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# Linking functional traits and demography to model species-rich communities

## Abstract

It has long been anticipated that relating functional traits to species demography would be a cornerstone for achieving large-scale predictability of ecological systems. If such a relationship existed, species demography could be modeled only by measuring functional traits, transforming our ability to predict states and dynamics of species-rich communities with process-based community models. Here, we introduce a new method that links empirical functional traits with the demographic parameters of a process-based model by calibrating a transfer function through inverse modeling. As a case study, we parameterize a modified Lotka–Volterra model of a high-diversity mountain grassland with static plant community and functional trait data only. The calibrated trait–demography relationships are amenable to ecological interpretation, and lead to species abundances that fit well to the observed community structure. We conclude that our new method offers a general solution to bridge the divide between trait data and process-based models in species-rich ecosystems.

## Introduction

Trait-based ecology has emerged in the last decades as an important avenue for improving our understanding of community assembly and dynamics1. Functional traits are canonically defined as “measurable morphological, physiological or phenological features of species that impact their fitness via their effects on demographic features”2. The second part of the definition implies that functional traits should be linked to species’ demographic rates and by extension to species ecological niches3,4. Yet, while the field has made major progress in collecting, organizing, and analyzing functional trait data for a large number of species from a variety of ecological communities5, attempts to formally demonstrate broad and consistent links between functional traits and species demographic rates across ecosystems have been less successful6,7.

The absence of a predictive link from traits to demographic rates not only challenges our ecological understanding, but also poses important practical limitations to our ability to predict community structure and dynamics with process-based community models. Here, by “community model”, we refer to any process-based model that predicts community structure and/or dynamics as a consequence of population-level processes such as growth, resource acquisition, mortality, and species interactions8. Model processes can be formulated across a range between phenomenological to more mechanistic descriptions9, but are generally specified by demographic rate parameters that vary across species. By predicting features such as species abundance, community structure, and dynamics over time, ecologists have argued that community models avoid many limitations of correlative models10,11,12, and would represent an important step towards predictions of local biodiversity responses to environmental changes13,14,15.

A major setback for the agenda of using process-based models for community ecology is that even conceptually simple community models, such as Lotka–Volterra models, are challenging to calibrate. This is because the number of demographic rate parameters of such models increases rapidly with the number of species13,16,17. Consequently, the use of process-based community models has been limited to systems with low to moderate complexity and diversity where a direct measurement of these parameters requires less effort, such as annual plant communities18,19 or laboratory communities20; and they have yet to be transferred to natural, species-rich systems.

Here, we propose a new method for calibrating process-based community models via functional traits. Functional traits are far easier to measure across a large number of species than are demographic rate parameters; therefore the ability to parameterize community models via functional trait data would expand their utility considerably. We aim to establish a transfer function that links empirically measured functional traits with the parameters of community models that describe species demography, hereafter called “demographic parameters”. The approach assumes that the demographic parameters of the modeled species can be predicted from their functional traits3. However, while the idea that form determines function is widely accepted, it would be challenging to predict the nature of this relationship only from a priori assumptions. For example, combinations of functional traits rather than single traits may be necessary to predict demographic parameters7,21. Instead, we use one additional data type, empirical community abundance data, to calibrate these trait–demography relationships through an inverse modeling approach16,22. In the end, the number of parameters in the transfer function scales only with the number of functional traits and the complexity of the transfer function. This is a critical advantage compared to a direct estimation of the demographic parameters, which scales with the number of species (Fig. 1).

The proposed method can support the calibration of any process-based community model (e.g. competition models, trophic models23,24) to empirical ecosystems as long as a relationship between traits and demographic parameters exists and sufficient functional trait data and community data are available. Here we illustrate the potential of our methodology by calibrating a simple community model with data from 18 communities distributed along a temperature gradient in the French Alps. The communities are composed of 118 mountain grassland plant species characterized by eight functional traits. More specifically, we test the hypothesis that the plant community structure follows the stress-dominance hypothesis: community composition is determined by plant ability to tolerate abiotic stress at the stressful end of the gradient (here at cold temperatures), and by plant ability to withstand competition at the benign end of the stress gradient (here at warm temperatures)25.

We formulate a community model derived from a Lotka–Volterra competition model that mimics these processes with four demographic parameters for each of the 118 species, three of which could be estimated with static community data. To do so, we use a transfer function to predict those 354 demographic parameters of the model from empirical functional traits through generalized linear functions using a total of only eleven parameters. Second, after the model calibration, we examine the quality of predictions against a set of null models calibrated from randomized trait data to demonstrate that empirical functional trait data bring significant improvement to the calibration. Third, we interprete the calibrated transfer function in the context of the ecological theory regarding the link between species demographic strategy and functional traits in mountain environments26,27.

## Results

### The community model

We formulated a simple community model derived from a Lotka–Volterra competition model (Eq. 1) that mimics the processes of the stress-gradient framework: plant biomass growth is modeled as a function of temperature and plant competition.

$$\frac{1}{B_{ij}}\frac{dB_{ij}}{dt}=g_{i}\times \left(\theta_{j} -\theta min_{i} \right) -c_{i} B_{ij} - l_{i} \mathop{\sum}\limits_{k}\, B_{kj}$$
(1)

Bij represent the biomass of species i in the site j characterized by temperature θj. The model includes the following ecological processes:

Temperature-dependent growth: $$g_{i}\times (\theta_{j} - \theta min_{i})$$ is the intrinsic relative biomass growth rate—incorporating reproduction, mortality, and individual biomass growth—of a species i in a site j. It is formulated as a positive linear function of the standardized temperature θj of the site. It is controlled by the parameter θmini that can be interpreted as the minimum temperature above which species j can have a positive relative biomass growth rate; below that threshold species j goes extinct. It is controlled by a slope parameter gi that can be interpreted as the within-species variability of the growth rate along the temperature gradient.

Sensitivity to surrounding biomass: the relative biomass growth rate of a species i in a site j decreases linearly with the total plant biomass of site j, $$\mathop{\sum}\limits_{k}\,B_{kj}$$, at a rate determined by demographic parameter li. Compared to a classical Lotka–Volterra model, this equation simplifies the formulation of biotic interactions by assuming that a given species is equally sensitive to all other competitors and only affected differently by its own biomass.

Intraspecific competition: the relative biomass growth rate of a species i in a site j decreases linearly with the biomass of its conspecifics at an additional rate determined by parameter ci. This term is based on the theoretical expectation and the empirical finding that intraspecific competition tends to be superior to interspecific competition28. When further accounting for a species’ general sensitivity to surrounding biomass, species relative growth rate decreases linearly at an overall rate of ci + li with the biomass of its conspecifics.

Depending on the strength of trade-offs among demographic parameters, the model can return different species abundance patterns along a temperature gradient going from the dominance of a single species along the gradient to strong species turnover12,29. The model was implemented as an ordinary differential equation system model (ODE).

### Estimating demographic parameters with a transfer function

The modeled processes are controlled by four unknown demographic parameters for each of the 118 species: minimum tolerated temperature, within-species variability of the growth rate along the temperature gradient, sensitivity to surrounding biomass, and intraspecific competition. In the absence of temporal data about our studied plant communities, we assumed that they were at equilibrium and could be modeled from the ODE equilibrium. Because of this, within-species variability of the growth rate along the temperature gradient could not be estimated and was thus kept constant across species (see Methods and Supplementary information 1.4). To estimate the remaining 354 demographic parameters, we used generalized linear functions between each demographic parameters and three PCA trait axes that summarized 67% of total trait variance. The eleven parameters of those transfer functions were then calibrated with a Markov-Chain Monte Carlo algorithm (see Methods for more details).

### Model validation

The calibrated model predicted the expected pattern of species turnover along an environmental gradient (Fig. 2a). Nagelkerke’s pseudo R2 value30 of the calibrated model was 0.590 (median value across the posterior, 95%CI: [0.587, 0.592]) and the deviance information criterion31 (DIC) was 6476.4. In contrast, the pseudo R2 at the median of the posterior of each of 200 null models with randomized traits was always less than 0.179, and we found a mean difference in DIC of 1844 in favor of the calibrated model (DIC of the null models was distributed within the 95%CI: [7719.4, 9205], against 6476.4 with real functional trait data). For 15 out of 18 plots, the calibrated model performed better than the null models (i.e. for those plots, the pseudo R2 values was superior to 95% of the null pseudo R2, Fig. 2b). In comparison, species distribution models and joint species distribution models similarly capture the abundance of well-sampled (aka dominant) species (see Supplementary information 3) but differ in the modeling of species that were undersampled along transects32. Owing to a much larger number of parameters (between 354 and 1062 for the SDM and the jSDMs vs. 11 for our approach), they also fit more closely the transect data (Nagelkerke’s pseudo R2: [0.760, 0.961], DIC: [3182.82, 13269]) but have a similar performance to predict plant species presence/absence (AUC: [0.68–0.79] against 0.68 for our approach).

### Trait–demography relationships

To study the output of the calibrated community model, we assessed Pearson’s correlation coefficients among the calibrated demographic parameters, the correlation between demographic parameters and observed functional traits across the posterior distribution. Then we tested if the distribution of calibrated demographic parameters differs among functional groups at the median of the posterior distribution. Functional groups were forbs (76 species), grasses (26 species), legumes (11 species), and shrubs (5 species). We related demographic parameters to the observed species functional trait values rather than to the PCA trait axes used for the calibration to facilitate the ecological interpretation of our results. The posterior distribution of the transfer function parameters is available in the Supplementary Information (Supplementary Fig. 5).

The estimated demographic parameters showed ecologically sensible trade-offs (Fig. 3b): the minimum tolerated temperature (θmin) was strongly negatively correlated with the sensitivity to surrounding biomass (l) (95%CI: −0.977 < r(θmin, l) < −0.959). We predicted demographic strategies26,33 along an axis characterized by “competitive” species with a weak tolerance to low-temperature stress, but less sensitivity to competition on the one end, and “stress-tolerant” species with the opposite demographic strategy on the other end26,33.

Those two demographic parameters were correlated with functional traits as predicted by ecological theory4,27,34. Plant species with a tall stature or with tender leaves were associated with a high minimum tolerated temperature and lower sensitivity to surrounding biomass26,35: minimum tolerated temperature (θmin) and sensitivity to biomass (l) were strongly correlated with vegetative height (across the posterior distribution, 95% CI, 0.861 < r(θmin) < 0.883, −0.843 < r(l) < −0.806, Fig. 3a), reproductive height (95% CI, 0.774 < r(θmin) < 0.810; −0.832 < r(l) < −0.792), and moderately correlated with specific leaf area (95% CI, 0.482 < r(θmin) < 0.519; −0.538 < r(l) < −0.488) and LDMC (95% CI, −0.445 < r(θmin) < −0.399; 0.436 < r(l) < −0.494). To an extent, these calibrated relationships match the trait-environment linkages suggested by a fourth corner analysis36 (see Supplementary information 3.2.2). This suggests that these two demographic rates depend on functional traits that vary strongly along the temperature gradient. Our model further estimated a lower minimum tolerated temperature and higher sensitivity to biomass for shrub species compared to other functional groups (Fig. 4, θmin: Forbs, t = 2.518, p = 0.013; Grasses, t = 2.658, p = 0.009; Legumes: t = 2.712, p = 0.008; l: Forbs, t = −3.204, p = 0.002; Grasses, t = −3.279, p = 0.001; Legumes: t = −2.644, p = 0.009). This suggests that shrubs (the latter category includes five alpine low shrub species) were more resistant to low temperature but had lower competitive ability than other species across the temperature gradient.

The intraspecific competition parameters (c) were moderately correlated with the minimum tolerated temperature parameters (95%CI: −0.741 < r(θmin) < −0.593) and sensitivity to competition (95%CI: −0.400 < r(l) < −0.560). They were further negatively correlated with vegetative height (95% CI, −0.787 < r(c) < −0.679, Fig. 3a) and reproductive height (95% CI, −0.616 < r(c) < −0.481), showing that tall plant species were associated with higher competitive ability than small species37. It was further strongly correlated with leaf δ15N (95% CI, −0.801 < r(c) < −0.761) and, to a lesser extent, leaf nitrogen content (95% CI, −0.557 < r(c) < −0.463). Hence, species with a low leaf nitrogen content as well as a low leaf δ15N ratio (that usually characterizes mycorrhizal dicots38) were also associated with higher intraspecific competition. The model parametrization associated a higher intraspecific competition rate with forb species compared to grass and legume species (Fig. 4, Grass: t = −2.625, p = 0.001; Legumes: t = −3.101, p = 0.002, Shrub: t = −0.554, p = 0.58). This association likely reflects the higher competitive ability of grasses and legumes over the other functional groups, which may explain their dominance in these grassland communities.

## Discussion

### General approach

Combining functional trait data with static community abundances, our approach allowed us to successfully fit a process-based community model to a species-rich ecosystem. Compared to other trait-based modeling methods39, our approach is not tied to any particular framework of community assembly. It rather establishes, based on empirical data, a plausible link between measured functional traits and the unknown demographic parameters of a community model. To do so, it leverages a fundamental insight of trait-based ecology: species functional attributes vary only across a limited number of dimensions40, and these should be relatable to species demography. When translating this insight into a transfer function between empirical traits and the demographic parameters of community models, the number of parameters to be calibrated is much reduced, which makes it possible to fit the combined model to static data despite the large number of species. An important point is that our approach does not rely on a priori assumptions about the demographic trade-offs and relevant trait–demography relationships, they rather emerge from the inverse modeling approach.

In our case study, we made the conjecture that species demography follows the stress-dominance hypothesis, and we quantified the support for this theory using the estimated demographic trade-offs. These corresponded to (i) a competition-stress tolerance trade-off that matched closely traits associated with the fast-slow leaf-economics trade-off34 and (ii) an axis of variation in intraspecific competition intensity that matched traits related to plant size. These relationships relate to existing knowledge about the global spectrum of plant trait variation and its relationship to plants’ ecological niche and demography3,4,41. While the functional trait–demography relationships calibrated by our approach are unsurprising, they are ecologically sensible and allow us to validate the fit of the transfer function a posteriori. They further suggest that, when moving towards more complex demographic models, a lack of unanimous knowledge about demography-trait relationships may not be an insurmountable obstacle.

### Developing more realistic community models

Beyond the scope of our particular community model, this new method paves the way to study more complex—and potentially better-performing models—of community assembly and dynamics in diverse ecosystems. We foresee no fundamental difficulties in using transfer functions to calibrate a broader variety of ecological processes. The only requirement to implement new processes is that they should be formulated as a function of attributes linked to species-specific (or individual-specific) functional traits. Some processes, typically biotic interactions, are typically modeled as attributes of two or more species19 and would need to be reformulated. Several theoretical studies have shown a path forward, for instance by expressing pairwise interaction coefficients from Lotka–Volterra competition models as functions of species’ ability to acquire resources42,43. Empirical studies have further suggested how pairwise interactions among species can be modeled from their functional traits6,44, which could inspire new theoretical frameworks that consider other features of biotic interactions such as niche partitioning among species23, facilitation45 or trophic interactions24.

### Towards dynamic community models

We assumed that the observed plant communities were at equilibrium, a common assumption in spatial community modeling46. Furthermore, we chose a globally-stable community model (Supplementary information 1.3), which assumes that, for a given set of demographic parameters values and environmental conditions, a unique stable community structure will emerge. These assumptions make the analysis more practical, but they may not always hold in nature12,46. In our case study, assuming community equilibrium led us to fix one demographic rate across species (see Methods), thus limiting our ability to fully model the interspecific variability of species’ responses to temperature and biotic stress. While this is arguably reasonable when modeling a relatively small static dataset without temporal data, it would be less adequate to model larger spatio-temporal datasets where it would be more essential, and feasible, to distinguish and calibrate detailed demographic processes.

Theoretical ecology has produced a vast corpus of models that study communities in terms of spatio-temporal dynamics, population structure, stability, and alternative stable states47,48, with some approaches making explicit references to functional trait theory49,50. With adequate community models, our transfer function approach could be used to move away from the equilibrium assumption, and to model spatial and temporal variation of species-rich communities. Besides the important data requirements, however, more complex models would create new analytical challenges as simulation approaches (e.g. ODEs) might not be appropriate to efficiently characterize community dynamics and equilibrium states51. In this context of increasing model and data complexity, assuming a priori that species demographic parameters depend on a limited number of functional traits will likely be a critical asset to study the model behaviors and reduce the complexity of its calibration at the onset.

### Comparing alternative modeling frameworks

Our case study uses environmental data, species abundances, and functional traits to test the stress-dominance hypothesis. An interesting question is how our framework compares with alternative, existing methods. One such approach to analyze static community data is (joint) species distribution models ((j)SDMs), which directly fit a relationship between environmental predictors and species occurrences or abundances10, without considering species traits or underlying demographic processes. Alternatively, trait-based community analyses, such as the fourth corner method36, also relate environment variables to species traits on the basis of species occurrence while other trait-based methods go a step further and use trait–environment relationships to predict local species abundances52,53.

(j)SDMs and the fourth corner method detected, to an extent, the same patterns as the transfer function approach (see Results and Supplementary information 3.2.1). A key difference, however, is that they cannot infer dynamic community parameters, while our method explicitly establishes a link between traits, demography, and abundance. Furthermore, our approach is explicit about the hypotheses underlying the modeled demographic processes. We were directly able to evaluate whether community assembly follows the stress-dominance hypothesis and quantify the support from the data.

A possible drawback of a process-based approach is that it is constrained by both the presumed functional form of the mechanism, and the existence of a statistical link between the processes and the available functional traits. We suspect that this explains why, in our case study, (j)SDMs achieve a closer fit to the data (Supplementary materials 3.2.1) than our approach: (j)SDMs have a much larger number of parameters, and are thus far more flexible, and they do not require a connection between environmental preferences and the available functional traits. We speculate, however, that our approach could be improved by confronting multiple alternative transfer functions and community models. In doing so, it opens new avenues to empirically test and compare alternative process-based models with various degrees of complexity as long as there is a link between the studied demographic processes and the available functional traits. As process-based models imply causal explanations for observed patterns8,54, this can lead to a promising hypothesis-driven confrontation of community models with different structures.

The key advantage of our approach is that it allows parametrization of process-based models with functional trait data and static community data, two data types that are more easily accessible than labor-intensive demographic measurements. We acknowledge that the proposed transfer function cannot predict the full intricacies of species demographic responses55, but we believe that it constitutes a reasonable approximation to build dynamic models of species-rich ecosystems in which ecologists are unlikely to ever obtain demographic measurements for every individual species. If the link between demography and functional traits is reasonably strong, we can draw on all the advantages of functional trait data: they are standardized, reasonably easy to measure across species and environments, and increasingly collected into large open databases5. With our approach, the main challenge of developing process-based models shifts from data gaps towards developing appropriate statistical frameworks that transfer functional traits into model parameters. We expect that our approach will promote the use of parameterized process-based community models to explore current theories of community assembly and ultimately better predict the dynamic of biodiversity in a changing world.

## Methods

### Dataset

#### Study area

The study was conducted in the central French Alps (45.12°N, 6.40°E) during the 2012 summer season. Nine sites, each containing two 100 m2 quadratic plots, were studied along a continuous elevation gradient (1858–2724 m) on a single south-east facing slope in a cow-grazed pasture. Mean annual air temperature varied between 0.8 °C and 3.8 °C and annual precipitations between 919 and 1186 mm56. Subalpine grasslands dominated the bottom of the gradient while sparsely vegetated alpine meadows characterized higher elevations. The nine sites were evenly distributed along the elevation gradient that represents the main local abiotic gradient. Mean annual soil temperature was estimated in each site using soil thermocaptors and varied between 4.55 °C and 8.26 °C. Temperature values were then standardized.

#### Community data

In each square-shaped plot, 101 plant individuals were sampled in July 2012 along two transects that followed its diagonals57. 118 species were sampled at least once and on average, 28.0 species were sampled in each plot. The 118 species included 26 grass species (Poales), 11 legumes species (Fabaceae), five shrubs species (Vaccinium, Daphne, and Salix sp.), and 76 forb species (remaining species). This sampling scheme tended to favor the sampling of locally dominant species over low-abundance species32. Botanical surveys were further done on the same plots simultaneously to collect more exhaustive data on species occurrences.

#### Functional traits

For each sampled individual, we identified the species and measured eight functional traits using standardized protocols58. (i) Reproductive and (ii) vegetative height are associated with plant competitive ability37. (iii) Specific leaf area (SLA) is usually correlated positively with plant growth rate and negatively with leaf lifespan3. (iv) Leaf dry matter content (LDMC) is related to the average density of leaf tissues and tends to scale negatively with SLA. (v) Leaf nitrogen concentration (LNC) quantifies the allocation of available nitrogen to photosynthetic enzymes in leaf chloroplasts59. (vi) Leaf carbon concentration (LCC) represents investment in structural tissues60. (vii) Leaf carbon isotopic ratio (δ13C) provides a time-integrated measure of stomatal conductance61. (viii) Leaf nitrogen isotopic ratio (δ15 N) reflects the isotope signature of nitrogen sources of the plant and thus provides a measure of the plant’s nitrogen acquisition strategy38. Details about the traits and the measurement protocol are available in the Supplementary information.

#### Definition of the species pool’s functional trait space

Trait measurements were averaged by species. All traits except δ13C and δ15N were log-transformed to better approach a normal distribution. To optimize the number of parameters used in the transfer function, we ran a Principal Component Analysis (R-package ade462) on the species-trait matrix, retained and scaled the three first orthogonal empirical functional trait axes (t1, t2, and t3) that collectively explain 67.0% trait variance. The three trait PCA axes can be described as follows (Supplementary Table 1 and Supplementary Figs. 12):

- t1: this axis represents 29.5 % of the total trait variance. It is negatively correlated with vegetative height, reproductive height, SLA, leaf nitrogen content, foliar δ15N and positively related to LDMC and foliar δ13C. The analysis of species distribution along this axis indicated that species typical of subalpine grasslands were associated with negative scores while species typical of alpine grasslands (including all five shrub species) were associated with positive scores.

- t2: this axis represents 19.5% of the total variance. It is positively correlated with SLA and negatively related to vegetative height, LDMC, leaf carbon content, foliar δ15N, and foliar δ13C. The analysis of species distribution along this axis indicated that positive scores were associated with forbs species.

- t3: this axis represents 17.9% of the total variance. It is positively related to reproductive height and LDMC and negatively related to leaf nitrogen content and foliar δ15N. The analysis of species distribution along this axis indicated that legumes and shrub species were associated with negative scores while positive scores were associated with grass species.

### Transfer function

In principle, any mathematical relationship between traits and demographic parameters could be specified as a transfer function. Given a matrix of N empirical functional traits T = {ti,n}, where ti,n is the known value of functional trait n of species i, and a matrix of unknown demographic parameters D = {di,m}, where di,m is the value of demographic parameter m of species i, the transfer function specifies the mathematical link between the two. In our case, we used a linear function of the functional traits to specify θmin as it can take both positive and negative values across species and a log-linear function for l and c, which can take only positive values.

We used a hyperspherical parameterization of the regression coefficients of the linear and log-linear functions63,64. This formulation defines the relationship between demographic parameters and functional traits with two sets of parameters: a first set that controlled the link between demographic parameters and functional traits, and a second set that controlled the mean and standard deviation of demographic parameters across species.

1—The regression coefficients of the linear expressions linking the species demographic parameters values di,m to the functional trait values ti,n can be viewed as coordinates lying on a unit hypersphere of dimension N. We express them using the multidimensional extension of the transformation of Cartesian coordinates to polar coordinates. In that scheme, the coefficients are defined by N-1 angle parameters {φm,n}. This parameterization samples efficiently all possible correlations between each demographic parameter and functional trait while keeping constant the mean and standard deviation of the demographic parameters (Supplementary information 1.4.4). For each demographic parameter m of species i, we calculate the link scale Ei,m that depends on the trait matrix T containing N traits through N-2 parameters {φm,n} that vary within the range [0, π] and one parameter φi,N-1 that varies within the range [0, 2π].

$$E_{i,m} = \cos\,\varphi_{m,1}\times t_{i,1} + \cdots + \mathop{\prod}\limits_{k}^{n-1} \sin\,\varphi_{m,k} \times \cos\,\varphi_{m,n}\times t_{i,n} + \cdots + \mathop{\prod}\limits_{k=1}^{N-1}\, \sin\,\varphi_{m,k}\times t_{i,N}$$
(2)

In the situation, where all traits are orthogonal and follow a standard normal distribution across species, the link scale Ei,m also follows a standard normal distribution across species.

2—We proposed two functions to transform the above expression to an appropriate distribution for the demographic parameter di,m across species. However, any monotonically increasing transformation of Ei,m could be employed depending on the data structure. Both expressions depend on additional parameters am and bm that together control the mean and standard deviation of di,m across species.

1) If di,m only takes positive values, we used a log-linear formulation (e.g., c and l):

$$d_{i,m} = e^{a_{m}\times E_{i,m}+b_{m}}; a_{m}\ge 0$$
(3)

2) If di,m can take both positive and negative values, we used a linear formulation (e.g., θmin):

$$d_{i,m} = a_{m}\times E_{i,m} + b_{m}; a_{m} \ge 0$$
(4)

This hyperspherical parameterization is mathematically equivalent to a classical linear combination, albeit less intuitive. Compared to the latter, it allowed us to set regularizing priors65 on the mean and variance of the demographic parameters (through parameters of am and bm). This ensured the convergence of the ODE model (Supplementary Information 1.4.3) and controlled parameter trade-offs, but also allowed us to maintain uninformative priors on the parameters {φm,n} and avoid making prior assumptions on trait–demography relationships.

### Bayesian Inference of the transfer function parameters

#### Likelihood function

For any given set of parameter values and for each plot temperature θj, we initialized the ODE with random positive biomass for all species. The ODE was run for a fixed amount of time steps sufficient to characterize the equilibrium across the prior distribution of the parameters (see Supplementary information 1.4.3). Because of the structure of the pairwise interaction matrix, the ODE model was globally stable for any parameter values: a single simulation was thus sufficient to characterize the equilibrium (see Supplementary information 1.3). We then calculated the likelihood of obtaining the observed set of sampled plant individuals in plot j given the equilibrium state of the ODE model. We assumed that the likelihood follows a multinomial distribution where each species has a probability to be sampled equal to its relative biomass in the community.

#### Priors

We set uninformative priors for all angle parameters {φm,n}. We further set regurlarizing priors for all the parameters am and bm which together control the mean and variance of demographic parameters (Supplementary Table 2). Because we calibrated our model using static relative abundances data and we used a likelihood function that follows a multinomial distribution, the demographic parameter gi and the parameter bc associated with the demographic parameter vector c, were not identifiable (see Supplementary information 1.4). In consequence, they were fixed (for all species i, gi = 10−3.43 and bc = 10-3.8, Eqs. 1 and 3). Those parameters would have been identifiable if the dataset included absolute abundance data (parameter bc) or temporal data (demographic parameter gi).

#### Posterior estimation

We used a Differential-Evolution Markov-Chain Monte Carlo algorithm (DEzs MCMC, R-package BayesianTools66) to estimate the posterior distribution of the transfer function parameters. We ran eight independent DEzs MCMC chains in parallel for 50000 steps, the posterior was estimated on the last 15000 steps. Convergence of the posterior distribution was assessed with Gelman–Rubin diagnostics31 (multivariate psrf was equal to 1.01). Posterior distribution data of the transfer function parameters are available in the Source data.

### Performance assessment

To evaluate whether the calibrated model performs better than random, we compared its fit to the fits of 200 replicate null models for which the trait data was randomized across species. To generate random functional trait datasets, we shuffled the PCA trait axes values among species and then ran the calibration procedure using these randomized trait data. One chain was run for each randomized dataset. After the burn-in phase (35 000 steps), 95% of the chains displayed a convergence criterion inferior to 1.48. The fits of the calibrated model and of the null models were compared using two metrics: the Deviance information criterion (DIC)31 and Nagelkerke’s pseudo R2 metric30 which lends itself well to multinomial models and gives an indication of the variance they explain. Nagelkerke’s pseudo R2 was calculated from the ratio of a model’s posterior likelihood and the likelihood of the hypothesis that all sampled species have equal relative abundance in each plot, as well as the sampling effort. We also computed this pseudo R2 metric at the median of the posterior for each plot individually to access the power of the calibrated model relative to the null model along the studied gradient. Finally, we tested the ability of the calibrated model to predict plant presence/absence in the studied plots by characterizing their ROC curves and AUC scores. To do so, we used the botanical surveys because they are more accurate to infer plant presence/absence compared to transect data32.

### Comparative analysis

To illustrate the usefulness of our approach, we compared it to existing correlative approaches. First, we analyzed species abundances along the mean annual temperature gradient using a species distribution model and a set of joint species distribution models67,68. The performance of those models was also assessed with the Deviance information criterion, Nagelkerke’s pseudo R2, ROC curves, and AUC scores. Second, we analyzed the relationship between mean annual temperature and plant functional traits using the fourth corner analysis36. Detailed methods and results are available in the Supplementary information.

All analyses were carried out using the software R 3.5.3, R-packages ade4 1.7.16 and BayesianTools 0.1.7 using the resources of ETH Zürich and the University of Wyoming Advanced Research Computing Center69.

### Reporting summary

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

## Data availability

Data are archived in the code repository https://doi.org/10.5281/zenodo.4682287Source data are provided with this paper.

## References

1. 1.

McGill, B. J., Enquist, B. J., Weiher, E. & Westoby, M. Rebuilding community ecology from functional traits. Trends Ecol. Evol. 21, 178–185 (2006).

2. 2.

Violle, C. et al. Let the concept of trait be functional! Oikos 116, 882–892 (2007).

3. 3.

Adler, P. B. et al. Functional traits explain variation in plant life history strategies. Proc. Natl. Acad. Sci. 111, 740–745 (2014).

4. 4.

Pierce, S., Brusa, G., Vagge, I. & Cerabolini, B. E. Allocating C. S. R. plant functional types: the use of leaf economics and size traits to classify woody and herbaceous vascular plants. Funct. Ecol. 27, 1002–1010 (2013).

5. 5.

Kattge, J. et al. TRY - a global database of plant traits. Glob. Change Biol. 17, 2905–2935 (2011).

6. 6.

Kunstler, G. et al. Plant functional traits have globally consistent effects on competition. Nature 529, 204–207 (2016).

7. 7.

Yang, J., Cao, M. & Swenson, N. G. Why functional traits do not predict tree demographic rates. Trends Ecol. Evol. 33, 326–336 (2018).

8. 8.

Connolly, S. R., Keith, S. A., Colwell, R. K. & Rahbek, C. Process, mechanism, and modeling in macroecology. Trends Ecol. Evol. 32, 835–844 (2017).

9. 9.

Dormann, C. F. et al. Correlation and process in species distribution models: bridging a dichotomy. J. Biogeogr. 39, 2119–2131 (2012).

10. 10.

Thuiller, W., Pollock, L. J., Gueguen, M. & Münkemüller, T. From species distributions to meta-communities. Ecol. Lett. 18, 1321–1328 (2015).

11. 11.

Zurell, D. et al. Benchmarking novel approaches for modelling species range dynamics. Glob. Change Biol. 22, 2651–2664 (2016).

12. 12.

Alexander, J. et al. Lags in the response of alpine plant communities to climate change. Glob. Change Biol. 24, 563–579 (2018).

13. 13.

Evans, M. E., Merow, C., Record, S., McMahon, S. M. & Enquist, B. J. Towards process-based range modeling of many species. Trends Ecol. Evol. 31, 860–871 (2016).

14. 14.

Urban, M. C. et al. Improving the forecast for biodiversity under climate change. Science 353, aad8466 (2016).

15. 15.

Paine, C. E. T., Deasey, Anna, Bradley, DuthieA. & Ken, Thompson Towards the general mechanistic prediction of community dynamics. Funct. Ecol. 32, 1681–1692 (2018).

16. 16.

Hartig, F. et al. Connecting dynamic vegetation models to data–an inverse perspective. J. Biogeogr. 39, 2240–2252 (2012).

17. 17.

Levins, R. The strategy of model building in population biology. Am. Sci. 54, 421–431 (1966).

18. 18.

Kraft, N. J., Godoy, O. & Levine, J. M. Plant functional traits and the multidimensional nature of species coexistence. Proc. Natl. Acad. Sci. 112, 797–802 (2015).

19. 19.

Mayfield, M. M. & Stouffer, D. B. Higher-order interactions capture unexplained complexity in diverse communities. Nat. Ecol. Evol. 1, 0062 (2017).

20. 20.

Carrara, F., Giometto, A., Seymour, M., Rinaldo, A. & Altermatt, F. Experimental evidence for strong stabilizing forces at high functional diversity of aquatic microbial communities. Ecology 96, 1340–1350 (2015).

21. 21.

Laughlin, D. C. & Messier, J. Fitness of multidimensional phenotypes in dynamic adaptive landscapes. Trends Ecol. Evol. 30, 487–496 (2015).

22. 22.

Curtsdotter, A. et al. Ecosystem function in predator–prey food webs—confronting dynamic models with empirical data. J. Anim. Ecol. 88, 196–210 (2019).

23. 23.

Chesson, P. Mechanisms of maintenance of species diversity. Annu. Rev. Ecol. Syst. 31, 343–366 (2000).

24. 24.

Terborgh, J. W. Toward a trophic theory of species diversity. Proc. Natl. Acad. Sci. 112, 11415–11422 (2015).

25. 25.

Swenson, N. G. & Enquist, B. J. Ecological and evolutionary determinants of a key plant functional trait: wood density and its community-wide variation across latitude and elevation. Am. J. Bot. 94, 451–459 (2007).

26. 26.

Körner, C. Alpine Plant Life. (Springer, 2003).

27. 27.

Westoby, M. A leaf-height-seed (LHS) plant ecology strategy scheme. Plant Soil 199, 213–227 (1998).

28. 28.

Adler, P. B. et al. Competition and coexistence in plant communities: intraspecific competition is stronger than interspecific competition. Ecol. Lett. 21, 1319–1329 (2018).

29. 29.

Chalmandrier, L., Albouy, C. & Pellissier, L. Species pool distributions along functional trade-offs shape plant productivity–diversity relationships. Sci. Rep. 7, 15405 (2017).

30. 30.

Nagelkerke, N. J. A note on a general definition of the coefficient of determination. Biometrika 78, 691–692 (1991).

31. 31.

Gelman, A., Hwang, J. & Vehtari, A. Understanding predictive information criteria for Bayesian models. Stat. Comput. 24, 997–1016 (2014).

32. 32.

Godínez-Alvarez, H., Herrick, J. E., Mattocks, M., Toledo, D. & Van Zee, J. Comparison of three vegetation monitoring methods: Their relative utility for ecological assessment and monitoring. Ecol. Indic. 9, 1001–1008 (2009).

33. 33.

Grime, J. P. Evidence for the existence of three primary strategies in plants and its relevance to ecological and evolutionary theory. Am. Nat. 111, 1169–1194 (1977).

34. 34.

Reich, P. B. The world-wide ‘fast–slow’ plant economics spectrum: a traits manifesto. J. Ecol. 102, 275–301 (2014).

35. 35.

de Bello, Fde et al. Hierarchical effects of environmental filters on the functional structure of plant communities: a case study in the French Alps. Ecography 36, 393–402 (2013).

36. 36.

Dray, S. et al. Combining the fourth-corner and the RLQ methods for assessing trait responses to environmental variation. Ecology 95, 14–21 (2014).

37. 37.

Violle, C. et al. Competition, traits and resource depletion in plant communities. Oecologia 160, 747–755 (2009).

38. 38.

Michelsen, A., Schmidt, I. K., Jonasson, S., Quarmby, C. & Sleep, D. Leaf 15N abundance of subarctic plants provides field evidence that ericoid, ectomycorrhizal and non-and arbuscular mycorrhizal species access different sources of soil nitrogen. Oecologia 105, 53–63 (1996).

39. 39.

Laughlin, D. C., Joshi, C., van Bodegom, P. M., Bastow, Z. A. & Fulé, P. Z. A predictive model of community assembly that incorporates intraspecific trait variation. Ecol. Lett. 15, 1291–1299 (2012).

40. 40.

Laughlin, D. C. The intrinsic dimensionality of plant traits and its relevance to community assembly. J. Ecol. 102, 186–193 (2014).

41. 41.

Díaz, S. et al. The global spectrum of plant form and function. Nature 529, 167–171 (2016).

42. 42.

Letten, A. D., Ke, P.-J. & Fukami, T. Linking modern coexistence theory and contemporary niche theory. Ecol. Monogr. 87, 161–177 (2017).

43. 43.

O’Dwyer, J. P. Whence Lotka-Volterra? Theor. Ecol. 11, 441–452 (2019).

44. 44.

Pichler, M., Boreux, V., Klein, A.-M., Schleuning, M. & Hartig, F. Machine learning algorithms to infer trait-matching and predict species interactions in ecological networks. Methods Ecol. Evol. 11, 281–293 (2020).

45. 45.

Maestre, F. T., Callaway, R. M., Valladares, F. & Lortie, C. J. Refining the stress-gradient hypothesis for competition and facilitation in plant communities. J. Ecol. 97, 199–205 (2009).

46. 46.

Damgaard, C. A critique of the space-for-time substitution practice in community ecology. Trends Ecol. Evol. 34, 416–421 (2019).

47. 47.

Arnoldi, J.-F., Loreau, M. & Haegeman, B. The inherent multidimensionality of temporal variability: how common and rare species shape stability patterns. Ecol. Lett. 22, 1557–1567 (2019).

48. 48.

May, R. M. Patterns of species abundance and diversity. in Ecology and Evolution of Communities (eds Cody, M. L. & Diamond, J. M.) 81–120 (Harvard University Press, 1975).

49. 49.

Falster, D. S., Brännström, Å., Westoby, M. & Dieckmann, U. Multitrait successional forest dynamics enable diverse competitive coexistence. Proc. Natl. Acad. Sci. 114, E2719–E2728 (2017).

50. 50.

Boulangeat, I., Georges, D. & Thuiller, W. FATE-HD: a spatially and temporally explicit integrated model for predicting vegetation structure and diversity at regional scale. Glob. Change Biol. 20, 2368–2378 (2014).

51. 51.

Lischke, H. & Löffler, T. J. Finding all multiple stable fixpoints of n-species Lotka–Volterra competition models. Theor. Popul. Biol. 115, 24–34 (2017).

52. 52.

Jamil, T., Ozinga, W. A., Kleyer, M. & ter Braak, C. J. Selecting traits that explain species–environment relationships: a generalized linear mixed model approach. J. Veg. Sci. 24, 988–1000 (2013).

53. 53.

ter Braak, C. J. New robust weighted averaging-and model-based methods for assessing trait–environment relationships. Methods Ecol. Evol. 10, 1962–1971 (2019).

54. 54.

Pielou, E. C. Ecological Diversity. (Wiley, New York, 1975).

55. 55.

Gibert, A., Gray, E. F., Westoby, M., Wright, I. J. & Falster, D. S. On the link between functional traits and growth rate: meta-analysis shows effects change with plant size, as predicted. J. Ecol. 104, 1488–1503 (2016).

56. 56.

Durand, Y. et al. Reanalysis of 44 Yr of climate in the French Alps (1958–2002): methodology, model validation, climatology, and trends for air temperature and precipitation. J. Appl. Meteorol. Climatol. 48, 429–449 (2009).

57. 57.

Chalmandrier, L. et al. Spatial scale and intraspecific trait variability mediate assembly rules in alpine grasslands. J. Ecol. 105, 277–287 (2017).

58. 58.

Cornelissen, J. H. C. et al. A handbook of protocols for standardised and easy measurement of plant functional traits worldwide. Aust. J. Bot. 51, 335–380 (2003).

59. 59.

Reich, P. B. et al. Generality of leaf trait relationships: a test across six biomes. Ecology 80, 1955–1969 (1999).

60. 60.

Poorter, H. & Bergkotte, M. Chemical composition of 24 wild species differing in relative growth rate. Plant Cell Environ. 15, 221–229 (1992).

61. 61.

Farquhar, G. D., O’leary, M. H. & Berry, J. A. On the relationship between carbon isotope discrimination and the intercellular carbon dioxide concentration in leaves. Funct. Plant Biol. 9, 121–137 (1982).

62. 62.

Thioulouse, J. et al. Multivariate Analysis of Ecological Data with ade4. (Springer, 2018).

63. 63.

Rapisarda, F., Brigo, D. & Mercurio, F. Parameterizing correlations: a geometric interpretation. IMA J. Manag. Math. 18, 55–73 (2007).

64. 64.

Blumenson, L. E. A derivation of n-dimensional spherical coordinates. Am. Math. Mon. 67, 63–66 (1960).

65. 65.

Banner, K. M., Irvine, K. M. & Rodhouse, T. The use of bayesian priors in ecology: the good, the bad, and the not great. Methods Ecol. Evol. 00, 1–8 (2020).

66. 66.

Hartig, F., Minunno, F. & Paul, S. BayesianTools: General-Purpose MCMC and SMC samplers and tools for bayesian statistics. R package (2017).

67. 67.

Warton, D. I. et al. So Many Variables: Joint Modeling in Community Ecology. Trends Ecol. Evol. 30, 766–779 (2015).

68. 68.

Pichler, M. & Hartig, F. A new method for faster and more accurate inference of species associations from novel community data. Preprint at https://arxiv.org/abs/2003.05331 (2020).

69. 69.

Advanced Research Computing Center (ARCC). Teton Computing Environment. https://doi.org/10.15786/m2fy47 (2018).

## Acknowledgements

LC acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 840946 (Project “CLIMB”). Data acquisition leading to these results received funding from the European Research Council under the European Community’s Seven Framework Program FP7/2007–2013 Grant Agreement no. 281422 (TEEMBIO) granted to W. Thuiller (CNRS-Univ. Grenoble Alpes). We thank T. Ott (University of Regensburg) and B. Flück (ETH Zürich) for coding assistance. LP acknowledges funding from the Swiss National Science Foundation (SNSF) project no. 162604.

## Author information

Authors

### Contributions

LC, FH and LP conceived and designed the study. LC implemented the models and conducted the main data analysis. LC and HL studied the mathematical properties of the model. LC and MP conducted the comparative analysis. LC, FH, DL, HL, MP, DS and LP discussed all aspects of the research and contributed to writing and revising the paper.

### Corresponding authors

Correspondence to Loïc Chalmandrier or Loïc Pellissier.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Chalmandrier, L., Hartig, F., Laughlin, D.C. et al. Linking functional traits and demography to model species-rich communities. Nat Commun 12, 2724 (2021). https://doi.org/10.1038/s41467-021-22630-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-021-22630-1

• ### A Partition Theorem for a Randomly Selected Large Population

• Arni S. R. Srinivasa Rao

Acta Biotheoretica (2022)

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text:
Question concerning probablity

1. Aug 20, 2007

Mins

How can I calculate the probablity of 2 same numbers being right next to each other, when 100 random numbers chosen from '1,2,3.... 6' form a line?

2. Aug 20, 2007

CRGreathouse

That's pretty well certain. To five decimal places, S = 100%.

To see why, and to get a more precise answer:
Let S denote a success, that is at least two in a row are the same. Consider instead the smaller problem with only two such dice in a row:

(1-6)(1-6)

S is just the chance that the two are the same, which is 6/36 = 1/6.

For three dice:

(1-6)(1-6)(1-6)

The middle die has a 1/6 chance of being the same as the one before it, and the last die has a 1/6 chance of being like the die before it. That's 1/6 + 1/6, except that now you're double counting when all three are the same, so it's 1/6 + 1/6 - 1/36.

This is easier if you calculate 1-S, which is 5/6 with two dice and (5/6)^2 for three dice.

3. Aug 20, 2007

Mins

Thanks. Another question

How can you calculate the probability of 'n' pairs of same numbers being right next to each other?

4. Aug 21, 2007

CRGreathouse

I'm not sure what you mean, give an example.

5. Aug 21, 2007

Mins

10 numbers will be chosen from 1,2,3...6, and they will form a line, like this

1442345662

There are 2 pairs of sets, as underlined above.

What I want to find out is the probability of not just only one, but 2 or more sets(2 same numbers being right next to each other) appearing in a line.

Last edited: Aug 21, 2007
6. Aug 21, 2007

CRGreathouse

Does 1112345 count as having two "sets"? Does 1111234?

7. Aug 21, 2007

Mins

First one has one "set"(I defined the word, think you should know).
And second one has two "sets".
But if ruling those out simiplify your calculation, think it's OK.

8. Aug 21, 2007

Mins

What I want is a method to calculate those :

two sets appearing in a line of 50 numbers
three sets appearing in a line of 50 numbers
four sets appearing in a line of 50 numbers
etc.

But a solution that can be used in similar situations will help me.

Last edited: Aug 21, 2007

format: plain


text:
## Essential University Physics: Volume 1 (3rd Edition)

$\frac{M}{4 \pi R^3(1-2e^{-1})}$
We know: $\rho(R)=\int_0^R \rho_0 e^{\frac{-r}{R}}dr$ We also know that $\rho=\frac{M}{4\pi R^3}$. Thus, we take the integral and simplify to obtain: $\rho=\frac{M}{4\pi R^3}=\rho_0(1-2e^{-1})$ $\rho_0=\frac{M}{4\pi R^3(1-2e^{-1})}$


text:
# How do you determine if y=3(5/2)^x is an exponential growth or decay?

Nov 26, 2016

It is an exponential growth

#### Explanation:

Because the number that is raised to the power of $x$ is greater than one. If it was between zero and one it will be an exponential decay.


text:
# add section to toc with number

I have some scanned pdf documents which I want to put together in a latex file and structure them with sections which appear only in the TOC. This has the advantage, that I can jump to the page using hyperref.

However with this code only text is added to the toc, but I would like them to be numbered as if I had written a normal \section.

\phantomsection\addcontentsline{toc}{chapter}{SI-System}
\includepdf[pages=1-3,pagecommand={\thispagestyle{plain}}]{02-SI.pdf}

-

You can use the addtotoc option for the \includepdf command; this option has the syntax

addtotoc={<page number>,<section>,<level>,<heading>,<label>}

where <page number> is the page number of the inserted document that will be linked to from the ToC, <section> is the LaTeX sectioning name (e.g., section, subsection,...), <level> denotes depth of section (e.g., 1 for section level, 2 for subsection level,...), <heading> is the title inserted in the table of contents, and <label> is the name of the label which can be referred to with \ref and \pageref.

With your current approach you can use \numberline to add the number of the corresponding sectional unit to the ToC; for example, you can say something like:

\phantomsection\addcontentsline{toc}{chapter}{\protect\numberline{\thechapter}SI-System}

Here's an example or the use of addtotoc option (kindly provided by Marco Daniel):

\RequirePackage{filecontents}
\begin{filecontents}{\jobname-a5.tex}
\documentclass[english,paper=a5]{scrartcl}
\usepackage{geometry}
\usepackage{babel,blindtext}
\begin{document}
\Blinddocument
\end{document}
\end{filecontents}

\documentclass{scrartcl}
\immediate\write18{pdflatex \jobname-a5.tex}
\usepackage{pdfpages,lipsum}
\includepdfset{frame,noautoscale}
\setlength{\fboxrule}{5pt}

\begin{document}
\tableofcontents
\section{foo}
\lipsum[1]
\end{document}

save the file as, for example, test.tex and run it using

pdflatex --shell-escape test

-
addtotoc does not work for me because I do not know the pagenumber and \protect\numberline{\thechapter} adds a number, but that is always 0, since there is nothing which could change it. –  Matthias Pospiech Dec 15 '11 at 18:36
@MatthiasPospiech: but you can always use 1 for <page number>; this way, the first page of the included document will be linked from the ToC. –  Gonzalo Medina Dec 15 '11 at 20:09
Ok, now I understand the meaning of the 'number' –  Matthias Pospiech Dec 15 '11 at 20:41

I solved it now by defining the numbering myself:

\renewcommand{\chapter}[1]{
\setcounter{section}{0}
{\protect\numberline{\thechapter}#1}
}

\renewcommand{\section}[1]{

I will provide an example which uses the command addtotoc. –  Marco Daniel Dec 15 '11 at 19:18
The page number of addtotoc is the number of the page which will be linked. –  Marco Daniel Dec 15 '11 at 19:23


text:
# I Angular momentum of an Odd-Odd nucleus (in its ground state)

#### JD_PM

Summary
Issue explaining why sometimes the table just gives one of the possible values for the total AM of an O/O nucleus.
What I know is the following:

The total angular momentum of the nucleus is just the total sum of the angular momentum of each nucleon.

If the nucleons are even the total angular momentum in the ground state will simply be $0+$.

If the odd number of nucleons is close to one of the magic numbers then the shell model can explain what total angular momentum we'll get.

There are 3 cases to consider:

1) E/E nucleus (40-Ca for instance): Both proton and neutron numbers are even so the total angular momentum of the nucleus (I have read in other sources they call it Isospin) in the ground state will be $0+$.

2) E/O nucleus (41-Ca for instance): The proton number is even. Thus the contribution by protons to the final angular momentum is none because $j_p = 0$. What about neutrons? The unpaired neutron will be located in the 1f7/2 shell, then $j_n = 7/2-$. Then the final angular momentum of 41-Ca in its GS will be $I = 7/2-$.

Here comes my issue:

3) O/O nucleus (14-N for instance): here both proton and neutron numbers are odd so we expect that both contribute to the final angular momentum of 14-N.

Proton contribution: The unpaired proton will be located in the 1p1/2 shell, then $j_p = 1/2-$.

Neutron contribution: The unpaired neutron will be located in the 1p1/2 shell, then $j_n = 1/2-$.

Now it is about adding both AMs up to get $I$:

$$I = 1+, 0+$$

But in the table just 1+ is given. Why?

Related High Energy, Nuclear, Particle Physics News on Phys.org

#### snorkack

Summary: Issue explaining why sometimes the table just gives one of the possible values for the total AM of an O/O nucleus.

Now it is about adding both AMs up to get $I$:

$$I = 1+, 0+$$

But in the table just 1+ is given. Why?
Because the states of different angular momenta are not, in general, of equal energy.
Therefore only one of the different states is the ground state. The others must be excited states.
If the table must be that of ground states then it has to give just one, and namely the one which is the ground state.
Though this sometimes causes problems of relevance. Notoriously so with Ta-180.

#### JD_PM

Therefore only one of the different states is the ground state.
I see, but how can we predict theoretically which of the states is the GS?

In my example how can we know which is the GS? (I = 1+, 0+; the table says 1+ but I guess that is an experimental result).

"Angular momentum of an Odd-Odd nucleus (in its ground state)"

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text:
निरंजन
I liked the way watermark is set for the [documentation of package background](http://texdoc.net/texmf-dist/doc/latex/background/background.pdf). I saw how it is done in its source and made this code -

#
\documentclass[border=2cm]{standalone}
\usepackage[placement=top]{background}
\SetBgContents{abcd}
\SetBgAngle{0}
\SetBgScale{3}
\SetBgHshift{65}
\SetBgVshift{-5}

\begin{document}
Hello world
\end{document}

This doesn't give me the watermark, but in \documentclass{article} it actually works. I want to set a watermark which will automatically be adjusted to the given page-size. Is there any alternative to this package?
marmot
In case this is in any way related to your previous question on frames around standalalone documents, and also if it is not, then there is good news: you can enhance the boxes from the previous tcolorbox answer to allow for watermarks. Ironically, you really basically need to add the enhanced key to get access to these features (and load the skins library). In fact, tcolorbox comes with an extended support for watermarks, which are described in section 10.3 of manual v4.31. Since there is no mention of a frame, for the time being I switched it off with colframe=white (there are other ways, of course).

\documentclass[border=3mm]{standalone}
\usepackage[skins]{tcolorbox}
\newtcolorbox{mybox}[1][]{hbox,colback=white,colframe=white,
watermark color=blue,watermark opacity=0.3,
enhanced,boxsep=3cm,#1}
\standaloneenv{mybox}
\begin{document}
\begin{mybox}[watermark text=pft]
Hello world!
\end{mybox}
\begin{mybox}[watermark text=quack]
Hello ducks!
\end{mybox}
\end{document}

![ani.gif](/image?hash=ffc1d8d7172446226ee16f6541823de9271170502753861425bfee41ffb363a3)

You can then create whatever watermark you like. Just to give an (admittedly not overly nice) example, you can typeset the watermark text along a circle.

\documentclass[border=3mm]{standalone}
\usepackage[skins]{tcolorbox}
\usepackage{tikz}
\usetikzlibrary{decorations.text}
\newtcolorbox{mybox}[1][]{hbox,colback=white,colframe=white,
watermark color=blue,watermark opacity=0.3,
enhanced,boxsep=3cm,#1}
\tcbset{circular watermark/.style={watermark tikz={%
\pgfmathsetmacro{\myr}{0.2cm+1*width("#1")/2/pi}%
\pgfmathsetmacro{\myR}{\myr+height("#1")-2pt}%
\path[decorate,decoration={text along path,text align={align=center},
raise=-0.15ex,text={|\sffamily|#1}}]
(0,-\myr pt) arc[start angle=270,end angle=-90,radius=\myr pt];
% necessary to get the bounding box right
\path (0,-\myR pt) arc[start angle=270,end angle=-90,radius=\myR pt];
}}}
\standaloneenv{mybox}
\begin{document}
\begin{mybox}[circular watermark=pft]
Hello world!
\end{mybox}
\begin{mybox}[circular watermark=meow]
Hello cats!
\end{mybox}
\begin{mybox}[circular watermark=quack]
Hello ducks!
\end{mybox}
\end{document}

![ani.gif](/image?hash=0383a0551c14206b582da561e92bf084f6eea39a80912b4e7461957b4e4d82cb)

Skillmon
There are 3 code versions in this answer:

1. using the development version of LaTeX
1. using eso-pic instead of the development version of the kernel
1. an example with an auto-scaling watermark

## Development kernel version

The following uses the new hook-mechanism provided in the development version of the LaTeX kernel (so you need to compile this with pdflatex-dev, lualatex-dev or xelatev-dev).

It also requires xfp for the placement, both xcolor and graphics are only needed for the used contents of \mywatermark.

\documentclass{standalone}

\usepackage{xcolor}
\usepackage{graphics}
\usepackage{xfp}

\newcommand*\mywatermark{\rotatebox{30}{\sffamily\color{gray}\huge watermark}}

{%
\setlength\unitlength{1pt}%
\put(\fpeval{1in+0.5*\paperwidth},\fpeval{-1in-0.5*\paperheight})
{%
\sbox0{\makebox[0pt][c]{\mywatermark}}%
\raise.5\dimexpr\dp0-\ht0\relax\box0
}%
}

\begin{document}
\parbox[t][5cm]{10cm}{Hello world}
\end{document}

![test-1.png](/image?hash=0ee9b3c6cc4e4b42f866f43eb4747f70eb5b97ea51c167384c467090864a57a0)

----

## eso-pic version

Without the development versions, you can use eso-pic with the option texcoords (just so I don't have to change the rest of the code). So the following works in current versions of LaTeX:

\documentclass{standalone}

\usepackage{xcolor}
\usepackage{graphics}
\usepackage{xfp}
\usepackage[texcoord]{eso-pic}

\newcommand*\mywatermark{\rotatebox{30}{\sffamily\color{gray}\huge watermark}}

{%
\setlength\unitlength{1pt}%
\put(\fpeval{1in+0.5*\paperwidth},\fpeval{-1in-0.5*\paperheight})
{%
\sbox0{\makebox[0pt][c]{\mywatermark}}%
\raise.5\dimexpr\dp0-\ht0\relax\box0
}%
}

\begin{document}
\parbox[t][2cm]{5cm}{Hello world}
\end{document}

(results are the same as above, so no new picture)

-----------

## Auto-scaling example

Just to show, that auto-scaling is possible, the following uses adjustbox and its max totalsize key for the that.

\documentclass{standalone}

\usepackage{xcolor}
\usepackage{graphics}
\usepackage{xfp}
\usepackage[texcoord]{eso-pic}

\newcommand*\mywatermark
{%
{max totalsize={\paperwidth}{\paperheight}}%
{\rotatebox{30}{\sffamily\color{gray}\huge watermark}}%
}

{%
\setlength\unitlength{1pt}%
\put(\fpeval{1in+0.5*\paperwidth},\fpeval{-1in-0.5*\paperheight})
{%
\sbox0{\makebox[0pt][c]{\mywatermark}}%
\raise.5\dimexpr\dp0-\ht0\relax\box0
}%
}

\begin{document}
Hello World!
\end{document}

![watermark-1.png](/image?hash=0743b5c7df6decec35a6ed2cd58087805ac84d2ed3ead5c62d28cbb463822d70)

Enter question or answer id or url (and optionally further answer ids/urls from the same question) from

Separate each id/url with a space. No need to list your own answers; they will be imported automatically.


text:
# How can I prevent 2D teleportation from teleporting through solid objects?

In my 2D game, the player is able to teleport to any place on the screen with the click of the mouse button. What I am trying to prevent is the ability to teleport across solid objects. If the point from the player and the mouse click position is across an object then I want the player to teleport next to that object instead of across it.

I am currently trying to achieve this with raycasting. The code I wrote somewhat works but only if the mouse click is on an object, if the mouse click is across another object, it still teleports me accross the object without it registering that there is an object between the player and the mouse position.

My code:

Vector3 player_position = Camera.main.WorldToScreenPoint(transform.position);
Vector3 to_object_vector = transform.position - Camera.main.transform.position;
Vector3 linear_distance_vector = Vector3.Project(to_object_vector, Camera.main.transform.forward);

float distance_to_camera = linear_distance_vector.magnitude;
Vector3 mouse_position = Input.mousePosition;
float dist = Vector2.Distance(player_position, mouse_position);
Vector3 dir = (mouse_position - player_position).normalized;
mouse_position.z = distance_to_camera;

RaycastHit2D hit = Physics2D.Raycast(new Vector2(player_position.x, player_position.y), new Vector2(dir.x, dir.y), dist, 0);
if (hit.collider != null) {
Debug.Log(hit.point);
mouse_position = new Vector3(hit.point.x, hit.point.y, mouse_position.z);
}

Is there a better approach to this problem?

• sorry I couldn't understand what you actually want yo ask. Can you please post your question with some snaps that what is currently going on and what you want, that'd be very easy to understand for all the community members Mar 14 '16 at 11:05
• my fault it is hard for me to explain. Here is a gyazo of what I am trying to prevent: gyazo.com/2674861c985b6e845a4bb8aed06a4a12 . As you can see I can teleport across this object but when I click on the object itself it prevents me. I want to prevent myself from teleporting across that object. So imagine a line if any collider interrupts that line then teleport next to that object instead of across it.
– Sol
Mar 14 '16 at 11:18
• So you want if you click on block in middle, but player should teleport as you click in empty space next to block? Mar 14 '16 at 11:28
• I dont want the player to teleport across it at all. What I have so far is stopping the player from teleporting in the wall but I also want to stop them from teleporting across it as well.
– Sol
Mar 14 '16 at 11:39
• Then I'd suggest to move the player instead of assigning new position. As disable SpriteRenderer then assign velocity to that direction and then check for the collision Mar 14 '16 at 11:44

One thing you should do, and this is useful for all of your Unity projects, is using Debug.DrawLine (if you have origin and destination) or Debug.DrawRay if you only have the direction vector.

I haven't used 2D mode in Unity, but I guess you can still use 3D in Scene inspector, right? If so, then use the functions above to throw rays from player to the hit.point position. You should do it into Update function, since it could draw a ray just for 1 frame if you don't and you wouldn't see it.

Things to check if working properly:

1. Check if the ray is drawing always trough the object when you click behind it, if it does, then you're not setting the nearest position properly.
2. Check that those objects you don't want to go through are not in a Ignore raycast layer.

Now, talking about how to move your character, I would try to use Physics AddForce. Even though it is 3D, you can just use 0 into x/z axis and use Physics to move character. If you multiply the vector for a high thrust, I think you could get a teleport simulation (deactivating MeshRenderer before doing it, for example). Remember you need a Rigidbody to use that physics, and probably need to mark it as Kinetic in order to avoid it bouncing.

Hope this helps you.

• Thanks I have tried drawline but I was unable to see the actual line because maybe it was drawn for one frame, but I'll try it again! And yeah using physics addforce is a good idea to simulate teleportation! That's a great idea, but I really want to solve this particular problem since I spent so much time on it. And yes the objects are not on the ignore raycast layer i checked.
– Sol
Mar 14 '16 at 10:59
• I added force but I am still able to go through solid objects if I run up to it and addforce against the wall, which is very unfortunate
– Sol
Mar 14 '16 at 23:14
• Did you set Dynamic Continuous inside Rigidbody options? Mar 14 '16 at 23:53
• no i didn't I did not realize there was an option for that! Even if the collider is slim it still doesn't allow me to go through!! That worked thank you so much!
– Sol
Mar 15 '16 at 0:46
• You're welcome. You have enough information on what this option is made for on the net, it's something we all must know. Good luck with your game Mar 15 '16 at 11:19

Well since raycasting cannot solve my specific issue which is unfortunate, since it should return the end point of any solid objects that interrupts that line I was forced to expand my colliders far enough to where the player can't click over the object like so:

Since my if statement is checking if the player is clicking on a collider I will always teleport to the hit.point preventing the player from going outside the map indefinitely!


text:
# Joint pdf of Order Statistics

Let $$X_1, ... X_n \stackrel{i.i.d} \sim Unif(0,2).$$ Find $$P(Y_1 < \frac12 < Y_n),$$ where $$Y_1 = \min \{X_1, ..., X_n\}$$ and $$\ Y_n = \max\{X_1, ..., X_n\}$$

When I attempted this problem, I decided that $$n>2 \implies Y_1$$ and $$Y_n$$ are independent, which leads me to the answer

$$P(Y_1 < \frac12 < Y_n) = \left(1-\left(\frac14\right)^n\right) \cdot \left(1-\left(\frac34\right)^n\right)$$

I have since decided is untrue. Clearly, if I know $$Y_n = 1,$$ then $$Y_1 \neq 1.5.$$ I have also discovered that in my notes there is a joint pdf of two order statistics of a given sample:

$$f_{Y_i,Y_j}(u,v) = \cfrac{n!} {(i-1)!(j-i-1)!(n-j)!} [F_X(u)]^{i-1} [F_X(v)-F_X(u)]^{j-i-1} [1-F_X(v)]^{n-j} f_X(u)f_X(v), \forall \ u

I am looking first of all for the derivation and perhaps some intuition for this equation. For context, I am familiar with the derivation of the cdf/pdf of a single order statistic, but this one is throwing me for a loop. Second of all, I would like to know if I am using this equation correctly:

$$P(Y_1<\frac12 \frac12) \\ = \int_0^\frac12 \int_\frac12^2 \cfrac{n!}{0!(n-2)!0!} \left[\frac u2\right]^{0} \left[\frac v2 - \frac u2 \right]^{n-2} \left[1- \frac v2 \right]^{0} \left(\frac12 \right) \left(\frac12 \right)\mathrm{d}v\mathrm{d}u \\ = \frac14 \int_0^\frac12 \int_\frac12^2 n(n-1) \left[\cfrac{v-u}2 \right]^{n-2} \mathrm{d}v\mathrm{d}u$$

which I don't know how to solve explicitly, but is still useful as it can be solved given a value of $$n$$.

• Clearly $Y_1$ and $Y_n$ are not independent, since $Y_1$ cannot exceed $Y_n$. – Glen_b Apr 22 '20 at 5:40
• The integral you have set up in the end misses the $u<v$ restriction. If this is some sort of homework, please add self-study tag. – StubbornAtom Apr 22 '20 at 6:31
• See stats.stackexchange.com/questions/225867 for a more complicated version of this problem, because its solution easily applies here. Concerning the issue of independence, see stats.stackexchange.com/questions/208658. A formula for the joint pdf is given at stats.stackexchange.com/a/69582/919 and, in the special case of a uniform distribution, at stats.stackexchange.com/a/207596/919. – whuber Apr 22 '20 at 13:46
• @StubbornAtom How does my integral miss the restriction? No $u$ value is above any $v$ value in the integration. Also, this is a former exam question not assigned to me in any capacity - let me know if that still qualifies it as self-study – jeremy909 Apr 22 '20 at 15:42
• Yes, all exam questions etc. qualify as self-study: link through to the tag wiki for details. As far as computing your integral goes, it's far easier than it might look: you're just integrating polynomials, after all. – whuber Apr 22 '20 at 16:25

I don't care to evaluate the integral mechanically (although Wolfram Alpha will). And nor should anyone else, because the pedagogical usefulness of that exercise is doubtful.

Instead, considering that the event $$Y_{1} \le 1/2 \le Y_{n}$$ is the complement of the union of events

$$\left(Y_{1}\gt 1/2\right)\ \cup\ \left(Y_{n}\lt 1/2\right)$$

and, because $$Y_1$$ cannot exceed $$Y_n$$ those events are disjoint, it follows that

$$\Pr(Y_{1}\le 1/2\le Y_{n}) = 1 - \Pr(Y_{1} \gt 1/2) - \Pr(Y_{n} \lt 1/2).$$

These probabilities are simple to evaluate because the first is the chance that all the $$X_i$$ exceed $$1/2,$$ the second is the chance that all the $$X_i$$ are less than $$1/2,$$ and the $$X_i$$ are independent.

Even if your interest is only on whether you are doing the integration correctly, this approach provides a way to check your answer.


text:
# Parallel Guide¶

This section expands on the Keras-to-Sherpa tutorial in that it goes more into detail about the configuration options. An optimization in SHERPA consists of a trial-script and a runner-script.

## Trial-script¶

The trial-script trains your machine learning model with a given parameter-configuration and sends metrics to SHERPA. To get a trial, use the Client:

import sherpa

client = sherpa.Client()
trial = client.get_trial()

The client will connect to the MongoDB instance created by the Runner-script (more below). From that it obtains a hyperparameter configuration i.e. a trial. The trial contains the parameter configuration for your training:

# Model training
num_iterations = 10
for i in range(num_iterations):
pseudo_objective = trial.parameters['param_a'] / float(i + 1) * trial.parameters['param_b']
client.send_metrics(trial=trial, iteration=i+1,
objective=pseudo_objective)

During training send_metrics is used every iteration to return objective values to SHERPA i.e. send them to the MongoDB instance. When using Keras the client also has a callback Client.keras_send_metrics that can be used directly.

## Runner-script¶

The runner-script defines the optimization and runs SHERPA. Parameters are defined as a list of Parameter-objects. For a list of the available parameters see here.

import sherpa
parameters = [sherpa.Choice(name="param_a",
range=[1, 2, 3]),
sherpa.Continuous(name="param_b",
range=[0, 1])]

Once you decided on the parameters and their ranges you can choose an optimization algorithm.

algorithm = sherpa.algorithms.RandomSearch(max_num_trials=10)

Schedulers allow to run an optimization on one machine or a cluster:

scheduler = sherpa.schedulers.LocalScheduler()

The optimization is run via :ref:sherpa.optimize <optimize-api>:

results = sherpa.optimize(parameters=parameters,
algorithm=algorithm,
lower_is_better=True,
filename=filename,
output_dir=tempdir,
scheduler=scheduler,
max_concurrent=2,
verbose=1)

The code for this example can be run as python ./examples/runner_mode.py from the SHERPA root.


text:
A positive semidefinite programming problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:22:53Z http://mathoverflow.net/feeds/question/85031 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85031/a-positive-semidefinite-programming-problem A positive semidefinite programming problem KOMA 2012-01-06T05:49:52Z 2012-01-07T05:07:20Z
Dear all,

I've got a SDP problem as follows:

$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,

where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell me how to solve this SDP problem? Thank you very much in advance.

[Added] Thanks for Suvrit to point out some issues. I add one more parameter $\lambda$ (to be pre-defined) and assume ${\bf b}={\bf a}$ for the second term in the above problem as:

$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - \lambda{\bf a}^{\top}{\bf H}{\bf a}$.
 http://mathoverflow.net/questions/85031/a-positive-semidefinite-programming-problem/85043#85043 Answer by S. Sra for A positive semidefinite programming problem S. Sra 2012-01-06T11:06:57Z 2012-01-06T11:06:57Z
Your problem has no solution. Here is why.

Let $H$ be $2 \times 2$. Let $a=(2, 0)$ and $b=(1, 0)$. Then, since $a^THb=\mbox{tr}(Hab^T)$, the objective function of your problem can be rewritten as $\mbox{tr}(H-Hab^T) = \mbox{tr}(HC)$, where $$C = I-ab^T = \begin{bmatrix} -1 & 0\\ 0 & 1\end{bmatrix}.$$ Now you can see that if we set \begin{equation*} H=\begin{bmatrix} \alpha & 0\\ 0 & 0 \end{bmatrix}, \end{equation*} then as $\alpha\to\infty$, your objective function goes to $-\infty$. Thus, in general, there is no solution.

Even if you let $a=b$, the same example above shows that there is no solution. You need to restrict $H$ to lie in a compact set.


text:
# Bug Tracker

ID: 16 🔗 Closed (Fixed) glossaries 3.04 2013-01-13 02:22:36 \pagenumbering causes an error in \printglossaries

## Report

Hello Nicola,

since one of the last MiKTeX updates \printglossaries causes an error, when \pagenumbering{Roman} is used before.

Best regards, Stefan

### MWE

% ----- minimal example -----
\documentclass{scrbook}
\usepackage[
acronym,
nolong,nolist,notree,
style=super,
]{glossaries}
\newacronym[
description={dummy text (\emph{dummy:} \protect\acrshort{test})},
]{dummy}{dummy}{just a dummy}
\newacronym{test}{test}{just a test}
\makeglossaries
\begin{document}
\gls{dummy}
\pagenumbering{Roman}   % <-- comment me and it will work
\printglossaries
\end{document}
% ---------------------------

## Evaluation

This is now fixed in version 3.05.

Comment from Uwe Müller
Date: 2013-01-26 03:47:33 CST

Hello Nicola,

I have the same problem as Stefan. I changed the pagenumbering to "litte" roman numbers and it works. There seems to be a problem, when using "big" roman numbers.

I posted my problem to a german LaTeX-forum and got a short analysis from Rainer. It is the second posting in the thread at http://www.mrunix.de/forums/showthread.php?t=74623. Maybe this will help you to solve the problem.

Regards,
Uwe

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text:
Publications
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CONTENTS
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# A Simple, Fast and Low-Cost Method for in situ Monitoring of Topographical Changes and Wear Rate of a Complex Tribo-System Under Mixed Lubrication

## B. Meylan, P. Dogan, D. Sage, K. Wasmer

### Wear—An International Journal on the Science and Technology of Friction, Lubrication and Wear, vol. 364-365, pp. 22-30, October 15, 2016.

This contribution presents a simple, fast, and low-cost method to track in situ and in real time the evolution of the surface topography. The method combines an optical method (collimated light) with image analysis. The method was validated using a complex tribo-system; a reciprocal sliding of a rough cast iron counter-piece under mixed lubrication.

We demonstrated that the optical method is well suited to observe the evolution of the contact areas during sliding. We also proved that the contact area occurs on the highest peaks of the surface roughness. Finally, we estimated the wear rate by combining the information of the contact area and the Abbott-Firestone curve obtained by profilometry. The wear rate was found to be 40 times higher in the early stage of sliding as compared to the steady-state. The running-in of this particular system was found to be approximately 10 h.

In addition to be simple, fast and low-cost, the proposed method has other advantages. It is adaptable for real industrial tests conditions. The fast data analysis allows the wear rate of a tribo-system being determined in real-time. This method characterise the real contact areas of the observed piece, which is difficult, if not impossible, to obtain by profilometric observation alone.

@ARTICLE(http://bigwww.epfl.ch/publications/meylan1601.html,
AUTHOR="Meylan, B. and Dogan, P. and Sage, D. and Wasmer, K.",
TITLE="A Simple, Fast and Low-Cost Method for {\textit{in situ}}
Monitoring of Topographical Changes and Wear Rate of a Complex
Tribo-System Under Mixed Lubrication",
JOURNAL="Wear---An International Journal on the Science and Technology
of Friction, Lubrication and Wear",
YEAR="2016",
volume="364-365",
number="",
pages="22--30",
month="October 15,",
note="")

© 2016 Elsevier. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from Elsevier.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.


text:
E15-13 (Stock Split and Stock Dividend) The common stock of Warner Inc. is currently selling at $110 per share. The directors wish to reduce the share price and increase share volume prior to a new issue. The per share par value is$10; book value is $70 per share. Five million shares are issued and outstanding. Instructions Prepare the necessary journal entries assuming the following. (a) The board votes a 2-for-1 stock split. (b) The board votes a 100% stock dividend. (c) Briefly discuss the accounting and securities market differences between these two methods of increasing the number of shares outstanding. (Kieso 882) ## Answers (2) • (a) No entry—simply a memorandum note indicating the number of shares has increased to 18 million and par value has been reduced from$10 to $5 per share.  (b) Retained Earnings ($10 X 9,000,000)................ 90,000,000 Common Stock Dividend Distributable....... 90,000,000 Common Stock Dividend Distributable............ 90,000,000 Common Stock............................................. 90,000,000

(c)      Stock dividends and splits serve the same function with regard to the securities markets. Both techniques allow the board of directors to increase the quantity of shares and reduce share prices into a desired “trading range.”

For accounting purposes the 20%–25% rule reasonably views large stock dividends as substantive stock splits. In this case, it is necessary to capitalize par value with a stock dividend because the number of shares is increased and the par value remains the same. Earnings are capitalized for purely procedural reasons

• (a) No entry—simply a memorandum note indicating the number of shares has increased to 18 million and par value has been reduced from $10 to$5 per share.

(b)
Retained Earnings (\$10 X 9,000,000)................
90,000,000

Common Stock Dividend Distributable.......

90,000,000

Common Stock Dividend Distributable............
90,000,000

Common Stock.............................................

90,000,000

(c) Stock dividends and splits serve the same function with regard to the securities markets. Both techniques allow the board of directors to increase the quantity of shares and reduce share prices into a desired “trading range.”

For accounting purposes the 20%–25% rule reasonably views large stock dividends as substantive stock splits. In this case, it is necessary to capitalize par value with a stock dividend because the number of shares is increased and the par value remains the same. Earnings are capitalized for purely procedural reasons
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