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# Soil net nitrogen mineralisation across global grasslands

## Abstract

Soil nitrogen mineralisation (Nmin), the conversion of organic into inorganic N, is important for productivity and nutrient cycling. The balance between mineralisation and immobilisation (net Nmin) varies with soil properties and climate. However, because most global-scale assessments of net Nmin are laboratory-based, its regulation under field-conditions and implications for real-world soil functioning remain uncertain. Here, we explore the drivers of realised (field) and potential (laboratory) soil net Nmin across 30 grasslands worldwide. We find that realised Nmin is largely explained by temperature of the wettest quarter, microbial biomass, clay content and bulk density. Potential Nmin only weakly correlates with realised Nmin, but contributes to explain realised net Nmin when combined with soil and climatic variables. We provide novel insights of global realised soil net Nmin and show that potential soil net Nmin data available in the literature could be parameterised with soil and climate data to better predict realised Nmin.

## Introduction

Soil nitrogen (N) availability is one of the most important drivers of ecosystem productivity1,2,3 and microbial decomposition4, and is key in regulating N cycling. During the breakdown and depolymerisation of organic material to monomers and inorganic N, plants and microbes compete for these N resources5,6,7. The net balance of N mineralisation and immobilisation (soil net Nmin) is mediated by soil physico-chemical properties, aboveground and belowground litter input, plant and microbial nutrient demand and climatic factors5,8,9,10,11,12,13, and is regarded as a good index of overall soil N availability5. Soil net Nmin usually is greater in well-aerated soils from more humid climates at lower latitudes14,15,16, reflecting the controls of soil temperature, moisture and oxygen content over microbial activity. However, climatic conditions also shape soil properties over long time-scales17,18, so understanding the impact of climate and soil properties for soil net Nmin is crucial to achieve robust estimates of soil N availability, and ultimately, productivity in terrestrial ecosystems.

Soil net Nmin is commonly estimated either in the field or laboratory. Field measures represent realised soil net Nmin constrained by site macro-climatic and micro-climatic conditions (Fig. 1) and are typically collected at local to regional scales12. In contrast, studies at the continental15,16 or global scale14,19 generally rely on laboratory incubations that estimate potential soil net Nmin. Laboratory incubations use homogenised soil samples incubated under optimised and controlled temperature and moisture conditions to allow the comparison of samples collected from different locations in a standardised way. Yet, homogenisation disrupts the soil structure and removes plant residues, which may affect these estimates20,21. Laboratory measures also fail to account for soil micro-climatic differences found under field conditions. Although potential may in some cases effectively predict realised soil net Nmin, we do not know whether, or under what conditions this is the case. Successfully identifying the environmental drivers connecting the two indices could greatly enhance the use of potential soil net Nmin data to model and predict global patterns in realised soil net Nmin. This would facilitate our understanding of how global soil N availability may respond to future global anthropogenic influences such as climate change or eutrophication22,23.

We conducted coordinated measurements of realised and potential soil net Nmin, and assessed soil properties and climatic variables across 30 grasslands on six continents that span a globally relevant range of climatic and edaphic conditions (Fig. 2; Supplementary Tables 1 and 2). We focused on grassland ecosystems because they cover approximately one-third of the Earth's terrestrial landscape24,25, are threatened by global change22,23,26, and provide important ecosystem services intricately linked with Nmin. Notably, they store 20–30% of the terrestrial C, mostly in the soil24,25,27. Here, we describe the global spatial patterns in realised and potential soil net Nmin and the relationship between them. We then explore the key drivers of each soil net Nmin index separately. Finally, we use structural equation modelling (SEM) to build a system-level understanding of how these specific climate and soil variables together with potential soil net Nmin could be used to predict realised soil net Nmin. This final step provides a basis upon which extensively available data on potential soil net Nmin could be leveraged to improve predictions of realised soil net Nmin at a global scale.

First, we expect opposite global spatial patterns in realised and potential soil net Nmin: Namely, decreasing realised soil net Nmin with increasing distance to the equator due to colder growing season temperatures and therefore lower mineralisation. Concurrently, we expect increasing potential soil net Nmin with increasing distance to the equator, because higher amounts of labile organic material accumulated under colder conditions are mineralised when incubated under standardised conditions in the laboratory. Second, climate variables and the presence of plant residues should predict realised soil net Nmin through their effects on soil properties and the activity of soil biota11,12,13. In contrast, soil chemical properties, soil texture and the activity/abundance of soil biota may be more important for potential soil net Nmin than climatic variables11,12,13. Nonetheless, as climatic conditions impact soil formation17,18, we expected that climate provides additional predictive information to explain potential soil net Nmin. Third, we expect that realised soil net Nmin can be estimated by combining key environmental drivers with potential soil net Nmin.

All 30 sites (Fig. 2, Supplementary Tables 1 and 2) are part of the Nutrient Network globally distributed experiment (NutNet [https://nutnet.umn.edu/])28. We incubated one soil sample per plot in the field to assess realised soil net Nmin (Fig. 1)29 and collected additional samples to measure potential soil net Nmin and soil properties, i.e., water holding capacity, bulk density, C and N content, texture, pH, pore space, microbial biomass, and archaeal (AOA) and bacterial (AOB) ammonia oxidiser abundance using identical materials and methods at each site. The field incubation averaged 42 days (range 36–57) and ended at peak plant biomass at each site. Soil moisture was kept at 60% of the field capacity of each sample and 20 °C for the 42-day laboratory incubations. Climate data for each site were obtained from global data sets30.

We dropped correlated variables prior to analyses31, calculated the correlation between realised and potential soil net Nmin and then explored the global spatial patterns in realised and potential soil net Nmin with linear mixed effects models (LMMs). Then, we used LMMs and multilevel inference32 to determine the drivers of realised and potential soil net Nmin. Based on the LMMs and existing literature, we developed a conceptual model of causal relationships among environmental drivers, potential soil net Nmin and realised soil net Nmin (Fig. 3 and Table 1)33.

Here, we find that realised correlates only weakly with potential soil net Nmin, and that different environmental parameters drive the two measures. However, potential soil net Nmin contributes to explain realised Nmin when combined with soil and climatic variables. We provide new insights for realised soil net Nmin and show how potential soil net Nmin data could be parameterised to better predict realised Nmin in global grasslands.

## Results and discussion

### Global patterns and drivers of soil net N mineralisation

Across our 30 grassland sites, realised and potential soil net Nmin were weakly positively correlated (r = 0.21, p = 0.052, df = 83, Supplementary Fig. 1). Individual values of realised soil net Nmin ranged from −0.14 to 1.40 mg N kg soil−1 day−1, and from −0.31 to 2.85 mg N kg soil−1 day−1 for potential soil net Nmin (Fig. 4a, b). Among sites, average realised soil net Nmin ranged from −0.02 mg N kg soil−1 day−1 at Bariloche, Argentina (bari.ar) to 1.09 mg N kg soil−1 day−1 at Las Chilcas, Argentina (chilcas.ar), whereas average potential soil net Nmin ranged from −0.19 mg N kg soil−1 day−1 at Podocarpus, Ecuador (podo.ec) to 2.54 mg N kg soil−1 day−1 at Lancaster, UK (lancaster.uk, Fig. 4a, b). Within-site range of realised soil net Nmin was lowest at Sheep Station, UT, USA (shps.us; Δ = 0.02 mg N kg soil−1 day−1) and highest at Kilpisjärvi, Finland (kilp.fi; Δ = 1.01 mg N kg soil−1 day−1); for potential soil net Nmin, the range was lowest at Mt. Caroline, Australia (mtca.au; Δ = 0.03 mg N kg soil−1 day−1) and highest at Bogong, Australia (bogong.au; Δ = 1.08 mg N kg soil−1 day−1; Fig. 4a, b).

Contrary to our predictions, realised soil net Nmin was not related to distance to the equator (F1,28 = 0.057, p = 0.81, Supplementary Fig. 2A), even when we corrected the values for growing season length for each site (Supplementary Fig. 3A, B). However, potential soil net Nmin significantly increased with increasing distance (F1,28 = 22.86, p < 0.001, Supplementary Fig. 2B), which supports our hypothesis but contrasts with meta-analyses14,15 reporting decreased potential soil net Nmin with increasing distance to the equator. However, the meta-analyses may not be comparable to our more closely controlled study from a single vegetation type, because they included a wide range of data from different land-use and vegetation types (croplands, wetlands, forests, shrublands, grasslands) and incubation conditions (duration, temperature, and soil moisture).

The variation in realised soil net Nmin across our 30 grassland sites was jointly explained by positive effects of temperature of the wettest quarter and microbial biomass (Model 1; Table 2); by microbial biomass alone (Model 2, Table 2), clay alone (Models 3, Table 2) or by a positive effect of temperature of the wettest quarter combined with a negative effect of soil bulk density (Model 4; Table 2). Many studies consider soil organic C as one of the main drivers of soil net Nmin. In our study, soil organic C was highly correlated with soil microbial biomass (r = 0.85). When replacing microbial biomass with soil organic C the model selection process yielded similar results (Models 1–3; Supplementary Table 3). Together our findings suggest that with higher temperature of the wettest quarter and more microbial biomass (or soil organic C), more organic matter was mineralised by soil microbes. The greater explanatory power of temperature of the wettest quarter, as opposed to MAT or MAP, shows that annual averages may be less useful for predicting ecosystem processes than more temporally specific climatic variables30,34; higher temperatures may only promote soil biological activity if soil moisture levels are sufficiently high35. Our results also suggest that sites with higher soil clay content and lower soil bulk density likely featured more conducive soil micro-climatic conditions for soil microbes to thrive and allowed for higher realised soil net Nmin. In contrast to microbial biomass, soil clay content and soil organic C, soil bulk density is usually not considered a key predictor of soil N mineralisation11,12. Here, we show that bulk density as a measure of favourable soil structure improved predictions of realised soil net Nmin. Future soil net Nmin studies and simulation studies for soil N cycling may benefit from including bulk density.

Potential soil net Nmin was higher when more AOB were present at the start of the incubation (Models 5, 6, Table 2). However, there is no mechanistic link between AOB and potential soil net Nmin, because AOB only transform ammonium to nitrate, but do not drive net production of total inorganic nitrogen in the soil. Yet, AOB abundance was positively correlated with potential net nitrification (Supplementary Fig. 4), which is similar to previous findings36,37,38,39. Further, potential soil net Nmin was positively influenced by temperature seasonality and was negatively affected by temperature of the wettest quarter (Models 5, 6, 8, 9, Table 2). The same models were selected when we replaced soil microbial biomass with organic C (Models 4–7, Supplementary Table 3). However, higher potential soil net Nmin was also explained by higher soil microbial biomass alone (Model 7; Table 2) and microbial biomass was positively correlated with potential soil net nitrification (Supplementary Fig. 5). Our results agree with findings of a recent meta-analysis that identified soil microbial biomass as an important driver of potential soil net Nmin19. In addition, the effect of microbial biomass on potential soil net Nmin indicates that a quantitatively improved understanding of the soil microbial community could likely improve soil biogeochemical models40,41. In addition, the selection of the two climate variables rather than the expected individual soil physical and chemical variables as predictors of potential soil net Nmin suggests that there is a long-term legacy effect of climate on these grasslands that we were not able to capture with the soil physico-chemical variables that we measured17,18.

Soil C:N ratio is often regarded as an important predictor of soil net Nmin as it determines the transition from net N immobilisation to net N mineralisation16,42. However, soil C:N was not important in our study as all but one (temple.us) of our grassland soils had C:N ratios below the critical threshold of 20. Also contrary to our expectations, realised and potential soil net Nmin were both partially constrained by climatic variables. Interestingly, temperature of the wettest quarter positively influenced realised soil net Nmin, but had a negative effect on potential Nmin (Table 2, also see Fig. 5). This pattern suggests that N mineralisation may ‘acclimatise’ along climate gradients43,44: greater mineralisation occurring in a warmer and wetter climate may lead to the depletion of easily available organic N pools compared to soils from cooler climates. When incubated under constant temperature in the laboratory, mineralisation rates from substrate-depleted soils from warmer climates were less than those from soils of cold regions where labile organic N has accumulated for centuries43,44. Alternatively, physical disturbance and disruption of the soil structure caused by sieving and sample homogenisation may have more profoundly affected the samples from warmer and wetter climates. Finally, it could be that the soil microbes did not perform as well as expected in the laboratory because the 20 °C incubation temperature was considerably lower than the field temperatures during peak growing season at the warmer and wetter sites. Thus, the potentials we measured may not have represented full potential mineralisation for these sites.

### Estimating soil net N mineralisation in grasslands worldwide

By combining the identified main drivers from the LMMs and potential net Nmin (Figs. 3 and 5, Table 1), we produced a SEM model that explained 33% (marginal R2) of the variation in realised soil net Nmin across these grasslands (Fig. 5a). This is similar to the explained variability in potential soil net Nmin measured in other studies14,16,19. The model revealed a new system-level understanding of the controls on global-scale patterns in realised net Nmin by showing that temperature of the wettest quarter, soil microbial biomass, and potential soil net Nmin (positive effects) can be directly related to realised soil net Nmin. Soils with higher clay content, which have higher soil microbial biomass, have higher potential soil net Nmin, altogether having a positive effect on realised soil net Nmin (Fig. 5a, b). The negative effect of temperature of the wettest quarter and the positive effect of temperature variability represent a legacy effect of climate on soil properties that affect potential soil net Nmin. Again, our findings were very similar when we substituted soil microbial biomass with soil organic C (Supplementary Fig. 6).

This study is the first to directly and simultaneously compare realised and potential soil net Nmin across a globally relevant range of biotic and climatic conditions in grasslands. The two indices were only weakly related across these grasslands, highlighting the uncertainty in using laboratory measurements of soil net Nmin to predict rates actually occurring in the field. By combining potential soil net Nmin with specific climate and soil property data, we produced more robust estimations of realised soil net Nmin across our global set of grasslands. Thus, our results provide a first insight into how potential soil net Nmin data that is widely available in the literature could be leveraged to learn more about large-scale N mineralisation processes under field conditions. Accurately quantifying realised N mineralisation is crucial for estimating the role of increasing reactive N in ecosystem functioning. Mis-estimation of these processes could lead to errors in predicting how N limitation affects ecosystem functioning. Given the global extent of grasslands24,25 this could, in turn, substantially affect our predictions of global change-driven impacts on C cycling45. Overall, our findings suggest that management activities that alter soil compaction, nutrient content, or microbial community function may interact with future changes in temperature and precipitation regimes to severely impact the amount of N that is mineralised in grassland soils22,46.

## Methods

### Study sites and experimental design

The 30 study sites are part of the Nutrient Network Global Research Cooperative (NutNet [https://nutnet.umn.edu/]; Fig. 2, Supplementary Table 1 and 2). At each site, the effects of nutrient addition and herbivore exclusion treatments are examined via a random-block design28. This block design is replicated three times at the majority of the sites. For four sites, we only had data from one (1 site) and two blocks, respectively. This study is restricted to data collected from the untreated control plots (n = 85). Each 5 m × 5 m plot is divided into four 2.5 m × 2.5 m subplots. Each subplot is further divided into four 1 m × 1 m square sampling plots, one of which is set aside for soil sampling28. Plots are separated by at least 1 m wide walkways. Mean annual temperature of our sites ranged from −4 to 19 °C, mean annual precipitation from 252 to 1592 mm, and elevations from 6 to 4241 m above sea level (Fig. 2, Supplementary Table 1). Soil organic C varied from 0.32% to 22.30%, soil total N from 0.03% to 1.25% and the soil C:N ratio from 9.07 to 23.64 across our 30 sites. Also soil clay content (3.0–53.3%) and soil pH (3.25–7.71) spanned a large gradient across the 30 sites (Supplementary Table 2). Thus, our 30 sites cover a wide range of grasslands globally that are typical for the respective region (Fig. 2, Supplementary Tables 1 and 2).

### Soil net N mineralisation and soil properties

Each site received an identical package shipped from the Swiss Federal Institute for Forest, Snow and Landscape Research (WSL) with material to be used for sampling and on-site incubations (steel cores and rings, resin bags, caps, gloves, etc.). For the field incubation, we followed the protocol by Risch et al.29. Briefly, at randomised locations in each plot we clipped the vegetation and then we drove a 5 × 15 cm (diameter × depth) steel cylinder 13.5 cm deep into the soil so that 1.5 cm on top of the cylinder remained empty. To capture incoming N from run-off and/or deposition, we placed a polyester mesh bag (mesh-size 250 µm) filled with 13.2 ± 0.9 g of acidic and alkaline exchanger resin (1:1 mixture; ion-exchanger I KA/ion-exchanger III AA, Merck AG, Darmstadt) into the upper 1.5 cm space of the cylinder. The bag was fixed in place with a metal Seeger ring (Bruetsch-Rüegger Holding, Urdorf, Switzerland). Thereafter, we removed 1.5 cm soil at the bottom of the cylinder and placed another resin bag into the cylinder to capture N leached from the soil column. We made sure that the exchange resin was saturated with H+ and Cl prior to filling the bags by stirring the mixture for 1 h in HCl 1.2 M and then rinsing it with demineralised water until the electrical conductivity of the water reached 5 µS/cm. The cylinders including the soil core and the resin bags were then re-inserted into the soil, at the same location where the sample was collected, flushed with the soil surface, and incubated for 42 days (range 36–57, see also Fig. 1a). Each site coordinator chose the timing of incubation so that it started 6 weeks prior to peak plant biomass production. All the incubations were completed between February 2015 and January 2016. At the end of the incubation, the cylinders were re-collected and immediately put in a cool box for transport to the home institution, where they were immediately packed in an insulated box together with cool packs or blue ice to halt further mineralisation, and overnight-shipped to WSL. Gloves were worn at all times to avoid contamination of the samples. Immediately upon arrival at the laboratory at WSL, the resin bags and a 20 g subsample of sieved soil (4 mm) from the cylinders were separately extracted in a 100 ml PE-bottle with 80 ml 1 M KCl for 1.5 h on an end-over-end shaker and filtered through ashless folded filter paper (DF 5895 150, ALBET LabScience). We measured NO3 (colorimetrically47) and NH4+ concentrations (flow injection analysis; FIAS 300, Perkin Elmer) on these filtrates.

At the start of the field incubation, we additionally collected two soil cores of 5 cm diameter and 12 cm depth with a steel core at each sampling plot for potential soil net Nmin, soil chemical and biological analyses (see below). We composited the two samples and then re-used the steel cylinder to collect one additional sample (5 × 12 cm) to assess soil physical properties. This third sample remained within the steel core and both ends were tightly closed with plastic caps. The capped steel cores were then gently packed to avoid further disturbance and together with the composited soil samples overnight-shipped to the laboratory at WSL.

From the composited samples, we extracted an equivalent of 20 g dry soil with KCl as described above and NO3 and NH4+ concentrations were measured. Realised soil net Nmin was then calculated as the difference between the inorganic N content of samples collected at the end of the incubation (plus N extracted from the bottom resin bag) and the N content at the beginning of the incubation and scaled to represent daily mineralisation rates (mg N kg−1soil day−1)29. Note that our values represent soil net Nmin for an average period of 42 days prior to peak biomass, i.e., typically during the highest biological activity, and not for the entire year.

A second subsample of the composited sample was used to determine potential soil net Nmin in the laboratory. For this purpose, we weighed duplicated samples of soil equivalent to 8 g dry soil into 50-ml Falcon tubes. Soil moisture was brought to 60% of the field capacity of each individual plot (see methods for determining field capacity below). The Falcon tubes were tightly closed and incubated at 20 °C for 42 days (6 weeks) in a dark room. Every week the Falcon tubes were opened and ventilated. At the end of the incubation, the soil samples were extracted the same way as described above and NO3 and NH4+ determined. Potential soil net Nmin was calculated as the difference between the N content before and after the incubation and scaled to represent daily values. We also calculated both realised and potential soil net Nmin on an area basis correcting for soil depth and bulk density, expressed as kg N ha−1 12 cm−1 day−1 and compared these values against our soil net Nmin expressed in mg N kg−1 soil day−1. The two values correlated well for both realised (R2 = 0.899, p < 0.001) and potential (R2 = 0.900, p < 0.001) soil net Nmin. For comparability among studies we decided to express our soil net Nmin indices in mg N kg−1soil day−1.

A third subsample from the composited sample, was sieved (4 mm) and used to assess soil biological properties. Metagenomic DNA was isolated from 0.25 g of bulk soil by using the Qiagen Power soil DNA isolation kit and the extracted DNA was quantified with Pico Green48. The abundances of archaeal (AOA) and bacterial (AOB) ammonia oxidisers were quantified using real-time PCR. Functional marker genes encoding archaeal and bacterial ammonia mono-oxygenase (archaeal amoA, bacterial amoA) were quantified by real-time PCR using primers and thermocycling conditions as previously described for bacterial amoA (AmoA-1F and AmoA-2R)49 and archaeal amoA (Arch-amoAF and Arch-amoAR)50. All quantitative PCR assays were carried out with equal amounts of template DNA (2 ng DNA) in 20 µl reactions using QuantiTect SYBR Green PCR master mix (Qiagen, Hirlen, Germany) on an ABI 7500 (PE Applied Biosystems) real-time PCR instrument48. The specificity of the amplification products was confirmed by melting-curve analysis, and the expected sizes of the amplified fragments were checked in a 1.5% agarose gel stained with ethidium bromide. Abundances of bacterial and archaeal amoA genes refer to copy numbers per gram dry soil. Standards for qPCR were generated by serial dilution of stocks containing a known number of plasmids carrying the respective functional gene as an insert51. Reaction efficiencies of qPCRs were 95% (±2) for archaeal amoA and 93% (±3) for bacterial amoA. R2 values were 0.99 for all runs.

A fourth subsample of the composite sample was sieved (2 mm) and microbial biomass (μg Cmic g−1 soil dry weight) was estimated by measuring the maximal respiratory response to the addition of glucose solution (4 mg glucose per gram soil dry weight dissolved in distilled water; substrate-induced respiration method) on ~5.5 g of soil52. The rest of the composited sample was dried at 65 °C for 48 h and ground to pass a 2 mm mesh. A subsample was finely ground and analysed for organic C53 and for total C and N concentrations (Leco TruSpec Analyser, Leco, St. Joseph, MI, USA). We pretreated subsamples with HCl to remove inorganic C prior to soil organic C determination for soils with pH > 753. Mineral soil pH was measured potentiometrically in 10 mm CaCl2 (soil:solution ratio = 1:2, equilibration time 30 min).

The intact sample within the capped steel core was used to assess field and water holding capacity, fine fraction bulk density, density of the solid phase, soil porosity, and soil texture. For this purpose, the steel core was weighed (without plastic caps), covered with a nylon mesh (SEFAR NITEX 03-60-32, 60 µm mesh) at the bottom and placed in a water bath to saturate the soil core from the bottom-up. The water-saturated soil core was then weighed and placed on a water saturated silt/sand bed with a suction of 60 hPa (pF 1.8; pendant water column of 60 cm) to drain the soil to field capacity. The time elapsing from saturation to drainage of the soil was dependent on soil texture and ranged from 17 and 135 h. After drainage, we weighed the soil (at field capacity), dried it at 105 °C to constant weight and recorded the dry weight. The difference between these two weights corresponds to field capacity. The difference between the weight of the saturated and dried cylinder corresponds to water holding capacity. To calculate the mass of the soil sample, we also weighed the empty steel cylinder and the dried nylon mesh. The density of the solid phase was determined with pycnometers and the pore space was calculated as $$\varphi = 1 - \frac{\rho }{{\rho _0}}$$, where φ = pore space, ρ = fine fraction bulk density, ρ0 = density of the solid phase. Soil texture was determined with the pipette method54. Of the three variables sand, silt and clay, we retained soil clay content as an explanatory variable. A list of all soil chemical, physical and biological properties measured can be found in Supplementary Table 4.

Climate data: We selected the following bioclimatic variables that we expected to be important for explaining variability in soil net Nmin from WorldClim30 ([http://www.worldclim.org/current], Supplementary Table 4): mean annual temperature (bioclim.T.ann), mean annual precipitation (bioclim.P.ann), the seasonality of annual temperature (bioclim.Tvar), calculated as the standard error of monthly means × 100, temperature of the warmest quarter (bioclim.T.q.warm), temperature of the wettest quarter (bioclim.T.q.wet), temperature of the driest quarter (bioclim.T.q.dry), and precipitation of the wettest quarter (bioclim.P.q.wet; Supplementary Table 4). To obtain a rough estimate of realised soil net N mineralisation over the course of the entire growing season, we obtained daily temperature, precipitation [ftp://ftp.cdc.noaa.gov/Datasets/] and potential evapotranspiration (PET [https://earlywarning.usgs.gov/fews/datadownloads]) data to calculate the total number of growing days for each site. A growing day was defined to have daily precipitation > 0.5*PET and daily minimum temperature > 0.5 °C55. All growing days per site were summed to the total length of the growing season. We then multiplied our realised soil net Nmin with growing season length to assess whether growing season differences among our sites would help to explain latitudinal differences in realised soil net Nmin.

### Numerical calculations and variable selection

We examined the distributions of our explanatory variables. Some of them were highly skewed and therefore were log transformed. We centred and scaled all explanatory variables to have a mean of zero and variance of one. We then filtered our variables to avoid collinearity between them. For this purpose, we performed a correlation analysis (Supplementary Fig. 7). If variables were strongly correlated (Pearson’s |r| > 0.70)31, we selected the ones that allowed us to minimise the number of variables (Supplementary Fig. 7, Supplementary Table 5). In case of the highly correlated variables soil bulk density, soil total C, soil organic C, soil total N, soil pore space and microbial biomass, we chose a soil physical, chemical and biological variable each for use: First, soil bulk density as it is easy and inexpensive to measure and therefore likely to be more commonly available in other studies, second, microbial biomass as the only soil biological variable of the group, and third, soil organic C as it is most often thought to drive soil net N mineralisation. In summary, temperature of the warmest quarter, temperature of the driest quarter, precipitation of the wettest quarter, soil total C, soil total N, and soil pore space were removed from the dataset (Supplementary Table 5). We also transformed our two response variables realised and potential soil net Nmin (square root transformation) to account for a highly skewed data distribution (yt = sign(y)sqrt|y|; negative values in the data set meant log transformation was not possible).

### Statistical analyses

To assess the spatial patterns in soil net Nmin, we used linear-mixed effect models (LMMs) fitted by likelihood maximisation using the R nlme package56 (version 3.131.1) and lme function (R version 3.4.4; R Foundation for Statistical Computing). Realised or potential soil net Nmin, respectively, was the dependent variable, distance to the equator the fixed factor and site identity a random effect. To determine global drivers of grassland soil net Nmin, we used multi-model inference32 and LMMs. We separately assessed how our variables explained realised soil net Nmin and potential soil net Nmin. Site identity was used as a random effect in these models. We first calculated the full models including all predictor variables (Supplementary Table 5) and then used the MuMin package57 (version 1.42.1) to select the simplest models that explained the most variation based on Akaike’s information criterion (AIC; model.avg function). We used the corrected AIC (AICc) to account for our small sample size32,58 and selected the top models that fell within 2 AICc units (delta AICc < 2). We present all our top models rather than model averages. We calculated all our models using either microbial biomass or soil organic C as these two measures represent a very similar soil feature and were highly correlated (r = 0.85). The models with soil microbial biomass are included in the main text, the ones with soil organic C in the Supplementary Information.

Based on findings from the LMM analyses and the literature we developed an a priori causal conceptual model of relationships among environmental drivers, potential and realised soil net Nmin to test with SEM using a d-sep approach33,59 (Fig. 3, Table 1). The variables included in this model were the best climatic (temperature of the wettest quarter, temperature seasonality), soil texture (clay content) and soil microbial (microbial biomass) predictor variables from the LMM analyses (Figs. 3, 5a, Table 1). We also calculated the same models using soil organic C instead of microbial biomass (Supplementary Fig. 6). In our LMMs, the temperature of the wettest quarter and/or temperature variability predicted realised and potential soil net Nmin. The direct links between climate properties (Tvar, T.q.wet) and potential soil net Nmin may represent legacy effects of climate on soil properties that we did not directly measure (see main text). Soil clay content was, in turn, predicted to affect microbial biomass (or soil organic C content), realised and potential net Nmin. As we determined microbial biomass prior to incubating the samples in the laboratory or field, we assume that microbial biomass impacts N process rates and not vice versa. Further, as it was the goal of this study to explore if potential soil net Nmin in combination with other properties could be used to predict realised soil net Nmin, we added a link from potential to realised soil net Nmin. We tested our conceptual model (Fig. 3, Table 1) following a d-sep approach using the piecewiseSEM package (version 2.0.2)59 in R (3.4.0), in which a set of linear structured equations are evaluated individually. This approach allows us to account for nested experimental designs and also to overcome some of the limitations of standard structural equation models such as small sample sizes33,59. We first used the lme function of the nlme package to model response variables, including site as a random factor. Good fit was assumed when Fisher’s C values were non-significant (p > 0.05). Although the abundance of AOB explained some of the variability in potential soil net Nmin, we did not include this variable in our SEMs as there is no mechanistic basis to rationalise that AOB drives the total accumulation of inorganic nitrogen in soil, only its partitioning between ammonium and nitrate. However, we calculated an SEM including AOB to assess if we can predict realised soil net nitrification using our predictors as well as potential soil net nitrification (Supplementary Fig. 8). While we were able to explain potential soil net nitrification well, the model fits rather poorly for realised soil net nitrification (Supplementary Fig. 8).

### Reporting summary

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

## Data availability

The data will be available at www.envidat.ch. https://doi.org/10.16904/envidat.87, Source data for Figs. 2, 4, 5, Supplementary Figs. 18 can be found in the source data file.

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## Acknowledgements

We appreciate the many helpful comments from two anonymous reviewers that greatly improved the manuscript. This work was conducted within the Nutrient Network (http://www.nutnet.org) experiment, funded at the site-scale by individual researchers. The soil net Nmin add-on study was funded by two internal competitive WSL grants to A.C.R., B.M., M.Sc., F.H. and S.Z. as well as to B.F., A.C.R. and S.Z. Coordination and data management have been supported by funding from the National Science Foundation Research Coordination Network (NSF-DEB-1042132) to E.T.B. and E.W.S., and from the Long-Term Ecological Research (LTER) programme (NSF-DEB-1234162), and the Institute on the Environment at the University of Minnesota (DG-0001-13). We also thank the Minnesota Supercomputer Institute for hosting project data, and the Institute on the Environment for hosting Network meetings. We are grateful to Roger Köchli and Simon Baumgartner for their help with sample processing and analyses, and to Benjamin R. Fitzpatrick for support with calculating growing season lengths and statistical advise. In addition, A.di V. thanks the Nature Conservancy, Gustavo Iglesias and People from Fortin Chacabuco Ranch for access to the field plots to conduct field work there. L.Y. was supported by Universidad de Buenos Aires and Agencia Nacional de Promocion Cientifica y Tecnologica (PICT 2014-3026), and S.M.P. thanks Georg Wiehl for technical assistance, Denise and Malcolm French for use of Mt. Caroline and the support from the TERN Great Western Woodlands SuperSite. N.E. and J.S. acknowledge support of the German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig funded by the German Research Foundation (FZT 118).

## Author information

A.C.R., S.Z., M.Sc., F.H. and B.M. developed the idea. A.C.R., R.O.-H. and B.M. analysed the data and M.Sc., J.L.F., F.H., P.B.A. contributed to data analyses. A.C.R. wrote the paper with input of S.Z., R.O.H., M.Sc., B.F., J.L.F., P.A.F., F.H., E.T.B., E.W.S., W.S.H., J.M.H.K., R.L.McC., A.A.D.B., C.J.S., M.L.S., P.B.A., S.B., L.A.B., J.M.B., C.S.B., M.C.C., S.L.C., P.D., A.di V., A.Eb, N.E., E.E., A.Es, N.H., Y.H., K.P.K., A.S.McD., J.L.M., S.A.P., S.M.P., C.R., M.Sa., J.S., K.L.S., P.M.T., R.V., L.Y. and M.B. S.Z., B.F. and J.S. analysed the samples. All authors but S.Z., R.O.-H., B.F., F.H., J.S. and B.M. are NutNet site coordinators, collected and shipped the soil data and samples. E.T.B., E.W.S. and W.S.H. are NutNet network coordinators. A detailed listing of author contributions can be found in the author contribution matrix (Supplementary Table 6).

Correspondence to A. C. Risch.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Xiaorong Wei and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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# [GIF] A to Z (Image morph)

Posted 2 years ago
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 A to ZI've mentioned before that there's a construction which gives a correspondence between points on the Grassmann manifold $G_2(\mathbb{R}^n)$ of 2-dimensional linear subspaces of $\mathbb{R}^n$ and planar $n$-gons; details at 1, 2, 3, 4, 5, 6. In particular, this gives a way of morphing between any two (discrete) shapes you like; this animation shows the shortest path from A to Z.Here's the code; notice that I use the smootheststep function: ProjectionBasis[{A_, B_}, {C_, D_}] := Normalize[#] & /@ Eigenvectors[ Transpose[Transpose[{A, B}].{A, B}.Transpose[{C, D}].{C, D}], 2]; PlaneGeo[{A_, B_}, {C_, D_}, t_] := Module[{a, b, c, d, cPerp, dPerp, dist1, dist2}, {a, b} = ProjectionBasis[{C, D}, {A, B}]; {c, d} = ProjectionBasis[{A, B}, {C, D}]; {cPerp, dPerp} = {Normalize[c - (c.a)*a], Normalize[d - (d.b)*b]}; dist1 = ArcCos[a.c]; dist2 = ArcCos[b.d]; {Cos[t*dist1]*a + Sin[t*dist1]*cPerp, Cos[t*dist2]*b + Sin[t*dist2]*dPerp} ]; MakePoly[data_] := Normalize[#] & /@ Transpose[{Re[#], Im[#]} & /@ Sqrt[Table[#[[n]][[1]] + I*#[[n]][[2]], {n, 1, Length[#]}] &[ RotateLeft[#] - # &[data]]]]; RawAData = {{-3., .9}, {1.07, 8.18}, {1.85, 8.2}, {5.96, 0.86}, {4.89, 0.86}, {1.77, 6.6}, {1.18, 6.6}, {.12, 4.58}, {2.87, 4.6}, {3.35, 3.76}, {-.18, 3.76}, {-1.72, .94}}; RawZData = {{0.82, 7.26}, {7.61, 7.32}, {7.65, 6.04}, {1.92, -.06}, {7.76, -.02}, {7.8, -1.16}, {.71, -1.12}, \ {.71, .02}, {6.36, 6.12}, {.85, 6.12}, {.82, 6.12}, {.82, 7.26}}; ToPol[frame_] := Accumulate[ReIm[(Complex @@ #)^2] & /@ Transpose[frame]]; smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4; DynamicModule[{cols = RGBColor /@ {"#f77e5e", "#3dbd5d", "#303030"}, centeredpoints, s}, Manipulate[ s = smootheststep[t]; centeredpoints = # - ConstantArray[Mean[#], Length[#]] &[ RotationTransform[(26 - 44 s) Degree][ Prepend[ToPol[ PlaneGeo[MakePoly[RawAData], MakePoly[RawZData], s]], {0, 0}]]]; Graphics[{Thickness[.008], JoinForm["Round"], Blend[cols[[;; 2]], s], Line[centeredpoints]}, ImageSize -> {540, 540}, PlotRange -> {{-.26 + .02 s, .29 + .02 s}, {-.28 - .065 s, .32 - .065 s}}, Background -> cols[[-1]]], {t, 0, 1}] ]
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Posted 2 years ago
 This is interesting: this animation shows the shortest path from A to Z But "shortest" in what sense, what is the measure? Perhaps the area that boundary traces? This is an attempt to visualize the "integral change". If we modify just the part of code starting from DynamicModule, we get (note .SVG gives nice vector look): Module[{cols = RGBColor /@ {"#f77e5e", "#3dbd5d", "#303030"}, centeredpoints, s}, tmp = Table[s = smootheststep[t]; centeredpoints = # - ConstantArray[Mean[#], Length[#]] &[ RotationTransform[(26 - 44 s) Degree][Prepend[ToPol[ PlaneGeo[MakePoly[RawAData], MakePoly[RawZData], s]], {0, 0}]]]; Graphics[{Thickness[.008], Opacity[.5], JoinForm["Round"], Blend[cols[[;; 2]], s], Line[centeredpoints]}, ImageSize -> {540, 540}, PlotRange -> {{-.26 + .02 s, .29 + .02 s}, {-.28 - .065 s, .32 - .065 s}}, Background -> cols[[-1]]], {t, 0, 1, .005}]]; Show[tmp, PlotRange -> All]
Posted 2 years ago
 By the way, I really like your image. Here's my version: Module[{cols = RGBColor /@ {"#f77e5e", "#3dbd5d", "#303030"}, centeredpoints, s}, tmp = Table[s = smootheststep[t]; centeredpoints = # + ConstantArray[{.6 s, .36 s}, Length[#]] &[ RotationTransform[(26 - 44 s) Degree][ Prepend[ToPol[ PlaneGeo[MakePoly[RawAData], MakePoly[RawZData], s]], {0, 0}]]]; Graphics[{Thickness[.008], Opacity[.5], JoinForm["Round"], Blend[cols[[;; 2]], s], Line[centeredpoints]}, Background -> cols[[-1]]], {t, 0, 1, .005}]]; Show[tmp, PlotRange -> {{-.05, .95}, {-.2, .55}}, ImageSize -> {1000, 700}]
Posted 2 years ago
 Very cool. But your GIF has letter "A" straight. And you Manipulate does not. Do you have the code that makes Z - > A with both letters straight, as on GIF?
Posted 2 years ago
 Yes, there often is some interesting math hiding behind my animations, and you're right that I should probably do a better job pointing it out when it's there. Often when I finish an animation (which can sometimes take a long time), the last thing I want to do is spend another hour explaining the ideas behind it, but I will try to be more conscientious about that.
Posted 2 years ago
 - another post of yours has been selected for the Staff Picks group, congratulations! We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!
Posted 2 years ago
 Neat image!
Posted 2 years ago

This is a complete mystery. Below the Manipulates for both codes you provided on

$Version "11.0.1 for Mac OS X x86 (64-bit) (September 21, 2016)"  ## 1st ## 2nd Answer Posted 2 years ago  Sorry, the Markdown parser let me down; I've edited to put in the links to more detailed descriptions of the connection between polygons and Grassmannians.In any case, the upshot is that one can view$n$-gons as points in the Grassmannian$G_2(\mathbb{R}^n)$, so two different polygons correspond to two different points. The special orthogonal group$SO(n)\$ naturally acts on the Grassmannian, and the Grassmannian has a unique (up to scale) left-invariant Riemannian metric. When I say "shortest path", I really mean the shortest path between the two points in the Grassmannian with respect to this Riemannian metric.Now, what this means at the level of polygons is a little unclear. In the usual way, we can think of geodesics in the Grassmannian as critical points of some energy functional. We can interpret the energy as a functional on the space of polygons, and the path I showed is a critical point of this functional. Unfortunately, I do not (yet) know of a nice polygonal interpretation of this functional: I would absolutely love to prove that it's something like the area the boundary traces (though it presumably can't actually be that because the Grassmannian interpretation of polygons is orientation-independent, whereas the area surely depends on how you register the images to begin). If you have any ideas of how to interpret it, I would certainly be very interested.
Posted 2 years ago
 Ah, okay, I'm on a different version, so my guess is that this is some internal change in how eigenvectors are computed. How about this (which completely obscures what's going on, but eliminates the eigenvector computations)? ToPol[frame_] := Accumulate[ReIm[(Complex @@ #)^2] & /@ Transpose[frame]]; smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4; DynamicModule[{cols = RGBColor /@ {"#f77e5e", "#3dbd5d", "#303030"}, centeredpoints, s, AZPath}, AZPath[s_] := {{-0.6183 Cos[0.6237 s] - 0.05281 Sin[0.6237 s], -0.1936 Cos[0.6237 s] - 0.05875 Sin[0.6237 s], -0.4726 Cos[0.6237 s] + 0.1089 Sin[0.6237 s], -0.05233 Cos[0.6237 s] - 0.7733 Sin[0.6237 s], -0.3966 Cos[0.6237 s] + 0.2414 Sin[0.6237 s], -0.03886 Cos[0.6237 s] + 0.1973 Sin[0.6237 s], -0.1054 Cos[0.6237 s] - 0.07660 Sin[0.6237 s], -0.3627 Cos[0.6237 s] - 0.3582 Sin[0.6237 s], -0.1609 Cos[0.6237 s] + 0.3523 Sin[0.6237 s], -0.09504 Cos[0.6237 s] + 0.1416 Sin[0.6237 s], -0.1219 Cos[0.6237 s] - 0.05366 Sin[0.6237 s], 0.05338 Cos[0.6237 s] - 0.07418 Sin[0.6237 s]}, {-0.1933 Cos[1.476 s] - 0.06772 Sin[1.476 s], 0.04221 Cos[1.476 s] + 0.1317 Sin[1.476 s], 0.4471 Cos[1.476 s] + 0.4660 Sin[1.476 s], -0.2261 Cos[1.476 s] - 0.05784 Sin[1.476 s], -0.4140 Cos[1.476 s] + 0.1673 Sin[1.476 s], -0.1679 Cos[1.476 s] - 0.5314 Sin[1.476 s], 0.3220 Cos[1.476 s] - 0.2102 Sin[1.476 s], 0.08257 Cos[1.476 s] - 0.3301 Sin[1.476 s], 0.1509 Cos[1.476 s] - 0.4968 Sin[1.476 s], -0.4106 Cos[1.476 s] + 0.003600 Sin[1.476 s], 0.3832 Cos[1.476 s] - 0.2160 Sin[1.476 s], 0.2482 Cos[1.476 s] - 0.02369 Sin[1.476 s]}}; Manipulate[s = smootheststep[t]; centeredpoints = # - ConstantArray[Mean[#], Length[#]] &[ RotationTransform[(26 - 44 s) Degree][ Prepend[ToPol[AZPath[s]], {0, 0}]]]; Graphics[{Thickness[.008], JoinForm["Round"], Blend[cols[[;; 2]], s], Line[centeredpoints]}, ImageSize -> {540, 540}, PlotRange -> {{-.26 + .02 s, .29 + .02 s}, {-.28 - .065 s, .32 - .065 s}}, Background -> cols[[-1]]], {t, 0, 1}]]
Posted 2 years ago
 Beautiful. And thanks for the explanation. I am finding myself going on a loop reading through it ;-) So there is basically so much more math behind this animation. I was watching your animation posts for quite awhile already. And I got a sense that despite visual simplicity there is always some math complexity to it. And as this case proves, there is more to it than meets the eye. I think many people here would be interested in hearing the more detailed mathy explanations when you post animations. It is very cool to try to understand what is really happening. I'm looking forward to more food for thought and eyes ;-)
Posted 2 years ago
 Yep this works perfectly. Thanks.
 The Manipulate gives a straight A on my computer, so I'm not sure what the problem is. Maybe this will work? ProjectionBasis[{A_, B_}, {C_, D_}] := Normalize[#] & /@ Eigenvectors[ Transpose[Transpose[{A, B}].{A, B}.Transpose[{C, D}].{C, D}], 2]; PlaneGeo[{A_, B_}, {C_, D_}, t_] := Module[{a, b, c, d, cPerp, dPerp, dist1, dist2}, {a, b} = ProjectionBasis[{C, D}, {A, B}]; {c, d} = ProjectionBasis[{A, B}, {C, D}]; {cPerp, dPerp} = {Normalize[c - (c.a)*a], Normalize[d - (d.b)*b]}; dist1 = ArcCos[a.c]; dist2 = ArcCos[b.d]; {Cos[t*dist1]*a + Sin[t*dist1]*cPerp, Cos[t*dist2]*b + Sin[t*dist2]*dPerp}]; AFrame = {{0.5587691672676915, 0.1981237081903733, 0.561282439075137, 0., 0.29303258430728735, 0., 0.17528178862450003, 0.37198264830477284, 0.1908285679285995, 0., 0.205163066218776, 0.00396483829652537}, {0.3277750387491631, 0.00253963018178132, -0.3289966965729807, 0.23203064786359323, 0.49280250767258005, 0.17229769584575513, -0.28992889821878054, 0.0013526462896582667, -0.11074211838066708, 0.421444913379544, -0.34580148914295383, -0.25381158645873014}}; ZFrame = {{0.5395712230915538, 0.16826135725211128, 0.2378635055662534, 0.500401524060885, 0.15909799999674124, 0.0015553022620756472, 0.1563317110647911, 0.547153066305452, 0., 0., 0.15633171106479107, 0.}, {0.0023839205766322895, -0.16308532865529982, -0.5497830309003788, 0.0017136837511977539, -0.15361352050274618, 0.5513590391833545, 0.1563317110647911, 0.23900682840695045, 0.4860550423730885, 0.035864954221169684, 0.15633171106479107, 0.}}; ToPol[frame_] := Accumulate[ReIm[(Complex @@ #)^2] & /@ Transpose[frame]]; smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4; DynamicModule[{cols = RGBColor /@ {"#f77e5e", "#3dbd5d", "#303030"}, centeredpoints, s}, Manipulate[s = smootheststep[t]; centeredpoints = # - ConstantArray[Mean[#], Length[#]] &[ RotationTransform[(26 - 44 s) Degree][ Prepend[ToPol[PlaneGeo[AFrame, ZFrame, s]], {0, 0}]]]; Graphics[{Thickness[.008], JoinForm["Round"], Blend[cols[[;; 2]], s], Line[centeredpoints]}, ImageSize -> {540, 540}, PlotRange -> {{-.26 + .02 s, .29 + .02 s}, {-.28 - .065 s, .32 - .065 s}}, Background -> cols[[-1]]], {t, 0, 1}]]

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# Making random draws from an arbitrarily defined pdf

July 23, 2014
By

(This article was first published on Econometrics by Simulation, and kindly contributed to R-bloggers)

I recently found myself in need of a function to sample randomly from an arbitrarily defined probability density function. An excellent post by Quantitations shows how to accomplish this using some of Rs fairly sophisticated functional approximation tools such as integrate and uniroot. The only problem with this excellent post was that the machine cost was enormous with samples of 1000 draws taking 5 seconds on my machine with repeated samples of 100,000+ draws (which I was after) clearly being unworkable.

Thus I decided to take my own crack at it. First let us review the basics of drawing random variables from non-uniform distributions. The standard method I think most algorithms use works as follows:

Assumptions
1. You can draw pseudo-random uniform variable u
2. You can integrate the pdf to construct a cdf
$$p = F(x) = \int_{-\infty}^\infty f(x) dx$$
3. You can invert the cdf in order to solve for p
$$G(F(x))=F^{-1}(F(x))=F^{-1}(p)=x$$

The method thus relies upon the somewhat simple method of calculating x by drawing u and plugging into G (the inverse of F).

Now let us assume that we are in a bit of a bind. We can neither integrate f(x) nor invert F(x). The previously mentioned post by Quantitations demonstrates how to do the operation in this case by directly approximating the cdf followed by approximating the inverse. This process is computationally intensive for simple functions and extremely time consuming for complex functions.

In contrast I approximate the cdf by drawing x values and associated probabilities from the pdf along a user specified range. I create a matrix of bins over which the approximate cdf is divided into. After that I apply some function (mean or median) over the range of x which corresponds to each bin providing an x point value for each cdf bin value.  The gacdf returns a list of results from this process.

The ricdf function then takes the list of returns and is able to draw values from the approximated inverse cdf. Once the gicdf has completed its operation, ricdf is able to generate variables nearly as fast as that of standard non-uniform random variables.

As a matter of comparison, I define the funciton f as the pdf of the normal (dnorm) in R and draw from it 1000 time. Using rnorm, ricdf [defined in this post], and samplepdf [defined at Quantitations] I graphically see how the three draws compare below.
 Three random sampling procedures for the random normal.
On the graph, Black type=0, is rnorm. Dark blue type=1, is ricdf. Light blue, type=2 is samplepdf. Redrawing the graph several times, visually I could not tell the difference between the three methods.

Comparing run times of the three methods over ten repetitions rnorm used no seconds, ricdf (with ficdf) used .8 seconds, and samplepdf used 5. Because ricdf and gicdf are separate functions with gicdf setting up the table to draw from, we only need to set it up once per pdf so ricf can be benchmarked separately from gicfd. Comparing 1000 replications of rnorm and ricdf, rnorm took .1 seconds and ricdf took .11 seconds. I did not compare with samplepdf which I expected would take 5200 seconds (87 minutes).

For fun I have generated several other strange 'proportional' pdfs.  I say proportional because strictly speaking pdfs are required to integrate to 1. However, gicdf forces the probabilities to sum to 1 across the range making the formal requirement not necessary.

I define a piecewise pdf

And sample from it.

I also define a multimodal  pdf

And sample from it.

Find the R code below or the gist on github.

gicdf <- function(fun,                  min=-3.5,                  max=3.5,                  bins=1000,                  nqratio=10,                  grouping=mean,                  ...) {  # Generate an inverse CDF of an arbitrary function  fun <- match.fun(fun)  grouping <- match.fun(grouping)   # Number of points to draw  nq=nqratio*bins   # Draw indexes  qdraw <- seq(min, max,length.out=nq)   # Calculate proportional probability of each draw  pdraw <- fun(qdraw,...)   # Rescale probability sum to 1, rescale  pdraw <- pdraw/sum(pdraw)   # Calculate the cumulative probability at each qdraw  cpdraw <- cumsum(pdraw)   # Caculate the cumulative probability at each bin  pbin <- (1:bins)/bins  xbin <- NA*(1:bins)   for (i in 1:bins) {    xbin[i] <- grouping(qdraw[cpdraw<pbin[i]&cpdraw>0], na.rm = TRUE)    cpdraw[cpdraw<pbin[i]] <- 2  }   (draw.set <- list(digits=floor(log10(bins)), xbin=xbin, pbin=pbin))} # Draw from acdfricdf <- function(N, draw.set) {  digits <- draw.set$digits pdraws <- ceiling(runif(N)*10^digits)/10^digits draw.set$xbin[match(pdraws,draw.set$pbin)]} # Define an arbitrary pdf f: to compare lets start with normal pdff <- function(x) dnorm(x) library(ggplot2) # set the number to samplensamples <- 1000 # Draw from rnormsample0 <- cbind(draw=rnorm(nsamples), type=0) # Draw inverted cdf distribution information for range between -4 and 4# using mean approachsample1 <- cbind(draw=ricdf(nsamples, gicdf(f,min=-4,max=4)), type=1) # samplepdf and endsign defined on Quantitations blog# begin Quantitations codeendsign <- function(f, sign = 1) { b <- sign while (sign * f(b) < 0) b <- 10 * b return(b)}samplepdf <- function(n, pdf, ..., spdf.lower = -Inf, spdf.upper = Inf) { vpdf <- function(v) sapply(v, pdf, ...) # vectorize cdf <- function(x) integrate(vpdf, spdf.lower, x)$value  invcdf <- function(u) {    subcdf <- function(t) cdf(t) - u    if (spdf.lower == -Inf)       spdf.lower <- endsign(subcdf, -1)    if (spdf.upper == Inf)       spdf.upper <- endsign(subcdf)    return(uniroot(subcdf, lower=spdf.lower, upper=spdf.upper)\$root)  }  sapply(runif(n), invcdf)}# end Quantitations codesample2 <- cbind(draw=samplepdf(nsamples, f), type=2) # join the three samples togethersamples <- as.data.frame(rbind(sample0, sample1, sample2)) ggplot(samples, aes(x=draw, color=type))+  geom_density(aes(group=type, size=-type)) library(rbenchmark)# Let us do some speed comparisonsbenchmark(rnorm(nsamples),           ricdf(nsamples, gicdf(f,min=-4,max=4)),          samplepdf(nsamples, f),          replications=10)# test replications elapsed relative user.self#2 ricdf(nsamples, gicdf(f, min = -4, max = 4))   10    7.97#1                              rnorm(nsamples)   10    0.00#3                       samplepdf(nsamples, f)   10   52.39 # This sets the entire process of generating and drawing from ricdf as about 7 times # faster than samplepdf. However, if we seperate the gicdf function from the # ricdf function which only need be run each time a new pdf is input, then we can# considerably increase run speeds. myicdf <- gicdf(f,min=-4,max=4)# Sets up the initial icdf to draw from.  benchmark(rnorm(nsamples), ricdf(nsamples, myicdf), replications=1000)# elapsed time for rnorm = .10 seconds and ricdf = .11# thus drawing from ricdf can be extremely fast # Now let's look at an unusual proportional pdf distributionf <- function(x) {  p <- rep(0, length(x))  p[x>0&x<1] <- x[x>0&x<1]^2  p[x>2&x<3] <- 3-x[x>2&x<3]^.5  p[x>4&x<5] <- x[x>4&x<5]  p} x <- seq(-1,6,.01)plot(x,f(x), type='l',      xlab='Proportional Probability',     main='Proportional pdf')# A key thing to note is that the pdf does not need to integrate to 1.# The gicdf function will rescale it to ensure it does.# However, it should all be positive # I define bins as equal to 10,000 when normally they are equal to 1000myicdf <- gicdf(f,min=-1,max=6, bins=1000)samples <- data.frame(draws=ricdf(100000,myicdf)) ggplot(samples, aes(x=draws))+  geom_histogram(binwidth=.1)+  aes(y = ..density..)  # Okay here is one more probabilityf <- function(x) dnorm(x)*abs(1+x^4) x <- seq(-5,5,.01)plot(x,f(x), type='l',      xlab='Proportional Probability',     main='Proportional pdf2')  myicdf <- gicdf(f,min=-5,max=5, bins=1000)samples <- data.frame(draws=ricdf(100000,myicdf)) ggplot(samples, aes(x=draws))+  geom_histogram(binwidth=.2)+  aes(y = ..density..) # I will leave you to define your own pdfs to draw from.# If you end up using this method, please cite!
Created by Pretty R at inside-R.org


text:
# zbMATH — the first resource for mathematics

Partial regularity for minimisers of certain functionals having nonquadratic growth. (English) Zbl 0698.49001
Summary: We prove partial regularity results for local minimisers of $\int_{\Omega}(G^{\alpha \beta}(x,u)g_{ij}(x,u)D_{\alpha}u^ iD_{\beta}u^ j)^{p/2},\quad where\quad p\geq 2.$

##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 49J45 Methods involving semicontinuity and convergence; relaxation
##### Keywords:
partial regularity
Full Text:
##### References:
 [1] E.Evans,Quasi convexity and partial regularity in the calculus of variations, University of Maryland, preprint (1984). [2] Fusco, H.; Hutchinson, J., C^1,α partial regularity of functions minimising quasiconvex integrals, Manuscripta Mathematica, 54, 121-143 (1985) · Zbl 0587.49005 [3] M.Giaquinta,Multiple Integrals in the Calculus of Variations, Annals of Mathematics Studies,105, Princeton University Press (1983). · Zbl 0516.49003 [4] Giaquinta, M.; Giusti, E., Nonlinear elliptic systems with quadratic growth, Manuscripta Mathematica, 24, 323-349 (1978) · Zbl 0378.35027 [5] Giaquinta, M.; Giusti, E., The singular set of the minima of certain quadratic functionals, Annali della Scuola Normale Sup. di Pisa, 11, 45-55 (1984) · Zbl 0543.49018 [6] M.Giaquinta - G.Modica,Partial regularity of minimisers of quasiconvex integrals, University of Florence, preprint (1985). · Zbl 0594.49004 [7] Schoen, R.; Uhlenbeck, K., A regularity theory for harmonic maps, J. Diff. Geom., 17, 307-335 (1982) · Zbl 0521.58021 [8] Tolksdorf, P., Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Annali di Matematica Pura ed Applicata, 134, 241-266 (1983) · Zbl 0538.35034 [9] P.Tolksdorf,Regularity for a more general class of quasilinear elliptic equations, Journal of Diff. Equations. · Zbl 0488.35017 [10] Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030 [11] Giaquinta, M.; Modica, G., Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Mathematica, 57, 53-99 (1986) · Zbl 0607.49003 [12] S.Luckhaus,Partial Hölder continuity of energy minimizing p-harmonic map between Riemannian manifolds, C.M.A. preprint, Canberra (1986). [13] S.Luckhaus, 24-02regularity for energy minimizing Hölder continuous p-harmonic map between Riemannian manifolds, C.M.A. preprint, Canberra (1986).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# The most common theorems taught in Abstract Algebra

I am self learning abstract algebra. I want to know which theorems are a must to understand.

Now these are limits I have to deal with (please consider when answering):

1. I have limited internet access
2. Few mathematical books written in English are available.
3. I can not afford to order books from abroad．

I just want to know what is the core knowledge (theorems, lemmas, etc) of any decent graduate level abstract algebra class.

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Why not just look at the table of contents (via Amazon or Google book, for example) of some graduate-level abstract algebra books? For that matter, beyond "doing the undergrad stuff more seriously," I don't think there's going to be much in common between everyone's ideal notion of a decent such course (commutative algebra vs. advanced group theory vs. homological algbera vs. category theory, etc.) –  Cam McLeman Feb 3 '12 at 11:46
And, of course, the web is full of resources. For example, there's these course notes from Paul Garrett www.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf –  Cam McLeman Feb 3 '12 at 11:52
@CamMcLeman One of my problems is that each book seems to be in a different branch, I was wondering what would be the fundamental information. I will give your idea a try. –  yiyi Feb 3 '12 at 11:58
I agree with Prof McLeman: most sources for the traditional group/ring/field course are going to hit the same stuff. You'll prefer some books over others, but a list of theorems doesn't really help there. The important thing for you as a student is to gain a feel for the definitions and think about a lot of examples, and while texts are helpful for this I think they're often minimally so. –  Dylan Moreland Feb 3 '12 at 14:11
This is to say that while I sympathize with the limitations you list, the best thing to do might be to pick up some book (my impression is that you can get books not written in English) and start enjoying the material. It is hard to imagine that it won't hit the highlights mentioned in, say, Kannappan's answer. Good luck! –  Dylan Moreland Feb 3 '12 at 14:18

Assuming you are aiming for a Groups/Rings/Fields, at a level one might expect from someone who has taken a year's undergraduate sequence but not much more:

Groups

1. Definition, lots of examples.
2. Subgroups, homomorphisms, normal subgroups
3. Cosets; Lagrange's Theorem.
4. Group actions.
5. Permutation groups. Symmetric and alternating groups.
6. Cayley's Theorem.
7. The Isomorphism Theorems.
8. Cauchy's Theorem and the Sylow Theorems.
9. Abelian groups.
10. The Fundamental Theorem of Finitely Generated Abelian Groups.

Rings

1. Definition, lots of examples.
2. Homomorphisms.
3. Ideals.
4. The Isomorphism Theorems.
5. Field of quotients of integral domains.
6. Polynomial rings.
7. Euclidean rings, Principal Ideal Domains, Unique Factorization Domains
8. Gauss's Lemma and polynomials over Unique Factorization Domains.

Fields

1. Field extensions.
2. Algebraic extensions. Dedekind's Product Theorem.
3. Primitive Element Theorem.
4. Separability.
5. Fundamental Theorem of Galois Theory (finite case).
7. Finite fields.

I think Herstein's Topics in Algebra is a good resource, though it is a bit old-fashioned (it has almost nothing on group actions, for example).

With a more time, I would add the following topics:

Groups

1. Introduction to Semigroups.
2. Semidirect products; extensions.
3. Jordan-Höder Theorem.
4. Commutator subgroup; solvability (this will have been covered somewhat when discussing solvability by radicals under Fields; study it in more detail).
5. Basic results on $p$-groups.
6. Divisible and reduced abelian groups.
7. Structure Theorem of Divisible Abelian Groups.
8. Free groups, free products, free products with amalgamation.
9. Basics of representation theory of groups.

Rings

2. Wedderburn's Theorem.
3. Wedderburn-Artin Theory.
4. Division rings.
5. Localizations.
6. Group rings.
7. Basic theory of modules.
8. Modules over PIDs.

Fields

1. Inseparable extensions.
2. Transcendental extensions.
3. Kummer Theory.
4. Normal Basis theorem.
5. Fundamental theorem of Galois Theory (infinite case).

Lang's Algebra is a good resource, but is not easy to learn from. For Ring Theory, Lam's First Course in Noncommutative Ring Theory is very good.

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TOPICS is a book near and dear to my heart-as I've said here many times,it's the book I learned algebra from. I doubt it's that old fashioned that it would be a problem-and the exercises are the best of any standard textbook I know. –  Mathemagician1234 Feb 3 '12 at 17:22
@Mathemagician1234: My undergraduate courses were from Herstein as well, and I certainly don't feel I was ill-served by them. But I think that having very little on group actions is a serious defect. –  Arturo Magidin Feb 3 '12 at 17:46
I started with Herstein, and I stand by that book to this day. (Not the best ever, but a great start). –  mixedmath Feb 3 '12 at 17:51
Isaacs's book seems to be gaining popularity. [These days it is possible to obtain books without paying for them.] It's very "filled out", if that makes any sense. –  Dylan Moreland Feb 3 '12 at 19:14
@Dylan: I've heard good things about Isaac's Group Theory book, though I have not had the leisure of reading it through. However, as I recall it is intended to be a sort of "Second Course in Group Theory" book... But perhaps you mean his abstract algebra book. –  Arturo Magidin Feb 3 '12 at 19:42

Warning.

I am an undergraduate student in Math. I have no idea whatsoever about a graduate level Abstract Algebra course, but, I thought I'll sketch a list of resources one should master before graduate School (in my opinion).

Group Theory:

• Group action (with as many examples and applications as you can).
• Sylow's Theorems
• The concept of semidirect product (with as many examples, and it's use in construction of groups)
• Lots of examples of groups ($M_n$, $GL_n(\mathbb C)$, $GL_n(\mathbb F_q)$ $SL_n$, $PSL_n$, etc). By this, I mean, as many theorems and non-trivial facts about these groups as you can.
• Permutation groups (inside out)
• Free groups
• Solvable groups (Hall's Theorem), Nilpotent group, Central series

Where Should I learn these from?

Well, this is a question to which there is no right answer always. So, here is my choice not in order .

1. Dummit and Foote (The Modern encyclopedia of Abstract Algebra)
2. M.Hall -Theory of Groups
3. M.Aschbacher-Finite Group Theory

I would like to emphasize that you must have good command over examples and counter examples for exceptions, theorems (in which some hypothesis plays a crucial role) and so on, either from books or through self-Construction.

(P.S.: This is an advice I received from a Senior of mine who is now a Ph.D student and I have edited it suitably (for the forum) and added my own experience through the learning process.)

For other branches of Abstract Algebra, I'd write it up as I get time. Please bear with me!

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You leave quite a few restrictions. You don't have books, can't get books, and can't use the internet much. So, there isn't really much for us to tell you. You HAVE to have some form of book. You can't learn from nothing but a list of a few theorems. So, it seems the least restrictive of your list is the internet. Obviously, you can use it some.

By the way, you can't just learn the most important theorems. The rest of the stuff in the book leads up to these theorems and the theorems make no sense, or can not be proven, unless you have all the rest.

This should get you a lot of the basics. But, it's on the internet. As I said before, you basically left no possibilities for help, so I have to delete one of your restrictions.

http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/contents.html

Other than that, search for undergraduate abstract algebra online. "Intro" or "elementary" are words that tell you it's going to be undergrad level. Here are lecture notes from Hungerford's undergrad book, which is the book I used as an undergrad. If you could at least print this off, or something like it, that would solve the internet access problem.

http://www.math.msu.edu/~meier/Classnotes/M411S06/M411S06notes.pdf

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text:
Problem on the angle between vectors in 3d

Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
• Solution

First, let's sketch all unit vectors in 3d.
Recall that
For visual simplicity, we will sketch vectors as single points, as opposed to arrows pointing from the origin.
The set of 3d points with distance 1 from the origin forms a sphere:
Now we select which of these points has angle $\pi/6$ with respect to $\bfi$.
Recall that
Because we are considering vectors as points, all our vectors share the origin as their base.
The angle of $\pi/6$ can be swept in any plane containing $\bfi$.
The set of vectors that form an angle with $\pi/6$ from $\bfi$ form a cone opening toward $\bfi$ in 3d:
The intersection of the sphere with this cone is a circle.
Hence there is a whole circle of 3d unit vectors that have angle $\pi/6$ from $\bfi$: