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## Introduction

Cells are exposed to hydrostatic pressure driven by the difference between inner and outer osmolarity. For example, at the boundaries of tissues in higher eukaryotes, cells often encounter a wide range of osmotic changes due to cytokines, hormones, etc. Similarly, unicellular organisms such as the eukaryotic model organism Saccharomyces cerevisiae or yeast, proliferate under a wide range of osmotic conditions caused, for example, by periods of rain or drought. In the presence of these changing conditions yeast has evolved strategies to maintain cellular integrity, ranging from regulating intracellular osmolarity to constructing elastic scaffolds such as the cytoskeleton or the cell wall. Water flow over the cell membrane follows the osmotic and hydrostatic pressure differences1 and, therefore, impacts cell size, according to the cellular deformability. Therefore, yeast on the one hand, has to adapt its internal osmotic pressure to external conditions2,3,4 to prevent bursting as well as critical shrinking, on the other hand has to regulate its growth rate. The uptake and subsequent metabolization of nutrients provides not only building blocks and energy for the synthesis of new cell material, but also change the internal osmolarity and thereby can drive inward water flux, which in turn can lead to an increase in cell size. In walled cells, such as Baker's yeast or plant cells, the difference between internal and external osmotic pressures are counteracted by turgor pressure arising from elastic expansion of cell wall material. Turgor pressure prevents exaggerated swelling and maintains cell shape. Although reported values of turgor pressure in yeast range from 0.1 to 1.0 MPa,5,6 more recent single-cell measurements suggested a value of 0.2 MPa.7

Several studies have already addressed aspects of osmo-regulation and single cell growth concomitantly, however, the mutual influence of both processes remained poorly understood. In a previous model, thermodynamic descriptions of volume and pressure changes were integrated within the osmotic stress response system, i.e. the high osmolarity glycerol (HOG) signaling pathway, metabolism, and gene expression.3 This integrative model permitted predictions regarding the effect of several gene-knockouts on volume dynamics. Another model integrated further published data with biophysical and mechanical properties of yeast to describe the loss in volume immediately after osmotic stress.4 Both models explain volume regulation following a hyperosmotic shock, but are not designed to describe the small and steady volume variations during normal growth.

Although various volume regulation models have been proposed, a unified understanding of the interplay between cell mechanics, turgor, volume, and metabolism during growth and perturbations, e.g. osmotic shocks, is still missing. Earlier approaches focused solely on animal cells, where cellular integrity is maintained by the cytoskeleton.8,9 However, mammalian cells can also face high osmotic pressure changes and cell integrity of certain species is supported by external structures, such as matrix, mucus or wax, which fulfill similar functions as a cell wall.

Here, we present a single-cell growth model (SCGM), which focuses on the interplay of three thermodynamic quantities: cell volume, osmolarity, and turgor pressure, and which covers growth and budding of single yeast cells as well as the response to external osmotic variations. We further tested the model against single-cell growth data from brightfield microscopy images and used atomic force microscopy (AFM) to gain information on the cell wall elasticity during budding. The model combines different concepts, such as cell wall mechanics in yeast10,11,12,13,14,15 rheology, a subfield of continuum mechanics and broadly used in plant physiology16,17,18,19 and applied to fungi,20,21 thin shell theory,22,23,24 water homeostasis and dynamics,1,25 and osmoregulation (in general or exemplified by HOG).3,26,27 The SCGM is capable of describing both drastic volume variations caused by hyperosmotic or hypoosmotic shocks, as well as relatively small but steady gains in cell size during growth. To demonstrate that the SCGM can be combined with models for cellular signaling and metabolism, we introduced the HOG signaling cascade model27 as an exemplary pathway that plays a major role in yeast osmoregulation.2

## Results

### The SCGM combines formalisms for turgor pressure, osmo-regulation, and cell wall mechanics

Cellular volume varies according to material accumulation and water flux across the cell membrane, which follows the osmotic and hydrostatic pressure gradient. For volume flux and the conversion from osmolarity to osmotic pressure, we considered established formalisms described by Kedem–Katchalsky and Boyle van’t Hoff.1,3,25,27 To this end, we defined total cell size Vt = Vos + Vb as the sum of the osmotic volume Vos, which is sensitive to osmotic or hydrostatic pressures changes, and the solid volume Vb, which is not affected by water dynamics, e.g. volume occupied by macromolecules (see Fig. 1a). In a confined system, such as a cell, the outward-directed water flux over the boundary, the cytoplasmic membrane, must equal the negative change in volume of this system over time:

$$\dot V_{{\mathrm{os}}}= - J_{\mathrm{w}}$$

According to Kedem–Katchalsky, water flux Jw across the membrane is proportional to the causative force, i.e. pressure differences, and is thus defined as

$$J_{\mathrm{w}} = L_{\mathrm{p}}G({\Pi}_{\mathrm{t}} + {\Pi}_{\mathrm{e}} - {\Pi}_{\mathrm{i}}),$$

where Lp denotes the hydraulic conductivity of the membrane per unit area, G is the area of the cell surface, Πt is the turgor pressure and Πe and Πi are the external and internal osmotic pressures. Turgor pressure is typically calculated under a steady-state assumption of negligible water fluxes (Jw = 0) and, hence, equals the difference between external and internal osmotic pressures. To capture the dynamics of volume variation, we went a step further by neither constraining Jw nor the total number of particles in the system, thereby establishing distinct mathematical descriptions for the three main quantities: volume, turgor pressure, and osmolarity. The osmotic pressures Πe and Πi depend on the the internal and external osmolyte concentrations ci and ce according to Boyle van’t Hoff’s equation:

$${\Pi}_{\mathrm{i}} = c_{\mathrm{i}}{RT}\;{\mathrm{and}}\;{\Pi}_{\mathrm{e}} = c_{\mathrm{e}}{RT},$$

where R is the gas constant and T the temperature. While ce was assumed to be unaltered by the processes of a single cell, ci changes with the uptake and dilution of osmolytes, such as nutrients and ions. Particularly, osmolyte uptake was assumed to be proportional to the cell surface G, over which osmolytes must be transported, while consumption of osmolytes was assumed to be proportional to the cell volume V. Further, volume expansion must lead to a dilution of osmolytes, hence to a decrease in ci. Combined, we obtain the following simplified description of osmolyte dynamics within the cell:

$$\dot c_{\mathrm{i}} = \frac{1}{V}(k_{{\mathrm{uptake}}}G - k_{{\mathrm{consumption}}}V - c_{\mathrm{i}}\dot V),$$

with the proportionality constants kuptake and kconsumption. This basic description can be extended, if a more detailed metabolism model is at hand, as we exemplify below for osmotic stress response by including the HOG pathway and glycerol accumulation.

Having derived descriptions for the dynamics of the volume and inner osmolarity, we required further a description for the turgor pressure. Given that the turgor pressure equals hydrostatic pressure acting on the cell wall, the main structural element of the yeast cell, the cell wall and its mechanical properties must be considered in our description. For simplification, we assumed the cell wall to be a thin spherical shell with radius r. The cell wall expands elastically or plastically depending on forces it is exerted to. While an elastically expanded volume relaxes to its initial size when the pressure vanishes, plastic expansion is irreversible. Taking a constant wall thickness into account, the irreversible expansion can also been interpreted as cell growth, where new cell wall material has to be provided by the cell wall synthesis machinery. Assuming a linear constitutive relationship for elasticity, the deformation of the cell volume, i.e. strain εHook, scales linearly with turgor pressure in a purely elastic cell wall.

$$\varepsilon _{{\mathrm{Hook}}} = \frac{{1 - \nu }}{E}\frac{{{\Pi}_{\mathrm{t}}r}}{{2d}}.$$

Here, E is the Young’s modulus, representing the elasticity of the cell wall, ν is the Poisson’s ratio, and d is the thickness of the cell wall. Consequently, water influx will lead to an increase in r and hence directly to larger strain inside the cell wall. Note that cell wall thickness is assumed to be constant and therefore depended neither on the radius r nor on the time t.

In contrast to elastic deformation, plastic deformation was assumed to occur only above the critical turgor pressure Πct and instead of the strain, the strain rate is proportional to the acting pressure with a factor ϕ. Whereby ϕ represents the extensibility of the cell wall. The strain rate reads

$$\dot \varepsilon _{{\mathrm{Bingham}}} = \frac{{\phi {\mkern 1mu} r}}{{2{\mkern 1mu} d}}f_{\mathrm{m}}({\Pi}_{\mathrm{t}},{\Pi}_{{\mathrm{ct}}}),$$
$${\mathrm{where}}\;\;\;f_{\mathrm{m}}({\Pi}_{\mathrm{t}},{\Pi}_{{\mathrm{ct}}}) = \left\{ {\begin{array}{*{20}{l}} {{\Pi}_{\mathrm{t}} - {\Pi}_{{\mathrm{ct}}}} \hfill & {{\mathrm{if}}\;\;\;{\Pi}_{\mathrm{t}} \ge {\Pi}_{{\mathrm{ct}}}} \hfill \\ 0 \hfill & {{\mathrm{else}}} \hfill \end{array}} \right..$$

To reflect elastic and plastic behavior of the cell wall, we used a mechanical model of the cell wall, in which a Hookean and a Bingham element, representing the elastic and plastic response, are coupled in series (Fig. 1a). Then the total strain is the sum of the strains of each element $${\mathrm{d}}\varepsilon = {\mathrm{d}}r_{{\mathrm{Hook}}}{\mathrm{/}}r + {\mathrm{d}}r_{{\mathrm{Bingham}}}{\mathrm{/}}r$$. Taking the time derivative $$\dot \varepsilon$$ of the total strain enabled us to combine descriptions for both mechanical elements in a single ODE for the change in turgor pressure over time $$\dot {\Pi}_{\mathrm{t}}$$ (for details see Supplementary Note 1):

$$\dot {\Pi}_{\mathrm{t}} = \frac{{2{\mkern 1mu} E{\mkern 1mu} d}}{{1 - \nu }}\frac{{\dot r}}{{r^2}} - {\Pi}_{\mathrm{t}}\frac{{\dot r}}{r} - \frac{{E\phi }}{{(1 - \nu )}}f_{\mathrm{m}}({\Pi}_{\mathrm{t}},{\Pi}_{{\mathrm{ct}}}).$$

Models that include two of the three terms of this equation have already been applied in other turgor-related models.16,18,19,28 A comparison with the previously proposed description of turgor in plant physiology19 revealed, that the additional term resulted from the geometric description of the cell as a thin spherical shell, in particular when calculating the time derivative of the elastic element.

Cell expansion is therefore mainly influenced by two processes: first, the control of internal osmolarity ci by uptake, dilution, and consumption of osmolytes, which together with turgor pressure Πt drive water influx and outflux Jw, and hence determine the water volume in the cell (Vos), and second, the elasto-plastic deformation of the cell wall due to turgor pressure. The water influx Jw < 0 expands the cell wall elastically and thereby increases turgor pressure. When turgor pressure exceeds the critical value Πct, the cell wall starts to yield, e.g. the cell wall expands irreversibly and the cell grows. Yielding leads to relaxation of stress in the cell wall and in turn to a decrease in turgor pressure, facilitating further influx of water. Hence, growth typically proceeds close to, but not at steady state (Jw = 0), where water and osmolyte influxes are balanced by turgor dynamics.

Summarizing, the basic SCGM comprises three coupled ODE’s for volume, turgor pressure, and internal osmolarity. Key parameters and initial values for simulation or analytic solution are listed in Table 1. The temperature was set to T = 303 K, i.e. the optimal growth temperature for yeast and our experimental standard condition. The external osmolyte concentration ce was set to 240 mM, the mean value for our standard growth medium.

Considering a cell growing without any osmotic perturbations, the SCGM can be reduced to a simpler description of cell size development. Simulations revealed that without osmotic stress, internal osmolarity ci and turgor pressure Πt approach steady states. In this case the SCGM can be solved analytically (Fig. 1b, Supplementary Fig. 1). We obtained an implicit differential equation F comprising the time derivative of the radius ṙ(t), the time-dependent function of cellular radius r(t), steady states and time:

$$F(\dot r,c_{\mathrm{i}}^{{\mathrm{SS}}}(r(t)),{\Pi}_{\mathrm{t}}^{{\mathrm{SS}}} = {\Pi}_{{\mathrm{ct}}},t) = 0,$$

where $$c_{\mathrm{i}}^{{\mathrm{SS}}}$$ is a quasi-steady state for internal osmolarity (quasi-, because still depends on r(t)) and $${\Pi}_{\mathrm{t}}^{{\mathrm{SS}}}$$ the steady state for turgor pressure, which equals critical turgor pressure Πct during growth. Solving F (Supplementary Note 3), we first, identified a limes of cellular radius for long times scales:

$$r_{{\mathrm{final}}} = \mathop {{\lim }}\limits_{t \to \infty } r(t) = 3\frac{{k_{{\mathrm{uptake}}}}}{{k_{{\mathrm{consumption}}}}}.$$

and second, obtained an analytical solution for r(t). Since the resulting expression for r(t) is rather complex, we approximated $$r(t) \approx \tilde r(t)$$ (for derivation see Supplementary Note 4), yielding a function for the radius of a growing cell:

$$\tilde r(t) = r_{{\mathrm{final}}} - \left( {r_{{\mathrm{final}}} - r_0} \right){\mathrm {e}}^{\frac{{ - k_{{\mathrm{uptake}}}RT}}{{r_{{\mathrm{final}}}\:({\Pi}_{{\mathrm{ct}}} + {\Pi}_{\mathrm{e}})}}(t - t_0)},$$

where rfinal is the final cellular radius for long time scales t, r0 is the initial radius at time t0. The solution points out that external osmotic pressure, critical turgor pressure and internal osmotic pressure, as well as kuptake and kconsumption, dictate the trajectory of cell growth. For our solution it was crucial to include the quasi-steady state $$c_{\mathrm{i}}^{{\mathrm{SS}}}(r(t))$$. In this way, the influences of kuptake and kconsumption on the quasi-steady state of internal osmolarity could be propagated to the radius description $$\tilde r(t)$$. As shown in Fig. 2c, our model suggests that final cell size depends exclusively on the ratio ku/c = kuptake/kconsumption, but not on cell wall-related quantities. Scaling both rate constants equally while keeping ratio ku/c constant, scales growth rate, but keeps final cell size constant. The growth dynamics $$\tilde r(t)$$ depends additionally on critical turgor pressure and external osmolarity. For hyperosmotic shock response, we derived another analytical solution for turgor dynamics, which is related to the Merritt–Weinhaus equation,29 i.e. the pressure-to-size relation of an ideal elastic thin shell (Supplementary Fig. 2 and Supplementary Note 2).

### Coupled SCGM permits the description of combined growth of mother and bud

Above, we provide a description of growth dynamics for an individual cell, which is only valid for yeast cells in G1 phase (single cell without a bud). In other cell cycle phases, cells consist of two compartments (mother and bud), connected by a neck and with different growth rates.30 To model these growth phases, we coupled two SCGMs, one each for mother and bud (Fig. 2a), in which both start with the same small initial volume, but with a time delay tbudstart for the bud, leaving time for the mother to grow. tbudstart represents the time between two successive bud formations and, hence, a characteristic length of the cell cycle. Both model instances were coupled by allowing water and osmolyte exchange according to the pressure and concentration differences. The coefficients for both fluxes were arbitrarily chosen such that water and osmolyte gradients vanish in the considered time scale. Apart from the mechanical cell wall properties and the initial geometry, both instances were similar, particularly in their osmolyte uptake and consumption rates, kuptake and kconsumption. However, it is very unlikely that the mechanical cell wall properties are identical in both compartments, as changes in cell wall structure during budding were reported.31 In particular, chitin incorporation into the lateral cell wall is delayed in buds until after septation. As a potentially discriminating cell wall property, we considered either the Young’s modulus E or the extensibility ϕ. Systematic analysis of the impact of varying E on bud growth showed that only a strongly decreased Ebud compared to Emother led to bud growth (Supplementary Fig. 3), though at the expense of the growth of the mother. In contrast, when varying ϕ, both compartments continued to grow (Fig. 2b), as long as ϕbud was at least 101.5 times higher than ϕmother. Note that we focused here on biophysical principles, instead of on the complex biochemical processes governing cell division, such as cell polarization and subsequent bud emergence.

### Measurements of local cell wall elasticity of mother and bud reveal a distinct difference between two asymmetric growth processes in yeast, budding and shmooing

As discussed above, a significantly lower elastic modulus of the bud cell wall could drive bud expansion in the model. Previously, we reported that such localized softening of cell wall material occurs during sexual conjugation in S. cerevisiae.7 To test whether both processes, budding and sexual conjugation, follow similar underlying principles, we applied multi-parametric AFM on living yeast cells32 as depicted in Fig. 3a. From the force-response curves of nano-indentation measurements (Fig. 3b), we obtained the local Young’s modulus E. To compare mother and bud, we selected regions of equal size at each compartment (Fig. 3c, d). We avoided strongly tilted regions, compared to the scanning plane, and regions of previous budding events, so-called bud scars, which contain a higher amount of chitin and are reported to be stiffer.33 Intriguingly, bud cell walls appeared to be stiffer, not softer, than their mothers’ cell walls (Fig. 3). From a linear fit, we estimated a 1.3 0.1-fold increase in the Young’s modulus from mother to bud (Supplementary Fig. 6). Although the apparent stiffness at the bud cell wall could be biased by different surface curvatures, a significant cell wall softening at the bud, as measured for sexual conjugation, can be rejected. This reveals a distinct difference between two of the asymmetric growth processes in yeast, budding and sexual conjugation. Consequently, we focused on the extensibility as distinguishing feature between mother and bud cell wall.

### Verification of the model against volume data from light microscopy

To test whether the coupled SCGM correctly describes the growth of single yeast cells and to estimate otherwise inaccessible parameters, we measured the growth trajectories of single yeast cells and used these to constrain model parameters. Cells were grown as monolayer in a microfluidic device, allowing observation of individual cells for long time periods (>15 h). We tracked single cells in microscopic bright field images using the software CellStar34 and determined the first bud emergence manually for each cell (Fig. 4a). A bud neck marker (Cdc10-mKate2) was used to identify pairs of mothers and corresponding buds and to reconstruct the lineage. Knowing the lineage enabled us to collect 21 coupled volume trajectories for mother and bud. Since the time between first appearance of a cell and its first budding event varied considerably, we did not fit the model to mean growth data but to single volume trajectories instead. Figure 4b and Supplementary Fig. 7 show that the coupled SCGM is able to describe the growth pattern of single cells growing at different rates. For the growth of mother and bud, six independent parameters (mean ± s.d.) have been determined (Fig. 4c). In addition to two individual parameters (r0 and tbudstart), we considered two global parameters (kuptake and ku/c) and two compartment-dependent parameters (ϕmother and q = ϕbud/ϕmother). r0 was the fitted radius belonging to the initial osmotic volume of the later mother. With $$\bar t_{{\mathrm{budstart}}} = 117 \pm 33\,\min$$ the estimated mean cell cycle length differed remarkably from the time when a new bud was clearly recognizable for the first time (first green data point in Fig. 4b). When analyzing estimated rates constants for osmolyte uptake and consumption, we found that the growth rate determining kuptake showed some variation between single cells (1.59 × 10−16 ± 0.72 × 10−16 mmol μm−2 s−1). In contrast ku/c, which limits the maximum volume, showed very little variation (0.82 ± 0.07 μm). Focusing on cell wall extensibility, we estimated a value of 0.57 ± 0.34 kPa−1 s−1 for the mother ϕmother and found that ϕbud needs to be at least 100 times higher. We could not determine a precise value for the extensibility ratio ϕbud/ϕmother. However, profile likelihood and sensitivity analysis (Supplementary Fig. 8) revealed that variation of ϕbud/ϕmother has no significant impact on other parameters.

For the Δ cdc42 mutant a probability for the emergence of two buds was reported. Our model suggests that this probability (occurrence of multiple buds) is not only influenced biochemically (e.g. by Cdc42p) but also by the the passive dynamics in cell size described above. Therefore, the constellation of two small volumes connected to a main volume establishes an additional layer of filtering out stunted buds.

### Analysis of volume trajectories from Garmendia-Torres et al. 35

So far the analysis of growth parameters was based on 21 volume–trajectory pairs (dataset 1), which provided valuable information on the dimension of each parameter. However, for the extensibility ratio ϕbud/ϕmother only a lower boundary could be determined and the relative small sample size did not allow for decisive statements on the parameter distribution. To increase the sample size and further challenge the cSCGM we searched for single-cell data with time-resolved volume information for mother and bud and found a recent study by Garmendia-Torres et al.35

They reported a new microscopy-based method to investigate yeast cell-cycle and cell-size progression in parallel, by monitoring volume and histone levels of individual yeast cells. Using an automatized experimental setup they followed up to 15,000 cell cycles for each of the investigated 22 cell-cycle-related mutants. This innovative approach of tracking the fluorescence of fast maturating HTB2-sfGFP fusion proteins over time and assigning its intensity to distinct cell cycle phases, allowed them to relate the volumes of mother and bud at certain cell cycle stages to the time spent in each phase. Although this analytic approach revealed intriguing relations between cell cycle and cell size, it simplifies the dynamics of the cell size progression in the data. In particular, they assumed linear growth during the budded and unbudded phase. To shed light on this cell-size dynamics and to further challenge the cSCGM, the model was fitted to the experimental data provided by Gilles Charvin (dataset 2). This allowed the comparison between completely independent data sets, from different laboratories, and the analysis of data with a drastically increased sample size.

### Data and data selection

The advantage of dataset 2, besides its dimension, is the resemblance of the used experimental and analytic approach used for dataset 1. In particular, yeast strain background and culture medium, BY4741 and synthetic medium supplemented by amino acids and glucose, were similar. Furthermore, the proliferation of single cells, confined to a plane using a microfluidic device, was followed at the same sampling frequency (every 3 min) using bright-field and fluorescence microscopy and volumes were calculated from cell contours, assuming ellipsoidal geometry. In contrast to dataset 1, dataset 2 relied on a super-folding GFP fused to one of the histone 2B loci (HTB2-sfGFP), instead of a bud-neck marker, to monitor the cell cycle stage and to discriminate between buds and new born daughter cells. Additionally, a self-developed MATLAB software Autotrack for automatized cell segmentation and linage tracking, enabled to track and analyze thousands of cells in dataset 2.

The cSCGM was fitted to the provided volume trajectories pairs of WT yeast cells, in the same manner as described above. For better comparison with dataset 1, dataset 2 was limited to the first cell cycle of new born daughter cells (N = 6079), i.e. cells with a replicative age of 0. Cells, defined as outliers in the data set, due to incorrect segmentation, tracking, or bud assignment to mother cells and due to insufficient fitting of histone level curves, were also neglected for further analysis. In contrast to dataset 1, tbudstart was no fitting parameter but provided by the dataset 2. For 97% of the 6079 cells, parameter sets were found for which χ2 was sufficiently small (Supplementary Fig. 10). Exemplary volume trajectories, representing data and model fits, are shown in Fig. 5c and Supplementary Fig. 9. Further analysis was restricted to a reduced data set (N = 4680), discarding all failed optimization runs, which yield ku/c < 1 and χ2 < 50. The resulting distribution of the fitting parameters and χ2 is shown in Fig. 5.

### Parameter distributions

Except for ϕmother the free fit parameter showed a unimodal, though partially skewed, distribution and their medians as well as their interquartile range (IQR) are listed in Table 2, together with parameter estimates from dataset 1. Although tbudstart was not fitted for dataset 2, as it was included in the data, the parameter is listed in Table 2 for comparison.

The extensibility ϕ, which reflects the capability of the cell for plastic cell wall expansion and adjustment of the turgor pressure in the SCGM, varied drastically between dataset 1 and dataset 2. While ϕmother values obtained from dataset 2 followed a clear bimodal-distribution, values obtained from dataset 1 showed no indication of a multimodal-distribution. Both maxima, max(ϕmother|1) = 9.6 × 10−8 Pa−1 s−1 and max(ϕmother|2) = 7.8 × 10−7 Pa s−1 were at least two magnitudes smaller than estimates from dataset 1 and thereby closer to reported extensibility values for plant cells (~1 × 10−10 Pa−1 s−1).19,36 For each maximum of ϕmother, parameter sets were selected, for which ϕmother was close to this maximum, and examined for differences (Supplementary Fig. 13). Nevertheless no difference in parameter distribution could be observed. Furthermore, screening of volume trajectories for which the estimated ϕmother was either close to max(ϕmother|1) or maxmother|2) (Supplementary Figs. 14 and 15) revealed no common pattern for the two groups. Whether the bimodal distribution of ϕmother results from numerical issues of the optimization algorithm or reflects two distinct population could not yet be finally clarified.

Interestingly, the ratio ϕbud/ϕmother (median (IQR)) was comparable between dataset 2 (220(150–300)) and dataset 1 (230(130–8.1 × 105), underscoring that ϕbud needs to be at least 100 times higher than ϕmother to facilitate bud expansion, regardless the magnitude of ϕ. Osmolyte uptake rates kuptake and ratios kc/u estimated from dataset 2 were slightly higher than estimates from dataset 1, with kuptake = 3.0(1.9–4.2)10−16 mmol μm−1 s−1 and kc/u = 1.4(1.3–1.5) μm−1, indicating a higher single-cell growth rate and maximal volume for dataset 2.

The initial volume of the new born daughter, i.e. a mother with replicative age 0, is represented by its radius r0. With r0,1 = 1.2(1.0–1.3) μm this radius was slightly smaller for dataset 1 than for dataset 2, where r0,2 = 1.4(1.3–1.5) μm. There are several possible reasons for the discrepancy between the r0,1, r0,2: differences in the genome of investigated strains, experimental setup or the initial growth phase of the population. In contrast to the initial volume, the time until bud emergence tbudstart was increased for dataset 1 compared to dataset 2, with tbudstart,1 = 93(93–143) min and tbudstart,2 = 56(41–74) min. Between all fit parameters and the volume maximas of mother and bud (maxVmother maxVbud), the Pearson’s correlation coefficient was calculated and displayed in a correlogram (Fig. 5b). The correlation coefficients can vary by definition between −1 and 1, indicating a negative or positive correlation, whereby values close to zero represent no correlation. Except for a small correlation (0.19) for kc/u and kuptake, the fit parameter (upper left quadrant) were uncorrelated. In contrast, the experimental volume measures and r0 (lower right quadrant) were all positively correlated. The only correlation between volume measures and fitting parameters was found for kc/u. Thereby kc/u correlated negatively to all volume measures, though the correlation was much stronger for maxVmother (−0.55) than for maxVbud (−0.27) or r0 (−0.12). This can be explained by the cSCGM, since kc/u defines the maximal volume of the cell compartment, which is not reached for the bud until after cell separation.

### Comparing active and passive osmotic shock responses

Adaptation to changing osmotic conditions is vital for cells. In S. cerevisiae, adaptation to high osmolarity is mainly achieved via increased production and retention of the small osmolyte glycerol, coordinated by the HOG pathway, through transcriptional and post-translational regulation of metabolism. We extended the SCGM by a reported description of this HOG cascade (Supplementary Note 6), comprising signaling dynamics and the activation of osmolyte production via Hog1.27 Kinetics in the HOG cascade model were scaled according to the recently measured lower turgor pressure.7 Additionally, we chose r0 to be at least 1.2 μm to avoid artificially low Hog1 concentrations. Utilizing the augmented model we compared the response of the SCGM to hyperosmotic shock with and without active shock response mechanisms mediated by HOG signaling (blue and orange lines in Fig. 6).

When unperturbed (Fig. 6, dashed lines), both SCGM and SCGM with HOG establish steady states for internal osmolarity and turgor pressure while sharing comparable growth rates. However, if we simulate a hyperosmotic shock as depicted in Fig. 6a, models show an identically fast decrease in volume and turgor and a drastic increase in ci (Supplementary Fig. 16), but the following trajectories diverge. The active response of the HOG cascade leads to production and accumulation of glycerol, which contributes to fast reestablishment of the former turgor pressure and cell size (blue line). Without the HOG cascade, cells still manage to adapt volume and turgor, but much slower. A systematic analysis of the effects of strength and timing of osmotic shock is provided in Supplementary Figs. 17 and 18.

The SCGM with HOG can reproduce previous measurements of growing yeast cells that undergo a hyperosmotic shock. In, ref. 26 cellular volume and Hog1 localization over time in response to hyperosmotic shock was measured for individual yeast cells, resulting in a phase plot that is reproduced by our model simulations (Fig. 6e). In particular, the characteristics throughout the four different phases that are reflected correctly by our approach, validate that not only the HOG response model or our volume model are functional by themselves but, more importantly, that interplay of both models is captured properly. Further, we tested whether the model also copes with hypoosmotic shock (Fig. 6a, c). Again model integrity is restored following a first hyperosmotic shock and, thus, regains osmolarity and volume after a second, now hypoosmotic, shock. Obviously, hypoosmotic shock results in a short overshoot of cell size (Fig. 6c), as previously reported in ref. 37 Hyperosmotic shock exclusively leads to elastic deformation, while hypoosmotic shock leads to negligibly small plastic deformation. Both types of osmotic shock induce elastic behavior of the cell wall, hence the modeled cell would not grow effectively through step-like osmotic changes of the environment.

The reference volume (Fig. 6f and Supplementary Note 1 for definition) quantifies growth as it integrates plastic deformation of the cell wall under normal conditions reflecting a elasto-viscoplastic behavior (EP) (Fig. 6f, white area). In contrast, a hyperosmotic shock induces immediate cell shrinkage and stops growth, as indicated by the reference volume. The reference volume remains constant, since the elasto-viscoplastic regime is left and the model behaves ideal-elastically (E) (Fig. 6f, gray area).

## Discussion

Knowledge of single-cell growth dynamics is crucial for the understanding of comprehensive biological systems, particularly for model systems such as S. cerevisiae. Here, we introduced and justified a volume model of yeast by integrating different theoretical concepts of cell mechanics, integrity, and osmolyte dynamics in combination with experimental data on single cell growth and cell wall elasticity. The model applies to several scenarios, where cell size variation plays a role: from growth of an individual bud connected to the mother cell, the case of a two-bud growing mutant of yeast, and the direct and long-term volume response after hyperosmotic shock.

In previous studies, two modes of single-cell growth were considered for yeast, linear and exponential.38,39,40,41 Based on our model we proposed a more complex picture of the specific growth pattern of a single yeast cell (Fig. 1b). We derived an exact analytic solution for the cellular radius over time r(t) for a single undisturbed-growing cell. Prospectively, this solution may support future work as it relates relevant parameters, which originate from formerly distinct concepts, such as cell wall mechanics, water homeostasis, and osmoregulation, in the form of a single expression describing the growth of an individual cell overall. This expression allows us to deduce properties of the described system in a closed form. Specifically, we exemplified such a deduction by calculating the limes of the radius for long time spans rt→∞. The approach revealed that the maximal cell size is determined by the ratio ku/c of osmolyte or nutrient uptake and consumption, but is independent of mechanical cell wall properties, such as Young’s modulus or extensibility. We assumed yeast cells to be spherical, though their true shape might be better described as rotational ellipsoid. Introducing such ellipsoidal geometry would drastically complicate the model, as the assumption of a uniform stress inside the cell wall does not hold anymore and local stresses have to be considered. We argue even for rotational-ellipsoidal cells with rather small differences between major and minor axes, the approximation of a sphere is sufficient to capture the geometric impact on cell growth. Besides the cell wall mechanic, an ellipsoidal geometry would impact the osmolyte uptake, as it scales with the cell surface. Although the surface of a rotational ellipsoid is always bigger than that of a sphere, both surface-volume ratios follow the same trend (Supplementary Fig. 4). Therefore, the fitted values for kuptake might be overestimated, but the growth dynamic is preserved.

The SCGM works fully without a connection to the cell cycle machinery comprising the action of cyclins, CDKs, and CK1. Nevertheless, the coupled SCGM is not independent of cell cycle, since the characteristic length of first cell cycle is integrated via tbudstart. Parameterizing the SCGM with measured volume trajectories we found that this characteristic cell cycle length varied significantly between individual cells.

Yeast cells proliferate by budding, a process in which a larger mother and a smaller bud grow simultaneously but with different growth rates. To describe this coordinated growth, two instances of the SCGM with different initial volumes were coupled by allowing water and osmolyte fluxes between the instances. For simplicity, we neglected the specific bud localization at the mother, which would include the complex polarization process.42,43 This coupled SCGM can be interpreted as an augmented model compared to a previous description of interconnected soap bubbles or balloons.29,44 However, those models lack the exchange of matter with their environment which is crucial for biological applications. In the coupled SCGM we allowed fluxes in-between the compartments and between compartments and medium such that our model operates near equilibrium instead of being at equilibrium. Plastic and elastic cell wall expansion depend on the lateral stresses, which scales with radius and acting pressure. To allow expansion of the smaller of the two volumes, the cell wall of that volume must be either more elastic (E) or more extendable (ϕ).

Utilizing AFM, we could reject the hypothesis that elasticity is the distinguishing cell wall property between mother and bud. The measured local Young’s modulus at the bud was not lower but similar or even slightly higher compared to the mother. Therefore, we inferred, from the SCGM, that ϕ must be significantly higher at the bud to facilitate bud expansion. Fitting the coupled SCGM to the growth data suggests that ϕmother is three orders of magnitude higher than reported values for plants45,46 and ϕbud needs to be at least two magnitudes higher than at mother cell wall. Furthermore, we have shown that significantly increased ϕbud compared to ϕmother can explain the observed higher growth rates of buds.30 The higher extensibility of the bud cell wall might be caused by incorporation of new more expandable material, which maturates over time or by subsequent alterations of the already formed cell wall. Plant enzymes are reported to alter mechanical cell wall properties,47,48 thereby influencing growth and cell shape.49 Further reports also suggest that cell wall plasticity in fungi is controlled by hydrolytic enzymes, like chitinases or glucanases.5,50,51,52 The higher extensibility of the bud cell wall could also reflect the orientation of the cytoskeleton towards the emerging bud,53 which directs new cell wall material predominantly to that bud. Fitting the coupled model to single-cell volume trajectories from two completely independent data sets enabled us to estimate crucial growth parameters, such as the global osmolyte uptake rate kuptake and the ratio ku/c. Both parameters appeared to be very sensitive, as kuptake had the strongest impact on individual growth rate and ku/c on the maximum cell volume. From comparison of both data sets, two main observations can be stated: First, cell expansion of S. cerevisiae, under non-limiting conditions, requires a total osmolyte uptake rate of 1.0–4.0 mmol μm−2 s−1 and a consumption rate which is 1.2–1.5 μm−1 times higher. Second, the extensibility of the bud cell wall has to be at least 100 higher than the mother’s, to facilitate bud expansion.

Intriguingly, the estimated osmolyte uptake rates kuptake were only two to five times higher than the average glucose uptake rate per surface area under aerobic conditions reported by Jouhten et al. 54 (see Supplementary Note 5). Under these conditions glucose would account for 30–60% of the osmolytes taken up by the cell, revealing a new angle on the impact of glucose onto cellular growth: So far the impact of glucose onto cell growth based exclusively on the fact that glucose is a main nutrient source, providing chemical energy and building blocks for macromolecules, and hence can be regarded as fuel of the cell. While this study indicates, that additionally to those aspects, the massive glucose import could drive water influx and hence contribute significantly to cell expansion.

In general, the cSCGM described very well the volume expansion of different single-cell trajectories, which have been measured in microfluidic growth experiments. Using the cSCGM, we could also recapitulate earlier experimental data presented by Soifer et al. 39 and resulting relations between division time and ratio of volumes at division and birth (Supplementary Fig. 5), despite different interpretation for the growth dynamics of single cells.

By integrating the osmotic stress response, specifically the HOG-signaling pathway and metabolic adaptation from ref. 27, in the SCGM, we could compare the passive volume adaptation with the active response arising from the HOG pathway. This model was able to capture the different time scales ranging from steady growth to the drastic shrinkage of cell size followed by an osmotic shock. To our knowledge, this is the first time that the joint-dynamics of Hog1 enrichment versus volume following a hyperosmotic shock26 could be reproduced and explained by a mathematical model. Additionally, our model can describe the response to hypoosmotic shock similarly to previous reports.37 The simulations illustrate the functionality of the coupled model over a wide range of cell sizes. However, validity of the combined model is limited in cases of tiny initial volume (e.g. 1 fL), since the Zi-model assumes fixed numbers of Hog1 molecules. During growth, increasing volume results in dilution of all species within the HOG model and, hence, would lead to non-physiologically small concentrations for tiny initial volumes. To overcome such limiting effects, production rates for all species of the HOG model need to be introduced and parameterized to counteract the dilution process. Instead, we chose a simplistic implementation of coupling both models, in order to preserve their original properties.

In this study, we focused on single-cell growth of newly budded yeast cells and their volume dynamics. Extending the cSCGM to cover several cell cycles with several consecutive budding events and test it against experimental data might help to understand the interplay of morphogenesis and aging of S. cerevisiae. A next step would be the combination of the SCGM with models focusing on cell cycle regulation or polarization into larger models to investigate the interplay between regulatory networks and growth. Mathematical models for growth dynamics at the population level are often based on greatly simplified single cell volume models.55,56 The presented SCGM might help to improve such models and increase our understanding of population growth dynamics.

## Methods

### Yeast strains

For the microfluidic growth analysis we used a S. cerevisiae strain based on BY4742 (MATαhis3Δ1 leu2Δ0 lys2Δ0 ura3Δ0),57 in which the bud neck marker Cdc10 has been genomically labeled with mKate2. The sequence of mKate2 was cloned into the plasmid pUG72 (Euroscarf) and Cdc10 was labeled using PCR-based homologous recombination. The necessary Ura3 marker cassette has been removed using the Cre-loxP recombination system.58 In the nano-indentation experiments we used the strain BY4741.57

### Yeast cell culture

For the microfluidic growth experiments, cells were grown at 30 °C overnight in synthetic medium (SD; 0.17% yeast nitrogen base without amino acids, 0.5% ammonium sulfate, 2% glucose, 55 mg/L adenine, 55 mg/L l-tyrosine, 55 mg/L uracil, 20 mg/L l-arginine, 10 mg/L l-histidine, 60 mg/L l-isoleucine, 60 mg/L l-leucine, 40 mg/L l-lysine, 10 mg/L l-methionine, 60 mg/L phenylalanine, 50 mg/L l-threonine, and 40 mg/L l-tryptophan). Medium osmolarity was measured with an osmometer (gonotec, Berlin, Germany). Before the experiment, 500 μL of overnight culture were diluted in 5 mL SD medium and grown for 2.5 h at 30 °C.

### Microfluidic growth experiments

We used the CellASIC ONIX microfluidic platform (Merck Millipore, Darmstadt, Germany) with the haploid yeast plates (Y04C) for growth analysis. Plates were primed with SD medium and cells were loaded as described in the ONIX yeast protocol. Flow control pressure was set to 2 psi. Cells were observed using a Visitron Visiscope inverted spinning disc laser confocal microscope (Visitron, Puchheim, Germany). The temperature of the microfluidic plate was controlled to 30 °C using a temperature control chamber (OL IX73/83 cellVivo, PeCon GmbH, Erbach, Germany). We used a 150× oil immersion objective (Olympus UPlanSApo 150X/1.47, Oil, TIRM) and the Photometrics Evolve 512 EMCCD camera (Photometrics, Tuscon, USA). Fluorescent illumination was provided by a 561 nm diode laser and a multibad dichroic filter (405/488/559/635 nm) together with a 600/50 nm emission filter for detection. A brightfield image was taken every 3 min, while a fluorescent image was only taken every 12 min, to minimize fluorescent exposure of cells. Each image was acquired as a z-stack (six z-positions, 0.5 μm distance).

### Image analysis

For each z-stack, the sharpest plane was determined using the ImageJ plugin “Find focused slices” by Quingzong Tseng. Brigthfield images were segmented and cells were tracked with the CellProfiler59 plugin Cellstar.34 Tracking was manually checked, and mis-tracked cells were removed from the analysis. For each cell, bud appearance time was manually defined as the time when a bud was first visible in the brightfield image. The lineage was manually determined using the fluorescent images of the bud neck marker Cdc10-mKate2. Cell area was converted to volume by assuming a spherical cell shape.

### Cell wall nano-indentation

In brief, log-phase yeast cells were mechanically trapped in porous polycarbonate membrane, with pore size of 5 μm.60 The cell-containing membrane was subsequently immobilized on the bottom of a liquid chamber and probed with an AFM (Nanowizard III, JPK, Germany). All AFM measurements were conducted in liquid solution of synthetic media using MLCT-E cantilever (Bruker). Prior to all experiments, the spring constant was determined via thermal noise method. Multiparametric imaging of cells was done using the provided QI-Mode, whereby maximal applied force was set to 1 nN, corresponding to an indentation depth of ~35 nm. For further analysis we used JPK data-processing software and the Python packages numpy, scipy61 or pandas.62 For a more detailed description see ref. 7


text:
# $E$ is a nonempty subset of an ordered set. If $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound, then $\alpha \leq \beta$.

I think I may be citing the law of transitivity incorrectly in this proof.

Theorem. Let $E$ be a nonempty subset of an ordered set. Suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.

Proof. Since $\alpha$ is a lower bound of $E$, and $E$ is nonempty, we have that $\forall x \in E, x \geq \alpha$. Similarly, since $\beta$ is an upper bound, we have $\forall x \in E, x \leq \beta$. Stringing these together yields $\forall x \in E, \alpha \leq x \leq \beta$, which by the transitivity property of ordered sets yields $\alpha \leq \beta$.

Here are my questions on this:

(a) Is it a fair assumption that a subset of an ordered set is also ordered? I wasn't sure whether I ought to 'prove' this first, though it seems almost too trivial to prove.

(b) I use transitivity a bit differently from Rudin's definition, based on past formulations I've seen. He seems to define this as $x < y \wedge y < z \implies x < z$, i.e., with strict inequalities. Should I break this into cases, wherein I consider all possible cases of strict inequalities as well as equalities between $\alpha$, $x$, and $\beta$? Or is this a natural extension of this property?

Thanks.

• It seems good. Regarding your second question, it is a natural extension. – Dog_69 Aug 18 '18 at 17:15

For your first question, your ordering actually relies on the ordering on the larger set, since the upper and lower bounds need not be members of the subset $E$ itself! For instance, if the original set is $\mathbb{R}$ and the subset is $(0,1)$, your lower bound could be $0$. So really you rely on the fact that the ordering is defined on the original set.

For your second question, to be completely rigorous, the transitivity of $\leq$ follows from transitivity of $<$ and $=$. Personally, I think it is fine to simply leave it as you have, but perhaps your professor wants you to add in additional detail if this is a first course in proofs or something.

You're proof is almost correct.

• The fact that E is nonempty is useless to state that ∀x∈E, x≥α (it would also be the case if E were empty!). Same for ∀x∈E, x≤β. These two relations are the definition of lower and upper bounds, respectively.

• However, you need the fact that E is nonempty to prove that α≤β, because ∀x∈E,α≤x≤β would be also true if E were empty, but you couldn't argue that α≤β (because in this case α≤x≤β would never happen). A correct statement would be "E is nonempty so ∃x∈E. So α≤x and x≤β so α≤β."

(a) Yes, you can prove it easily.

(b) "<" and ">" are not orders because they do not check the reflexivity property of an order

• They can be considered as strict orders. In fact, some books define orders as $<$ instead of $\leq$. – Dog_69 Aug 18 '18 at 17:28
• Would you mind explaining why they don't satisfy antisymmetry? It seems that $a < b$, for example, cannot coexist with $b < a$. Surely $<$ wouldn't be reflexive, though, whereas $\leq$ would be. – Matt.P Aug 18 '18 at 17:33
• @Matt.P Oh, very good point. I think he/s wanted to say reflexivity. – Dog_69 Aug 18 '18 at 17:47
• Yes, my mistake I wanted to say reflexivity. I fixed it, thx – Edouard Berthe Aug 20 '18 at 5:48

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text:
## 7.15 Topoi

Here is a definition of a topos which is suitable for our purposes. Namely, a topos is the category of sheaves on a site. In order to specify a topos you just specify the site. The real difference between a topos and a site lies in the definition of morphisms. Namely, it turns out that there are lots of morphisms of topoi which do not come from morphisms of the underlying sites.

Definition 7.15.1 (Topoi). A topos is the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of sheaves on a site $\mathcal{C}$.

1. Let $\mathcal{C}$, $\mathcal{D}$ be sites. A morphism of topoi $f$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is given by a pair of functors $f_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $f^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ such that

1. we have

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, f_*\mathcal{F})$

bifunctorially, and

2. the functor $f^{-1}$ commutes with finite limits, i.e., is left exact.

2. Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be sites. Given morphisms of topoi $f :\mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $g :\mathop{\mathit{Sh}}\nolimits (\mathcal{E}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the composition $f\circ g$ is the morphism of topoi defined by the functors $(f \circ g)_* = f_* \circ g_*$ and $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.

Suppose that $\alpha : \mathcal{S}_1 \to \mathcal{S}_2$ is an equivalence of (possibly “big”) categories. If $\mathcal{S}_1$, $\mathcal{S}_2$ are topoi, then setting $f_* = \alpha$ and $f^{-1}$ equal to a quasi-inverse of $\alpha$ gives a morphism $f : \mathcal{S}_1 \to \mathcal{S}_2$ of topoi. Moreover this morphism is an equivalence in the $2$-category of topoi (see Section 7.36). Thus it makes sense to say “$\mathcal{S}$ is a topos” if $\mathcal{S}$ is equivalent to the category of sheaves on a site (and not necessarily equal to the category of sheaves on a site). We will occasionally use this abuse of notation.

Two examples of topoi. The empty topos is topos of sheaves on the site $\mathcal{C}$, where $\mathcal{C}$ has a single object $\emptyset$ and a single morphism $\text{id}_\emptyset$ and a single covering, namely the empty covering of $\emptyset$. We will sometimes write $\emptyset$ for this site. This is a site and every sheaf on $\mathcal{C}$ assigns a singleton to $\emptyset$. Thus $\mathop{\mathit{Sh}}\nolimits (\emptyset )$ is equivalent to the category having a single object and a single morphism. The punctual topos is the topos of sheaves on the site $\mathcal{C}$ which has a single object $pt$ and one morphism $\text{id}_{pt}$ and whose only covering is the covering $\{ \text{id}_{pt}\}$. We will simply write $pt$ for this site. It is clear that the category of sheaves $=$ the category of presheaves $=$ the category of sets. In a formula $\mathop{\mathit{Sh}}\nolimits (pt) = \textit{Sets}$.

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Note that $f_*$ commutes with all limits and that $f^{-1}$ commutes with all colimits, see Categories, Lemma 4.24.5. In particular, the condition on $f^{-1}$ in the definition above guarantees that $f^{-1}$ is exact. Morphisms of topoi are often constructed using either Lemma 7.21.1 or the following lemma.

Lemma 7.15.2. Given a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ corresponding to the functor $u : \mathcal{C} \to \mathcal{D}$ the pair of functors $(f^{-1} = u_ s, f_* = u^ s)$ is a morphism of topoi.

Proof. This is obvious from Definition 7.14.1. $\square$

Remark 7.15.3. There are many sites that give rise to the topos $\mathop{\mathit{Sh}}\nolimits (pt)$. A useful example is the following. Suppose that $S$ is a set (of sets) which contains at least one nonempty element. Let $\mathcal{S}$ be the category whose objects are elements of $S$ and whose morphisms are arbitrary set maps. Assume that $\mathcal{S}$ has fibre products. For example this will be the case if $S = \mathcal{P}(\text{infinite set})$ is the power set of any infinite set (exercise in set theory). Make $\mathcal{S}$ into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section 3.11). We claim that $\mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ is equivalent to the category of sets.

We first prove this in case $S$ contains $e \in S$ which is a singleton. In this case, there is an equivalence of topoi $i : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ given by the functors

7.15.3.1
$$\label{sites-equation-sheaves-pt-sets} i^{-1}\mathcal{F} = \mathcal{F}(e), \quad i_*E = (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E))$$

Namely, suppose that $\mathcal{F}$ is a sheaf on $\mathcal{S}$. For any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) = S$ we can find a covering $\{ \varphi _ u : e \to U\} _{u \in U}$, where $\varphi _ u$ maps the unique element of $e$ to $u \in U$. The sheaf condition implies in this case that $\mathcal{F}(U) = \prod _{u \in U} \mathcal{F}(e)$. In other words $\mathcal{F}(U) = \mathop{Mor}\nolimits _{\textit{Sets}}(U, \mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor

$i_* : \textit{Sets} = \mathop{\mathit{Sh}}\nolimits (pt) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{S}), \quad E \longmapsto (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E))$

is an equivalence of categories, and its inverse is the functor $i^{-1}$ given above.

If $\mathcal{S}$ does not contain a singleton, then the functor $i_*$ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty $\tilde e \in S$ and a map $\varphi : \tilde e \to \tilde e$ whose image is a singleton. For any sheaf $\mathcal{F}$ set

$\mathcal{F}(e) := \mathop{\mathrm{Im}}( \mathcal{F}(\varphi ) : \mathcal{F}(\tilde e) \longrightarrow \mathcal{F}(\tilde e) )$

and show that this is a quasi-inverse to $i_*$. Details omitted.

Remark 7.15.4. (Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Although we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose $\mathcal{C}$ and $\mathcal{D}$ are given sites. Consider a functor $\Phi : \mathcal{C} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$. Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on $\Phi$.

1. Is it true, given a sheaf $\mathcal{F}$ on $\mathcal{D}$, that the rule $U \mapsto \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\Phi (U), \mathcal{F})$ defines a sheaf on $\mathcal{C}$? If so, this defines a functor $\Phi _* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

2. Is it true that $\Phi _*$ has a left adjoint? If so, write $\Phi ^{-1}$ for this left adjoint.

3. Is it true that $\Phi ^{-1}$ is exact?

If the last question still has the answer “yes”, then we obtain a morphism of topoi $(\Phi _*, \Phi ^{-1})$. Moreover, given any morphism of topoi $(f_*, f^{-1})$ we may set $\Phi (U) = f^{-1}(h_ U^\# )$ and obtain a functor $\Phi$ as above with $f_* \cong \Phi _*$ and $f^{-1} \cong \Phi ^{-1}$ (compatible with adjoint property). The upshot is that by working with the collection of $\Phi$ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of $\Phi$ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of conditions (2) and (3) above; we do this in the case of points of topoi in Section 7.32.

Remark 7.15.5. (Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$ (see Remark 7.14.9) gives rise to a morphism of topoi $f$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ exactly as in Lemma 7.15.2.

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# Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ with a semileptonic tagging method

Abstract : The experimental results on the ratios of branching fractions $\mathcal{R}(D) = {\cal B}(\bar{B} \to D \tau^- \bar{\nu}_{\tau})/{\cal B}(\bar{B} \to D \ell^- \bar{\nu}_{\ell})$ and $\mathcal{R}(D^*) = {\cal B}(\bar{B} \to D^* \tau^- \bar{\nu}_{\tau})/{\cal B}(\bar{B} \to D^* \ell^- \bar{\nu}_{\ell})$, where $\ell$ denotes an electron or a muon, show a long-standing discrepancy with the Standard Model predictions, and might hint to a violation of lepton flavor universality. We report a new simultaneous measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$, based on a data sample containing $772 \times 10^6$ $B\bar{B}$ events recorded at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB $e^+ e^-$ collider. In this analysis the tag-side $B$ meson is reconstructed in a semileptonic decay mode and the signal-side $\tau$ is reconstructed in a purely leptonic decay. The measured values are $\mathcal{R}(D)= 0.307 \pm 0.037 \pm 0.016$ and $\mathcal{R}(D^*) = 0.283 \pm 0.018 \pm 0.014$, where the first uncertainties are statistical and the second are systematic. These results are in agreement with the Standard Model predictions within $0.2$, $1.1$ and $0.8$ standard deviations for $\mathcal{R}(D)$, $\mathcal{R}(D^*)$ and their combination, respectively. This work constitutes the most precise measurements of $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ performed to date as well as the first result for $\mathcal{R}(D)$ based on a semileptonic tagging method.
Keywords :
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-02350157
Contributor : Inspire Hep <>
Submitted on : Tuesday, November 5, 2019 - 11:44:31 PM
Last modification on : Thursday, November 14, 2019 - 6:01:15 AM

### Citation

G. Caria, P. Urquijo, I. Adachi, H. Aihara, S. Al Said, et al.. Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ with a semileptonic tagging method. 2019. ⟨hal-02350157⟩

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# B Is string theory a deterministic hidden variable theory?

#### Aidyan

There are several interpretations of QM which differ from being deterministic or non deterministic, with or without hidden variables and local or non-local. As I understand it ST poses the existence of vibrating strings, that is, physical objects with definite properties, moving along a world sheet coordinate system having determinate position and momentum and where interactions and correlations are expresses as local string dynamics. Does this imply that it is a deterministic and local theory? And where form comes quantum randomness in ST? Is it a consequence of the strings vibration? If so, could one then say that it is a hidden variable theory?

#### atyy

String theory is a hidden variable theory (strings), but it is a quantum hidden variable theory, so it does not have much to say beyond standard quantum theory about about the issues of locality/nonlocality and determinism/randomness involved in violation of a Bell inequality.

There does seem to be another form of nonlocality in string theory related to the Holographic principle. Here is a Scientific American article about the holographic principle in string theory by Juan Maldacena, who first concretely raised the conjecture: https://www.scientificamerican.com/article/the-illusion-of-gravity-2007-04/

#### Demystifier

2018 Award
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.

#### haushofer

Just to add: non-locality is a problem when you want to quantize branes. The reason is intuitively as follows: stretchin a string costs energy, so strings are localized in space. But branes, e.g. 2 branes, can be stretched indefinitely without energy cost. Consider for example a long peak on a 2 brane, which you make arbitrarily narrow. As such you can make the corresponding area (and hence energy!) arbitrarily small. With this, the brane can "peak out" arbitrarily far without energy cost. As such the quantization is troublesome. To be honest, I'm not sure to which extent the quantization of branes is well-defined nowadays, but I like this intuitive picture.

#### MathematicalPhysicist

Gold Member
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.
I find it hard to grasp a theory such as string theory called either Classical or Quantum, where an alleged theory of QGR should be neither quantum nor classical; obviously in the appropriate limits we should get back both theories, but in the intermediate we should get something different from both, otherwise it's not a different theory but a patch of differing theories.

Now, this is akin the question what are those limits?

#### stevendaryl

Staff Emeritus
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.
I had the idea that string theory was nonlocal even classically, although the nonlocality is only significant over short distances.

https://arxiv.org/pdf/1310.4957.pdf

#### PAllen

I find it hard to grasp a theory such as string theory called either Classical or Quantum, where an alleged theory of QGR should be neither quantum nor classical; obviously in the appropriate limits we should get back both theories, but in the intermediate we should get something different from both, otherwise it's not a different theory but a patch of differing theories.

Now, this is akin the question what are those limits?
I don't think it is a given that the QG solution is neither quantum nor classical. For example, asymptotic safety would be a pure quantum approach that would (if it worked) fully solve quantum gravity and include a complete theory of matter (unlike e.g. LQG, which in addition to other issues, has not yet accommodated anything like the standard model). That AS (asymptotic safety) does nothing to resolve various other issues in physics (the observations bundled in "the dark matter problem", the cosmological constant problem, etc.) is irrelevant. There is no a priori expectation that a QG solution should resolve these.

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#### Demystifier

2018 Award
an alleged theory of QGR should be neither quantum nor classical
Where did you get it from?

2018 Award

#### MathematicalPhysicist

Gold Member
@Demystifier I assumed that just like GR isn't Newtonian Gravity and QM isn't CM, but in the appropriate limits they coincide, i.e. when $\hbar \to 0$ QM becomes CM, and GR becomes NG in an appropriate limit which I am not sure what it is, then also QGR should obey the same behaviour, otherwise why call it a different theory than QM or GR?

#### Demystifier

2018 Award
@Demystifier I assumed that just like GR isn't Newtonian Gravity and QM isn't CM, but in the appropriate limits they coincide, i.e. when $\hbar \to 0$ QM becomes CM, and GR becomes NG in an appropriate limit which I am not sure what it is, then also QGR should obey the same behaviour, otherwise why call it a different theory than QM or GR?
Just because a theory is not quantum mechanics doesn't mean that this theory isn't quantum.

#### MathematicalPhysicist

Gold Member
Just because a theory is not quantum mechanics doesn't mean that this theory isn't quantum.
Then what makes a theory quantum?

I have seen the trichotomy in Zelevinsky's textbook of Quantum Mechanics, Quantum Physics and another term which I have forgotten.
Shouldn't a new theory transcend beyond the quantum?

#### Demystifier

2018 Award
Then what makes a theory quantum?

I have seen the trichotomy in Zelevinsky's textbook of Quantum Mechanics, Quantum Physics and another term which I have forgotten.
Shouldn't a new theory transcend beyond the quantum?
For instance, quantum field theory is not quantum mechanics, but is quantum. Quantum mechanics is a quantum theory of pointlike particles.

#### MathematicalPhysicist

Gold Member
For instance, quantum field theory is not quantum mechanics, but is quantum. Quantum mechanics is a quantum theory of pointlike particles.
OK, so string theory is quantum of strings which aren't fields nor points.

So the thing that makes something 'quantum' is what exactly?

#### MathematicalPhysicist

Gold Member
Obviously I know the maths of QM and QFT, but what makes these two theories quantum I am not sure.
Maybe because no technical book that I read discuss this matter.

#### Demystifier

2018 Award
So the thing that makes something 'quantum' is what exactly?
If observables are represented by non-commuting operators, then it's quantum. If probabilities are given by something of the form $|\psi|^2$, then it's quantum.

#### MathematicalPhysicist

Gold Member
If observables are represented by non-commuting operators, then it's quantum. If probabilities are given by something of the form $|\psi|^2$, then it's quantum.
In QM $\psi$ are wave-functions, in QFT they are wave-functionals.
So only these two conditions are enough to declare a theory as quantum.

#### Demystifier

2018 Award
In QM $\psi$ are wave-functions, in QFT they are wave-functionals.
So only these two conditions are enough to declare a theory as quantum.
Superficially speaking, yes. But if you want a deeper insight, see my paper https://arxiv.org/abs/quant-ph/0505143 where classical mechanics is represented by a quantum-like formalism.

#### stevendaryl

Staff Emeritus
Are you one of the authors of that paper?
No! Did I give that impression, somehow?

I actually was trying to find a John Baez comment from years ago saying that string theory was slightly nonlocal. I say "slightly", because the nonlocality was confined to a region the size of the length of the string. I didn't remember the argument for why.

#### stevendaryl

Staff Emeritus
OK, so string theory is quantum of strings which aren't fields nor points.

So the thing that makes something 'quantum' is what exactly?
The difference between a classical field theory and a quantum field theory is that whereas a classical field theory is deterministic--the configuration of the field and its sources at one time determines the configuration at later times---a quantum field theory is nondeterministic. The equations of quantum field theory describe not the fields themselves but probability amplitudes for the field.

The relationship between classical field theory and quantum field theory is very similar to the relationship between Newtonian physics and the Heisenberg equations of motion for quantum mechanics. The Newtonian equations of motion (or actually, the Euler-Lagrange equations, which are equivalent) for the harmonic oscillator, say, are:

$\frac{dx}{dt} = \frac{p}{m}$
$\frac{dp}{dt} = -K x$

Those are exactly the same as the Heisenberg equations of motion for a harmonic oscillator, but in the Heisenberg equations, $x$ and $p$ are operators, rather than real-valued functions of time.

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# Specific strength

The specific strength is a material's strength (force per unit area at failure) divided by its density. It is also known as the strength-to-weight ratio or strength/weight ratio. In fiber or textile applications, tenacity is the usual measure of specific strength. The SI unit for specific strength is (N/m2)/(kg/m3) or more commonly N·m/kg.

Another way to describe specific strength is breaking length, also known as self support length: the maximum length of a vertical column of the material (assuming a fixed cross-section) that could suspend its own weight when supported only at the top. For this measurement, the definition of weight is the force of gravity at the Earth's surface applying to the entire length of the material, not diminishing with height. This usage is more common with certain specialty fiber or textile applications.

The materials with the highest specific strengths are typically fibers such as carbon fiber, glass fiber and various polymers, and these are frequently used to make composite materials (e.g. carbon fiber-epoxy). These materials and others such as titanium, aluminium, magnesium and high strength steel alloys are widely used in aerospace and other applications where weight savings are worth the higher material cost.

Note that strength and stiffness are distinct. Both are important in design of efficient and safe structures.

## Examples

Specific tensile strength of various materials
Material Strength
(MPa)
Density
(g/cm³)
Specific Strength
(kN·m/kg)
Breaking length
(km)
source
Concrete 10 2.30 4.35 0.44
Rubber 15 0.92 16.3 1.66
Copper 220 8.92 24.7 2.51
Brass 580 8.55 67.8 6.91 1
Nylon 78 1.13 69.0 7.04 2
Oak 60 0.69 87.0 8.86 3
Polypropylene 80 0.90 88.9 9.06 4
Magnesium 275 1.74 158 16.1 5
Aluminium 600 2.80 214 21.8 6
Stainless Steel 2000 7.86 254 25.9 6
Titanium 1300 4.51 288 29.4 6
Bainite 2500 7.87 321 32.4 7
Balsa (axial load) 73 0.14 521 53.2 8
Scifer steel wire 5500 7.87 706 71.2 7
carbon-epoxy composite 1240 1.58 785 80.0 9
spider silk 1400 1.31 1069 109
Silicon carbide 3440 3.16 1088 110 10
Glass Fiber 3400 2.60 1307 133 6
Basalt fiber 4840 2.70 1790 183 11
1 μm iron whiskers 14000 7.87 1800 183 7
Vectran 2900 1.40 2071 211 6
Carbon fiber (AS4) 4300 1.75 2457 250 6
Kevlar 3620 1.44 2514 256 12
Dyneema (UHMWPE) 3600 0.97 3711 378 13
Zylon 5800 1.54 3766 384 14
Carbon nanotube (see note below) 62000 .037-1.34 46268-N/A 4716-N/A 1516
Colossal carbon tube 6900 .116 59483 6066 17

The data of this table is from best cases, and has been established for giving a rough figure.

• Note: Multiwalled carbon nanotubes have the highest tensile strength of any material yet measured, with labs producing them at a tensile strength of 63 GPa,15 still well below their theoretical limit of 300 GPa. The first nanotube ropes (20 mm long) whose tensile strength was published (in 2000) had a strength of 3.6 GPa, still well below their theoretical limit.18 The density is different depending on the manufacturing method, and the lowest value is 0.037 or 0.55(solid).16
1. ^ RoyMech: Copper Alloys
2. ^ Goodfellow: Polyamide - Nylon 6
3. ^ Delft University of technology: Oak wood
4. ^ Goodfellow: Polypropylene
5. ^ eFunda: Magnesium Alloys
6. ^ a b c 52nd Hatfield Memorial Lecture: "Large Chunks of Very Strong Steel" by H. K. D. H. Bhadeshia 2005
8. ^ McGRAW-HILL ENCYCLOPEDIA OF Science & Technology, 8th Edition, (c)1997, vol. 1 p 375
9. ^ Specialty Materials, Inc SCS Silicon Carbide Fibers
10. ^ http://www.albarrie.com/techfabrics/continuousfiber.aspx
11. ^ Network Group for Composites in Construction: Introduction to Fibre Reinforced Polymer Composites
12. ^ "Dyneema Fact sheet". DSM (Company). 1 January 2008.
14. ^ a b Yu, Min-Feng; Lourie, O; Dyer, MJ; Moloni, K; Kelly, TF; Ruoff, RS (2000). "Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load". Science 287 (5453): 637–640. Bibcode:2000Sci...287..637Y. doi:10.1126/science.287.5453.637. PMID 10649994.
15. ^ a b K.Hata. "From Highly Efficient Impurity-Free CNT Synthesis to DWNT forests, CNTsolids and Super-Capacitors" (free download PDF).
16. ^ Peng, H.; Chen, D.; et al., Huang J.Y. et al. (2008). "Strong and Ductile Colossal Carbon Tubes with Walls of Rectangular Macropores". Phys. Rev. Lett. 101 (14): 145501. Bibcode:2008PhRvL.101n5501P. doi:10.1103/PhysRevLett.101.145501. PMID 18851539.
17. ^ "Tensile strength of single-walled carbon nanotubes directly measured from their macroscopic ropes" by F. Li, H. M. Cheng, S. Bai, G. Su, and M. S. Dresselhaus. doi:10.1063/1.1324984

## Relation to velocity

Since N·m/kg (the unit of specific strength) expands to (kg·m/s2)·m/kg or (m/s)2, a specific strength can also be understood as the square of a speed, hence represented by it square root, a speed. Further, the structural analysis for whether a particular material is strong enough for some purpose can sometimes be expressed in terms of comparing this speed to the speed of a relevant part of the structure. For example, a spinning cylindrical shell (as in an idealized flywheel or an idealized cylinder space ship) is only feasible if built of a material whose square root of specific strength is at least the speed at which the spinning shell moves.

$\sqrt{\tfrac{E}{\rho}}\ge v$

## The Yuri and Space Tethers

The International Space Elevator Consortium has proposed the Yuri (unit) as a unit of specific strength useful for describing space tether designs. 1 Yuri = 1 (m/s)2 = N/(kg/m)1 and 1 MYuri (capital M denoting mega- and capital "Y" being typical for this non-SI unit) is equal to 1 N/Tex = 1 GPa-cc/g.2 A functional space elevator would require a tether of 30-80 MYuri.3


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### Bayes estimators of a multivariate generalized hyperbolic partial regression model

Document Type : Research Paper

Authors

1 Statistician at the Nineveh Agriculture Directorate, Mosul, Iraq

Abstract

The matrix-variate generalized hyperbolic distribution belongs to the family of heavy-tailed mixed
probability distributions and is considered to be one of the continuous skewed probability distributions. This distribution has wide applications in the field of economics, especially in stock modeling.
This paper includes estimation the parameters of the multivariate semi-parametric regression
model represented by the multivariate partial linear regression model when the random error follows the matrix-variate generalized hyperbolic distribution, using the Bayesian method when noninformative prior information is available and under the assumption that the shape parameters and
the skewness matrix are known. In addition, the bandwidth parameter is estimated by a suggested
way based on the normal distribution rule and the proposed kernel function based on the mixed
Gaussian kernel function and studying the findings on the generated data in a way suggested for
the model, comparing the estimators depending on the criterion of the mean sum of squares error.
The two researchers concluded that the proposed kernel function is better than the Gaussian kernel
function in estimate the parameters.

Keywords


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# zbMATH — the first resource for mathematics

Logarithmic series and Hodge integrals in the tautological ring. With an appendix by Don Zagier. (English) Zbl 1090.14005
Let $$M_g$$ denote the coarse moduli space of non-singular complex algebraic curves of genus $$g\geq 2$$. One of the most useful compactifications of $$M_g$$ is given by the moduli space $$\overline M_g$$ of Deligne-Mumford stable curves, which was constructed by P. Deligne and M. Mumford [Publ. Math., Inst. Hautes Étud. Sci. 36, 75–100 (1969; Zbl 0181.48803)]. The intersection theory in this so-called Deligne-Mumford compactification $$\overline M_g$$ of $$M_g$$, that is, the study of the Chow ring $$A^*(\overline M_g)$$ of $$\overline M_g$$, has been a subject of central interest in algebraic geometry ever since, beginning with D. Mumford’s pioneering work [in: Arithmetic and Geometry, II, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)]. In the past two decades, many remarkable properties of these Chow rings have been discovered, ranging from the existence of certain “tautological classes” in them up to striking numerical results related to (and predicted by) quantum field theories in physics. E. Witten’s famous conjecture in topological gravity theory [in: Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243–310 (1991; Zbl 0757.53049)], M. Kontsevich’s spectacular proof of it [Commun. Math. Phys. 147, 1–23 (1992; Zbl 0756.35081)], and the avalanche of recent work in quantum cohomology theory, on Gromov-Witten invariants, and on the mirror symmetry phenomenon [D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry (1999; Zbl 0951.14026)] not only utilized the enumerative geometry on compactified moduli spaces as a basic framework, but also propelled its rapid development in a crucial way.
In this vein, the main goal of the paper under review is to describe a new perspective on the intersection theory of $$\overline M_g$$ that combines advances from both the classical theory of degenerations in algebraic geometry and topological quantum field theory in mathematical physics.
The central object of study is the so-called tautological ring $$R^*(\overline M_g)$$ of $$\overline M_g$$, i.e., the subring of $$A^*(\overline M_g)$$ generated by the so far known tautological classes, together with its analogues $$R^*(\overline M_{g,n})$$ for the moduli spaces $$\overline M_{g,n}$$ of $$n$$-pointed stable curves. The authors consider the moduli filtration $$\overline M_g\supset M^c_g\supset M_g\supset\{[X_g]\}$$, where $$X_g$$ is a fixed smooth curve of genus $$g$$ and $$M^c_g$$ denotes the moduli space of stable curves with compact generalized Jacobian. There is an associated sequence of successive quotient maps of rational Chow rings, $A^*(\overline M_g)\to A^*(M^c_g)\to A^*(M_g)\to A^*(\{[X_g]\})\cong \mathbb{Q},$ and the authors basically develop a uniform approach to the study of all these rings, and their related tautological rings with respect $$n$$-punctures, respectively, by successive descent methods. In fact, they show that the corresponding tautological rings in this sequence of quotient maps exhibit several parallel structures given by certain Hodge integrals of tautological classes, and that those can be effectively computed by a successive cutting-down method.
Some special cases of these computational results can be found in the authors’ earlier paper [Invent. Math. 139, No. 1, 173–199 (2000; Zbl 0960.14031)], whereas the subsequent paper under review largely completes the overall picture. The three main theorems of the present paper establish concrete integral formulae on $$\overline M_g$$ which are of greatest significance. Namely, these formulae settle some earlier conjectures in this context reprove some related results in a unified way, strengthen evidence for Faber’s conjectures on the structure of the tautological rings $$R^*(\overline M_g)$$ as formulated in his previous paper [C. Faber, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)], and reveal some surprising new combinatorial identies involving Stirling numbers of the second kind and Bernoulli numbers. In an appendix to the present paper, entitled “Polynomials arising from the tautological ring”, D. Zagier gives purely combinatorial proofs of these geometrically derived new identities.
Without any doubt, the novel approach to the study of the tautological ring of the moduli spaces $$\overline M_g$$, especially the amazing reduction of their enumerative geometry to polynomial combinatorics, has the character of a very promising new research program in this direction, including both algebraic geometry and quantum field theory.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C15 (Equivariant) Chow groups and rings; motives 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 11B73 Bell and Stirling numbers
Full Text:


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# How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series:

$$\sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$

As hardmath says in his comment, the series converges. I tried with Mathematica which gives for the sum 0.00526589. Is it possible to obtain some analytical result/approximation for the sum of the series?

Moreover, I would like to simplify the following sum:

$$\sum_{n=2}^{\infty}\frac{(n-1)^{2n-3}}{n(n+1)^{2n+1}}\left[ \sum_{m=0}^{n-1}\left(\frac{n+1}{n-1}\right)^m\right]^2$$

Such sums appear in quantum mechanics when dealing with second order perturbation theory (the first mentioned case comes out when dealing with the Coulomb potential 1/r perturturbed by a k^2/r term). Any hints appreciated.

Update: As hardmath says in his comment, in the "second series the finite inner sum (over m) is geometric, so it can be replaced by an explicit expression in terms of n".

This means that we can simplify the double sum into a single one:

$$\frac1{4}\sum_{n=2}^{\infty}\frac1{n(n^2-1)}\left[1-2\left(\frac{n+1}{n-1}\right)^n+\left(\frac{n+1}{n-1}\right)^{2n}\right]$$

which is closer in form to the first mentioned.

• What motivated these Questions? Some context would be helpful to a Reader willing to provide useful information. – hardmath Dec 21 '14 at 3:49
• @hardmath Motivation added:-) – wondering Jan 22 '15 at 12:26
• Thanks. The first series converges (e.g. by comparison with the sum of $1/n^3$), but I suspect you want further information about "what could be done wit[h] this series". Can you be more specific about what is known and what additional analysis would be useful to you? – hardmath Jan 22 '15 at 12:33
• For your first sum, I get 0.00526589019306362497, did you write it correctly? ISC found nothing for 0.00526589019306362497, so perhaps closed form evaluation will not be possible. – GEdgar Jan 22 '15 at 13:32
• In your second series the finite inner sum (over $m$) is geometric, so it can be replaced by an explicit expression in terms of $n$. Have you explored this? – hardmath Jan 24 '15 at 14:14


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### Author Topic: Quiz2 LEC0101 C  (Read 337 times)

#### Xinqiao Li

• Full Member
• Posts: 15
• Karma: 1
##### Quiz2 LEC0101 C
« on: October 03, 2020, 05:47:44 AM »
Questions: Find all points of continuity of the given function: $f(z)=(Imz-Rez)^{-1}$

Solutions:

Let $z=x+iy$ where x,y are real numbers.

Then $f(z)=f(x,y)=(Im(x+iy)-Re(x+iy))^{-1}=(y-x)^{-1}=\frac{1}{y-x}$

The function is not valid when the denominator equals 0, that is, $y=x$.

Therefore, the function is discontinuous only at all points of the line $y=x$.


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# Linear second order equations with constant coefficients

Up a level : Differential Equations
Previous page : Graphing all solutions
Next page : LSO -Two different real roots

Let us have a look at equations of the form

$y'' + ay' + by = 0$

(This, by the way, is also called a homogenous equation.) Except for the first term it is has the same form as

$ay' + by = 0$

We can rewrite that as

$\frac{{dy}}{y} = - \frac{b}{a}dx$

Integrating this gives us

$\ln |y| = - \frac{b}{a}x + {C_1}$

or

$|y| = {e^{ - \frac{b}{a}x + {C_1}}} = {C_2}{e^{ - \frac{b}{a}x}}$

The sign can be absorbed into the constant, finally giving us

$y = C{e^{ - \frac{b}{a}x}}$

So it may be an idea to test if the same type of solution could be solving our initial equations. Let us assume we have a solution of the form

$y = C{e^{rx}}$

We then have

$y' = Cr{e^{rx}},\quad y'' = C{r^2}{e^{rx}}$

Substituting this into our initial equation gives us

$C{r^2}{e^{rx}} + aCr{e^{rx}} + bC{e^{rx}} = 0$

This could be factorized as

$C{e^{rx}}\left( {{r^2} + ar + b} \right) = 0$

So if r have a value such that the second factor is 0 then we have a solution. The equation

${r^2} + ar + b = 0$

is called the characteristic equation, and the solutions of our original equation will depend on the solutions to the characteristic equation.

An example

Say we have

$y'' + 2y' - 15y = 0$

That gives the characteristic equation

${r^2} + 2r - 15 = 0$

We can solve that by (for example) splitting -15 into two factors that add up to 2. Doing so we get

$(r - 3){(r + 5)^2} = 0$

with the solutions r=3 and r=-5. We will thus have as solutions to our differential equation

$y = C{e^{3x}},\quad y = D{e^{ - 5x}}$

where C and D are constants. Matter a fact, as you can quite easily verify, we have

$y = C{e^{3x}} + D{e^{ - 5x}}$

as a solution (matter a fact it is the general solution).  But a quadratic equation, in this case the characteristic equation) may have two real roots, two equal real roots or two complex roots. So what does these different cases correspond to? Let us have a look in the following few pages.

Up a level : Differential Equations
Previous page : Graphing all solutions
Next page : LSO -Two different real rootsLast modified: Mar 26, 2019 @ 07:48