. For the crop yield data and soil properties, the mean values were calculated for each measurement, and analysis of variance was used to compare the effects of different treatments on the measured variables, with comparisons performed using the LSD test (P ≤ 0.05). LSD is the product of t0.05 and the square root of 2s2/r, where t0.05 is the tabular value of Student’s t for error degrees of freedom at a probability of 0.05, s2 is the pooled error variance, and r is the number of observations per mean. All LSD tests were two tailed, and no adjustments were made for multiple comparisons. Experimental data were analysed with the IBM SPSS statistical package v.20.0 (SPSS, Inc.), and the figures were generated using Origin 2015 (Sys Software, Inc.).

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


text:
Configurations

Changing Verdict configuration

1. Creating a configuration file (i.e., verdict.properties) in the config directory. In order for the options in this file to be effective, Verdict should be build again using a mvn package command.
2. Setting environment variables: When Verdict’s Java instance is created, it scans all environment variables and stores them in Verdict’s configuration if they start with verdict. For setting environment variables, periods (.) in option keys must be replaced with underscores (_). For example, verdict.loglevel option can be set by verdict_loglevel=debug.
3. Passing parameters when making a connection
• Spark: You can pass parameters using the --conf argument. For example
$spark-shell --jars target/ --conf verdict.loglevel=warn  will set the Verdict’s log level to warn. • Hive, Impala: You can pass the parameters in the JDBC connection string. For example $ veeline/bin/veeline -h "impala://host:port/default;verdict.loglevel=warn"

will set the Verdict’s log level to warn.

4. Using a Verdict query: You can issue a set key=value query after a Verdict instance is created.

The higher-numbered items take higher precedences.

Verdict configuration options

Basic options

Key Default values Description
verdict.bypass false If set true, Verdict simply passes all the query to the underlying database and returns the result to the user without any modifications. The only exception is set bypass=false.
verdict.loglevel info Determines the log level; should be one of “debug”, “info”, “warn”, “error”. loglevel is a synonym for verdict.loglevel.
verdict.relative_target_cost 10% Keeps the cost of Verdict’s query processing under a certain level. Verdict chooses a set of the largest samples under this limit.

JDBC options

Key Default values Description
verdict.jdbc.user   If set, pass as a “user” value when making a JDBC connection to an underlying database.
verdict.jdbc.password   If set, pass as a “password” value when making a JDBC connection to an underlying database.
verdict.jdbc.kerberos_principal n/a If set, makes an appropriate Kerberos authentication when making a JDBC connection to an underlying database. principal is a synonym for verdict.jdbc.kerberos_principal.
verdict.jdbc.ignore_user_credentials false If set to true, “user” and “password” values are not passed even if they are set in verdict.jdbc.user and verdict.jdbc.password.
verdict.jdbc.dbname   One of “impala”, “hive2”. The database name specified in the JDBC connection string is set.
verdict.jdbc.host 127.0.0.1 The host that will be used for a JDBC connection.
verdict.jdbc.port   The port number that will be used for a JDBC connection.
verdict.jdbc.schema default The default schema (or database) that a JDBC connection will connect to.
verdict.jdbc.hive2.default_port 10000 The default port number that will be used for a hive2 connection if the port number is not explicitly set in the passed JDBC connection string.
verdict.jdbc.hive2.class_name com.cloudera.hive.jdbc4.HS2Driver The class name of the JDBC driver that will be used for a hive2 connection.
verdict.jdbc.impala.default_port 21050 The default port number that will be used for an impala connection if the port number is not explicitly set in the passed JDBC connection string.
verdict.jdbc.impala.class_name com.cloudera.impala.jdbc4.Driver The class name of the JDBC driver that will be used for an impala connection.

Spark options

Key Default values Description
verdict.spark.cache_samples true (Lazily) cache the sample tables created by Verdict using Spark’s DataFrame#cache() function. The sample tables are typically significantly smaller than the original tables. Caching sample tables will lower query response times.

Key Default values Description
verdict.meta_data.meta_database_suffix _verdict When set to a non-empty string, Verdict creates a separate database (or called a schema or catalog in some systems) for storing its sample tables and related metadata. If the original database’s name is “default”, all the sample tables for that database are stored in the database named “default_verdict”. The user must have the privilege to create a new database unless the database already exists. Also, the user who wants to create samples must have the privilege to create new tables in the database.
verdict.meta_data.meta_name_table verdict_meta_name A table for storing the names of the sample tables.
verdict.meta_data.meta_size_table verdict_meta_size A table for storing the sizes of the sample tables.
verdict.meta_data.refresh_policy per_session How often to refresh metadata. One of “per_session”, “per_query”, “manual”. If set to “manual”, “refresh” query must be used for updating metadata.

Error bound options

Key Default values Description
verdict.error_bound.confidence_internal_probability 95% The probability with which Verdict’s error bounds hold. Should be one of 80%, 85%, 90%, 95%, 99%, 99.5%, 99.9%.
verdict.error_bound.error_column_pattern err bound of %s Column names for error bounds. Must include a single “%s”.
verdict.error_bound.method subsampling A method for error bound computation. One of “subsampling”, “bootstrapping” (only for research), “analytic”, “no_error_bound”.
verdict.error_bound.subsampling.partition_column __vpart A column name internally used by subsampling for partitioning tuples.
verdict.error_bound.subsampling.probability_column __vprob A column name internally used by subsampling for sampling probabilities.
verdict.error_bound.subsampling.partition_count 100 The number of partitions used by subsampling for computing error bounds.
verdict.error_bound.bootstrapping.random_value_column_name verdict_rand only for research
verdict.error_bound.bootstrapping.bootstrap_multiplicity_colname __mul only for research
verdict.error_bound.bootstrapping.num_of_trials 100 only for research

Note that changing subsampling-related parameters affect sample creation process.

Passing database-specific options through Verdict

Verdict-on-Spark

Simply launch Spark (or PySpark) as usual. Verdict runs as an imported module on Spark; thus, it does not affect Spark’s regular interface.

Verdict-on-{Hive, Impala}

Specify the parameters in the JDBC connection string. Or, if you are programmatically connecting to Verdict using the JDBC interface, you can pass java.util.Properties as an optional argument. For example, the following command will set your Hive applications to use a specific job queue.

\$ veeline/bin/veeline -h "hive2://host:port/default;mapred.job.queue.name=root.example_queue"

Verdict passes all key-value parameters (except for the ones that start with verdict) when it makes an internal JDBC connection to the database it works with.

format: plain


text:
## 1. Buffon's Problems

Buffon's experiments are very old and famous random experiments, named after comte de Buffon. These experiments are considered to be among the first problems in geometric probability.

### Buffon's Coin Experiment

Buffon's coin experiment consists of dropping a coin randomly on a floor covered with identically shaped tiles. The event of interest is that the coin crosses a crack between tiles. We will model Buffon's coin problem with square tiles of side length 1—assuming the side length is 1 is equivalent to taking the side length as the unit of measurement.

#### Assumptions

First, let us define the experiment mathematically. As usual, we will idealize the physical objects by assuming that the coin is a perfect circle with radius $$r$$ and that the cracks between tiles are line segments. A natural way to describe the outcome of the experiment is to record the center of the coin relative to the center of the tile where the coin happens to fall. More precisely, we will construct coordinate axes so that the tile where the coin falls occupies the square $$S = \left[ -\frac{1}{2}, \frac{1}{2} \right]^2$$.

Now when the coin is tossed, we will denote the center of the coin by $$(X, Y) \in S$$ so that $$S$$ is our sample space and $$X$$ and $$Y$$ are our basic random variables. Finally, we will assume that $$r \lt \frac{1}{2}$$ so that it is at least possible for the coin to fall inside the square without touching a crack.

Next, we need to define an appropriate probability measure that describes our basic random vector $$(X, Y)$$. If the coin falls randomly on the floor, then it is natural to assume that $$(X, Y)$$ is uniformly distributed on $$S$$. By definition, this means that

$\P[(X, Y) \in A] = \frac{\area(A)}{\area(S)}, \quad A \subseteq S$

Run Buffon's coin experiment with the default settings. Watch how the points seem to fill the sample space $$S$$ in a uniform manner.

#### The Probability of a Crack Crossing

Our interest is in the probability of the event $$C$$ that the coin crosses a crack.

The probability of a crack crossing is $$\P(C) - 1 - (1 - 2 r)^2$$.

Proof:

Note that in terms of the basic variables, $$C^c = \left\{r - \frac{1}{2} \lt X \lt \frac{1}{2} - r, r - \frac{1}{2} \lt Y \lt \frac{1}{2} - r\right\}$$.

In Buffon's coin experiment, change the radius with the scroll bar and watch how the events $$C$$ and $$C^c$$ and change. Run the experiment with various values of $$r$$ and compare the physical experiment with the points in the scatterplot. Compare the relative frequency of $$C$$ to the probability of $$C$$.

The convergence of the relative frequency of an event (as the experiment is repeated) to the probability of the event is a special case of the law of large numbers.

Solve Buffon's coin problem with rectangular tiles that have height $$h$$ and width $$w$$.

$1 - \frac{(h - 2 \, r)(w - 2 \, r)}{h \, w}, \quad r \lt \min \left\{ \frac{h}{2}, \frac{w}{2} \right\}$

Solve Buffon's coin problem with equilateral triangular tiles that have side length 1.

Recall that random numbers are simulation of independent random variables, each with the standard uniform distribution, that is, the continuous uniform distribution on the interval $$(0, 1)$$.

Show how to simulate the center of the coin $$(X, Y)$$ in Buffon's coin experiment using random numbers.

$$X = U - \frac{1}{2}$$, $$Y = V - \frac{1}{2}$$, where $$U$$ and $$V$$ are random numbers.

### Buffon's Needle Problem

Buffon's needle experiment consists of dropping a needle on a hardwood floor. The main event of interest is that the needle crosses a crack between floorboards. Strangely enough, the probability of this event leads to a statistical estimate of the number $$\pi$$!

#### Assumptions

Our first step is to define the experiment mathematically. Again we idealize the physical objects by assuming that the floorboards are uniform and that each has width 1. We will also assume that the needle has length $$L \lt 1$$ so that the needle cannot cross more than one crack. Finally, we assume that the cracks between the floorboards and the needle are line segments.

When the needle is dropped, we want to record its orientation relative to the floorboard cracks. One way to do this is to record the angle $$X$$ that the top half of the needle makes with the line through the center of the needle, parallel to the floorboards, and the distance $$Y$$ from the center of the needle to the bottom crack. These will be the basic random variables of our experiment, and thus the sample space of the experiment is $S = [0, \pi) \times [0, 1) = \{(x, y): 0 \le x \lt \pi, \; 0 \le y \lt 1\}$

Again, our main modeling assumption is that the needle is tossed randomly on the floor. Thus, a reasonable mathematical assumption might be that the basic random vector $$(X, Y)$$ is uniformly distributed over the sample space. By definition, this means that $\P[(X, Y) \in A] = \frac{\area(A)}{\area(S)}, \quad A \subseteq S$

Run Buffon's needle experiment with the default settings and watch the outcomes being plotted in the sample space. Note how the points in the scatterplot seem to fill the sample space $$S$$ in a uniform way.

#### The Probability of a Crack Crossing

Our main interest is in the event $$C$$ that the needle crosses a crack between the floorboards.

The event $$C$$ can be written in terms of the basic angle and distance variables as follows: $C = \left\{ Y \lt \frac{L}{2} \, \sin(X) \right\} \cup \left\{ Y \gt 1 - \frac{L}{2} \, \sin(X) \right\}$

The curves $$y = \frac{L}{2} \, \sin(x)$$ and $$y = 1 - \frac{L}{2} \, \sin(x)$$ on the interval $$0 \le x \lt \pi$$ are shown in blue in the scatterplot of Buffon's needle experiment, and hence event $$C$$ is the union of the regions below the lower curve and above the upper curve. Thus, the needle crosses a crack precisely when a point falls in this region.

The probability of a crack crossing is $$\P(C) = 2 L / \pi$$.

Proof:

From symmetry and simple calculus it follows that $$\area(C) = 2 L$$. Of course $$\area(S) = \pi$$.

In the Buffon's needle experiment, vary the needle length $$L$$ with the scroll bar and watch how the event $$C$$ changes. Run the experiment with various values of $$L$$ and compare the physical experiment with the points in the scatterplot. Compare the relative frequency of $$C$$ to the probability of $$C$$.

The convergence of the relative frequency of an event (as the experiment is repeated) to the probability of the event is a special case of the law of large numbers.

Find the probabilities of the following events in Buffon's needle experiment. In each case, sketch the event as a subset of the sample space.

1. $$\{0 \lt X \lt \pi / 2, \; 0 \lt Y \lt 1 / 3\}$$
2. $$\{1 / 4 \lt Y \lt 2 / 3\}$$
3. $$\{X \lt Y\}$$
4. $$\{X + Y \lt 2\}$$
1. $$\frac{1}{6}$$
2. $$\frac{5}{12}$$
3. $$\frac{1}{2 \pi}$$
4. $$\frac{3}{2 \pi}$$

#### The Estimate of $$\pi$$

Suppose that we run Buffon's needle experiment a large number of times. By the law of large numbers, the proportion of crack crossings should be about the same as the probability of a crack crossing. More precisely, we will denote the number of crack crossings in the first $$n$$ runs by $$N_n$$. Note that $$N_n$$ is a random variable for the compound experiment that consists of $$n$$ replications of the basic needle experiment. Thus, if $$n$$ is large, we should have $$\frac{N_n}{n} \approx \frac{2 L}{\pi}$$ and hence $\pi \approx \frac{2 L n}{N_n}$ This is Buffon's famous estimate of $$\pi$$. In the simulation of Buffon's needle experiment, this estimate is computed on each run and shown numerically in the second table and visually in a graph.

Run the Buffon's needle experiment with needle lengths $$L \in \{0.3, 0.5, 0.7, 1\}$$. In each case, watch the estimate of $$\pi$$ as the simulation runs.

Let us analyze the estimation problem more carefully. On each run $$j$$ we have an indicator variable $$I_j$$, where $$I_j = 1$$ if the needle crosses a crack on run $$j$$ and $$I_j = 0$$ if the needle does not cross a crack on run $$j$$. These indicator variables are independent, and identically distributed, since we are assuming independent replications of the experiment. Thus, the sequence forms a Bernoulli trials process.

The number of crack crossings in the first $$n$$ runs of the experiment is $N_n = \sum_{j=1}^n I_j$ which has the binomial distribution with parameters $$n$$ and $$2 L / \pi$$.

The mean and variance of $$N_n$$ are

1. $$\E(N_n) = n \frac{2 L}{\pi}$$
2. $$\var(N_n) = n \frac{2 L}{\pi} \left(1 - \frac{2 L}{\pi}\right)$$

With probability 1, $$\frac{N_n}{2 L n} \to \frac{1}{\pi}$$ as $$n \to \infty$$ and $$\frac{2 L n}{N_n} \to \pi$$ as $$n \to \infty$$.

Proof: a

These results follow from the strong law of large numbers.

Thus, we have two basic estimators: $$\frac{N_n}{2 L n}$$ as an estimator of $$\frac{1}{\pi}$$ and $$\frac{2 L n}{N_n}$$ as an estimator of $$\pi$$. The estimator of $$\frac{1}{\pi}$$ has several important statistical properties. First, it is unbiased since the expected value of the estimator is the parameter being estimated:

The estimator of $$\frac{1}{\pi}$$ is unbiased: $\E \left( \frac{N_n}{2 L n} \right) = \frac{1}{\pi}$

Proof:

This follows from the results above for the binomial distribution and properties of expected value.

Since this estimator is unbiased, the variance gives the mean square error: $\var \left( \frac{N_n}{2 L n} \right) = \E \left[ \left( \frac{N_n}{2 L n} - \frac{1}{\pi} \right)^2 \right]$

The mean square error of the estimator of $$\frac{1}{\pi}$$ is $\var \left( \frac{N_n}{2 L n} \right) = \frac{\pi - 2 L}{2 L n \pi^2}$

The variance is a decreasing function of the needle length $$L$$.

Thus, the estimator of $$\frac{1}{\pi}$$ improves as the needle length increases. On the other hand, the estimator of $$\pi$$ is biased; it tends to overestimate $$\pi$$:

The estimator of $$\pi$$ is positively biased: $\E \left( \frac{2 L n}{N_n} \right) \ge \pi$

Proof:

The estimator of $$\pi$$ also tends to improve as the needle length increases. This is not easy to see mathematically. However, you can see it empirically.

In the Buffon's needle experiment, run the simulation 5000 times each with $$L = 0.3$$, $$L = 0.5$$, $$L = 0.7$$, and $$L = 0.9$$. Note how well the estimator seems to work in each case.

Finally, we should note that as a practical matter, Buffon's needle experiment is not a very efficient method of approximating $$\pi$$. According to Richard Durrett, to estimate $$\pi$$ to four decimal places with $$L = \frac{1}{2}$$ would require about 100 million tosses!

Run the Buffon's needle experiment until the estimates of $$\pi$$ seem to be consistently correct to two decimal places. Note the number of runs required. Try this for needle lengths $$L = 0.3$$, $$L = 0.5$$, $$L = 0.7$$, and $$L = 0.9$$ and compare the results.

Show how to simulate the angle $$X$$ and distance $$Y$$ in Buffon's needle experiment using random numbers.

$$X = \pi U$$, $$Y = V$$, where $$U$$ and $$V$$ are random numbers.
Neil Weiss has pointed out that our computer simulation of Buffon's needle experiment is circular, in the sense the program assumes knowledge of $$\pi$$ (you can see this from the simulation result above).
Try to write a computer algorithm for Buffon's needle problem, without assuming the value of $$\pi$$ or any other transcendental numbers.


text:
+0

# Question

0
272
2

What is 2/3 - 3/2

May 23, 2017

#2
+2340
+2

$$-\frac{5}{6}=-0.8\overline3$$

$$\frac{2}{3}-\frac{3}{2}$$

Above, there is the original expession. To evaluate this, we must have common denominators. To do this, we must multiply the numerator and the denominator. If you multiply 3 by 2 and 2 by 3, then you will have a common denominator:

$$\frac{2}{3}*\frac{2}{2}=\frac{4}{6}$$

$$\frac{3}{2}*\frac{3}{3}=\frac{9}{6}$$

Notice how I am not actually changing the value of the fraction. I'm multiplying both fractions by 1, so I am not changing the value of the fraction, just the way the number is represented:

$$\frac{4}{6}-\frac{9}{6}=-\frac{5}{6}=-0.8\overline3$$

.
May 23, 2017

#1
0

-5/6

May 23, 2017
edited by Guest  May 23, 2017
#2
+2340
+2

$$-\frac{5}{6}=-0.8\overline3$$

$$\frac{2}{3}-\frac{3}{2}$$

Above, there is the original expession. To evaluate this, we must have common denominators. To do this, we must multiply the numerator and the denominator. If you multiply 3 by 2 and 2 by 3, then you will have a common denominator:

$$\frac{2}{3}*\frac{2}{2}=\frac{4}{6}$$

$$\frac{3}{2}*\frac{3}{3}=\frac{9}{6}$$

Notice how I am not actually changing the value of the fraction. I'm multiplying both fractions by 1, so I am not changing the value of the fraction, just the way the number is represented:

$$\frac{4}{6}-\frac{9}{6}=-\frac{5}{6}=-0.8\overline3$$

TheXSquaredFactor May 23, 2017


text:
# Consider the equation k/x=12 where k is any number between 20 and 65 and x is a positive integer. What are the possible values of x?

Dec 2, 2016

The possible values of $x$ are $2 , 3 , 4 , 5$.

#### Explanation:

$\frac{k}{x} = 12 \mathmr{and} 12 x = k$, "k" lies between $20 \mathmr{and} 65$

So $20 \le 12 x \le 65 \mathmr{and} \frac{20}{12} \le x \le \frac{65}{12} \mathmr{and} \frac{5}{3} \le x \le \frac{65}{12}$

"x" lies between $\frac{5}{3} \mathmr{and} \frac{65}{12}$. But x is a positive integer
So $2 \le x \le 5$

The possible values of $x$ are $2 , 3 , 4 , 5$ [Ans]


text:
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 The present state of investigations in the foundations of mathematicsA. Mostowski 3 Some problems of the qualitative theory of differential equationsV. V. Nemytskii 39 Some problems of nonlinear analysisM. A. Krasnosel'skii 57 Scientific notes and problems On a method of integration of partial differential equations of the first orderI. S. Arzhanykh 115 Extension of the method of characteristics to simultaneous partial differential equations of the first orderI. S. Arzhanykh 119 On the method of iterationsV. M. Dubrovskii 127 The determination of a completely additive function by its values on half-spacesP. O. Kostelyanets, Yu. G. Reshetnyak 135 On the solution of Cauchy's problem for regular systems of linear partial differential equationsA. G. Kostyuchenko, G. E. Shilov 141 On an indeterminate equation in Hilbert space and its application in potential theoryS. G. Krein 149 On implicit functionsL. D. Kudryavtsev 155 On the ring of functions with absolutely convergent Fourier seriesZ. I. Leibenzon 157 On the signs of the solution of a system of linear equationsV. S. Mikhel'son 163 On the continuous dependence of the solution of Cauchy's problem upon the initial dataA. D. Myshkis, U. K. Grinfel'd 171 On the curves and two-dimensional surfaces along which a solution of the wave equation can have a discontinuityI. G. Petrovskii, L. A. Chudov 175 On some criteria of locally nilpotent groupsB. I. Plotkin 181 On the mutual disposition of the lines of regressionN. A. Sapogov 187 On the variation of lengths and directions in pseudo-conformal mappingsB. A. Fuks 193 On conditions of pseudo-conformality of a mapping of four-dimensional spaceB. A. Fuks 201 Determination of the value of the measure for a region of $n$-dimensional Euclidean space from its values for all half-spacesA. A. Khachaturov 205 Methodical notes A simple proof of the mean-value theorem for harmonic functionsB. P. Demidovich 213 In the Moscow Mathematical Society Meetings of the Moscow Mathematical Society 215 Mathematical Events in the USSR Sergei Pavlovich Finikov (photo) Sergei Pavlovich Finikov (on his seventieth birthday)G. F. Laptev 245 Meetings of the Town's Mathematical Seminar in Leningrad 253 XVI Moscow School Mathematical CompetitionD. E. Men'shov, I. R. Shafarevich, E. A. Morozova, V. M. Zolotarev 257 School Mathematical Competition in KishinevI. A. Itskovich, K. S. Sibirskii 263 Reviews and Bibliography Historical-Mathematical Investigations, vols. IV, V (review)V. F. Rogachenko 267 Sh.  E. Mikeladze, Numerical methods of mathematical analysis (review)B. L. Ginzburg 274 On book “Numerical methods of mathematical analysis” by Sh. E. MikeladzeA. I. Zhukov 276 V. A. Unkovskii, Theory of probability (review)V. V. Petrov, Yu. V. Linnik 278 Letters to the Editor Comments on the paper “A three- and a four-dimensional analogue of Pascal's theorem” by D. D. Mordukhai-BoltovskiiI. Metel'ka 283 CorrectionF. R. Gantmakher 284 CorrectionG. P. Tolstov 284 Correction 285 To the paper “Classification of three-dimensional Riemannian spaces according to groups of motions”G. I. Kruchkovich 285


text:
The last modifications of this post were around 3 years ago, some information may be outdated!

👉 Note: UMAP & t-SNE.

## What?

Sometimes we need to "compress" our data to speed up algorithms or to visualize data. One way is to use dimensionality reduction which is the process of reducing the number of random variables under consideration by obtaining a set of principal variables. We can think of 2 approaches:

• Feature selection: find a subset of the input variables.
• Feature projection (also Feature extraction): transforms the data in the high-dimensional space to a space of fewer dimensions. PCA is one of the methods following this approach.

Figure 1. An idea of using PCA from 2D to 1D.

Figure 2. An idea of using PCA from 5D to 2D.

Questions: How can we choose the green arrows like in Figure 1 and 2 (their directions and their magnitudes)?

From a data points, there are many ways of projections, for examples,

Figure 3. We will project the points to the green line or the violet line? Which one is the best choice?

Intuitively, the green line is better with more separated points. But how can we choose it "mathematically" (precisely)? We need to know about:

• Mean: finds the most balanced point in the data.
• Variance: measures the spread of data from the mean. However, variance is not enough. There are many different ways in that we get the same variance.
• Covariance: indicates the direction in that data are spreading.
An example of the same mean and variance but different covariance.

Figure 4. Different data but the same mean and variance. That's why we need covariance!

### Algorithm

1. Subtract the mean to move to the original axes.

2. From the original data (a lot of features $x_1, x_2, \ldots, x_N$), we construct a covariance matrix $U$.

3. Find the eigenvalues $\lambda_1, \lambda_2,\ldots$ and correspondent eigenvectors $v_1, v_2, \ldots$ of that matrix (we call them eigenstuffs). Choose $K < N$ couples $\lambda$ and $v$ (the highest eigenvalues) and we get a reduced matrix $U_K$.

4. Projection original data points to the $K$-dimensional plane created based on these new eigenstuffs. This step creates new data points on a new dimensional space ($K$).

$Z = U_K^TX$
5. Now, instead of solving the original problem ($N$ features), we only need to solve a new problem with $K$ features ($K).

Figure 5. A big picture of the idea of PCA algorithm. "Eigenstuffs" are eigenvalues and eigenvectors. Source.

## Code

from sklearn.decomposition import PCAs = np.array([...])pca = PCA(n_components=150, whiten=True, random_state=42)# pca.fit(s)s1 = pca.fit_transform(s)print (pca.components_) # eigenvectorsprint (pca.explained_variance_) # eigenvalues

Some notable components (see full):

• pca.fit(X): only fit X (and then we can use pca for other operations).
• pca.fit_transform(X): Fit the model with X and apply the dimensionality reduction on X (from (n_samples, n_features) to (n_samples, n_components)).
• pca.inverse_transform(s1): transform s1 back to original data space (2D) - not back to s!!!
• pca1.mean_: mean point of the data.
• pca.components_: eigenvectors (n_components vectors).
• pca.explained_variance_: eigenvalues. It's also the amount of retained variance which is corresponding to each components.
• pca.explained_variance_ratio_: the percentage in that variance is retained if we consider on each component.

Some notable parameters:

• n_components=0.80: means it will return the Eigenvectors that have the 80% of the variation in the dataset.

When choosing the number of principal components ($K$), we choose $K$ to be the smallest value so that for example, $99\%$ of variance, is retained.[ref]

In Scikit-learn, we can use pca.explained_variance_ratio_.cumsum(). For example, n_components = 5 and we have,

[0.32047581  0.59549787  0.80178824  0.932976    1.]

then we know that with $K=4$, we would retain $93.3\%$ of the variance.

### Whitening

Whitening makes the features:

• less correlated with each other,
• all features have the same variance (or, unit component-wise variances).

PCA / Whitening. Left: Original toy, 2-dimensional input data. Middle: After performing PCA. The data is centered at zero and then rotated into the eigenbasis of the data covariance matrix. This decorrelates the data (the covariance matrix becomes diagonal). Right: Each dimension is additionally scaled by the eigenvalues, transforming the data covariance matrix into the identity matrix. Geometrically, this corresponds to stretching and squeezing the data into an isotropic gaussian blob.

If this section doesn't satisfy you, read this and this (section PCA and Whitening).

## PCA in action

• Example to understand the idea of PCA: html file -- open in colab.

• Plot points with 2 lines which are corresponding to 2 eigenvectors.
• Plot & choose Principal Components.
• An example of choosing n_components $K$.
• Visualization hand-written digits (the case of all digits and the case of only 2 digits -- 1 & 8).
• Using SVM to classifier data in the case of 1 & 8 and visualize the decision boundaries.
• Image compression: html file -- open in colab.

• When input is an image, the values of adjacent pixels are highly correlated.
• Import images from scipy and Google Drive or Github (with git).
• Compress grayscale images and colored ones.
• Plot a grayscale version of a colorful images.
• Save output to file (Google Drive).
• Fix warning Lossy conversion from float64 to uint8. Range [...,...]. Convert image to uint8 prior to saving to suppress this warning.
• Fix warning Clipping input data to the valid range for imshow with RGB data ([0..1] for floats or [0..255] for integers).
• Calculate a size (in KB) of a image file.
• PCA without scikit-learn: html file -- open in colab.


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# Computing document similarity in latent semantic analysis

I have a question regarding Latent Semantic Analysis - after performing SVD decomposition of term-document matrix and choosing some number of dimensions, I get the set of new document vectors.

Now, how can I calculate similarity between two documents? New document vectors contain negative values, and results produced by cosine similarity make no sense.

• What I mean is that if we have the following vectors: $(2, -4)$ and $(6,3)$, cosine similarity will be 0, indicating no relations between them. There is one more question, just to make sure - if I have 500 documents and 6000 terms, and the rank of term-document matrix equals 500, then the matrix $D$ in $TSD^{\top}$ will have size $500 \times 500$. Is that correct? Then, can I work with document vectors of size 500, or should I perform dimensionality reduction anyway? – user1315305 Aug 23 '13 at 22:16
• And also, in similarity computations, do I use document vectors taken directly from matrix $D$, or should I use $SD^{\top}$ (the second solution is given e.g. on Wikipedia). – user1315305 Aug 23 '13 at 22:25
• Since $S$ is a diagonal matrix multiplying a matrix by $S$ will only change the magnitudes of the vectors in the matrix, so cosine similarity between vectors in the matrix will not change. That is, you can use both of them. – Sanyo Mn Aug 23 '13 at 22:51
• I have one more question - how can I convert cosine similarity into distance value when using LSA? In traditional vector space model, it was $1 - cosine$, but I guess it won't work correctly in this case? – user1315305 Aug 24 '13 at 13:11


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# Adjusting to the new version of Pylab and Mayavi on Ubuntu 12.04

It seems the IPython and Pylab packages has also been updated in 12.04 and thus removing the old ipython -wthread flag that would ensure Mayavi plots to be run in a separate thread. Running with the flag causes this error to show up:

[TerminalIPythonApp] Unrecognized flag: '-wthread'

Without this flag, the Mayavi plots lock up the UI and hangs. If you want to get the possibility back to rotate and play around with the plots, just start IPython the following way from now on:

ipython --pylab=qt

This will launch IPython with the Qt backend and threading. Using only –pylab does not include threading. For easy and quick access, add the following to a file named .bashrc in your home folder:

alias pylab='ipython --pylab=qt'

From now on you can launch IPython just by typing

pylab

in a terminal.

# Using the same script on installs with different EPD versions

In the newest version of Enthought’s Python Distribution (EPD) on Ubuntu, the plotting package has been moved from enthought.mayavi.mlab to the shorter and more general mayavi.mlab. This does however mean that if you, like me, need to work with different versions of EPD on multiple systems, will experience the following error from time to time:

ImportError: No module named enthought.mayavi.mlab

Now, to avoid switching the import statement every time you switch systems, you can make Python check if one of the versions is installed during import. If it is not, we’ll tell it to try the other. This is done in this simple command:

try:
from enthought.mayavi.mlab import *
except ImportError:
from mayavi.mlab import *

Just replace any other similar import statements the same way and your code should once again be working across all your installations.

# Classical mechanics in HTML5 and Javascript

Ok, so we’re working on the home exam in classical mechanics and I decided to spend some time simulating the solution of the first exercise. Probably not the wisest way to spend time during an home exam, but what the heck. I’m learning some Javascript and HTML5, plus the fact that it’s always fun to verify results by looking at something moving!

So here you go. A pendulum on a cart.

I won’t go into details about deriving and solving the Lagrange equation for this system now, but I might get back to that at a later time. And of course there will be some added controls to this so you could set the initial positions, velocities and stuff. Sometime later, I hope.

# LaTeX It! (and a new blog)

Together with my good friend, Mikael, I’ve started a blog about physics, chemistry, maths and technology over at Mindseye.no. Some of my postings over there will be dual posted here. Especially those about technical notes and tutorials. This is the first one, about LaTeX equations in Thunderbird.

If you are using Thunderbird for e-mail and want to send mathematical formulas to your contacts, you should consider the LaTeX It! plugin or the Equations plugin. The former requires you to have LaTeX and ImageMagick installed, while Equations uses an external server to generate your images.

There are a few pros and cons to both approaches. If you are using the Equations plugin, your images won’t be integrated in your e-mail as files, but will simply be linked to. You will see the images, of course, but if the server should crash or no longer be operative, your LaTeX formulas will become invisible.

This will probably not happen in the nearest future, but if you want to see your formulas forever, it might be a good idea to use LaTeX It! instead.

The problem with LaTeX It! is that it requires your computer to have the right set of packages installed. In Ubuntu this is no problem – you just install the packages texlive-latex-base, texlive-binaries and imagemagick using either Synaptic or by issuing the following command in a Terminal:

sudo apt-get install texlive-latex-base texlive-binaries imagemagick

In Windows you need to install LaTeX and ImageMagick by using the automatic installers, and you will also need GhostScript. On a Mac, however, I’ve not yet got the installation to work. You could try to see if everything is installable from MacPorts, but I have been unlucky with the converting, and would suggest you use the Equations plugin if everything else fails.


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# Algebra

 1. (28 p.) Let $$a$$ and $$b$$ be positive real numbers such that $$ab=2$$ and $\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.$ Find $$a^6+b^6$$.

 2. (23 p.) Consider the set $$S\subseteq(0,1]^2$$ in the coordinate plane that consists of all points $$(x,y)$$ such that both $$[\log_2(1/x)]$$ and $$[\log_5(1/y)]$$ are even. The area of $$S$$ can be written in the form $$p/q$$ for two relatively prime integers $$p$$ and $$q$$. Evaluate $$p+q$$.

 3. (9 p.) Real numbers $$x,y,z$$ are real numbers greater than 1 and $$w$$ is a positive real number. If $$\log_xw=24$$, $$\log_yw=40$$ and $$\log_{xyz}w=12$$, find $$\log_zw$$.

 4. (28 p.) Let $$f:\mathbb N\rightarrow\mathbb R$$ be the function defined by $$f(1) = 1$$, $$f(n) = n/10$$ if $$n$$ is a multiple of 10 and $$f(n) = n+1$$ otherwise. For each positive integer $$m$$ define the sequence $$x_1$$, $$x_2$$, $$x_3$$, ... by $$x_1 = m$$, $$x_{n+1} = f(x_n)$$. Let $$g(m)$$ be the smallest $$n$$ such that $$x_n = 1$$. (Examples: $$g(100) = 3$$, $$g(87) = 7$$.) Denote by $$N$$ be the number of positive integers $$m$$ such that $$g(m) = 20$$. The number of distinct prime factors of $$N$$ is equal to $$2^u\cdot v$$ for two non-negative integers $$u$$ and $$v$$ such that $$v$$ is odd. Determine $$u+v$$.

 5. (9 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax


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## Tuesday, May 13, 2014

### The wave function 03

— 1.4. Normalization —

The Scroedinger equation is a linear partial differential equation. As such, if ${\Psi(x,t)}$ is a solution to it, then ${A\Psi(x,t)}$ (where ${A}$ is a complex constant) also is a solution.

Does this mean that a physical problem has an infinite number of solutions in Quantum Mechanics? It doesn't! The thing is that besides the The Scroedinger equation one also has condition 11 to take into account. Stating 11 for the wave function:

$\displaystyle \int_{-\infty}^{+\infty}|\Psi (x,t)|^2dx=1 \ \ \ \ \ (15)$

The previous equations states the quite obvious fact that the particle under study has to be in some place at a given instant.

Since ${A}$ was a complex constant the normalization condition fixes ${A}$ in absolute value but can't tell us nothing regarding its phase. Apparently once again one is haunted with the perspective of having an infinite number of solutions to any given physical problem. The things is that this time the phase doesn't carry any physical significance (a fact that will be demonstrated later) and thus we actually have just one physical solution.

In the previous discussion one is obviously assuming that the wave function is normalizable. That is to say that the function doesn't blow up and vanishes quickly enough at infinity so that the integral being computed makes sense.

At this level it is customary to say that these wave functions don't represent physical states but that isn't exactly true. A wave function that isn't normalizable because integral is infinite might represent a beam of particles in a scattering experiment. The fact that the integral diverges to infinity can then be said to represent the fact that beam is composed by an infinite amount of particles.

While the identically null wave function represents the absence of particles.

A question that now arises has to do with the consistency of our normalization and this is a very sensible question. The point is that we normalize the Schroedinger equation for a given time instant, so how does one know that the normalization holds for other times?

Let us look into the time evolution of our normalization condition 15.

$\displaystyle \frac{d}{dt}\int_{-\infty}^{+\infty}|\Psi (x,t)|^2dx=\int_{-\infty}^{+\infty}\frac{\partial}{\partial t}|\Psi (x,t)|^2dx \ \ \ \ \ (16)$

Calculating the derivative under the integral for the right hand side of the previous equation

{\begin{aligned} \frac{\partial}{\partial t}|\Psi (x,t)|^2&=\frac{\partial}{\partial t}(\Psi^* (x,t)\Psi (x,t))\\ &=\Psi^* (x,t)\frac{\partial\Psi (x,t)}{\partial t}+\frac{\partial \Psi^* (x,t)}{\partial t}\Psi (x,t) \end{aligned}}

The complex conjugate of the Schroedinger equation is

$\displaystyle \frac{\partial \Psi^*(x,t)}{\partial t}=-\frac{i\hbar}{2m}\frac{\partial^2\Psi^*(x,t)}{\partial x^2}+\frac{i}{\hbar}V\Psi^*(x,t) \ \ \ \ \ (17)$

Hence for the derivative under the integral

{\begin{aligned} \frac{\partial}{\partial t}|\Psi (x,t)|^2&=\frac{\partial}{\partial t}(\Psi^* (x,t)\Psi (x,t))\\ &=\Psi^* (x,t)\frac{\partial\Psi (x,t)}{\partial t}+\frac{\partial \Psi^* (x,t)}{\partial t}\Psi (x,t)\\ &=\frac{i\hbar}{2m}\left( \Psi^*(x,t)\frac{\partial^2\Psi(x,t)}{\partial x^2}-\frac{\partial^2\Psi^*(x,t)}{\partial x^2}\Psi (x,t)\right)\\ &=\frac{\partial}{\partial x}\left[ \frac{i\hbar}{2m}\left( \Psi^*(x,t)\frac{\partial\Psi(x,t)}{\partial x}-\frac{\partial\Psi^*(x,t)}{\partial x}\Psi(x,t) \right) \right] \end{aligned}}

Getting back to 16

$\displaystyle \frac{d}{dt}\int_{-\infty}^{+\infty}|\Psi (x,t)|^2dx=\frac{i\hbar}{2m}\left[ \Psi^*(x,t)\frac{\partial\Psi(x,t)}{\partial x}-\frac{\partial\Psi^*(x,t)}{\partial x}\Psi(x,t) \right]_{-\infty}^{+\infty} \ \ \ \ \ (18)$

Since we're assuming that our wave function is normalizable the wave function (and its complex conjugate) must vanish for ${+\infty}$ and ${-\infty}$.

In conclusion

$\displaystyle \frac{d}{dt}\int_{-\infty}^{+\infty}|\Psi (x,t)|^2dx=0$

In conclusion one can say that if one normalizes the wave equation for a given time interval it stays normalized for all time intervals.

 Exercise 1 At time ${t=0}$ a particle is represented by the wave function $\displaystyle \Psi(x,0)=\begin{cases} Ax/a & \text{if } 0\leq x\leq a\\ A(b-x)/(b-a) & \text{if } a\leq x\leq b \\ 0 & \text{otherwise}\end{cases} \ \ \ \ \ (19)$ where ${A}$, ${a}$ and ${b}$ are constants. Normalize ${\Psi}$. {\begin{aligned} 1&=\int_{-\infty}^{+\infty} |\Psi|^2\,dx\\ &=\int_0^a|\Psi|^2\,dx+\int_a^b|\Psi|^2\,dx\\ &=\dfrac{|A|^2}{a^2}\int_0^a|x^2\,dx+\dfrac{|A|^2}{(b-a)^2}\int_a^b(b-x)^2\,dx\\ &=\dfrac{|A|^2}{a^2}\left[ \dfrac{x^3}{3} \right]_0^a+\dfrac{|A|^2}{(b-a)^2}\left[ \dfrac{(b-x)^3}{3} \right]_a^b\\ &=\dfrac{|A|^2a}{3}+\dfrac{|A|^2}{(b-a)^2}\dfrac{(b-a)^3}{3}\\ &=\dfrac{|A|^2a}{3}+|A|^2\dfrac{b-a}{3}\\ &=\dfrac{b|A|^2}{3} \end{aligned}} Hence for ${A}$ it is $\displaystyle A=\sqrt{\dfrac{3}{b}}$ Sketch ${\Psi(x,0)}$ In ${0\leq x \leq a}$ ${\Psi(x,0)}$ is a strictly increasing function that goes from ${0}$ to ${A}$. In ${a \leq x \leq b}$ ${\Psi(x,0)}$ is strictly decreasing function that goes from ${A}$ to ${0}$. Hence the plot of ${\Psi(x,0)}$ is (choosing the following values ${a=1}$, ${b=2}$ and ${A=\sqrt{b}=\sqrt{2}}$): Where is the particle most likely to be found at ${t=0}$? Since ${x=a}$ is maximum of the ${\Psi}$ function the most likely value for the particle to be found is at ${x=a}$. What is the probability of finding the particle to the left of ${a}$? Check the answers for ${b=a}$ and ${b=2a}$. {\begin{aligned} P(x&= \int_a^b x|\Psi|^2\,dx\\ &=\dfrac{|A|^2}{a^2}\int_0^a x^3\,dx+\dfrac{|A|^2}{(b-a)^2}\int_a^b x(b-x)^2\,dx\\ &=\dfrac{|A|^2}{a^2}\left[ \dfrac{x^4}{4} \right]_0^a+\dfrac{|A|^2}{(b-a)^2}\left[ 1/2x^2b^2-2/3x^3b+x^4/4 \right]_a^b\\ &=\dfrac{2a+b}{4} \end{aligned}}

 Exercise 2 Consider the wave function $\displaystyle \Psi(x,t)=Ae^{-\lambda |x|}e^{-i\omega t} \ \ \ \ \ (20)$ where ${A}$, ${\lambda}$ and ${\omega}$ are positive real constants. Normalize ${\Psi}$ {\begin{aligned} 1&=\int_{-\infty}^{+\infty} |\Psi|^2\,dx\\ &=\int_{-\infty}^{+\infty} |A|^2e^{-2\lambda |x|}\,dx\\ &=2|A|^2\int_0^{+\infty}e^{-2\lambda |x|}\,dx \\ &=2|A|^2\int_0^{+\infty}e^{-2\lambda x}\,dx \\ &=-\dfrac{|A|^2}{\lambda}\left[ e^{-2\lambda x} \right]_0^{+\infty}\\ &=\dfrac{|A|^2}{\lambda} \end{aligned}} Hence it is $\displaystyle A=\sqrt{\lambda}$ Determine ${}$ and ${}$ {\begin{aligned} &=\int_{-\infty}^{+\infty} x|\Psi|^2\,dx\\ &=|A|^2\int_{-\infty}^{+\infty} xe^{-2\lambda |x|}\,dx\\ &=0 \end{aligned}} The integral is vanishing because we're calculating the integral of an odd function between symmetrical limits. {\begin{aligned} &=\int_{-\infty}^{+\infty} x^2|\Psi|^2\,dx\\ &=2\lambda\int_0^{+\infty} x^2e^{-2\lambda x}\,dx\\ &=2\lambda\int_0^{+\infty} \dfrac{1}{4}\dfrac{\partial^2}{\partial \lambda ^2}\left( e^{-2\lambda x} \right)\,dx\\ &=\dfrac{\lambda}{2} \dfrac{\partial^2}{\partial \lambda ^2}\int_0^{+\infty}e^{-2\lambda\,dx} x \,dx\\ &= \dfrac{\lambda}{2} \dfrac{\partial^2}{\partial \lambda ^2} \left[ -\dfrac{e^{-2\lambda\,dx}}{2\lambda} \right]_0^{+\infty}\\ &= \dfrac{\lambda}{2}\dfrac{\partial^2}{\partial \lambda ^2}\left(\dfrac{1}{2\lambda} \right)\\ &=\dfrac{\lambda}{2}\dfrac{\partial}{\partial \lambda}\left(-\dfrac{1}{\lambda ^2} \right)\\ &=\dfrac{\lambda}{2}\dfrac{1}{\lambda^3}\\ &= \dfrac{1}{2\lambda^2} \end{aligned}} Find the standard deviation of ${x}$. Sketch the graph of ${\Psi ^2}$. What is the probability that the particle will be found outside the range ${[-\sigma,+\sigma]}$? $\displaystyle \sigma ^2=-^2=\frac{1}{2\lambda ^2}-0=\frac{1}{2\lambda ^2}$ Hence the standard deviation is $\displaystyle \sigma=\dfrac{\sqrt{2}}{2\lambda}$ The square of the wave function is proportional to ${e^{-2\lambda |x|}}$. Dealing for piecewise definitions of the square of the wave function, its first derivative in order to ${x}$ and its second derivative in order to ${x}$ $\displaystyle |\Psi|^2=\begin{cases} e^{2\lambda x} & \text{if } x < 0\\ e^{-2\lambda x} & \text{if } x \geq 0 \end{cases} \ \ \ \ \ (21)$ $\displaystyle \dfrac{\partial}{\partial x}|\Psi|^2=\begin{cases} 2\lambda e^{2\lambda x} & \text{if } x < 0\\ -2\lambda e^{-2\lambda x} & \text{if } x \geq 0 \end{cases} \ \ \ \ \ (22)$ $\displaystyle \dfrac{\partial ^2}{\partial x ^2}|\Psi|^2=\begin{cases} 4\lambda ^2 e^{2\lambda x} & \text{if } x < 0\\ 4\lambda ^2 e^{-2\lambda x} & \text{if } x \geq 0 \end{cases} \ \ \ \ \ (23)$ As we can see the first derivative of ${|\Psi|^2}$ changes its sign on ${0}$ from positive to negative. Hence it was strictly increasing before ${0}$ and it is strictly decreasing after ${0}$. Hence ${0}$ is a maximum of ${|\Psi|^2}$. The second derivative is always positive so ${|\Psi|^2}$ is always concave up (convex). Hence its graphical representation is: The probability that the particle is to be found outside the range ${[-\sigma, +\sigma ]}$ is {\begin{aligned} P(-\sigma, +\sigma)&= 2\int_\sigma^{+\infty}|\Psi|^2\,dx\\ &= 2\lambda\int_\sigma^{+\infty}e^{2\lambda x}\\ &= 2\lambda\left[ -\dfrac{e^{2\lambda x}}{2\lambda} \right]_\sigma^{+\infty}\\ &=\lambda \dfrac{e^{2\lambda x}}{2\lambda}\\ &=e^{-2\lambda\dfrac{\sqrt{2}}{2\lambda}}\\ &=e^{-\sqrt{2}} \end{aligned}}