text:
## Introduction

Persistent shifts in forest composition, structure, and function depend largely on the demographic response of trees to changing environmental drivers and disturbance regimes1,2. Across temperate forests—representing ~25% of the world’s forested land area3—recent reports of increasing tree mortality have been attributed to complex interactions among climate, native insects and pathogens, and uncharacteristically severe wildfire4,5,6. Such pervasive changes in tree population dynamics can have substantial impacts on the ecosystem services provided by temperate forests, including carbon storage and sequestration7, climate regulation8, and provisioning of drinking water9. Sustained anthropogenic and natural stress is thus expected to result in broad-scale transformation of temperate forests and the services they provide10,11. As such, a key challenge for ecological research is to quantify the patterns and underlying drivers of changing tree populations to better inform forest management and help ease ecological transitions12.

Tree demographic rates (e.g., mortality) are important, widely used indicators of forest health12. However, tree demographic rates are confounded by stand development processes (i.e., stand aging)13,14 and do not yield a comprehensive depiction of tree population dynamics when considered individually (i.e., the net result of growth, recruitment, and mortality processes; tree abundance shifts)15,16,17. Thus, while recent reports of elevated tree mortality are suggestive of broad-scale changes in the composition and structure of temperate forests4,5,18, such conclusions should not be accepted in the absence of information regarding tree recruitment, growth, and stand development1,19. Previous efforts to quantify trends in tree population dynamics have often relied on observations from old forests to minimize the influence of stand development processes5,18. Still, patterns of tree population dynamics observed in old forests are seldom characteristic of those in younger forests20 and are thus unlikely to be representative of patterns emerging across forest landscapes (or forested regions) that are composed of a mosaic of stands in various stages of development21. As such, advancement in detection and prediction of forest health decline depends largely on the dissemination of methods that comprehensively describe tree population dynamics and account for variation in tree demography arising from stand development processes14,19.

To this end, we propose the forest stability index (FSI), a direct measure of temporal change in relative live tree density that is independent of stand development processes by design. Temporal change in absolute tree density (e.g., trees per hectare (TPH)) emerges from the joint demographic response of tree populations to endogenous (e.g., inter-tree competition) and exogenous drivers (e.g., wildfire)22. That is, change in absolute tree density is the net result of mortality, growth, and recruitment processes, and is thus a demographically comprehensive measure of tree population dynamics. Relative tree density may be defined as the proportion of absolute tree density observed in a stand relative to the maximum theoretical density the stand could achieve given its observed tree size-distribution. The maximum density that a population of trees may achieve is expected to decline as individuals grow in size, following well-established allometric scaling laws that drive stand development processes (i.e., self-thinning)15,16,17,23,24. Indices of relative tree density (i.e., ratio of observed and maximum theoretical tree density) are thus independent of stand development processes by definition, as allometric tree size-density relationships are explicitly acknowledged in their denominator.

The FSI is defined as the change in relative density observed in a population of trees over time (e.g., via remeasurement of forest inventory plots). Here stability is achieved when the relative density of a tree population is constant (e.g., FSI equal to zero, stand remains at 50% stocking over time), despite underlying changes in absolute tree density and size distribution. Stability in this sense is uncommon at the stand-scale, as stands progress toward maximum relative density in the absence of exogenous stress (positive FSI, e.g., tree growth exceeds mortality) and relative density is expected to decline given disturbance (negative FSI, e.g., disturbances act as thinning agents). At the landscape-scale, however, stability represents a balance between disturbance and tree growth processes, and deviations from this dynamic equilibrium may be indicative of pervasive changes in forest structure, composition, and function.

We use the FSI to identify patterns in relative tree density shifts of the eight most abundant tree species in the western United States (US) and determine the importance of major forest disturbances in driving the population dynamics of each species over the last two decades. Many forests in the western US have experienced recent increases in the extent, severity, and frequency of wildfire25,26, drought27,28, and insect-pest outbreaks29,30, owing in part to changing climate and past forest management (i.e., fire suppression). Likewise, large-scale tree mortality events5,31 and recruitment failures32,33 indicate that widespread forest change is already underway in the region. Such issues are not, however, unique to forests of the western US. Increased disturbance activity has been documented in other regions of the temperate biome in recent decades12,34,35, and the broad spatial and climatic domain encompassed by the western US suggests that patterns of forest change observed herein may be highly relevant to temperate forests across the globe.

We draw upon over 24,000 repeated censuses of US Forest Service Forest Inventory and Analysis (FIA) plots to address the following questions: (1) What is the current status of populations of the eight most abundant tree species in the western US (i.e., expanding, declining), and what do inter-specific differences in the rate of relative tree density shifts indicate about changes in forest composition? (2) Is the rate of relative tree density shifts size-dependent, and if so, what do these relationships indicate about changes in forest structure? (3) Does the rate of relative tree density shifts vary across space within species ranges, and what are the general patterns of change for each species? (4) How do major forest disturbances influence the populations of these dominant species, and what do these relationships suggest about species sensitivity to future changes in forest disturbance regimes?

Here, we show a majority of the most abundant tree species in the western US experienced significant population decline over the last two decades. Further, we show the magnitude of change in tree populations diverges strongly across species, species-size distributions, and species ranges, and the patterns of such change are generally inconsistent with broad-scale reversion of forests toward historical conditions. Altogether, we provide empirical evidence of widespread, yet spatially varying, changes in forest composition and structure over the last two decades in the western US.

## Results

We identified the eight most abundant tree species in the western US by their estimated total number of live stems (i.e., diameter ≥2.54 cm at 1.37 m above ground) across the region (Fig. 1). Top species represented six distinct genera and three families (Table 1). We categorized species by general ecosystem associations, including two woodland species (i.e., characteristic of mid-high-elevation desert/steppe), three subalpine species (i.e., commonly occurring in cool, moist high-elevation forests), and three montane species (i.e., characteristic of mid-elevation forests climatically bounded by woodland (hotter, drier) and subalpine ecosystems (cooler, wetter)). Together these top eight species accounted for 61.6% of all live trees across the study region (62.3% of total live basal area).

The FSI is defined as the average annual change in the relative density of a population of live trees, or the ratio of observed tree density to maximum potential tree density given site conditions and observed tree size-distributions. Hence, significant positive values of the FSI indicate increased relative density (i.e., population expansion), significant negative values indicate decreased relative density (i.e., population decline), and values of the FSI not significantly different from zero indicate population stability (i.e., no change in relative density). Herein, we treat changes in population range boundaries and within-range density shifts as functionally equivalent processes. For example, range expansion (population occurrence on a site where it was previously absent) is represented by the FSI as a positive change in relative density (i.e., where previous relative density is zero). Thus, when summarized across broad spatial domains the FSI represents a comprehensive measure of the net performance of a population of trees during the temporal frame of sampling (i.e., in terms of net changes in relative abundance).

### Broad-scale shifts in forest composition and structure

Species-level estimates of the mean FSI across the entire study region (i.e., range-average estimates) reveal broad-scale patterns of rapid change in the composition of western US forests (over 91 million hectares of forestland) over the last two decades (Fig. 2). Three of the eight most abundant species in the region (lodgepole pine, Engelmann spruce, and quaking aspen) exhibited average decreases in relative density (i.e., population decline) at rates exceeding 1% per year over the 18-year study period (2001–2018), whereas Douglas-fir increased in relative density at nearly the same rate. For reference, a %FSI equal to 1% is equivalent to an 18% change in relative density over the duration of the study period. Across all species, population decline occurred more frequently (five of eight of species) and with greater magnitude (Fig. 2; leftward skew) than population expansion.

Severe rates of population decline were apparent in all three subalpine species considered herein, with lodgepole pine and Engelmann spruce exhibiting the highest rates of decline among all species (lodgepole pine approaching decline of 2% annually, 36% over the entire study period). Douglas-fir and ponderosa pine, both montane species, exhibited the highest rates of range-wide population expansion. In contrast, quaking aspen populations declined at a rate exceeding the rate of expansion observed in other montane species. Woodland species exhibited the highest degree of population stability (range-average mean FSI values closest to zero) over the last two decades, although low rates (<0.20% annually, 3.6% over the duration of the study period) of population expansion and decline were observed for Utah juniper and common pinyon, respectively.

Variation in range-average estimates of the FSI across species-size distributions are indicative of extensive, complex shifts in the size-structure of forests of the western US over the period 2001–2018 (Fig. 3). Rapid population decline was evident in nearly all size-classes of subalpine tree species (except the smallest 10% of subalpine fir and lodgepole pine). Further, the rate of population decline appeared to increase with increasing tree size across all subalpine species (Fig. 3; downward trend in FSI), and this trend was most severe in lodgepole pine (largest 20% of trees declined at rates exceeding 3% annually, 54% over the duration of the study period). The opposite pattern appeared for Douglas-fir and ponderosa pine, where the largest trees generally outperformed the smallest trees of each species (i.e., higher or more positive changes in relative density).

Interestingly, population decline was evident among the smallest size-classes of Douglas-fir (lower 20%) and ponderosa pine (lower 50%), whereas the largest size-classes of both species exhibited population expansion. Patterns of change in relative density across the size-distribution of quaking aspen appeared to follow an approximately quadratic trend, with population expansion evident in the smallest and largest 10% of stems and severe population decline (approaching 3% annually, 54% over the duration of the study period) apparent near the median (50%) tree size class. The size distribution of woodland species (i.e., common pinyon and Utah juniper) appeared to be the most stable of all species examined, indicated by relatively low variation in relative density shifts across tree size-classes (Fig. 3; nearly flat trends).

### Early indications of shifting species distributions

Summaries of the FSI within ecoregion divisions revealed broad-scale spatial patterns of change in the relative density of each species over the last two decades in the western US (Fig. 4b). Population decline was spatially pervasive among all subalpine species and particularly severe for lodgepole pine in the mountain temperate steppe division (i.e., central and northern Rocky Mountains). Though interestingly, lodgepole pine populations increased in relative density (i.e., population expansion) in the mountain marine (i.e., coastal Pacific Northwest) and mountain Mediterranean (i.e., Sierra Nevada) divisions over the study period. Quaking aspen exhibited consistent rates of population decline across its range within the study region (~1% annually, 18% over the duration of the study period). Spatial patterns of relative density shifts of Douglas-fir and ponderosa pine were similar, with decline evident for both species in the mountain subtropical steppe division (i.e., southern Rocky Mountains) and expansion in the northwestern portion of the study region (particularly strong for Douglas-fir in the marine and mountain marine divisions). In contrast, we found general patterns of population stability across the ranges of both woodland species, with the majority of each species’ range characterized by FSI values near zero.

While the general spatial patterns of the FSI observed in subsection-level summaries (Fig. 4a) mirrored those of division-level summaries, subsection-level summaries revealed patterns of change in species relative density at finer spatial scales than division-level summaries. Qualitatively, the FSI appears to exhibit strong, positive spatial auto-correlation across ecoregion subsections, and such local-scale variation is not well represented in division-level summaries. Specifically, local regions of population expansion and regions of population decline (i.e., consisting of multiple adjoining ecoregion subsections with similar FSI values) emerge to varying degrees within the range of each species. However, spatial patterns of population performance were not always consistent among overlapping species, indicating shifts in local-scale forest composition.

### Forest disturbances as drivers of relative tree density shifts

Estimated effects of forest disturbances on relative density highlight disturbances as important drivers of tree population dynamics in the western US, though the magnitude of disturbance effects varied strongly by tree species and disturbance type (Fig. 5). Here, we present effects of forest disturbances as percent change in relative tree density estimated to be caused by each disturbance type at the population-level (i.e., product of disturbance severity and disturbance probability). Hence, high magnitude of estimated effects indicates a disturbance type is an important driver of the population dynamics of a tree species (e.g., Fig. 5, insect outbreaks in lodgepole pine).

Across all species, fire and insect outbreaks generally emerged as more important drivers (i.e., larger estimated effects) of relative density shifts than disease (Fig. 5), although the effects of disease exceeded those of fire and insects for quaking aspen. In most cases, disturbance was negatively associated with changes in relative density, however estimated effects of disease were positive in Engelmann spruce and lodgepole pine over the study period. Disturbance effects appeared to be most severe (i.e., highest magnitude, exceeding 0.2% annually) among subalpine species, where insect outbreaks were determined to be dominant drivers of relative density shifts in Engelmann spruce and lodgepole pine, and fire was of heightened importance for subalpine fir. Fire also emerged as the most important disturbance type affecting the relative density of Douglas-fir and ponderosa pine, both montane species. The effects of disturbance appeared to be least severe among Utah juniper and common pinyon, relative to other species (<0.1% annually).

## Discussion

Complex interactions between changing climate, forest disturbance regimes, and past forest management (e.g., fire suppression) are driving accelerated change in tree demography in temperate forests4,5,36. However, a great deal of uncertainty remains regarding the net result of such demographic change and its consequences for forest composition and structure across broad spatial domains1. Herein, we develop the FSI, a standardized demographic index that weights observed changes in live tree density and tree size against those expected given well-established allometric size-density relationships in forests. We then apply the FSI to over 24,000 remeasured forest inventory plots in the western US to quantify recent trends in tree population dynamics in the region. Our results indicate a majority of the most abundant tree species in the western US experienced significant population decline over the period 2001–2018 (five of eight species, representing 60.7% of all stems of study species), where population decline is indicated by a net decrease in relative live tree density (i.e., negative FSI). Furthermore, we found strong divergence in the magnitude of change in relative tree density across species (Fig. 2), species-size distributions (Fig. 3), and species ranges (Fig. 4). Such dramatic variation in relative density shifts provides empirical evidence of broad-scale, yet spatially varying, changes in forest composition and structure over the last two decades in the western US.

Importantly, the current composition and structure of many forests types in the western US differ drastically from their historic range of variability, arising as a legacy of widespread fire suppression (early 20th century to present day) and intensive harvesting (19th to early 20th century)37. Novel forest conditions (e.g., abundance of high density, closed-canopy forests dominated by fire-intolerant species) have interacted with changing climate to incite rapid increases in disturbance activity in the western US and other regions of the temperate biome29,34. At face-value, it is therefore not inherently surprising to observe a net decrease in the relative density of many top tree species in the western US, as disturbances act as natural thinning agents in tree populations. However, the examination of patterns in live tree density shifts across species ranges and size-distributions offers a sobering line of evidence that is inconsistent with broad-scale reversion of forests toward historical conditions. Instead, our results support the following general trends in western US forests over the last two decades: (1) severe, spatially pervasive decline of subalpine forests coinciding with density shifts towards smaller tree size-classes; (2) broad-scale expansion of large-diameter montane conifers and decline of small-diameter conspecifics; and (3) net population stability of woodland species.

The rate of population decline we observed in subalpine species is particularly severe (comprising 25.3% of all trees in the western US). Over the duration of the 18-year study period, the relative density of lodgepole pine declined 32.2% (±0.05%) across its range in the western United States, while Engelmann spruce and subalpine fir declined 21.3% (±0.05%) and 13.2% (±0.04%) across their respective ranges, respectively (Fig. 2). Decline appears spatially pervasive at broad scales across the study region (ecoregion divisions), however local regions of intense population decline and expansion emerge at finer spatial scales (ecoregion subsections) for all subalpine species (Fig. 4). Such patterns may indicate the primary drivers of subalpine tree population dynamics operate at sub-regional scales, likely responding to local-scale variation in climate, topoedaphic conditions, and disturbance history37. Previous studies have indicated that subalpine tree species are among the most vulnerable to future changes in climate and forest disturbance regimes38,39,40. Our results indicate this heightened vulnerability is already manifesting across the western US, serving as an early warning of potentially widespread, rapid decline of subalpine forests in other regions of the temperate biome.

Interestingly, we show that severity of population decline increased with tree size for each subalpine species during the study period (Fig. 3). That is, population decline was most severe among the largest trees of each species. This result adds to an increasing body of evidence, suggesting that large, old trees are at high risk of decline in forests across the globe41. Large, old trees generally occur at low density but are of high ecological significance, influencing the rates and patterns of regeneration and succession, moderating microclimate and water use, and contributing disproportionately to forest biomass and carbon cycling at a global scale42. Hence, the rapid decline of large-diameter trees we observe in subalpine tree species of the western US is of grave concern and may foreshadow broad-scale transformation in the structure and ecological function of subalpine forests.

In addition, we found population decline to be pervasive across all but the smallest size-classes of subalpine species (Fig. 3). Hence, our results indicate that subalpine forests of the western US have, on average, become younger and thinner over the last two decades. Of all species examined herein, the size-density distributions of subalpine fir and Engelmann spruce are arguably the most likely to exist within their historic range of variability as both species tend to occur in cool, moist forests characterized by infrequent stand-replacing fires (i.e., effects of fire suppression are marginal relative to dry forest)43. The pervasive, size-dependent decline we observe in subalpine fir and Engelmann spruce is thus particularly concerning, indicating the size-distribution of each species may be beginning to depart from historically stable conditions. In contrast, decades of fire suppression and intensive harvesting in the western US have resulted in an overabundance of mature, homogeneous lodgepole pine forest that is highly susceptible to native insect outbreaks37. As such, reversion to historical conditions would require an increase in heterogeneity in the size-distribution of lodgepole pine at the landscape-level. Instead, the size-dependent patterns of decline we observe in lodgepole pine is indicative of increased homogeneity in the species’ size-distribution, where small-diameter stems have become increasingly common relative to large-diameter stems despite a decline in relative tree density across nearly all tree size-classes (younger, thinner, more structurally homogeneous forests).

Our results further indicate that insect outbreaks were >2.5 times more important than other disturbance types in driving relative density shifts of lodgepole pine and Engelmann spruce over the study period (Fig. 5), likely linked to recent outbreaks of mountain pine beetle (Dendroctonus ponderosae)44, and spruce beetle (Dendroctonus rufipennis)45, respectively. As both mountain pine beetle and spruce beetle have shown preference for large hosts (i.e., large-diameter trees), heightened insect-pest activity may explain size-dependent patterns of decline in lodgepole pine and Engelmann spruce. In contrast, we determined wildfire to be more than 3 times more important than other disturbance types in driving relative density shifts of subalpine fir (Fig. 5), and recent increases in the extent and frequency of wildfire25,46 could potentially explain size-dependent decline observed in the species. Specifically, increases in fire probability (and disturbance probability more generally) are likely to coincide with reduced mean stand age21 and subsequent decline in populations of large trees across a landscape. Interestingly, we found the relative density of lodgepole pine and Engelmann spruce increase, on average, in response to disease outbreaks, opposite their response to other disturbance types. We argue this result may arise from inter-specific compensatory responses to host-specific pathogens and/or mortality complexes. That is, one species may benefit from the targeted mortality of a competing species within a stand. It is likely that diseases affecting quaking aspen (e.g., sudden aspen decline47) and subalpine fir (e.g., subalpine fir decline48) may result in a positive growth response of competing lodgepole pine and Engelmann spruce, thereby increasing their relative density within affected stands.

We observed the highest rates of range-average population expansion in Douglas-fir (17.1% ± 0.02 over the study period) and ponderosa pine (6.9% ± 0.03 over the study period; Fig. 2), widespread montane conifers that together represent 21.2% of all trees across the western US. It is important to note the population expansion observed for Douglas-fir and ponderosa pine may not be desirable in many settings, particularly in dry forests where both species occur frequently as canopy dominants49,50. Across the western US, decades of fire exclusion have created overstocked stand conditions that increase the probability of high-severity disturbance (i.e., wildfire, insect outbreak)51,52 and may degrade forest resilience37. In many cases, management aims to reduce tree density via stand thinning and fuels reduction. Hence high rates of relative density increases observed in Douglas-fir and ponderosa pine in the interior Pacific Northwest and portions of the Rocky Mountain region (Fig. 4), may be of substantial concern to forest managers. In contrast, patterns of decreased relative density of ponderosa pine and Douglas-fir observed in the Southern Rocky Mountains may be indicative of tree populations shifting nearer historic relative densities (Fig. 4).

Divergence in both sign and magnitude of relative density shifts across size-distributions of ponderosa pine and Douglas-fir is indicative of broad-scale shifts in the structure of montane coniferous forests in the western US. Specifically, a peak in population expansion is evident among the 50–75th percentile of tree size-classes in both species, and potentially arises as a legacy of widespread logging during the 19th and early 20th century (i.e., owing to simultaneous maturation of stands across a broad spatial domain)51. Furthermore, population decline in the smallest size classes of Douglas-fir and ponderosa pine may potentially be linked to heightened wildfire activity during the study period46, as fire was more than twice as important than other disturbance types in driving change in relative density in Douglas-fir, and more than five times as important in ponderosa pine (Fig. 5). That is, increased wildfire activity may have contributed to desirable change in the structure of montane coniferous forests (i.e., reduced relative density of small trees) over the last two decades in the western US, underscoring the potential for fire (both managed fire and wildfire) to foster forest resilience and contribute to restoration efforts in fire-adapted forests53.

We found that large-diameter populations of Douglas-fir and ponderosa pine outperformed (i.e., exhibited higher positive change in relative density) small-diameter populations of the same species between 2001 and 2018 in the western US (Fig. 3), opposite the pattern observed in subalpine species. In fact, we observed significant expansion among the largest-diameter populations of both species across the region. This surprising result contradicts previously described patterns of global decline of large-diameter trees41, suggesting that such patterns may vary strongly across species and ecosystem associations in temperate forests (e.g., subalpine vs. montane ecosystems). Yet, pervasive increases in the relative density of medium and large-diameter montane conifers may be highly undesirable in some settings. Specifically, the structure of many dry, fire prone forest types of the western US have shifted towards dense, closed-canopy stand conditions that are highly susceptible to crown fire and outbreaks of native insect and pathogens52. As Douglas-fir and ponderosa pine are common canopy dominants in dry forest types, our results may indicate that such systems have diverged further from their historic natural range of variability over the last two decades.

Elevated rates of population decline were evident for quaking aspen during the study period, pervasive across the range of the species (Fig. 4) and across its size-distribution (Fig. 3) in the western US. It is not inherently surprising to observe population decline in quaking aspen given recent spikes in mortality associated with sudden aspen decline47,54. However, the rate of population decline observed herein is particularly severe (20.2% ± 0.04% over the 18-year study period) and serves as a useful baseline to assess the performance of other species. Notably, the rates of range-average decline observed for Engelmann spruce and lodgepole pine exceed that of quaking aspen. Population decline of quaking aspen has received substantial attention in the forest ecology literature54,55,56, however decline of Engelmann spruce has received far less. The disproportionate focus on quaking aspen decline in the literature (and lack of focus on other declining species), emphasizes the need for large-scale, comprehensive assessments of changes in forest composition and structure in the western US and temperate forests elsewhere.

We found woodland species (i.e., Utah juniper and common pinyon) to exhibit the highest degree of population stability across their ranges in the western US, relative to other top species. Pinyon-juniper woodlands of the southwestern US, where both Utah juniper and common pinyon occur as dominant species50, have experienced dramatic expansions in tree density and equally dramatic tree mortality events over the past century57,58. However, inadequate understanding of historic disturbance regimes and tree population dynamics of pinyon-juniper woodlands make it difficult to conclude if such changes are beyond their natural range of variation57. It is important to note that our results do not preclude such striking change at spatial and/or temporal domains not addressed herein. Rather, our results indicate the net responses of Utah juniper and common pinyon populations are relatively stable across broad spatial domains (species’ ranges) and a relatively short period of time (18 years).

The development of methods to quantify the joint demographic response of tree populations to novel environmental and anthropogenic stressors is among the most important advances required to improve predictions of future change in forest composition, structure, and function, and hence inform the management of forest ecosystem services14,19,59. Herein, we present the FSI as one such method. The FSI is a standardized index of temporal change in relative live tree density that can be applied in forests of any community and or structural type. Specifically, the FSI weights observed changes in live tree density (e.g., absolute change in tree abundance) and tree size (e.g., tree basal area) against those expected under allometric size-density laws16,17. Hence, although most common indices of forest change (e.g., mortality rate) are confounded by transient dynamics in tree demography arising from metabolic scaling and density-dependent mortality15,16,20, the FSI is independent of these transient dynamics by design. As such, the FSI may yield simple, accurate measures of shifts in the structure and composition of forests when other common indices of forest change cannot.

The simplicity, flexibility, and highly informative nature of the FSI make it well suited for application in a wide range of ecological settings, and we expect the index to be applied in similar studies to assess broad-scale changes in forest composition and structure in other forested regions across the globe. The FSI relies on temporally replicated data (i.e., ideally from a large number of samples) to derive inference of forest change, however the form of data required by the index are quite simple. For example, all estimates presented herein were derived from basic measures of tree density (e.g., TPH), tree size (e.g., diameter at breast height; DBH), and binary codes indicating the presence or absence of disturbance at measurement sites. As such, the FSI is uniquely well suited for applications using data collected in large-scale national forest inventories. To this end, we provide a flexible implementation of the FSI in the publicly available R package, rFIA60, for application of the index using data collected by the US Forest Service FIA program.

Increased disturbance activity has been documented across temperate forests in recent decades12,34,35, and forests of the western US are no exception29,46. As disturbances modify forest structure and regulate tree population dynamics21, our observation of recent broad-scale shifts in relative tree density across the western US is not inherently surprising. However, the rate and pattern of change we observe across species, species-size distributions, and species ranges is alarming and in many cases undesirable. Furthermore, results of our efforts to quantify the importance of major forest disturbances in driving change in tree populations provide a unique opportunity to assess the vulnerability of tree species to sustained shifts in forest disturbance regimes, as expected under global climate change2,12. Importantly, the temporal frame of this study (18 years) is relatively short given the long life-spans of many tree species in the western US (i.e., individuals may live for multiple hundreds of years). As such, it is unclear how the patterns of change we observe herein will translate to long-term trends in forest dynamics in the region, highlighting an important challenge for future research. Nevertheless, our results offer an early warning of recent, widespread change in forest composition and structure across the western US, and suggest that sustained anthropogenic and natural stress are likely to result in broad-scale change of temperate forests globally.

## Methods

### Field observations

Since 1999, the FIA program has operated an extensive, nationally consistent forest inventory designed to monitor changes in forests across all lands in the US61. We used FIA data from 10 states in the continental western US (Washington, Oregon, California, Idaho, Montana, Utah, Nevada, Colorado, Arizona, and New Mexico) to quantify shifts in relative live tree density, excluding Wyoming due to a lack of repeated censuses (Fig. 1). This region spans a wide variety of climatic regimes and forest types, ranging from temperate rain forests of the coastal Pacific Northwest to pinyon-juniper woodlands of the interior southwest62. Although the spatial extent of the FIA plot network represents a large portion of the current range of all species examined in this study (Table 1), substantial portions of some species ranges (e.g., Douglas-fir) extend beyond the study region into Canada and/or Mexico and therefore were not fully addressed here.

The FIA program measures forest attributes on a network of permanent ground plots that are systematically distributed at a rate of ~1 plot per 2428 hectares across the US61. For trees, 12.7 cm DBH and larger, attributes (e.g., species, DBH, live/dead) are measured on a cluster of four 168 m2 subplots61. Trees 2.54–12.7 cm DBH are measured on a microplot (13.5 m2) contained within each subplot, and rare events such as very large trees are measured on an optional macroplot (1012 m2) surrounding each subplot61. In the event a major disturbance (i.e., >1 acre in size, resulting in mortality or damage to >25% of trees) has occurred between measurements on a plot, FIA field crews record the primary disturbance agent (e.g., fire) and estimated year of the event. In the western US, one-tenth of ground plots are measured each year, with remeasurements first occurring in 2011. Please see Data Availability for more information on forest inventory data accessibility.

### Forest stability index

Allometric relationships between size and density of live trees make it difficult to interpret many indices of forest change19. Live tree density is expected to decline as trees grow in size, owing to increased individual demand for resources and growing space (i.e., competition)16,23. The expected magnitude of change in tree density, given some change in average tree size, varies considerably across forest communities63, site conditions64, and stand age classes23. Thus, we posit it is useful to contextualize observed changes in live tree density relative to those expected given shifts in average tree size within a stand. To this end, we developed the FSI, a measure of change in relative live tree density that can be applied in stands of any forest community and/or structural type.

To compute the FSI, we first develop a model of maximum size-density relationships for tree populations in our study system (Fig. 6). This model describes the theoretical maximum live tree density ($${N}_{\max }$$; in terms of tree number per unit area) attainable in stands as a function of their average tree size ($$\overline{S}$$) and will be used as a reference curve to determine the proportionate live tree density of observed stands (i.e., observed density with respect to theoretical maximum density). We use average tree basal area as an index of tree size (one, however, could also use biomass, volume, or other indices of tree size). For stand-type i, the general form of the maximum tree size-density relationship is given by

$${N}_{\max }({\bar{S}}_{i})={a}_{i}\cdot {\bar{S}}_{i}^{\ {r}_{i}},$$
(1)

where a is a scaling factor that describes the maximum tree density at $$\bar{S}=1$$ and r is a negative exponent controlling the decay in maximum tree density with increasing average tree size. Such allometric size-density relationships (i.e., power functions) are widely accepted as quantitative law describing the behavior of even-aged plant populations under self-thinning conditions16,17, and have been used extensively to describe relative stand density in forests23,24. As detailed below, we allow both a and r to vary with stand-type i, as maximum size-density relationships have been shown to vary across forest communities and ecological settings63,65. Allowing a and r to vary by forest community type, for example, allows us to acknowledge that the maximum tree density attainable in a lodgepole pine stand is likely to differ from that of a pinyon-juniper stand with the same average tree size.

We next define an index of the relative density of a population of trees j (e.g., species, Pinus edulis) within a stand of type i (e.g., forest community type, pinyon/juniper woodland)

$${{\rm{RD}}}_{ij}=\sum _{h=1}\frac{{N}_{hij}}{{N}_{\max }({S}_{hi})},$$
(2)

where N is the density represented by tree h (in terms of tree number per unit area), and S is an index of individual-tree size (e.g., basal area, as used here). The denominator of Equation (2) represents the maximum tree density attainable in a stand of type i with average tree size equal to the size of tree h. We therefore express the relative density of a population j within stand-type i as a sum of the relative densities represented by individual trees within the stand. RD can be interpreted as the proportionate density, or stocking, of a population of trees within stand, where values range from 0 (population j is not present within a stand) to 1 (population j constitutes 100% of a stand and the stand is at maximum theoretical density given its size distribution). As we do in this study, one may apply any range of estimators to summarize the expected relative density of a population of trees j across a range of different stand-types (e.g., estimate the mean and variance of RDj across a broad region containing many different stand-types).

It is important to note that Equation (2) is approximately equal to a simpler method using aggregate indices (i.e., $$\frac{{\sum }_{h = 1}{N}_{hij}}{{N}_{\max }(\overline{{S}_{i}})}$$) when tree size-distributions are normally distributed (even age-structures). However, the use of aggregate indices introduces class aggregation bias that results in overestimation of relative density in stands with non-normal size distributions (i.e., uneven age-structures), consistent with other indices of relative tree density66. In contrast, summing tree-level relative densities eliminates such bias and allows RD to accurately compare density conditions across stands in very different structural settings (e.g., even-aged plantation vs. irregularly structured old forest). Furthermore, the partitioning of relative density into tree-level densities allows RD to be accurately summarized within tree size-classes66. That is, it is possible to explicitly estimate the contribution of tree size-classes to overall stand density using RD.

For a given population j within stand-type i, we define the FSI as the average annual change in relative tree density observed between successive measurements of a stand

$${\rm{FSI}}=\frac{{{\Delta }}{\rm{RD}}}{{{\Delta }}t},$$
(3)

where Δt is the number of years between successive measurement times t1 and t2 and ΔRD is the change in RD over Δt (i.e., $${{\rm{RD}}}_{{t}_{2}}-{{\rm{RD}}}_{{t}_{1}}$$). The FSI may also be expressed in units of percent change (%FSI), where average annual change in relative tree density is standardized by previous relative density

$$\% {\rm{FSI}}=\frac{100\cdot {\rm{FSI}}}{{{\rm{RD}}}_{{t}_{1}}}.$$
(4)

Here, stability is defined by zero net change in relative tree density over time (i.e., FSI equal to zero), but does not imply zero change in absolute tree density or tree size distributions. For example, a population exhibiting a decrease in absolute tree density (e.g., trees per unit area) may be considered stable if such decline is offset by a compensatory increase in average tree size. However, populations exhibiting expansion (i.e., $${{\rm{RD}}}_{{t}_{1}}<{\,}{{\rm{RD}}}_{{t}_{2}}$$) or decline (i.e., $${{\rm{RD}}}_{{t}_{1}}> {\,}{{\rm{RD}}}_{{t}_{2}}$$) in relative tree density will be characterized by positive and negative FSI values, respectively.

### Statistical analysis

We computed the FSI for all remeasured FIA plots in the western US (N = 24,229). We included plots on both public and private lands and considered all live stems (DBH ≥2.54 cm) in our analysis. As forest management can effect regional shifts in tree density, we excluded plots with evidence of recent (i.e., within 5 years of initial measurement) silvicultural treatment (e.g., harvesting, artificial regeneration, site preparation). All plot measurements occurred from 2001 to 2018, with an average remeasurement interval of 9.78 years (±0.005 years). For brevity, we restricted our analysis to consider the eight most abundant tree species in the western US. We identified the most abundant tree species using the rFIA R package60, defining abundance in terms of estimated total number of trees (DBH ≥ 2.54 cm) in the year 2018. We excluded species that exhibit non-tree growth habits (i.e., shrub, subshrub) across portions of the study region. All statistical analysis was conducted in Program R (4.0.0)67.

We developed a Bayesian quantile regression model to estimate maximum size-density relationships for stand-types observed within our study area. Here, we use TPH as an index of absolute tree density, average tree basal area ($$\overline{{\rm{BA}}}$$; equivalent to tree basal area per hectare divided by TPH) as an index of average tree size, and forest community type to describe stand-types. We produced stand-level estimates of TPH and $$\overline{{\rm{BA}}}$$ from the most recent measurements of FIA plots that (1) lack evidence of recent (within remeasurement period or preceding 5 years) disturbance and/or silvicultural treatment and (2) exhibit approximately normal tree diameter distributions (i.e., even-aged). Here we define an approximately normal tree diameter distribution as exhibiting Pearson’s moment coefficient of skewness between −1 and 1.

We transform the nonlinear size-density relationship to a linear function by taking the natural logarithm of TPH and $$\overline{{\rm{BA}}}$$, and use a linear quantile mixed-effects model to estimate the 99th percentile of TPH conditional on $$\overline{{\rm{BA}}}$$ (i.e., in log-log space) for all observed forest community types. We allowed both the model intercept and coefficient to vary across observed forest community types (i.e., random slope/intercept model), thereby acknowledging variation in the scaling factor (a) and exponent (r) of the maximum tree size-density relationship across stand-types. We place informative normal priors on the model intercept (μ = 7, σ = 1) and coefficient (μ = 0.8025, σ = 0.1) following the results of decades of previous research in maximum tree size-density relationships16,23,63,65.

The FIA program uses post-stratification to improve precision and reduce non-response bias in estimates of forest variables68, and we used these standard post-stratified estimators to estimate the mean and variance of the FSI for each species across their respective ranges within the study area (see Code Availability for all relevant code). Further, the FIA program uses an annual panel system to estimate current inventories and change, where inventory cycles consist of multiple panels, and individual panels are comprised of mutually exclusive subsets of ground plots measured in the same year within a region. Precision of point and change estimates can often be improved by combining annual panels within an inventory cycle (i.e., by augmenting current data with data collected previously). We used the simple moving average estimator implemented in the rFIA R package60 to compute estimates from a series of eight annual panels (i.e., sets of plots remeasured in the same year) ranging from 2011 to 2018. The simple moving average estimator combines information from annual panels with equal weight (i.e., irrespective of time since remeasurement), thereby allowing us to characterize long-term patterns in relative density shifts. We determine populations to be stable if the 95% confidence intervals for range-averaged FSI included zero. Alternatively, if confidence intervals of range-averaged FSI do not include zero, we determine the population to be expanding when the estimate is positive and declining when the estimate is negative.

To identify changes in species-size distributions, we used the simple moving average estimator to estimate the mean and variance of the FSI by species and size class across the range of each species within our study area. We assign individual trees to size-classes representing 10% quantiles of observed diameter distributions (i.e., diameter at 1.37 m above ground) of each species growing on one of seven site productivity classes (i.e., inherent capacity of a site to grow crops of industrial wood). That is, we allow size class definitions to vary among species and along a gradient of site productivity, thereby acknowledging intra-specific variation in diameter distributions arising from differences in growing conditions. The use of quantiles effectively standardizes absolute size distributions, simplifying both intra-specific and inter-specific comparison of trends in relative density shifts along species-size distributions.

We assessed geographic variation in species relative density shifts at two scales: ecoregion divisions and subsections69. Ecoregion divisions (shown for our study area in Fig. 1) are large geographic units that represent broad-scale patterns in precipitation and temperature across continents. Ecoregion subsections are subclasses of ecoregion divisions, differentiated by variation in climate, vegetation, terrain, and soils at much finer spatial scales than those represented by divisions. We again used the simple moving average estimator to estimate the mean and variance of the FSI by species within each areal unit (i.e., drawing from FIA plots within each areal unit to estimate mean and variance of the FSI). As a direct measure of changes in relative tree density, spatial variation in the FSI is indicative of spatial shifts in species distributions during the remeasurement interval (i.e., range expansion/contraction and/or within-range relative density shifts). That is, the distribution of populations shift toward regions increasing in relative density and away from regions decreasing in relative density during the temporal frame of sampling. We map estimates of the FSI for each areal unit to assess spatial patterns of changes in relative density and identify regions where widespread geographic shifts in species distributions may be underway.

We sought to quantify the average effect of forest disturbance processes on changes in the relative density of top tree species in the western US over the interval 2001–2018. To this end, we developed a hierarchical Bayesian model to determine the average severity and annual probability of disturbances (i.e., wildfire, insect outbreak, and disease) on sites where each species occurs. Average severity was modeled as

$${y}_{jk} \sim {\rm{normal}}({\alpha }_{j}+\sum _{l}{\beta }_{jl}\cdot {x}_{lk},\ {\varsigma }_{j}^{2}),$$
(5)

where yjk is the FSI of species j on plot k, αj is a species-specific intercept, βjl is a species-specific coefficient corresponding to the binary variable xlk that takes the value of 1 if disturbance l occurred within plot k measurement interval and 0 otherwise. The intercept and regression coefficients each received an uninformative normal prior distribution. The species-specific residual standard deviation ςj received a uninformative uniform prior distribution70.

On average, disturbance will occur at the midpoint of plot remeasurement periods, assuming temporal stationarity in disturbance probability over the study period. As plots in this study are remeasured on 10-year intervals, we assume that tree populations have, on average, 5 years to respond to any disturbance event. Hence, our definition of disturbance severity, βjl’s, cannot be interpreted as the immediate change in relative tree density resulting from disturbance. Rather, disturbance severity (as defined here) includes the immediate effects of disturbance, as well as 5 years of change in relative tree density prior to and following disturbance (where disturbance is assumed to be functionally instantaneous).

Annual probability of disturbance l on plot k was modeled as

$${x}_{lk} \sim {\rm{binomial}}({{\Delta }}{t}_{k},{\psi }_{jl}),$$
(6)

where Δtk is the number of years between successive measurements of plot k, viewed here as the number of binomial "trials,” and ψjl is the species-specific probability for disturbance which was assigned a beta(1,1) prior distribution. Hence, annual probability of disturbance is assumed to vary by species j and by disturbance type l.

We estimate the mean effect of forest disturbance processes on changes in species-specific relative tree density by multiplying the posterior distributions of βjl and ψjl. That is, we multiply species-specific disturbance severity by disturbance probability to yield an estimate of the mean change in relative density caused by disturbance over the study period. We then standardize these values across species by dividing by the average relative density of each species at the beginning of the study period. Thus, standardized values can be interpreted as the annual proportionate change in the relative tree density of each species resulting from disturbance over the period 2001–2018.

### Reporting summary

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


text:
# Phil and Linda are 25-year-old newlyweds and file a joint tax return. Linda is covered by a...

Phil and Linda are 25-year-old newlyweds and file a joint tax return. Linda is covered by a retirement plan at work, but Phil is not.

Assuming Phil’s wages were $55,000 and Linda’s wages were$70,000 for 2019 and they had no other income, what is the maximum amount of their deductible contributions to an IRA for 2019?

format: plain


text:
Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1100.30006
Abdulhadi, Z.; Muhanna, Y.Abu; Khuri, S.
On univalent solutions of the biharmonic equation.
(English)
[J] J. Inequal. Appl. 2005, No. 5, 469-478 (2005). ISSN 1029-242X/e

The biharmonic equation $$\Delta \Delta u = 0$$ occurs in many physical situations, most important applications are in fluid dynamics and elasticity problems [{\it St. Bergman, \ M.\ M. Schiffer}, Kernel Functions and Elliptic Differential Equations, (Academic Press, New York) (1953; Zbl 0053.39003)]. A continuous complex-valued function $F=u+iv$ in a domain $D \subset \mathbb{C}$ is biharmonic if $F$ satisfies the biharmonic equation. The paper deals with univalent and starlike biharmonic functions. If $F$ is biharmonic in any simply connected domain then it holds $F=r^2G+H,z=re^{i\varphi},$ where $G$ and $H$ are harmonic. It is proven for example that if $F$ is biharmonic in the unit disk and it holds $F(z) = r^2G(z), \vert z\vert <1,$ where $G$ is harmonic, then $F$ is starlike whenever $G$ is starlike.
[Bodo Dittmar (Halle)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
31A30 Biharmonic (etc.) functions and equations (two-dim.)

Keywords: biharmic equation; univalent and starlike solutions

Citations: Zbl 0053.39003

Cited in: Zbl 1156.31001

Highlights
Master Server


text:
# Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

Coralia Cartis, Nicholas I M Gould, Philippe Toint

Research output: Working paper

28 Downloads (Pure)

### Abstract

We establish or refute the optimality of inexact second-order methods for
unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould
and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of
their linesearch variants. For each method in this class and arbitrary
accuracy threshold $\epsilon \in (0,1)$, we exhibit a smooth objective
function with bounded range, whose gradient is globally Lipschitz continuous
and whose Hessian is $\alpha-$H\"older continuous (for given $\alpha\in [0,1]$), for which the method in question takes at least $\lfloor\epsilon^{-(2+\alpha)/(1+\alpha)}\rfloor$ function evaluations to generate a first iterate whose gradient is smaller than $\epsilon$ in norm. Moreover, we also construct another function on which Newton's takes $\lfloor\epsilon^{-2}\rfloor$ evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for $\alpha=1$, this lower bound is of the same order in $\epsilon$ as the upper bound on the worst-case evaluation complexity of the cubic %(and more generally $(2+\alpha)$) regularization method and other methods in a class of methods proposed in Curtis, Robinson and Samadi (2017) or in Royer and Wright (2017), thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton's method is suboptimal.
Original language English Arxiv 35 1709.07180 Published - 22 Sep 2017

### Keywords

• evaluation complexity
• nonlinear optimization
• second-order methods


text:
# Homework Help: Limit Calc Test Problem

1. Sep 5, 2010

### nick850

1. The problem statement, all variables and given/known data

This is a problem off a multiple choice practice test:
lim t->3 ( 1/(t^2-3t) - 2/(t^2-9) =

The solutions are:
(a) 0 (b) -1/9 (c) -1/18 (d)1 (e) 1/3

Can someone explain to me how to solve it? Any help would be very appreciated. You don't need to explain how to do limits. I just need to know how to manipulate the equation so that it's not indeterminate.

2. Relevant equations

NA

3. The attempt at a solution

I tried multiplying by the conjugate with no success.

2. Sep 5, 2010

### HallsofIvy

The first thing I would do is actually subtract the two fractions:
$$\frac{1}{t^2- 3t}- \frac{2}{t^2- 9}= \frac{1}{t(t- 3)}- \frac{2}{(t- 3)(t+ 3)}$$

Clearly the "common denominator" is t(t- 3)(t+ 3):
$$\frac{t+ 3}{t(t- 3)(t+ 3)}- \frac{2t}{t(t- 3)(t+ 3)}$$[tex]= \frac{t+ 3- 2t}{t(t- 3)(t+ 3)}= \frac{-t+ 3}{t(t- 3)(t+ 3)}= -\frac{t- 3}{t(t- 3)(t+ 3)}[/itex]

3. Sep 5, 2010

Thank you!!


text:
### The average income of A, B and C is Rs. 12,000 per month and average income of B, C and D is Rs. 15,000 per month. If the average salary of D be twice that of A, then the average salary of B and C is in Rs. :

A. 8,000

B. 18,000

C. 13,500

D. 9,000

$\begin{array}{l}A+B+C=12000\times3\\ B+C+D=15000\times3\\ \rightarrow D-A=3000\times3=9000\\ \text{ Also, }D=2A\\ \text{ Then, }D=18000\text{ and }A=9000\\ \text{Therefore},\\ \text{Average salary of B and C}\\ =\Large\frac{45000-18000}{2}\\ =13500\end{array}$


text:
0th

Percentile

##### Add edges to a graph

The new edges are given as a vertex sequence, e.g. internal numeric vertex ids, or vertex names. The first edge points from edges[1] to edges[2], the second from edges[3] to edges[4], etc.

##### Usage
add_edges(graph, edges, ..., attr = list())
##### Arguments
graph
The input graph
edges
The edges to add, a vertex sequence with even number of vertices.
...
attr
##### Details

If attributes are supplied, and they are not present in the graph, their values for the original edges of the graph are set to NA.

##### Value

• The graph, with the edges (and attributes) added.

Other functions for manipulating graph structure: +.igraph; -.igraph, igraph-minus; add.vertices, add_vertices; delete.edges, delete_edges; delete.vertices, delete_vertices; edge, edges; path; vertex, vertices

##### Aliases
g <- make_empty_graph(n = 5) %>%
plot(g)


text:
## Some more abuse of Gaussian processes …

Two arXivals today show us ways to mis-use or mis-apply Gaussian processes.  The first is another in this crazy series of papers trying to use GPs as extrapolators to estimate $H_0$ from $H(z)$ datapoints.  Yes, GPs can kind of extrapolate, but it’s really not their strength: as you get further and further from the domain of observed data you get the GP returning to the mean.  The only interesting thing is the rate at which it reverts, which is controlled by the GP kernel hyper-parameters: unfortunately, as we’ve discussed before, these are very difficult to estimate well for many types of experimental design.  As far as I can see from the scant details in this submission: the authors only look at how small the credible intervals on $H_0$ are from their GP fits, not at whether or not the true value from the model in the mocks is actually contained within them with any reasonable Frequentist coverage.  Interestingly, they do see that their credible intervals increase if they switch from a squared exponential kernel to various flavours of Matern, which recalls the observation from asymptotic theory that we should probably be using the Matern kernel if we actually want decent coverage.

The other GP paper today was presenting some ‘novel’ work that looks a lot like the Korean re-make version of an already published paper I was involved in via the COIN collaboration.  The idea there is to use spatial GPs as an interpolator to fill in gaps in irregularly sampled galaxy data (in our case it was IFU data) to construct a complete model ‘image’ (with uncertainties); plus maybe gain a little shrinkage at the observed locations.  Why I believe our version is a lot better is because we jointly fit a radial parametric function along with the GP, whereas today’s paper fits the radial term first and then separately fits the GP to the residuals.  There’s no computational need or theoretical justification to cut the fit at this point, especially if the errors in the first step are not going to be propagated coherently to the second.  Also, the first fit here is likely to give as sensible an estimate as most badly misspecified model fitting exercises; especially when you look at how irregular the ‘design’ of sampling locations is for some of these galaxies.


text:
# Two polynomials generating $\mathbb{C}(x,y^2)$

Let $\beta$ be the following involution (=automorphism of order two) on $\mathbb{C}[x,y]$: $\beta: (x,y) \mapsto (x,-y)$. Denote the set of symmetric elements w.r.t. $\beta$ by $S_{\beta}$, and the set of skew-symmetric elements w.r.t. $\beta$ by $K_{\beta}$.

It is not difficult to see that if $s \in S_{\beta}$, then $s= a_{2n}y^{2n}+a_{2n-2}y^{2n-2}+\cdots+a_2y^2+a_0$, where $a_{2n},a_{2n-2},\ldots,a_2,a_0 \in \mathbb{C}[x]$, while if $k \in K_{\beta}$, then $a_{2n+1}y^{2n+1}+a_{2n-1}y^{2n-1}+\cdots+a_3y^3+a_1y$, where $a_{2n+1},a_{2n-1},\ldots,a_3,a_1 \in \mathbb{C}[x]$.

Let $s_1,s_2 \in S_{\beta}=\mathbb{C}[x,y^2]$ satisfy: (1) Each of $s_1,s_2$ is of (total) degree $\geq 1$. (2) $s_1,s_2$ are algebraically independent over $\mathbb{C}$. (3) $\mathbb{C}(s_1,s_2)=\mathbb{C}(x,y^2)$.

For example: $s_1=x, s_2=xy^2$.

Is it possible to find a general form of such a pair $s_1,s_2$?

Remarks: (i) If I am not wrong, if we replace condition $(3)$ to $\mathbb{C}[s_1,s_2]=\mathbb{C}[x,y^2]$, then the answer is as follows: Define $h: (x,y) \mapsto (x,y^2)$. Then a general form for such a pair $s_1,s_2$ is $s_1=(hg)(x), s_2=(hg)(y)$ where $g$ is an arbitrary automorphism of $\mathbb{C}[x,y]$.

(ii) Denote the group of automorphisms of $\mathbb{C}(x,y)$, the Cremona group, by $C$. If my argument in $(1)$ is correct, then perhaps an analog argument will solve my question as follows: Define $h: (x,y) \mapsto (x,y^2)$ ($h$ is extended to the field of fractions). Then a general form for $s_1,s_2$ is $s_1=(hg)(x), s_2=(hg)(y)$ where $g$ is an appropriate automorphism of $\mathbb{C}(x,y)$.

(iii) If my argument in $(2)$ is correct, then I would like to know how to describe $g \in C$; I think that an answer can be found in one of the references in this question, but I am not sure if I fully understand it.

(iv) Notice that not all elements of $C$ will be relevant to my question, since $s_1,s_2$ are assumed to be polynomials; for example, $C \ni g: (x,y) \mapsto (\frac 1x ,y)$ would yield $s_1=\frac 1x \notin \mathbb{C}[x,y]$ and $s_2=y^2$. (Or perhaps we should allow other options for $h$, for example, $h: (x,y) \mapsto (x, \frac{1}{y^2})$ and then we can take arbitrary $g \in C$).

Thank you very much!

• Why does $\beta$ play a role? Aren't you asking for $s_1,s_2\in\mathbb{C}[x,u]$, satisfying 1),2) and 3) with $y^2$ replaced by $u$? – Mohan May 9 '18 at 16:07
• I guess you are right; do you know an answer to my question? The reason I mentioned $\beta$ is because I am also interested in the same question with $\alpha: (x,y) \mapsto (y,x)$ instead of $\beta$, but I wanted first to concentrate on $\beta$ hoping it is slightly easier to deal with than $\alpha$. – user237522 May 9 '18 at 16:49
• I am not aware of any simple answer to it. Birational generators are notoriously difficult to find. There might be some answer using resultants (of $s_1-a, s_2-b$ for general $a,b\in\mathbb{C}$). – Mohan May 9 '18 at 17:31
• Thank you for your comments. Please, could you elaborate a little your idea concerning resultants? – user237522 May 9 '18 at 18:04
• $s_1,s_2$ form birational generators if and only if for general $a,b\in\mathbb{C}$, $s_1-a,s_2-b$ have exactly one common zero. Resultants help you calculate the common zeroes of $s_1-a, s_2-b$, so may be something can be done. – Mohan May 9 '18 at 18:20


text:
# Moore Lab TBE

### Background

A TBE Saga

Traditional TBE recipes are weight-based, not molarity-based, which reduces the chance for preparation errors, but reinforces a mindless automaton mentality that plagues students and other lab workers. Also, there is no rationale for the amounts provided, so one is left guessing what the tolerances are for the buffering and divalent metal chelation system. This situation is analogous to LB (lysogeny broth), whose recipes range from the original published by Bertani ("Miller"), and lower NaCl versions ("Lennox" and "Luria").

It is common practice to use traditional TBE at 0.5X for routine agarose gel electrophoresis; not only does this formulation conserve buffer, but there is still plenty of buffering and the gels run cooler because there is less electrical current. We addressed the utility of the recipe and rounded the reagent concentrations to memorable values.

Many labs prepare or purchase 10X TBE, which is 890 mM Tris, 890 mM Borate, and 20 mM EDTA. At this high stock concentration, the reagents will crystallize into a hard mineral that is difficult to remove from storage containers and often throw off concentrations when remaking stocks using the same bottle.

5X is more stable, but still has a tendency to crystallize.

1X TBE would then be at respective concentrations of 89 mM of T and B and 2 mM of E, why? More recently, other established protocols figured out that 0.5X would render better gels, and that some of the previous reagent concentrations were built for lab situations which are now outdated. If gels run fine with 0.5X (45.5mM Tris, 45.5mM Borate, 1 mM EDTA) we thought, why not make the most commonly used form ("1 X") equal to something more convenient for daily use? Thus, our lab uses 1X to represent a TBE Mix of 50mM Tris, 50mM Borate, 1mM EDTA. These are rounded up from 45.5 in the original purely for convenience of calculations and do not appear to significantly alter the gels produced.

### Moore Lab - 5X TBE Stock Recipe

(per liter)

250 mM Tris base

250 mM Boric acid

5 mM EDTA (disodium)

- Put about 800 mLs of  diH2O  in a clean beaker with a stir bar;

- dump in the Tris base and stir;

- dump in the EDTA and wait until dissolved (the high pH of the Tris helps the EDTA go in faster);

- then dump in the boric acid and wait until dissolved. Near the end, squirt around the top of the beaker to rinse in any additional reagent.

- Transfer to a cyliner and top off with beaker-rinse to final volume. Transfer to storage bottle (can remove crystals while waiting for reagents to dissolve).

This is a good storage concentration, but obviously the whole point is to have an easily-accessible and effective working stock at 1X. Dilute it down from here to 1X and keep it nearby, in the event someone leaves you high and dry, it won't take too long to dilute it down from 5X.

Don't be lazy: look at the formula weights on the bottles, confirm the CAS numbers if needed, and calculate it. This method should be the norm for all reagent preparations. This way you won't accidentally use Tris-Cl or non-sodium EDTA.

If you are performing electrophoresis that requires the traditional "1X", just use 2X of this version (100mM Tris & Boric Acid concentrations), it buffers a little better and the current is a bit higher, but not enough to write home about.

Be sure to test your running buffer's conductivity in your setup with the gel you are using before running your protocol. Depending on your application and brand, the correct range may vary. In applications which require the higher amperage, you can just use 2X (on this scale) mentioned previously. Just be aware of your needs and your protocol, that way you can avoid just mindlessly making something that doesn't suit your method and having to backpedal.

### Storage & Cleaning

In the higher-concentration stock bottles, build-up of salt-like crystals will form and eventually cause inconsistencies in your gels.

When crystals are visually apparent or at a regular interval (whichever works for your lab), clean the bottle.

These formations do not effectively remove in water, but are easily dissolved in acetic acid. If bottles are used enough, it may be worth keeping a ~10% Acetic Acid container nearby, with which you can rinse and coat the interior of the bottle and threads of the cap. Typically a quick soak and rub-down with a gloved finger will do the trick but it'll depend on the amount of buildup you've accrued in the threads.

If crystals remain, they will eventually act as nucleation sites for later crystal formation, so really try to make sure you get as much as possible.

An ounce of prevention...

### References

1. Sambrook J, Russell SW, editors. Molecular Cloning: A Laboratory Manual. 3. Cold Spring Harbor Laboratory Press; Cold Spring Harbor, NY: 2001.

2. Sanderson, Brian A et al. “Modification of gel architecture and TBE/TAE buffer composition to minimize heating during agarose gel electrophoresis.” Analytical biochemistry vol. 454 (2014): 44-52. doi:10.1016/j.ab.2014.03.003


text:
## United States Postal Service: What to do?

We are the Borg.
EvilYeti
Posts: 9222
Joined: Sat Jun 05, 2004 5:55 am
Location: San Diego
Has thanked: 2 times
Been thanked: 10 times

### Re: United States Postal Service: What to do?

DrMatt wrote:And another thing. Roads. Who needs 'em? Any place you can go on a road that's worth going, you can go via webcam.
Totally!

Just like MIDI has replaced the need for real musicians and random number generators have replaced composers!

EvilYeti
Posts: 9222
Joined: Sat Jun 05, 2004 5:55 am
Location: San Diego
Has thanked: 2 times
Been thanked: 10 times

### Re: United States Postal Service: What to do?

In the interest of full disclosure, the post office put a parcel box at the end of my parents private drive. So this 'fixed' the late/missing deliveries.

Good example that the public sector fixes things, eventually.

Rob Lister
Posts: 19922
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
Has thanked: 593 times
Been thanked: 596 times

### Re: United States Postal Service: What to do?

EvilYeti wrote:In the interest of full disclosure, the post office put a parcel box at the end of my parents private drive. So this 'fixed' the late/missing deliveries.

Good example that the public sector fixes things, eventually.

Rob Lister
Posts: 19922
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
Has thanked: 593 times
Been thanked: 596 times

(Newser) – After deciding to close 491 post offices last year, the struggling US Postal Service will add as many as 2,000 additional branches to that tally beginning in March. Another 16,000 unprofitable ones will also be reviewed, as the USPS pushes Congress to allow them to close the worst performing, the Wall Street Journal reports (the law doesn't cite profitability as a reason for closing a post office). After a record $8.5 billion in losses last year, the USPS is arguing its network of 32,000 offices is outdated in the digital era, but a disproportionate number of the at-risk offices are in rural areas or small towns—and for the nearby residents, those offices are an integral part of staying connected to the rest of the nation. "When they close the post office, they probably won't even come up here anymore and clean the roads,” says one resident of a small Kentucky town that doesn’t yet have reliable cable, Internet, or cell phone service. Though residents of affected communities will still have delivery (either to their homes or to mailbox clusters constructed in town) and can use post offices in neighboring towns, many of the elderly or those with health problems rely on a nearby branch. Says one 62-year-old, who would need to drive 12 miles on a steep, winding mountain road to the nearest branch, “It will hurt us real bad. Just an update really. Rob Lister Posts: 19922 Joined: Sun Jul 18, 2004 7:15 pm Title: Incipient toppler Location: Swimming in Lake Ed Has thanked: 593 times Been thanked: 596 times ### Re: United States Postal Service: What to do? Postal Service warns of default as losses mount NEW YORK (CNNMoney) -- The U.S. Postal Service warned Wednesday that it may default on some of its financial obligations later this year after reporting yet another quarterly loss. The USPS, a self-supporting government agency that receives no tax dollars, said it suffered a loss of$329 million in the first quarter of federal fiscal year 2011. That compared with a loss of $297 million a year earlier. The agency has been suffering from an ongoing decline in mail volume, which has undercut revenues, while retiree health care costs have been straining its reserves. Excluding costs related to retiree benefits and adjustments to workers' compensation liability, the Postal Service said it had net income was$226 million in the first quarter, which ended Dec. 31.
double the price of stamps? that would only mean greater loss as those that don't absolutely have to snailmail would do so less. fire the slackers? good idea but, ya know, rights and shit. subsidize the loss? looks like that's the way it is going to be.

Rob Lister
Posts: 19922
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
Has thanked: 593 times
Been thanked: 596 times

### Re: United States Postal Service: What to do?

grayman
Posts: 5775
Joined: Sun Feb 11, 2007 3:43 am
Location: A little further north.
Has thanked: 92 times
Been thanked: 150 times

### Re: United States Postal Service: What to do?

There's talk of closing our post office. That would suck because it's the only thing we have in town, and usually during the winter mornings, this is where my neighbors hang out and chat. It's our social center.
Last edited by grayman on Mon Sep 05, 2011 5:10 pm, edited 1 time in total.
You will be electroplated. Resistance is potential divided by current! - Dr. Matt

Cogito, ergo Deus non est.

Rob Lister
Posts: 19922
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
Has thanked: 593 times
Been thanked: 596 times

### Re: United States Postal Service: What to do?

grayman wrote:There's talk of closing our post office. That would suck because it's the only thing we have in town, and usually during the winter mornings, this is where my neighbors hang out and chat. It's more of a social center than anything else.
perhaps a nice bar.

or an ihop.

WildCat
Posts: 13999
Joined: Tue Jun 15, 2004 2:53 am
Location: The 33rd Ward, Chicago
Has thanked: 33 times
Been thanked: 348 times

### Re: United States Postal Service: What to do?

grayman wrote:There's talk of closing our post office. That would suck because it's the only thing we have in town, and usually during the winter mornings, this is where my neighbors hang out and chat. It's our social center.
Perhaps if they charged each of you \$10/hr to hang out and chat they could stay open?
Do you have questions about God?

you sniveling little right-wing nutter - jj

Anaxagoras
Posts: 21745
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
Has thanked: 1410 times
Been thanked: 1234 times

### Re: United States Postal Service: What to do?

In Japan the post office is also a bank (the largest in Japan in terms of deposits) and an insurance company.

It's the poster boy of too big to fail.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare

ed
Posts: 33309
Joined: Tue Jun 08, 2004 11:52 pm
Title: Rhino of the Florida swamp
Has thanked: 451 times
Been thanked: 777 times

### Re: United States Postal Service: What to do?

grayman wrote:There's talk of closing our post office. That would suck because it's the only thing we have in town, and usually during the winter mornings, this is where my neighbors hang out and chat. It's our social center.
This is what drives me crazy. Seize the opportunity!!! Rent the property and keep it going. Sell stamps, take packages (for USPS, Fedex and UPS!!),. Sell Latte Sell Booze Sell Dope Sell p0rn!!!

WTF, you have an opportunity staring at you.
Wenn ich Kultur höre, entsichere ich meinen Browning!

grayman
Posts: 5775
Joined: Sun Feb 11, 2007 3:43 am
Location: A little further north.
Has thanked: 92 times
Been thanked: 150 times

### Re: United States Postal Service: What to do?

ed wrote:
grayman wrote:There's talk of closing our post office. That would suck because it's the only thing we have in town, and usually during the winter mornings, this is where my neighbors hang out and chat. It's our social center.
This is what drives me crazy. Seize the opportunity!!! Rent the property and keep it going. Sell stamps, take packages (for USPS, Fedex and UPS!!),. Sell Latte Sell Booze Sell Dope Sell p0rn!!!

WTF, you have an opportunity staring at you.
The property is in the front room of a private residence. There is a building next door that at one time was a general store. I always thought that would make a good place for a coffee shop.

Postal Service: Four county offices could close
You will be electroplated. Resistance is potential divided by current! - Dr. Matt

Cogito, ergo Deus non est.

sparks
Posts: 14085
Joined: Fri Oct 26, 2007 4:13 pm
Location: Friar McWallclocks Bar -- Where time stands still while you lean over!
Has thanked: 1966 times
Been thanked: 605 times

### Re: United States Postal Service: What to do?

Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
You can lead them to knowledge, but you can't make them think.

DrMatt
BANNED
Posts: 29811
Joined: Fri Jul 16, 2004 4:00 pm
Location: Location: Location!
Has thanked: 143 times
Been thanked: 70 times

### Re: United States Postal Service: What to do?

sparks wrote:Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
So proof of service of intent to sue in court will be accomplished by "I tried to e-mail them but didn't know their e-mail addresses." Or "I tried to Fedex them but Fedex said they don't serve that neighborhood."
Grayman wrote:If masturbation led to homosexuality you'd think by now I'd at least have better fashion sense.

Grammatron
Posts: 33586
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
Been thanked: 1747 times

### Re: United States Postal Service: What to do?

DrMatt wrote:
sparks wrote:Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
So proof of service of intent to sue in court will be accomplished by "I tried to e-mail them but didn't know their e-mail addresses." Or "I tried to Fedex them but Fedex said they don't serve that neighborhood."
That's a nice hypothetical, in reality USPS uses FedEx for some deliveries, never the other way around.

DrMatt
BANNED
Posts: 29811
Joined: Fri Jul 16, 2004 4:00 pm
Location: Location: Location!
Has thanked: 143 times
Been thanked: 70 times

### Re: United States Postal Service: What to do?

Grammatron wrote:
DrMatt wrote:
sparks wrote:Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
So proof of service of intent to sue in court will be accomplished by "I tried to e-mail them but didn't know their e-mail addresses." Or "I tried to Fedex them but Fedex said they don't serve that neighborhood."
That's a nice hypothetical, in reality USPS uses FedEx for some deliveries, never the other way around.
So when USPS ceases to exist, proof of service of intent to sue will be accomplished by citing FedEx as a government organ, or what?
Grayman wrote:If masturbation led to homosexuality you'd think by now I'd at least have better fashion sense.

Grammatron
Posts: 33586
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
Been thanked: 1747 times

### Re: United States Postal Service: What to do?

DrMatt wrote:
Grammatron wrote:
DrMatt wrote:
sparks wrote:Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
So proof of service of intent to sue in court will be accomplished by "I tried to e-mail them but didn't know their e-mail addresses." Or "I tried to Fedex them but Fedex said they don't serve that neighborhood."
That's a nice hypothetical, in reality USPS uses FedEx for some deliveries, never the other way around.
So when USPS ceases to exist, proof of service of intent to sue will be accomplished by citing FedEx as a government organ, or what?
I don't have a crystal ball that tells me hypothetical.

DrMatt
BANNED
Posts: 29811
Joined: Fri Jul 16, 2004 4:00 pm
Location: Location: Location!
Has thanked: 143 times
Been thanked: 70 times

### Re: United States Postal Service: What to do?

Grammatron wrote:
DrMatt wrote:
Grammatron wrote:
DrMatt wrote:
sparks wrote:Aside from delivering packages which can probably be done better by the private companies, we should simply move to a web based document exchange. In other words, let the USPS die a natural death. No heroic efforts, DNR.
So proof of service of intent to sue in court will be accomplished by "I tried to e-mail them but didn't know their e-mail addresses." Or "I tried to Fedex them but Fedex said they don't serve that neighborhood."
That's a nice hypothetical, in reality USPS uses FedEx for some deliveries, never the other way around.
So when USPS ceases to exist, proof of service of intent to sue will be accomplished by citing FedEx as a government organ, or what?
I don't have a crystal ball that tells me hypothetical.
Well, if we seriously propose to kill off the USPS, we ought to figure out how all its current use cases will be accomplished in its absence.
Grayman wrote:If masturbation led to homosexuality you'd think by now I'd at least have better fashion sense.

Grammatron
Posts: 33586
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
Been thanked: 1747 times

### Re: United States Postal Service: What to do?

DrMatt wrote: Well, if we seriously propose to kill off the USPS, we ought to figure out how all its current use cases will be accomplished in its absence.
No one is saying we shouldn't.

EvilYeti
Posts: 9222
Joined: Sat Jun 05, 2004 5:55 am
Location: San Diego
Has thanked: 2 times
Been thanked: 10 times

### Re: United States Postal Service: What to do?

DrMatt wrote: Well, if we seriously propose to kill off the USPS, we ought to figure out how all its current use cases will be accomplished in its absence.
Exactly the way they were before there was a USPS.

As it stand currently its just a means to fill my mailbox with junk.


text:
# Polar and cartesian issue

Tags:
1. Oct 15, 2015

### subinator

This is a problem that has been bugging me all day. While working with the well-known dydx = rdrdθ, where r is a function of θ I divided both sides of the equation by dxdθ to get dy/dθ = r(dr/dx)

For the left side, I use y = rsinθ and derive with respect to θ to get dy/dθ = sinθdr/dθ + rcosθ. For the left side, I use r^2 = y^2 + x^2, and derive both sides dx, to get

2r(dr/dx)=2y(dy/dx) + 2x, which simplifies to r (dr/dx)=y(dy/dx) + x.

I then put both of these equalities in the equation to get

sinθ(dr/dθ) + rcosθ = y(dy/dx) + x. Knowing that x = rcosθ, and that y/r = sinθ

I subtract x then divide y from both sides to get

(1/r)(dr/dθ) = (dy/dx).

However, this contradicts the proof that

dy/dx =((dr/dθ)sinθ+rcosθ)/((dr/dθ)cosθ-rsinθ).

I just want to know what went wrong. (I have checked mathematically, these two functions will NOT give the same results at most points.)

Last edited by a moderator: Oct 15, 2015
2. Oct 15, 2015

### andrewkirk

Where did you get this expression? In general it is not a valid expression. In certain tightly constrained contexts it might be able to be given a meaning specific to the context, in such a way that the equality holds. But we don't know from what context you took it, and whether the conditions of that context are preserved throughout the manipulations you do above.

A good general principle is to not treat differentials like dx like numeric quantities by which one can multiply and divide. They are not, and one can't. In some cases one can use such multiplication or division as a shorthand, but one needs to have a sense of what the shorthand is short for, and be confident that the abbreviation is valid.

3. Oct 15, 2015

### subinator

This expression is the Jacobian which the differentials dydx in cartesians are translated into polar differentials rdrdθ, and I am keeping it in the context of coordinates in a plane where it is valid (as that is where I make the comparison to the proof y'=(r'sinθ+rcosθ)/(r'cosθ-rsinθ), which simply equates dy/dx to (dy/dθ)/(dx/dθ), an algebraic transformation). I think I may see your point in the division of two differential numbers at once, which may violate the conditions of shorthand and there is where it becomes illegal, but I wish I knew for sure if that was the case or not.

4. Oct 15, 2015

### subinator

After checking further, this is indeed the case, as when there are two differential numbers this doesn't become a legal operation. The initial statement dydx = rdrdθ is valid, but changing it to dy/dθ=rdr/dx is not. (by this logic this would mean (dx/dr)(1/r) would have to equal dθ/dy, which is most certainly not true or how derivatives work. Thank you.)

5. Oct 15, 2015

### andrewkirk

OK, then the context where we are using Jacobians for area is that we have the following equation:

$$\iint_A f(x, y) \,dx \,dy = \iint_A f(r \cos \vartheta, r \sin \vartheta) \, r \, dr \, d\vartheta$$

That is not the same as saying that:

$$dx \,dy = r \, dr \, d\vartheta$$

The manipulations need to be done within the context of the first equation. As soon as they are taken out of that context, they lose validity.


text:
The Ess Kay Refrigerator Company is deciding to issue 2,000,000 of Rs 1,000, 14 per cent 7-year...

1 answer below »

The Ess Kay Refrigerator Company is deciding to issue 2,000,000 of Rs 1,000, 14 per cent 7-year debentures. The debentures will have to be sold at a discount rate of 3 per cent. Further, the firm will pay an underwriting fee of 3 per cent of the face value. Assume a 35% tax rate.

Calculate the after-tax cost of the issue. What would be the after-tax cost if the debenture were sold at a premium of Rs 30?

5 Ratings, (9 Votes)
Please find the solution...

Related Questions in Managerial Accounting

• Question 1: Case study You are a financial adviser and the following information is an extract of...

(Solved) 4 weeks ago

’s after-tax income for the year ended June 30 th 2018. Explain how David and Chloe could reduce their tax liability through splitting their income. Show the effect this strategy would have had, if they had split income for the tax year ended. B. Calculate Chloe’s capital gains liability if she were

Please find the attached complete solution of the Case study. The tax rates use for doing the solution is of 2018. Required: A. Calculate David and Chloe’s after-tax income for the year...
• CASE 11A–7 Transfer Pricing; Divisional Performance [LO5] Stanco, Inc., is a decentralized...

December 07, 2015

. . . . . . . . . . 10.00 Total cost per timing device . . . . . . . . . . . . . . . . . . . . $69.75 The manager of the Clock Division feels that she can’t quote a price greater than$70 per timing device to the oven manufacturer if her division is to get the job

• Refer to the data in Exercise 11—8. Assume that instead

(Solved) July 12, 2016

Refer to the data in Exercise 11—8. Assume that instead of producing 4,000 units during the month, the company produced only 3 ,000 units, using 14 ,750 pounds of material. (The rest of the material purchased remained in raw materials inventory.) Required: Compute the direct materials price

Notice in the arrangement beneath that the materials price variance is computed for the whole amount of materials purchased, while the materials quantity variance is computed just for the...
• Assignment 3: Professional Obligations Task Firstly, you are required to devise a series of...

(Solved) 4 weeks ago

Assignment 3 : Professional Obligations Task Firstly, you are required to devise a series of interview questions to investigate how ethics and professional behaviour affect individuals in their real work life. These interviews can be conducted face to face , by telephone, or (as a last choice

Introduction The report is a reflection of the accounting principles and ethics. For this purpose, some questions are prepared to investigate how the ethical and processional behaviours...
• Thinking of the Middle East, identify where you encounter products, services, or food items that...

(Solved) 4 weeks ago

Thinking of the Middle East, identify where you encounter products, services, or food items that should be relatively uniform and are based on standard inputs or costs . Identify the items, their standard components, and what the standards would probably be for each item. For example