text:
Chapter 229

Cardiovascular imaging plays an essential role in the practice of cardiology. Two-dimensional (2D) echocardiography is able to visualize the heart directly in real time using ultrasound, providing instantaneous assessment of the myocardium, cardiac chambers, valves, pericardium, and great vessels. Doppler echocardiography measures the velocity of moving red blood cells and has become a noninvasive alternative to cardiac catheterization for assessment of hemodynamics. Transesophageal echocardiography (TEE) provides a unique window for high-resolution imaging of posterior structures of the heart, particularly the left atrium, mitral valve, and aorta. Nuclear cardiology uses radioactive tracers to provide assessment of myocardial perfusion and metabolism, along with ventricular function, and is applied primarily to the evaluation of patients with ischemic heart disease. Cardiac MRI and CT can delineate cardiac structure and function with high resolution. They are particularly useful in the examination of cardiac masses, the pericardium, the great vessels, and ventricular function and perfusion. Gadolinium enhancement during cardiac MRI adds information on myocardial perfusion. Detection of coronary calcification by CT as well as direct visualization of coronary arteries by CT angiography (CTA) may be useful in selected patients with suspected coronary artery disease (CAD). This chapter provides an overview of the basic concepts of these cardiac imaging modalities as well as the clinical indications for each procedure.

### Two-Dimensional Echocardiography

#### Basic Principles

2D echocardiography uses the principle of ultrasound reflection off cardiac structures to produce images of the heart (Table 229-1). For a transthoracic echocardiogram (TTE), the imaging is performed with a handheld transducer placed directly on the chest wall. In selected patients, a TEE may be performed, in which an ultrasound transducer is mounted on the tip of an endoscope placed in the esophagus and directed toward the cardiac structures.

Table 229–1. Clinical Uses of Echocardiography

Current echocardiographic machines are portable and can be wheeled directly to the patient's bedside. Thus, a major advantage of echocardiography over other imaging modalities is the ability to obtain instantaneous images of the cardiac structures for immediate interpretation. Thus, echocardiography has become an ideal imaging modality for cardiac emergencies. A limitation of TTE is the inability to obtain high-quality images in all patients, especially those with a thick chest wall or severe lung disease, as ultrasound waves are poorly transmitted through lung parenchyma. Technology such as harmonic imaging and IV contrast agents (which traverse the pulmonary circulation) can be used to enhance endocardial borders in ...

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# Model formulation of microbial CO2 production and efficiency can significantly influence short and long term soil C projections

### Subjects

Models of soil C dynamics continue to be rapidly developed in a effort to improve projections of terrestrial C fluxes under a range of climate scenarios. Recently developed models of soil organic C (SOC) dynamics that incorporate microbes as the agents of decomposition [1,2,3,4,5] capture a greater number of relevant processes than their predecessors, but it remains unclear whether the mathematical formulations and parameterization of such models are accurate [6,7,8,9]. Different model structures, parameterizations, and functions describing responses to changing environmental conditions exert great influence on projections of SOC stocks and CO2 flux [1, 4, 9]. Discrepancies between projections of SOC stocks from microbial models and current Earth system model (ESM) soil C modules, which can span two orders of magnitude, are highly sensitive to assumptions regarding microbial respiration and efficiency [1, 4, 6].

Although physical structure, chemical composition, and edaphic conditions ultimately govern the accessibility of decomposable SOC [10], physiology dictates the fate of C once it is liberated and taken up from SOC by microbes, regardless of its original composition. A fundamental component of microbial physiology, heterotrophic respiration, which is the conduit for C flow from SOC to the atmosphere and one of the largest potential positive feedbacks to climate warming, is often incorporated in models of SOC dynamics heuristically in a term for efficiency. Alternatively, microbial respiration can be formulated more mechanistically to reflect the fact that heterotrophic respiration in soils is largely mass specific when microbes are actively decomposing SOC [11]. Because heterotrophic microbial respiration influences how much of the C liberated from SOC is retained in the soil and how much is lost to the atmosphere, it is important to understand both the short and long term dynamical consequences of formulating respiratory losses differently in models used to project future climate. Despite all the recent effort to develop better models, to date nobody has determined if the widespread, heuristic formulation of respiratory C losses gives rise to different dynamics than the more physiologically grounded formulations of microbial respiration, over a range of time scales. Here, we address this question for the first time.

Typically in terrestrial modules of ESMs, e.g., the community land model (CLM), respiratory C loss from the soil is phenomenologically defined as 1 minus the fraction of C transferred between C pools [8]. The fraction lost in these transformations is attributed primarily to heterotrophic respiration of soil microbes. Recently proposed microbial modules that are candidates for replacing current CLM belowground dynamics [1, 3,4,5, 12] incorporate respiratory C losses via a term for efficiency, commonly referred to as C use efficiency (CUE), that is a scalar coefficient of the uptake component of C flux into the microbial biomass pool, BC:

$$\frac{{dB_C}}{{dt}} = B_C\left[ {CUE \times U(S_C) - \lambda _B} \right]$$
(1)

Hereafter, we refer to Eq. 1 as the CUE formulation. In the CUE formulation, λB prescribes losses from the microbial biomass C pool that include allocation to enzyme production, C excretion, death or any additional generic proportional losses, and U(SC) is the function defining C substrate (SC) uptake, which is usually assumed to be of Michaelis-Menten form (i.e., substrate uptake rate is a saturating function of substrate availability).

The CUE formulation is conceptually rooted in the idea of microbial efficiency, which quantifies the partitioning of C flow through microbial biomass. Microbial efficiency is defined as the ratio of production or anabolism, P, to the sum of production and respiration (catabolism), R, $$\frac{P}{{P + R}}$$ [13], and quantifies the fraction of C liberated from SOC lost as CO2 from the soil and the fraction remaining in microbial biomass. As an integrative metric, microbial efficiency reflects the context for the partitioning of C flow through microbes and is typically used to characterize growth conditional on environmental conditions. Consequently, microbial efficiency should not be considered an inherent physiological trait, but rather a derivative of microbial growth and respiratory demand. Although the fundamental definition of efficiency, $$\frac{P}{{P + R}}$$, is straightforward and intuitive, making the measurements to quantify efficiency presents a significant challenge because both P and R must be simultaneously quantified [14] with a high degree of accuracy over a range of relevant conditions. Furthermore, P and R can vary independently in response to changing edaphic conditions, changes in substrate availability and composition, and microbial life history. Modelers have circumnavigated the dearth of independent empirical observations of P and R by treating $$\frac{P}{{P + R}}$$ as a single parameter varying from 0 to 1, CUE, that indirectly prescribes respiratory C loss from heterotrophic soil microbes.

Incorporating respiration in this manner, with CUE multiplying the SOC substrate uptake function, is convenient but can result in some logical inconsistencies. For example, respiratory losses are directly influenced by C availability in the environment, not only by cellular demand, and respiratory losses remain a constant fraction of substrate uptake. Consequently, maintenance respiration is a function of uptake rate and vanishes if uptake ceases. Incorporating all respiratory losses as in Eq. 1 means that microbial biomass can persist indefinitely without respiratory costs as long as C uptake is zero, which is unrealistic [11]. Thus, prescribing that all respiratory C losses result from the product of CUE and uptake is inconsistent with constitutive or baseline respiration. Furthermore, though it is obvious that C enters microbial biomass prior to being respired, C lost as CO2 in the CUE formulation is instantaneously respired as it is liberated from SOC, and is never depicted as having entered microbial biomass.

From basic physiology, we know that C must be taken into a cell before it is respired. Mathematically representing that respiratory C loss originates from an intracellular C pool necessitates the inclusion of a mass specific respiration rate (MSR; for simplicity it encompasses growth and maintenance respiration). This differs from the CUE formulation in that respiratory C losses only depend on microbial biomass, but C is lost via the same additional pathways as in the CUE formulation (e.g., allocation to enzyme production, C excretion—prescribed by λB in Eq. 1, and death). Aggregating mass specific growth and maintenance respiration into a single term results in no loss of generality because of linearity. Such a MSR term for maintenance respiration was recently incorporated by [11]. Dynamics with respiratory C losses prescribed in a mass specific manner assume the following generic form:

$$\frac{{dB_C}}{{dt}} = B_C\left[ {U(S_C) - MSR - \lambda _B} \right],$$
(2)

which we refer to as the MSR formulation. The components of efficiency for the CUE formulation are defined as: P = BC×CUE×U(SC), and R = BC×(1−CUEU(SC), whereas for the MSR formulation: P = BC×(U(SC)−MSR), and R = BC×MSR. Thus, the aggregate metric of microbial efficiency reduces to CUE and $$1 - \frac{{MSR}}{{U(S_C)}}$$ for the CUE and MSR formulations, respectively.

A key difference between the two formulations of respiratory C loss is that, for a particular set of environmental conditions, CUE is assumed constant for the CUE formulation and MSR is assumed constant for the MSR formulation. Consequently, for a given set of environmental conditions, the CUE formulation dictates that efficiency is constant irrespective of substrate availability. In contrast, the MSR formulation portrays microbial efficiency as a function of both MSR and substrate availability, i.e., a dynamic aggregate resulting from the interaction between microbial physiology and environmental conditions. Consequently, variation and covariation in MSR and in substrate availability, which has been recently demonstrated in controlled experimental conditions [15], can independently influence efficiency in the MSR formulation. The two formulations of respiratory losses can be seen as opposite ends of a continuum, with all respiratory losses proportional to uptake for a pure CUE formulation and proportional to microbial biomass C for a pure MSR formulation. We focus on the opposite ends of the continuum to elucidate the most pronounced dynamical consequences of different mathematical formulations of microbial respiratory C loss.

Gross C fluxes can be made equivalent for the CUE and MSR formulations for any level of SOC, but it is unclear whether the values of CUE and MSR that result in equivalence are mutually consistent. Efficiency can also be equivalent for both formulations if $$CUE = 1 - \frac{{MSR}}{{U(S_C)}}$$, but such equivalence necessitates that the ratio of $$\frac{{MSR}}{{U(S_C)}}$$ is constant for all SC. This underscores the fact that the CUE formulation treats efficiency as a pre-specified characteristic of microbes, rather than an emergent feature of microbial interactions with the environment, as in the MSR formulation. Explicitly including MSR in models may be more physiologically realistic, but it comes with a cost: measuring an additional parameter. The CUE formulation is appealing because CUE is an aggregate measure of efficiency, obviating the need to measure MSR, which is difficult in practice [14], and because it heuristically captures partitioning between anabolism and catabolism. The question of whether the more convenient, heuristic CUE and the more physiologically-based MSR formulations of respiratory C losses exhibit similar short term and long term dynamics has not been addressed, and needs to be answered before we continue moving forward with CUE-based microbial dynamics in soil biogeochemical modules of ESMs. Answering this question will tell us whether or not the more convenient CUE formulation faithfully portrays dynamics of a more physiologically realistic formulations of respiratory C losses from soil microbes.

Our goal here is to delineate the scenarios for which the CUE and MSR formulations of respiratory C loss result in significantly different soil C dynamics and when they agree. The different means of parsing C liberated from SOC into biomass and respired CO2 represented by Eqs. 1 and 2 can result in significantly different long term microbial biomass C and SOC pool sizes, and in significantly different dynamical responses of microbial biomass C, and associated soil respiration. To see this, we must explicitly couple SOC substrate (SC) dynamics to microbial biomass C dynamics. In the most abstract sense, the SOC decomposition process can be formalized with two general equations, one for SOC and one for microbial biomass C:

$$\begin{array}{l}\frac{{dS_C}}{{dt}} = I - B_CU(S_C) - S_C\lambda _S + B_C\lambda _B\varepsilon \\ \frac{{dB_C}}{{dt}} = \left\{ {\begin{array}{*{20}{l}} {B_C\left[ {CUE \times U(S_C) - \lambda _B} \right];} \hfill & {CUE\,{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & {} \hfill \\ {B_C\left[ {U(S_C) - MSR - \lambda _B} \right];} \hfill & {MSR\,{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$
(3)

The SOC pool increases as a function of external inputs, litter, (I) and C lost from microbial biomass (BC) and decreases as a result of microbial uptake (U) and leaching (note we consider microbial biomass distinct from SOC throughout). Microbial uptake, U(SC), microbial respiration rate, MSR, and other losses, λB, are all mass specific rates, and ε is the fraction of C lost from microbes recycled back to the substrate pool. We only consider a single SOC pool to highlight dynamical differences between different formulations of respiratory C loss. Such differences will be independent of substrate particulars such as quality or recalcitrance because our focus is on the fate of C fate after it is taken up. For the comparison of interest here, we do not need to address varying propensities of acquisition for different substrates. Differences in SOC composition will certainly influences rates of acquisition and microbial growth rate, but because our focus is on how C that is already liberated is lost as CO2, SOC chemistry will not influence the dynamical consequences of different mathematical formulations of microbial respiration. Although a soil enzyme C pool could have been explicitly included, we omitted enzymes for clarity. However, ε effectively prescribes that some C lost via λB increases the size of the substrate pool in a functionally similar manner to soil enzyme activity. The explicit inclusion of soil enzymes would shift C from either or both the SOC and biomass pools into an enzyme pool, but because soil enzyme C is usually assumed to be a small fraction of total soil C, the effect would be minimal. The same would hold for additional intermediate C pools, e.g., DOC [1].

The different formulations of respiratory C loss can result in three types of discrepancies: in equilibrium pool sizes for the same C pool, in the relative size of both C pools (biomass C and SOC), and in the response to perturbation, i.e., transient dynamics. For the first two, we examine equilibrium expressions for substrate C and biomass C:

$$\begin{array}{l}\hat S_C = \left\{ {\begin{array}{*{20}{l}} {U^{ - 1}\left( {\frac{{\lambda _B}}{{CUE}}} \right);} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {U^{ - 1}\left( {MSR + \lambda _B} \right);} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\\ \hat B_C = \left\{ {\begin{array}{*{20}{l}} {\frac{{I - U^{ - 1}\left( {\frac{{\lambda _B}}{{CUE}}} \right)\lambda _S}}{{\lambda _B(\frac{1}{{CUE}} - \varepsilon )}};} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{{I - U^{ - 1}\left( {MSR + \lambda _B} \right)\lambda _S}}{{MSR + (1 - \varepsilon )\lambda _B}};} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$
(4)

It is clear from Eq. (4) that discrepancies between equilibrium pool sizes will occur if $$\frac{{\lambda _B}}{{CUE}}$$ is substantially different from MSR + λB. The standard assumption is that the uptake function U(SC) is of Michaelis–Menten form, $$\frac{{vS_C}}{{(K + S_C)}}$$ with v equaling maximum substrate uptake rate and K the half-saturation constant. Explicit and straightforward steady-state expressions for SC in terms of the Michaelis-Menten uptake kinetics can easily be obtained and are:

$$\hat S_C = \left\{ {\begin{array}{*{20}{l}} {\frac{K}{{\left( {\frac{{vCUE}}{{\lambda _B}} - 1} \right)}};} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{K}{{\left( {\frac{v}{{MSR + \lambda _B}} - 1} \right)}};} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.$$
(5)

Steady state pool sizes ($$\hat S_C$$ in Eqs. 4 and 5) and microbial efficiency will be equal for both formulations if $$CUE = \frac{1}{{\frac{{MSR}}{{\lambda _B}} + 1}}$$, but if $$CUE \ne \frac{1}{{\frac{{MSR}}{{\lambda _B}} + 1}}$$ the steady state SOC pools for the two formulations will differ in size. Using the empirically observed and often assumed variation in CUE for soil microbes [1, 12, 14, 16], and recently quantified variation in MSR for a common soil microbe [15, 17], the potential for a discrepancy in steady-state SOC pool size between the two formulations can be assessed. In Fig. 1, the subset of parameter space corresponding to the assumed and empirically quantified ranges of CUE and MSR, shown as a shaded box, is superimposed on contour plots in which the contours denote the ratios of steady state SOC pool sizes for the CUE formulation relative to the MSR formulation. The lowest assumed value for λB found in the literature, 0.0002 h−1 [1], results in a discrepancy in SOC pool sizes of up to three orders of magnitude between the two formulations (the right edge of the shaded box in Fig. 1), whereas the highest assumed value, 0.05 d−1 [2], results in a less pronounced discrepancy (the left edge of the shaded box in Fig. 1) when combined with the lower portion of the empirical range for MSR, ≈0.05–0.2 h−1 [15, 17]. Smaller SOC pools for the MSR formulation relative to the CUE formulation are possible (lower left corner of each panel in Fig. 1, which corresponds to very low CUE and low MSR relative to λb), but the parameter space for this situation is restricted and is not supported by empirical evidence [18]. Thus, projections of microbial biomass C are likely to be significantly larger and projections of SOC are likely to be smaller using the CUE formulation with values in the current literature.

The two formulations also differ in how the steady state ratio of biomass C to SOC ($$\hat B_C/\hat S_C$$) varies as function of microbial physiology (Fig. 2). For ratios of v/λB above 50, lower values of CUE (≤0.2) than typically assumed, inferred from, or measured by experiments [1, 12, 15,16,17] are required for $$\hat B_C/\hat S_C$$ to be less than 0.05, the empirically observed upper limit [19]. This is especially true in environments characterized by large inputs to the SOC pool relative to non-respiratory losses from microbial biomass (darker envelopes in Fig. 2 corresponding to large values of I/λB). However, if v/λB is smaller in magnitude, assumed CUE values are consistent with $$\hat B_C/\hat S_C \le 0.05$$. In contrast, for large ratios of maximum uptake rate to non-respiratory loss rate (high values of v/λB), the MSR formulation is more consistent with low ratios of microbial biomass C to SOC. The v/λB ratio will likely be large in soils though, because realized uptake that depends on substrate availability must compensate for both respiratory and non-respiratory C losses from microbes. The consistency across columns in Fig. 2 reflects the fact that as one component of microbial physiology changes, e.g., v/λB, another, efficiency in this case, must simultaneously change to constrain microbial biomass within reasonable limits. Although incorporating an additional SOC pool would expand the parameter range consistent with $$\hat B_C/\hat S_C \le 0.05$$ for either model (sensu [4]), the qualitative differences between respiratory formulations, namely that the CUE formulation tends to generate greater microbial biomass C relative to SOC than the MSR formulation, will persist.

The discrepancies in steady state pool sizes for the CUE and MSR formulations are primarily the result of specific choices of parameter values. If the assumed values or the assumed ranges of values for CUE, MSR, non-respiratory loss rate (λB), and maximum substrate uptake rate (v) are incorrect, then discrepancies may be less distinct. Although the lowest assumed value for CUE in the soil literature is ≈0.3, aquatic microbes [13] and litterbag decomposition rates [12] are often characterized by CUE ≤ 0.2. Assuming a value ≤0.2 for CUE would indeed decrease the discrepancy in steady state soil C pool sizes, because lower values of CUE yield results more consistent with the MSR formulation and empirically observed ratios of $$\hat B_C{\mathrm{/}}\hat S_C$$. This fact is interesting because [14] have argued that CUE values reported in the soil literature are likely to overestimate true efficiency. Additionally, because the ratio of biomass C to SOC considered here and in recent microbial models [1,2,3,4,5, 12] is best thought of as the ratio of microbial biomass C to SOC accessible for microbial breakdown and assimilation, the upper limit of 0.05 that is accurate for total SOC may be an underestimate when considering the available SOC pool [19]. If CUE ≤ 0.2 and the upper limit for the ratio of microbial biomass C to accessible SOC is higher than 0.05, the CUE model will produce steady state results more in line with empirical observations. However, this will not affect the comparison of the CUE and MSR formulations if both incorporate a single SOC pool.

Transient dynamics, i.e., responses to perturbations or varying environmental conditions such as temperature, also differ between the two formulations of respiratory C loss. For a dynamical system, e.g., Eq. 3 here, transient dynamics are typically characterized by linear stability analysis, which involves computing the eigenvalues of the the linearized system. The signs of the eigenvalues prescribe qualitative stability and the magnitudes of the eigenvalues prescribe the rate at which the system returns to steady-state. Both the CUE and MSR systems have eigenvalues with negative real parts and are thus stable in the dynamical sense (see Appendix for details). As with the discrepancy between steady-state SOC pool sizes, the difference in the eigenvalues for the two formulations depends on the relationship between CUE and $$\frac{1}{{\frac{{MSR}}{{\lambda _B}} + 1}}$$. As above, if $$CUE = \frac{1}{{\frac{{MSR}}{{\lambda _B}} + 1}}$$, pool sizes will be identical for both formulations, but the CUE formulation will almost always have a larger magnitude dominant eigenvalue, and as a result will be more responsive to perturbation than the MSR formulation (see Appendix). Thus, even if steady state pools sizes are identical for both formulations, transient dynamics, i.e., the responses to punctuated or continued perturbations, will differ between the two formulations of respiratory C loss. In Fig. 3, the subset of parameter space corresponding to the assumed and empirically quantified ranges of CUE and MSR, shown by a shaded box as in Fig. 1, is superimposed on contour plots in which the contours denote ratios of dominant eigenvalues for the CUE formulation relative to the MSR formulation. For the same ranges of CUE and $$\frac{{MSR}}{{\lambda _B}}$$ delineated in Fig. 1, the dominant eigenvalue of the CUE formulation is approximately an order of magnitude larger than the dominant eigenvalue for the MSR formulation.

In general, the CUE formulation is characterized by larger magnitude negative eigenvalues (the shaded box in Fig. 3 overlaps with ratios substantially smaller than 1), meaning that CUE formulations will both return to steady-state more quickly after a perturbation (a faster approach to steady state in Fig. 4a, c) and will track fluctuations in the environment more closely (smaller magnitude oscillations in Fig. 4b, d) given the parameter values widely used in the literaure. In contrast, the MSR formulation results in a slower and less damped approach to steady state (red lines vs. blue lines in Fig. 4a, c) because of relatively smaller magnitude negative eigenvalues, and less of a tendency to respond to high frequency environmental variation (red lines vs. blue lines in Fig. 4b), depicted here for illustrative purposes in response to variation in temperature (see Appendix for details). For larger values of $$\frac{{MSR}}{{\lambda _B}}$$, the sensitivity of the MSR formulation decreases (the magnitude of the real part of the dominant eigenvalue decreases in absolute value) due to increased respiratory loss, and can lead to much longer transient dynamics (compare the red lines in Fig. 4a, b to the red lines in to Fig. 4c, d). However, for smaller values of $$\frac{{MSR}}{{\lambda _B}}$$, the MSR formulation can more closely track high frequency environmental variation (red lines in Fig. 4d) and will exhibit transient behavior similar to that of the CUE formulation. In general, prescribing respiratory losses solely as mass specific slows down the flow of C from SOC into the atmosphere relative to a CUE formulation because C must enter the biomass C pool and leave as CO2 after a period time dictated by the value of MSR. Oscillatory behavior is known to be a feature of non-linear models of SOC dynamics [20], but whether long period oscillations exist in nature is unknown. Perhaps more importantly though, the difference in responsiveness can result in different steady-state pool sizes for the MSR formulation in the presence of environmental variation (the difference between in the red dashed lines in panels a and b and between panels c and d in Fig. 4). As a consequence, the CUE formulation will therefore not generate the potentially significant changes in SOC pool size in response to environmental variation that result from the MSR formulation.

The way that microbial respiration is formulated in models of SOC dynamics influences projections of terrestrial C fluxes. Different formulations of microbial respiration can generate discrepancies in steady-state SOC and microbial biomass C pool sizes of more than two orders of magnitude, depending on assumed relative rates of respiratory and non-respiratory losses for microbes reported in the literature. Contrasting formulations of microbial respiration can also drive divergent transient responses to environmental perturbations, with CUE formulations exhibiting a reduced propensity for transient SOC and microbial biomass C dynamics than formulations based on mass specific respiratory losses. Thus, the distinction between efficiency based respiratory losses and mass specific respiratory losses is not merely syntactic or semantic. However, because there is significant uncertainty regarding parameters that influence discrepancies between formulations (e.g., λB and v), it is currently impossible to quantify the exact magnitude of these discrepancies in both short and long-term dynamics. Regardless, the results presented here highlight the critical dependence of modeled SOC stock dynamics on respiratory and non-respiratory C losses from microbes, on the way these fluxes are represented mathematically, and the need to more accurately quantify both fluxes. As a consequence, the growing community of soil modelers must reassess how microbial soil respiration is formally incorporated in models, and the parameter values that prescribe microbial efficiency.

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## Acknowledgements

We dedicate this paper to Dr. Henry Gholz, who was an inspiration and a supporter of our work on soil C dynamics. We wish to thank Chao Song and Stefano Manzoni for conversations and comments that improved the manuscript. We also thank Bob Sinsabaugh, Josh Schimel, Will Wieder, and an anonymous reviewer for comments on previous versions of the manuscript. This work was supported by a grant from the US National Science Foundation (DEB-0950095).

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Correspondence to Ford Ballantyne IV.

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## Mathematical appendix

### Mathematical appendix

To simplify analysis, we non-dimensionalized Eq. 3 to arrive at

$$\begin{array}{l}\frac{{dS_C^\prime }}{{d\tau }} = I^\prime - B_C^\prime \frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - S_C^\prime \lambda _S^\prime + B_C^\prime \varepsilon \\ \frac{{dB_C^\prime }}{{d\tau }} = \left\{ {\begin{array}{*{20}{l}} {B_C^\prime \left[ {CUE \times \frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - 1} \right];} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {B_C^\prime \left[ {\frac{{v\prime S_C^\prime }}{{(1 + S_C^\prime )}} - MSR^\prime - 1} \right];} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$

in which $$S_C^\prime = \frac{{S_C}}{K}$$, $$B_C^\prime = \frac{{B_C}}{K}$$, $$\tau = \lambda _Bt$$, $$I^\prime = \frac{I}{{K\lambda _B}}$$, $$v\prime = \frac{v}{{\lambda _B}}$$, $$\lambda _S^\prime = \frac{{\lambda _S}}{{\lambda _B}}$$, and $$MSR^\prime = \frac{{MSR}}{{\lambda _B}}$$.

The equilibria are given by

$$\begin{array}{l}\hat S_C = \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{\left( {v\prime CUE - 1} \right)}};} \hfill & {CUE\,{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{1}{{\left( {\frac{{v\prime }}{{MSR^\prime }} - 1} \right)}};} \hfill & {MSR^\prime \,{\mathrm{formulation}}} \hfill \end{array}} \right.\\ \hat B_C = \left\{ {\begin{array}{*{20}{l}} {\frac{{I^\prime - \left( {\frac{{\lambda _S^\prime }}{{v\prime CUE - 1}}} \right)}}{{\left( {\frac{1}{{CUE}} - \varepsilon } \right)}};} \hfill & {CUE\;{\mathrm{formulation}}} \hfill \\ {{\mathrm{or}}} \hfill & \, \hfill \\ {\frac{{I^\prime - \frac{{\lambda _S^\prime }}{{\left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)}}}}{{MSR^\prime + 1 - \varepsilon }};} \hfill & {MSR\;{\mathrm{formulation}}} \hfill \end{array}} \right.\end{array}$$
(6)

In the absence of life,

$$\hat S_C^\prime = \frac{{I^\prime }}{{\lambda _S^\prime }} = \frac{I}{{\lambda _S}},$$
(7)

and for biota to persist, v′ > CUE and $$\lambda _S^\prime$$ < I′(vCUE−1) for the CUE formulation, and v′ > MSR′ + 1 and $$\lambda _S^\prime < I^\prime \left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)$$ for the MSR formulation.

The ratio of equilibrium SOC for the MSR formulation to equilibrium SOC for the CUE formulation is given by

$$\frac{{\hat S_C^{\prime MSR}}}{{\hat S_C^{\prime CUE}}} = \frac{{\left( {v\prime CUE - 1} \right)}}{{\left( {\frac{{v\prime }}{{MSR^\prime + 1}} - 1} \right)}}.$$
(8)

The ratio will equal 1 if $$CUE = \frac{1}{{MSR^\prime + 1}}$$, denoted by the bold line in the figure panels.

To address the discrepancy in the response to perturbation, we analyze the characteristic equation resulting from the determinants of the Jacobians for the two formulations. The Jacobian evaluated at the equilibrium for both systems assumes the general form

$$\hat J_X = \left[ {\begin{array}{*{20}{c}} {\frac{{ - v\prime \hat B_C^\prime }}{{\left( {1 + \hat S^\prime_C } \right)^2}} - \lambda^{\prime} _{S}} & {\frac{{ - v\prime \hat S^\prime_C }}{{1 + \hat S^\prime_C }} + \epsilon } \\ {\frac{{v\prime \hat B_C^\prime X}}{{\left( {1 + \hat S^\prime_C } \right)^2}}} & 0 \end{array}} \right]$$
(9)

in which X = CUE or 1 and $$\frac{{ - v\prime \hat S^\prime }}{{1 + \hat S^\prime }} = \frac{1}{{CUE}}$$ or MSR′ + 1 for the CUE and MSR formulations, respectively. The eigenvalues for the associated characteristic equation are

$$\frac{{ - (A + \lambda _S^\prime ) \pm \sqrt {(A + \lambda _S^\prime )^2 + 4AX( \epsilon - B)} }}{2}$$
(10)

in which $$A = \frac{{v\prime \hat B_C^\prime }}{{\left( {1 + \hat S^\prime_C } \right)^2}}$$ for both formulations and $$B = \frac{1}{{CUE}}$$ or MSR′ + 1 for the CUE and MSR formulations, respectively. In the absence of life, we simply recover $$\lambda _S^\prime$$ as the only eigenvalue. Thus, the addition of living microbes to the soil system increases the absolute magnitude of the dominant eigenvalue, thereby decreasing return time to equilibrium and increasing system stability. The real parts of all eigenvalues for both formulations are negative, reflecting the stability of the steady-state. To most directly compare stability between formulations, we assume that the equilibria are the same for both, i.e $$CUE = \frac{1}{{MSR^\prime + 1}}$$. The discrepancy in stability between the two formulations is most easily seen for the case of no recycling, ε = 0, because the only difference in the eigenvalues arises in the AXB term. For the CUE formulation, AXB simply reduces to A because X = CUE and $$B = \frac{1}{{CUE}}$$, which also means that the response to perturbation is independent of CUE in the absence of recycling. For the MSR formulation, AXB = A(MSR′ + 1) because X = 1 and B = MSR′ + 1. A is the same by assumption, which means that in the case of no recycling the return to identical equilibrium will be faster for the CUE formulation. In general, increased rates of recycling, ε > 0, promote a faster return to equilibrium and shorter duration transient dynamics. MSR′ > 1, which is substantiated in other work [18], is a sufficient condition to guarantee the same damped response to perturbation and longer transient dynamics of the MSR formulation as for the case of no recycling. The influence of CUE on stability is more severely constrained because both ε and CUE are bounded by 0 and 1, whereas MSR′ can, and will often be significantly greater than 1.

Equations 3 were simulated with I′ = 1, v′ = 16, CUE = [0.2,0.5], MSR′ = [4,10], and $$\lambda _S^\prime$$ = 0.1 to generate temporal trajectories of SC and BC. To simulate the response to temperature variation, we made the standard assumption that v exhibits an Arrhenius temperature dependence [1], and incorporated variation in the following manner,

$$v^{\prime \prime } = v\prime \alpha e^{\frac{{ - E_a}}{{R\left( {288 + 5sin\frac{{\pi \tau }}{2}} \right)}}},$$
(11)

with α = 109, Ea = 50,000 and R = 8.134.

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Ballantyne IV, F., Billings, S.A. Model formulation of microbial CO2 production and efficiency can significantly influence short and long term soil C projections. ISME J 12, 1395–1403 (2018). https://doi.org/10.1038/s41396-018-0085-1

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text:
# Minimizing the cost of a path in a dynamic system

So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ is $c$, and my position $P_t$ changes in the following way:

$$P_t = (1-\gamma) P_{t-1} + \gamma S(t)$$ where $\gamma<1$ limits fast I can move. Which can be written $$\frac{P_t - P_{t-1}}{\gamma} = - P_{t-1} +S(t)$$

Sadly, I have a drift towards zero of $-P$, and $S$ has to drive me away from zero, towards $c$. So $S(t)$ is the "size" of my push towards $c$ at date $t$. The cost of $S$ is given by

$$C(T,S(t)) = \sum_1^{T} (S(t))^2$$ So I am penalized for large movements in my position, and big pushes towards $c$. I would like to calculate $$\min_{T,S(\cdot)} C(T,S(t))$$ subject to $P_0 =0$, $P_T = c$.

That is, if I am free to choose the number of steps it takes to get there, and the path can be whatever I want, what is the cost minimizing route from 0 to $c$?

Clearly, you could set $S(1) = c / \gamma$, so I arrive immediately, at cost $( c/\gamma )^2$. So that is an upper bound on the cost. I could do it in two steps; If I step $x$ the first period, I will need to step $$\frac{(c-(1-\gamma) \gamma x)}{\gamma}$$ the second period, for a total cost of of $$x^2 + (\frac{(c-(1-\gamma) \gamma x)}{\gamma})^2,$$ Which is minimized for $x=\frac{c (1-\gamma)}{\gamma (2-2 \gamma+\gamma^2 )}$, which therefore costs $$\frac{c^2 (1-\gamma)^2}{\gamma^2 (2-2 \gamma+\gamma^2)^2}+\frac{(c+\frac{c (1-\gamma) (-1+\gamma)}{2-2 \gamma+\gamma^2})^2}{\gamma^2}$$ which is a lower cost than the one-step option, since $\gamma<1$. It is not clear to me how to extend this analysis to large $T$ - is there a simple recursive form for the cost minimizing $k$-step path, and the problem reduces to choosing the best $k$?

-

For fixed $T$, or $k$ which is the same thing, it turns out that you are just minimizing a quadratic function with a linear constraint, and a simple solution exists. However, there is no best $k$ -- you can make the cost arbitrarily small by taking longer and longer to get there.
Let $s$ be the vector with components $s_i = S(i)$ for $i = 1, \ldots, k$. Then if you expand out the recursion in your first equation $P_k = \gamma s_k + (1-\gamma) P_{k-1}$, you will find that \begin{align} P_k &= \sum_{i=1}^k \gamma(1-\gamma)^{k-i} s_i + (1-\gamma)^k P_0 \\ &= b^Ts + 0 \end{align} where $b$ is a vector whose components are $b_i = \gamma(1-\gamma)^{k-i}$. So your problem is that of minimizing $C(s) = \sum_{i=1}^k s_i^2 = s^Ts$, subject to the constraint $b^Ts = c$. Using Lagrange multipliers, you can get a straightforward solution. However, you will find that the cost decreases asymptotically with $k$.


text: The range of a function is all actual output values. Let’s define $f \colon X \to Y$ to be a continuous, bijective function such that $X,Y \in \mathbb R$. Accelerated Geometry NOTES 5.1 Injective, Surjective, & Bijective Functions Functions A function relates each element of a set with exactly one element of another set. Let b 2B. So x 2 is not injective and therefore also not bijective and hence it won't have an inverse.. A function is surjective if every possible number in the range is reached, so in our case if every real number can be reached. Bijective. The Attempt at a Solution To start: Since f is invertible/bijective f⁻¹ is … is bijective, by showing f⁻¹ is onto, and one to one, since f is bijective it is invertible. Let f : A !B be bijective. The function f: ℝ2-> ℝ2 is defined by f(x,y)=(2x+3y,x+2y). A function is called to be bijective or bijection, if a function f: A → B satisfies both the injective (one-to-one function) and surjective function (onto function) properties. A bijective group homomorphism $\phi:G \to H$ is called isomorphism. Let f 1(b) = a. Thus, bijective functions satisfy injective as well as surjective function properties and have both conditions to be true. Then f has an inverse. Theorem 1. The codomain of a function is all possible output values. the definition only tells us a bijective function has an inverse function. The domain of a function is all possible input values. If we fill in -2 and 2 both give the same output, namely 4. I've got so far: Bijective = 1-1 and onto. I think the proof would involve showing f⁻¹. A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y. Alternatively, f is bijective if it is a one-to-one correspondence between those sets, in other words both injective and surjective. Since f is injective, this a is unique, so f 1 is well-de ned. 1. Now we much check that f 1 is the inverse … Let f: A → B. Bijective Function Examples. it doesn't explicitly say this inverse is also bijective (although it turns out that it is). it's pretty obvious that in the case that the domain of a function is FINITE, f-1 is a "mirror image" of f (in fact, we only need to check if f is injective OR surjective). This is equivalent to the following statement: for every element b in the codomain B, there is exactly one element a in the domain A such that f(a)=b.Another name for bijection is 1-1 correspondence (read "one-to-one correspondence).. We will de ne a function f 1: B !A as follows. 1.Inverse of a function 2.Finding the Inverse of a Function or Showing One Does not Exist, Ex 2 3.Finding The Inverse Of A Function References LearnNext - Inverse of a Bijective Function … A bijection of a function occurs when f is one to one and onto. Click here if solved 43 An example of a function that is not injective is f(x) = x 2 if we take as domain all real numbers. It means that each and every element “b” in the codomain B, there is exactly one element “a” in the domain A so that f(a) = b. Please Subscribe here, thank you!!! The above problem guarantees that the inverse map of an isomorphism is again a homomorphism, and hence isomorphism. In mathematics, a bijective function or bijection is a function f : A → B that is both an injection and a surjection. In order to determine if $f^{-1}$ is continuous, we must look first at the domain of $f$. Yes. Proof. Let f : A !B be bijective. Show that f is bijective and find its inverse. 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text:
# Solving logarithm problem for Y

1. Aug 23, 2008

### fr33pl4gu3

(For those values of x for which a solution exists), solve the following equation for y

3e3y-6 = 2x2-1

What does it mean by the values of x exists??

Last edited: Aug 23, 2008
2. Aug 23, 2008

### Hootenanny

Staff Emeritus
Re: logarithm

Because of the nature of the domain of the logarithm one can not evaluate $y=\ln x$ for all x. Specifically the domain of the logarithm is all positive numbers. Therefore, we say that no solution exists for $y=\ln x$ in the domain $x\in\left(-\infty, 0\right]$.

So the question is asking you to solve the equation for y, for all values of x which exist. I hope that makes sense.

Last edited by a moderator: Aug 23, 2008
3. Aug 23, 2008

### Swerting

Re: logarithm

Say you have $$ln(x)=y$$.
This could be seen as $$e^y=x$$
It is impossible to raise any real number to any power and have it equal 0 or any number below that (feel free to try, in fact, I encourage it). Since it can't exist as $$x\in(-\infty,0)$$, there are only certain numbers it can exist as.

4. Aug 23, 2008

### tiny-tim

Hi fr33pl4gu3!


text:
# Puiseux Series: a challenging computation

Recently we encountered an interesting computational problem that seems to be difficult for existing computer algebra systems, but which we have been able to complete with components of the Oscar computer algebra system we are implementing.

The problem came up in some research Daniel Schultz and I have been doing, to do with computing a new kind of modular equation. The theoretical details are not so important for the purposes of this blog, so I’ll only give a quick overview of where the problem came from.

The Dedekind eta function is given by the following infinite q-series product $\eta(q) = q^{1/24}\prod_{n=1}^\infty (1 - q^n),$ in the “nome” $q$. Naturally, we can only work with a finite precision approximation out to $O(q^N)$ for some precision $N$, on a computer.

Here, $q = exp(2\pi i\tau)$ for $\tau$ in the complex upper half plane $\mathbb{H}$. This makes $\eta$ a function of $\tau$ in the upper half plane. In fact, $\eta(\tau)$ is a modular form of weight $1/2$, if you know about such things.

We have been investigating identities for certain functions that are defined in terms of the Dedekind eta function, that hold for all $\tau \in \mathbb{H}$.

One of the classical Weber modular functions is $\mathfrak{f}_2(\tau) = \sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}.$

In this blog we work with a generalisation of this, defined by $\mathfrak{m}_{n, \infty}(\tau) = \sqrt{n}\frac{\eta(n\tau)}{\eta(\tau)}.$ Obviously $n = 2$ is the classical Weber function $\mathfrak{f}_2(\tau)$.

For any $m \in \mathbb{N}$ we define $x_{n, m}(\tau) = \frac{\mathfrak{m}_{n,\infty}(mn\tau)}{\mathfrak{m}_{n,\infty}(\tau)}, \;\; y_{n,m}(\tau) = \frac{\mathfrak{m}_{n,\infty}(n\tau)}{\mathfrak{m}_{n,\infty}(m\tau)}.$

We have been computing the minimum polynomial relation $P_{n,m}(x_{n,m}(\tau), y_{n,m}(\tau)) = 0$ that holds as an identity for all $\tau \in \mathbb{H}$. The bivariate polynomial $P_{n,m}(X, Y)$ has coefficients in $\mathbb{Z}$.

We call such a relation $P_{n,m}(x, y) = 0$ a modular equation of degree $m$ for level $n$ functions, though terminology varies wildly here, and these certainly don’t relate directly to the modular equations of Schlaefli, Weber, Jacobi, Russel, Ramanujan, etc.

As an example, the minimum relation for $n = 2$ (corresponding to the original Weber modular function) and $m = 3$ is given by $P_{2, 3}(x, y) = x^5y + xy^5 - 1 = 0,$ for all $\tau \in \mathbb{H}$.

## Computing the modular equations

There are various numerical ways to compute such equations, e.g. numerically compute the value of the eta function at various random values of $\tau$ in the upper half plane and try to find the relation between values of our functions using LLL.

This is not an efficient approach, and it also requires a lot of work to decide on a provable bound on numerical precision.

There are various algebraic approaches, one of which forms the basis of the paper we are working on. In particular, we can give what we believe to be sharp bounds on the degrees of $x$ and $y$ in our relations $P_{n, m}(x, y) = 0$.

We can currently only prove our bounds are sharp in some cases, but experimentally they seem to always be sharp.

For any given values of $m$ and $n$, we can do a computation to check they are sharp. This computation boils down to finding the nullspace of a certain matrix over $\mathbb{Q}$ and checking that its nullity is 1. If it is, the bounds are sharp.

If the nullity is $1$, then we can prove that the generator of the nullspace gives the coefficients of the minimal polynomial $P_{n, m}(X, Y) \in \mathbb{Z}[X, Y]$ that appears in our modular equation.

## The interesting computational challenge

The interesting challenge is not in computing the rank of the matrix (this can be done modulo a random prime, which allows us to bound the nullity above). The real challenge is in computing the entries of the matrix itself.

As an example of this challenge, we explicitly describe this matrix for the case $n = 11$ and $m = 4$. The resulting matrix is 9001x4816 over $\mathbb{Q}$.

First we define the functions $s(q) = x(q)/y(q),\;\; t(q) = x(q)^{12}.$

We then compute these out to $O(q^{9001})$. Although the q-series of $s(q)$ and $t(q)$ are in the nome $q$, they are defined in terms of functions in $q^{1/24}$. This means that naively the Puiseux series will be using an internal precision of $24\times 9001 = 216024$. In fact, all of the systems that we discuss in this blog, with true Puiseux series implementations, have to work to this precision initially.

Once we have computed $s(q)$ and $t(q)$ to this precision, we compute the q-series of $s^it^j$ for $i \in [0, 300]$ and $j \in [0, 15]$.

The entries in each column of our matrix are the q-series coefficients of one of the $s^it^j$.

Thus, there are $301\times 16 = 4816$ columns and $9001$ rows in the matrix, since the q-series must be computed out to $O(q^{9001})$.

This gives us the matrix for $n = 11$ and $m = 4$ which conjecturally has nullity 1.

## Computation of the q-series in Oscar

First we should write out how the computation can be done in Oscar (note that the $\sqrt{11}$ cancels in the functions we are using and can be omitted).

As we will be working with Puiseux series in $q^{1/24}$ we will need to work with $9001\times 24$ terms in order to compute everything out to $O(q^{9001})$.

Here is the code required to compute the functions $s$ and $t$.

using Nemo

R, q = PuiseuxSeriesRing(ZZ, 9001*24, "q")

m(q) = divexact(eta_qexp(q^11), eta_qexp(q))

x(q) = divexact(m(q^44), m(q))
y(q) = divexact(m(q^11), m(q^4))

s(q) = divexact(x(q), y(q))
t(q) = x(q)^12

This does not take any appreciable time, as nothing is actually computed.

The next step is to populate an array with the q-series expressions for $s(q)^i$ for $i \in [0, 300]$ and $t(q)^j$ for $j \in [0, 15]$ so that these powers are not recomputed over and over again. We do this efficiently with one multiplication for each entry.

T1 = typeof(q)
S = Array{T1, 1}(301)
T = Array{T1, 1}(16)

sq = s(q);
tq = t(q);

S[1] = sq^0
T[1] = tq^0

for i = 2:301
S[i] = S[i - 1]*sq
end

for i = 2:16
T[i] = T[i - 1]*tq
end

The final step is to compute all the $s^it^j$ for $i$ and $j$ in the required ranges and populate the matrix. To keep things simple, we just compute all the products and don’t bother computing the matrix.

for i = 0:300
for j = 0:15
p = S[i + 1]*T[j + 1]
end
end

This by far consumes the largest part of the total runtime in Oscar, at 605s.

## Computation of the q-series in Mathematica

The systems we decided to use for this blog are SageMath, Mathematica, Magma and Oscar.

Although we expected this to be a simple task, we encountered issues straight away.

We had some existing Mathematica code that we had been using in webMathematica, so it was only necessary to alter the code.

Unfortunately, due to a recent update in webMathematica, there is a parser error that causes newlines to be interpreted as products in some cases, meaning our code no longer works. Therefore it is not possible to use webMathematica until they fix this issue.

However, we do know that once bignums are encountered in power series, Mathematica switches to a generic representation, and things get really slow. We found that the computation was not really practical in Mathematica when the code was working.

In the end, we were able to write something that worked in desktop Mathematica.

Here are the basic definitions.

$Eta][q_] := q^(1/24)*QPochhammer[q,q]; f[q_] := \[Eta][q^11]/\[Eta][q]; x[q_] := f[q^44]/f[q]; y[q_] := f[q^11]/f[q^4]; s[q_] := x[q]/y[q]; t[q_] := x[q]^12;  Computing s and t to precision O(q^9001) takes about 4 minutes. The PowerExpand is necessary to enforce the rule (q^a)^(1/24) = q^(a/24), which is true in this case. This results in power series data types using only integral exponents. Timing[ sq=Series[PowerExpand[s[q]],{q,0,9000}]; tq=Series[PowerExpand[t[q]],{q,0,9000}]; ][[1]] 243  The timing here is in seconds. We next need to fill in powers of s and t, which the following code does in 3.5 hours. Timing[ sp[0] = 1 + O[q]^9001; Do[sp[i] = sp[i-1]*sq, {i,1,300}]; tp[0] = 1 + O[q]^9001; Do[tp[i] = tp[i-1]*tq, {i,1,15}]; ][[1]] 12306  The following code should then take at least 16 times as long as the code above, which puts a conservative estimate on the total time to fill in the matrix to be more than two days. Timing[Do[sp[i]*tp[j],{i,0,300},{j,0,15}]][[1]]  ## Computation of the q-series in SageMath The first issue we encountered in SageMath is that we were unable to figure out how to do Puiseux series. SageMath has power series and Laurent series, but raising to fractional powers doesn’t seem to be available. In this specific case, just because we are using eta quotients, it is possible to work out how to write the series in terms of Laurent series instead. This isn’t always convenient, but it works in our case after some minutes with a pen and paper (at least, the third time we checked our arithmetic, it worked). But then we encounter our next issue. The qexp_eta function in SageMath does not seem to work with Laurent series. It seems to require a power series ring. Of course, we can also work with power series, and if not, Sage can coerce to Laurent series, so this is just a minor annoyance. The next issue is that the qexp_eta function does not take a “nome” as input, but rather a ring in which you want to compute the expansion of the eta product, minus the leading q^{1/24}. This is fine if you actually want to compute \eta(q), but not if you want to compute \eta(q^n) for example, as we do. Trying to solve this leads to another problem that is subtle and that I was only able to detect because I have implemented numerous power series modules myself. Naively, what seems logical is that we ought to substitute q^n for q in the output of qexp_eta to get \eta(q^n). One can perform such substitutions in SageMath using either the subs function, or by functional evaluation of the q-series at q^n, i.e. by typing f(q^n), where f is the power series in question. However, SageMath has a notion of exact power series, i.e. power series given to infinite precision. The expression q^n is such an exact power series in Sage. Therefore, substituting it into the expression from eta_qexp results in an expression with about n times the precision. Because of the dense representation in Sage, this would kill performance. Of course we can replace q with q + O(q^{9002}) to emulate not working with infinite precision power series. However, this still isn’t sufficient. Both Magma and Oscar do not interpret eta(q^n) as substituting (q + O(q^{9002}))^n into the q-series for eta. They interpret it to mean that we want the q-series expansion of eta in a ring with relative precision 9001. This is subtly different, though it takes a little thought to see why. The precision that the user supplies when creating a power series ring in Sage is not applied as a precision cap (after all, one can have infinite precision power series in such a ring). It’s only used as a default precision for things that cannot be computed to infinite precision, and where no precision is implied by the computation, such as 1/(1 + q). Because of this completely different model of power series, it’s going to be hard to get an apples for apples comparison with Sage. As a first approximation, we can simply replace q with q + O(q^{9002}) in the Sage computations. But this will result in Sage working to higher precision than the other systems. We can truncate the power series to q^{9001}. This is technically equivalent to using an absolute cap, whereas we really want a relative precision cap, so this advantages Sage somewhat. We question whether we are just implementing a power series module on top of Sage, at this point. First we implement the functions we require. R.<q> = PowerSeriesRing(ZZ, default_prec=9001) def m(q): return qexp_eta(R, 9001)((q + O(q^9002))^11)/qexp_eta(R, 9001)(q + O(q^9002)) def x(q): return m((q + O(q^9002))^44)/m(q + O(q^9002)) def y(q): return m((q + O(q^9002))^11)/m((q + O(q^9002))^4) def s(q): return (q + O(q^9002))^15*x(q+O(q^9002))/y(q+O(q^9002)) def t(q): return (q + O(q^9002))^215*x(q+O(q^9002))^12  As usual, this takes no time as nothing is actually computed. Next we compute the arrays of powers of s and t. To ensure Sage is not disadvantaged, we truncate s(q) and t(q) to the same precision as it is computed in Oscar. sq = s(q) + O(q^9016); tq = t(q) + O(q^9216); S = [sq^0] T = [tq^0] for i in range(1, 301): S.append(S[i - 1]*sq) for i in range(1, 16): T.append(T[i - 1]*tq)  This takes around 190s. Finally, we compute the products of the powers. for i in range(0, 301): for j in range(0, 16): p = S[i]*T[j]  This takes around 1650s. ## Computation of the q-series in Magma Magma supports Puiseux series. As with Oscar, it requires one to specify the internal Laurent series precision, depending on the fractional power of q you are using. For example, to compute to O(q^n) with Puiseux series in q^{1/24} requires a Magma PuiseuxSeriesRing precision of at least 24n. The DedekindEta function computes the Puiseux expansion of the Dedekind eta function in Magma. But it immediately complains that the rationals are not contained in the coefficient ring of the argument. So we must work with Puiseux series over the rationals. After this, it’s easy to define the functions we require in Magma. R<q> := PuiseuxSeriesRing(Rationals(), 9001*24); function m(q) return DedekindEta(q^11)/DedekindEta(q); end function; function x(q) return m(q^44)/m(q); end function; function y(q) return m(q^11)/m(q^4); end function; function s(q) return x(q)/y(q); end function; function t(q) return x(q)^12; end function;  As in Oscar, this takes negligible time, as nothing is actually computed. At the next step we try to compute arrays of powers of s(q) and t(q). In order to ensure Magma is not at a disadvantage we truncate s(q) and t(q) to the same precision as it is computed in Oscar, though this makes little difference. sq := s(q) + O(q^9016); tq := t(q) + O(q^9216); S := [sq^0]; for i := 1 to 300 do S := Append(S, S[i]*sq); end for; T := [tq^0]; for i := 1 to 15 do T := Append(T, T[i]*tq); end for;  This takes approximately 104s. One has to be very careful to precompute s(q) and t(q). Otherwise this takes 10s each time each of these are computed. Without being careful about this, Magma could take over an hour to compute all the powers. In comparison, the total Oscar time does not change radically if we don’t precompute these q-series. Finally we compute the products of the powers of s and t. for i := 1 to 301 do for j := 1 to 16 do p := S[i]*T[j]; end for; end for;  This takes about 2488s. ## Summary Here is a table summarising the results. We give the times to compute the arrays of powers of s(q) and t(q) separately, since these show a different aspect of the performance comparison than the total times for the whole benchmark n = 11, m = 4 SageMath Mathematica Magma Oscar Setup 190s ~ 3.5 hours 104s 38s Total 1840s ~ 2 days 2592s 643s I’m not sure what we learn from this, other than that benchmarking complex calculations can be somewhat of a deep hole at times. Artificially truncating series and doing additional precomputation to keep times in check doesn’t seem like it reflects what a user would do. We should point out that we have heard that Magma has very fast nullspace code (very likely much faster than our own). So perhaps that would make a more reasonable benchmark for a future blog. ## Why are Oscar q-series fast Assuming that we accept that Oscar has relatively fast Puiseux series, we should say a few words about why that is. In Oscar, we store Puiseux series internally as Laurent series along with a common denominator for the exponents. The Puiseux series precision is automatically computed in each operation. When combining q-series in different nomes, the two expressions are first brought to expressions in the same nome. Laurent series themselves are also stored with a valuation, precision and a scale factor. Thus regularly spaced q-series such as q + q^4 + q^7 + O(q^8) would be stored with a scale factor of 3. This radically reduces the space required to store them, and dramatically speeds up computations. This scale factor can be used regardless of the valuation or precision, though this does make computation quite complex internally, as for example, two Laurent series with a scale factor of 3 may not sum to a Laurent series with a scale factor of 3, etc. Finally, Oscar makes use of fast power series arithmetic over \mathbb{Z}, implemented in Flint. ## An example modular equation The modular equation for n = 11, m = 4 is really too huge to reproduce here (and in fact, we have not bothered computing it). We were only interested in checking whether our bounds on the degrees are tight, for this benchmark. To finish this blog, we give an example of an actual modular equation that we have computed using our methods. In terms of the functions x_{n,m} and y_{n,m} described above, the modular equation for n = 5 and m = 8 is \[\begin{multline}x^{17}y - 4x^{16}y^2 + 10x^{15}y^3 + x^{15}y - 20x^{14}y^4 + 6x^{14}y^2 +\\ 35x^{13}y^5 - 35x^{13}y^3 + 2x^{13}y - 52x^{12}y^6 + 61x^{12}y^4 +\\ 13x^{12}y^2 + 68x^{11}y^7 - 59x^{11}y^5 - 39x^{11}y^3 + 3x^{11}y -\\ 80x^{10}y^8 + 42x^{10}y^6 + 2x^{10}y^4 - 12x^{10}y^2 + 85x^9y^9 -\\ 33x^9y^7 + 67x^9y^5 + 40x^9y^3 + 5x^9y - 80x^8y^{10} +\\ 30x^8y^8 - 52x^8y^6 - 58x^8y^4 - 10x^8y^2 + 68x^7y^{11} -\\ 33x^7y^9 + 16x^7y^7 - 9x^7y^5 - 8x^7y^3 - 7x^7y - 52x^6y^{12} +\\ 42x^6y^{10} - 52x^6y^8 + 78x^6y^6 + 37x^6y^4 + 10x^6y^2 +\\ 35x^5y^{13} - 59x^5y^{11} + 67x^5y^9 - 9x^5y^7 - 49x^5y^5 +\\ 12x^5y^3 - x^5y - 20x^4y^{14} + 61x^4y^{12} + 2x^4y^{10} - 58x^4y^8 +\\ 37x^4y^6 - 36x^4y^4 - 2x^4y^2 - x^4 + 10x^3y^{15} - 35x^3y^{13} -\\ 39x^3y^{11} + 40x^3y^9 - 8x^3y^7 + 12x^3y^5 + 5x^3y^3 - x^3y -\\ 4x^2y^{16} + 6x^2y^{14} + 13x^2y^{12} - 12x^2y^{10} - 10x^2y^8 +\\ 10x^2y^6 - 2x^2y^4 + 6x^2y^2 + xy^{17} + xy^{15} + 2xy^{13} +\\ 3xy^{11} + 5xy^9 - 7xy^7 - xy^5 - xy^3 + xy - y^4 = 0\end{multline}$

Verifying this as a q-series identity to high precision in Oscar takes 13s whereas it takes around 75s to the same precision in Magma (assuming we precompute the q-series expansions of x and y).


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# Getting Started

## Big Changes to O’Reilly’s Authoring Tools

If you worked with us on a book prior to fall 2012, you probably committed your files to a Subversion repository and used a special commit string to trigger PDF builds. You probably even accessed these very guidelines via an SVN repo. Well, our developers have been at work creating a new authoring platform called Atlas that is now available to you as a replacement for SVN. Atlas is a git-managed, wiki-like authoring platform for creating books, and we’re excited about it. See the Atlas guidelines for more info.

Although Atlas was originally designed for working with AsciiDoc files—because AsciiDoc is a lightweight language and can be easily written in a web interface—it can support DocBook files as well. There are several advantages to using Atlas with DocBook:

• Manage your files in git.

• Build PDFs that use the new CSS-based stylesheets.[1]

• More tolerant PDF builds.[2]

These guidelines are still very much relevant—you’ll still need to use an XML editor, make correct markup decisions, and deliver valid files to Production.

### How Atlas works

1. Receive an invite from an O’Reilly editor, Tools staff member, or another author, and create an account.

2. Write in DocBook as described in these guidelines.

3. Commit and push your files to your Atlas git repo.

4. When you’re ready, build your book. Atlas interfaces with O’Reilly’s publishing toolchain and can generate PDF, EPUB, and Mobi files for you on the fly. These files look just like those for sale on oreilly.com.

Please see the Atlas guidelines for more on the following:

The Atlas guidelines often refer to .asciidoc files, but you can do pretty much all the same steps using .xml files. One difference to note is that when working with XML, if you go to the Build tab, you will see the message, “This book has a book.xml file, so the index creator is disabled.” That’s nothing to worry about—just select the formats you wish to generate and click the “Build!” button.

If you have already started working in a Subversion repository and you’d like to switch to Atlas, just send a request to and we’ll move your files over. We can also convert your DocBook files to AsciiDoc if you’d like to switch formats.

## Why DocBook?

DocBook is an OASIS standard for XML that is ideal for writing long, technical documents that have complex structure and cross-references. With its semantic tagging, DocBook can be rendered as PDF for printing, HTML, man pages, audio, or even Braille. This versatility and reusability make DocBook 4.5 XML the preferred format for O’Reilly books.

Not only does this document explain how to start writing in DocBook, but it was created using the same markup and toolchain used for a typical O’Reilly book, so you can treat it as a model for your own manuscript. Chapter 2 contains information about markup.

We’ve got nothing against LaTeX, troff, or any other formatting markup system. But typesetting markup like LaTeX is inherently focused on formatting—font size, margins, alignment, etc. We’d rather you spend your time focused on the semantic structure of your book (how sections are divided, which information goes in a sidebar versus a note, and so on), and that’s where DocBook shines. In the same way most well-designed websites separate content from formatting using XHTML and CSS (XHTML for the content, and CSS for the formatting), DocBook lets us abstract formatting decisions away from content decisions.

Particularly as O’Reilly expands its efforts to provide content in multiple formats and at multiple stages of the content’s life cycle, cost-effectively generating multiple, distinct output formats from the same source document becomes critical, even though it means losing some degree of granular control over the output.

What does that mean for your book? It means that you’ll be able to view drafts of your book throughout the authoring process that are formatted much as they will be for print, and it means once your book is finished, it will go live almost immediately on Safari Books Online and other digital and ebook retail channels, rather than needing to first be converted into DocBook from another format, which can take a week or longer.

### Will What I See in the XML Editor Mirror the PDF Builds?

DocBook markup specifies the structure and semantics of your document, but not the appearance. DocBook isn’t a WYSIWYG format, so it will display differently in your XML editor than it will after rendering to PDF and other formats downstream. This means not only will fonts and formatting display differently, but line breaks may differ as well.

The O’Reilly toolchains that transform your DocBook into its final form (both print and downstream electronic formats) rely on semantic XML tags that you apply to the elements of your text. Those tags are then processed in combination with customized CSS stylesheets and a commercial processor to render your content.

### For More on DocBook

This guide is tailored to help authors get the most out of O’Reilly’s DocBook authoring toolchain. For more general information on DocBook XML, see Norm Walsh’s excellent DocBook: The Definitive Guide. You’ll find reference pages for each DocBook element, including a content model, a description of purpose, a list of parents that can contain it and children that can be nested in it, and allowed attributes. Other resources that may be helpful include:

[1] The Nutshell and Pocket Reference series are not compatible with Atlas at this time.

[2] If you’ve ever had to decipher an error log generated when an SVN commit-hook build fails, you’ll be happy about this one. One potential downside of this is there’s less protection against invalid markup. See “Validating Your XML” for more on this.


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# Moving average

34,203pages on
this wiki

In statistics, a moving average is one of a family of similar techniques used to analyze time series data. A moving average series can be calculated for any time series. Moving averages are used to smooth out short-term fluctuations, thus highlighting longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly.

Mathematically, each of these moving averages is an example of a convolution. These averages are also similar to the low-pass filters used in signal processing.

## Simple moving averageEdit

A simple moving average (SMA) is the unweighted mean of the previous n data points. For example, a 10-day simple moving average of closing price is the mean of the previous 10 days' closing prices. If those prices are $p_{M}$, $p_{M-1}$... $p_{M-9}$ then the formula is

$\textit{SMA} = { p_{M} + p_{M-1} + \cdots + p_{M-9} \over 10 }$

When calculating successive values, a new value comes into the sum and an old value drops out, meaning a full summation each time is unnecessary,

$\textit{SMA}_\mathrm{today} = \textit{SMA}_\mathrm{yesterday} - {p_{M-n+1} \over n} + {p_{M+1} \over n}$

In technical analysis there are various popular values for n, like 10 days, 40 days, or 200 days. The period selected depends on the kind of movement one is concentrating on, such as short, intermediate, or long term. In any case moving average levels are interpreted as support in a rising market, or resistance in a falling market.

In all cases a moving average lags behind the latest price action, simply from the nature of its smoothing. An SMA can lag to an undesirable extent, and can be influenced too much by old prices dropping out of the average. This is addressed by giving extra weight to recent prices, as in the WMA and EMA below.

One characteristic of the SMA is that if the data has a periodic fluctuation, then applying an SMA of that period will eliminate that variation (the average always containing one complete cycle). But a perfectly regular cycle is rarely encountered in economics or finance.[1]

## Weighted moving averageEdit

A weighted average is any average that has multiplying factors to give different weights to different data points. But in technical analysis a weighted moving average (WMA) has the specific meaning of weights which decrease arithmetically. In an n-day WMA the latest day has weight n, the second latest n-1, etc, down to zero.

$\textit{WMA}_{M} = { n p_{M} + (n-1) p_{M-1} + \cdots + 2 p_{M-n+2} + p_{M-n+1} \over n + (n-1) + \cdots + 2 + 1}$

When calculating the WMA across successive values, it can be noted the difference between the numerators of $WMA_{M+1}$ and $WMA_{M}$ is $n p_{M+1} - p_{M} - \cdots - p_{M-n+1}$. If we denote the sum $p_{M} + \cdots + p_{M-n+1}$ by $Total_{M}$, then

$Total_{M+1} = Total_{M} + p_{M+1} - p_{M-n+1}$
$\textit{Numerator}_{M+1} = \textit{Numerator}_{M} + n p_{M+1} - Total_{M}$
$\textit{WMA}_{M+1} = { \textit{Numerator}_{M+1} \over n + (n-1) + \cdots + 2 + 1}$

The denominator is a triangle number, and can be easily computed as $n(n+1)\over2$

The graph at the right shows how the weights decrease, from highest weight for the most recent days, down to zero. It can be compared to the weights in the exponential moving average which follows.

## Exponential moving averageEdit

An exponential moving average (EMA), sometimes also called an exponentially weighted moving average (EWMA), applies weighting factors which decrease exponentially. The weighting for each day decreases exponentially, giving much more importance to recent observations while still not discarding older observations entirely. The graph at right shows an example of the weight decrease.

The degree of weighing decrease is expressed as a constant smoothing factor α, a number between 0 and 1. α may be expressed as a percentage, so a smoothing factor of 10% is equivalent to α=0.1. Alternately, α may be expressed in terms of N time periods, where $\alpha={2\over{N+1}}$. For example, N=19 is equivalent to α=0.1.

The observation at a time period t is designated Yt, and the value of the EMA at any time period t is designated St. S1 is undefined. S2 may be initialized in a number of different ways, most commonly by setting S2 to Y1, though other techniques exist, such as setting S2 to an average of the first 4 or 5 observations. The prominence of the S2 initialization's effect on the resultant moving average depends on α; smaller α values make the choice of S2 relatively more important than larger α values, since a higher α discounts older observations faster.

The formula for calculating the EMA at time periods t≥2 is[2]

$S_{t} = \alpha \times Y_{t-1} + (1-\alpha) \times S_{t-1}$

This formulation is according to Hunter (1986)[3]; an alternate approach by Roberts (1959) uses Yt in place of Yt-1[4]

This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest price, but only by a proportion of the difference (each time),[5]

$\textit{EMA}_{\text{today}} = \textit{EMA}_{\text{yesterday}} + \alpha \times (\textit{price} - \textit{EMA}_{\text{yesterday}})$

Expanding out $\textit{EMA}_{\textit{yesterday}}$ each time results in the following power series, showing how the weighting factor on each price $p_1$, $p_2$, etc, decrease exponentially,

$\textit{EMA} = { p_1 + (1-\alpha) p_2 + (1-\alpha)^2 p_3 + (1-\alpha)^3 p_4 + \cdots \over 1 + (1-\alpha) + (1-\alpha)^2 + (1-\alpha)^3 + \cdots }$

In theory this is an finite sum, but because 1-α is less than 1, the terms become smaller and smaller, and can be ignored once small enough. The denominator approaches 1/α, and that value can be used instead of adding up the powers, provided one is using enough terms that the omitted portion is negligible.

The N periods in an N-day EMA only specifies the α factor. It isn't a stopping point for the calculation in the way N is in an SMA or WMA. The first N days in an EMA do represent about 86% of the total weight in the calculation though.

The power formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied.

The question of how far back to go for an initial value depends, in the worst case, on the data. If there are huge p price values in old data then they'll have an effect on the total even if their weighting is very small. If one assumes prices don't vary too wildly then just the weighting can be considered, and work out how much weight is omitted by stopping after say k terms. This is $(1-\alpha)^k + (1-\alpha)^{k+1} + \cdots$, which is $(1-\alpha)^k \times (1 + (1-\alpha) + (1-\alpha)^2 \cdots)$, ie. a fraction ${(1-\alpha)^k \over \alpha}$ out of the total weight.

Thus if the aim was to have 99.9% of the weight then $k={ \log (0.001 \times \alpha) \over \log (1-\alpha)}$ many terms should be used. And what's more it can be shown $\log\,(1-\alpha)$ approaches $-2 \over N+1$ as N increases, so this simplifies to (roughly) $k=3.45\times(N+1)$ for this example 99.9% weight.

### J. Welles Wilder Edit

Noted technical analyst J. Welles Wilder uses a different form for specifying the period of an EMA. For say 14 days he writes[6]

$\textit{EMA}_{\text{today}} = {1\over14} \times \text{price} + {13\over14} \times \textit{EMA}_{\text{yesterday}}$

So α=1/N rather than α=2/(N+1) as described above. The calculation and properties are all the same, it's just a different reckoning of the rate of smoothing. Clearly care must be taken with which is intended. A conversion can be easily made, for instance 14-days from Wilder is equivalent to 27-days in the above (conversion 2N-1).

## Other weightings Edit

Other weighting systems are used occasionally – for example, a volume weighting will weight each time period in proportion to its trading volume.

There are weighting systems designed using a combination of moving averages: The DEMA indicator (and TEMA indicator (Triple Exponential Moving Average) are unique composites of a single exponential moving average, a double exponential moving average, and in the latter case a triple exponential moving average that provides less lag than either of the three components individually. They were originally introduced January 1994 by Patrick Mulloy.[How to reference and link to summary or text]

The TRIX indicator uses a triple-EMA in its calculation. This ends up as just a certain set of weights on past data, and a set quite different from a plain EMA actually.

## Notes and referencesEdit

1. Statistical Analysis, Ya-lun Chou, Holt International, 1975, ISBN 0030894220 , section 17.9.
2. Moving Averages page at StockCharts.com
3. Relative Strength Index construction at IncredibleCharts.com


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Over a period of months, an adult male patient has taken eight blood tests for uric acid. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. The average weight for these birds is  = 3.15 grams. 2 - (a) Find an 80% confidence interval for the. Calculate the error bound.e. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. $\overline{x}=$ _____ii. D. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80. Please help! A study of the health of such children in later years examined a random sample of 219 children. Thank you. (Round your answers to two decimal places. (d) Size Find the sample size necessary for an $80 \%$ confidence level with a maximal margin of error $E=0.08$ for the mean weights of the hummingbirds. The population standard deviation is known to be 0.1 ounce.a.i. The average weight for these birds is x= 3.15 grams. A sample of 18 randomly selected boxes produced a mean weight of 9.87 oz.a. In words, define the random variable $X$ .d. Explain your choice.d. Twenty percent answered crime. ? 3) a) We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval. (1) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. The average weight for these birds is x = 3.15 grams. (Use whatever you find.) Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.20 gram. Say this value is 1.5. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 Allen's hummingbirds has been under study in Arizona. Define the random variables $X$ and $\overline{X}$ in words.c. E. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20. $\sigma=$ _____iii. Sciences, Culinary Arts and Personal Construct a 90$\%$ confidence interval for the population mean time to complete the tax forms.i. Based on previous studies, we can assume that the weights of Allen s hummingbirds have a normal distribution, with ? For a limited time, find answers and explanations to over 1.2 million textbook exercises for FREE! The average weight for these birds is x = 3.15 grams. $n=$ _____iv. As the confidence levels increase, do the confidence intervals increase in length? Assume that the underlying population distribution is normal.a.i. d) The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80. = 0.33 gram. how much money would i have if I saved up 5,200 for 6 years? Become a Study.com member to unlock this lower limit upper limit margin of error. 359,109 students got unstuck by Course Hero in the last week, Our Expert Tutors provide step by step solutions to help you excel in your courses. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 18 Allen's hummingbirds has been under study in Arizona. a) We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval. Question: Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with {eq}\sigma = 0.36 Allen's hummingbird (Selasphorus sasin x σ) has been studied by zoologist Bill Alther.† Suppose a small group of 11 Allen's hummingbirds has been under study in Arizona. $n-1=$ _____b. $n=28$. The sample mean is 23.6 hours. $s_{x}=$ _____iii. Whoops, there might be a typo in your email. Sketch the graph.iii. (Round up to the nearest whole number.) And if we use our table toe look that up. A newborn baby has extremely low birth weight $(\mathrm{ELBW})$ if it weighs less than 1000 grams. Look up the Z value corresponding to a p value of 0.9 (for the 90% upper cut-off point). (a) Find a $99 \%$ confidence interval for the population mean blood plasma volume in male firefighters. State the confidence interval.ii. Based on previous, studies, we can assume that the weights of Allen's. Calculate the error bound.f. Which distribution should you use for this problem? If the population s.d. Assume that $\sigma=7.50 \mathrm{ml} / \mathrm{kg}$ for the distribution of blood plasma.

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