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# A near-term to net zero alternative to the social cost of carbon for setting carbon prices

## Abstract

The social cost of carbon (SCC) is commonly described and used as the optimal CO2 price. However, the wide range of SCC estimates provides limited practical assistance to policymakers setting specific CO2 prices. Here we describe an alternate near-term to net zero (NT2NZ) approach, estimating CO2 prices needed in the near term for consistency with a net-zero CO2 emissions target. This approach dovetails with the emissions-target-focused approach that frames climate policy discussions around the world, avoids uncertainties in estimates of climate damages and long-term decarbonization costs, offers transparency about sensitivities and enables the consideration of CO2 prices alongside a portfolio of policies. We estimate illustrative NT2NZ CO2 prices for the United States; for a 2050 net-zero CO2 emission target, prices are US$34 to US$64 per metric ton in 2025 and US$77 to US$124 in 2030. These results are most influenced by assumptions about complementary policies and oil prices.

## Main

Economists overwhelmingly support pricing CO2 emissions1. How much to charge for each ton of emissions is perhaps the most important element of a carbon pricing policy, yet little consensus exists among economists about the appropriate level for CO2 prices2.

To find optimal CO2 prices, economists have long focused on a metric called the social cost of carbon (SCC), an estimate of the marginal damages of an additional ton of CO2 emissions. However, the SCC cannot be credibly estimated with sufficient precision to provide practical assistance to policymakers setting CO2 prices. The SCC approach is also disconnected from real-world policy discussions that position CO2 prices as one element of a strategy to avoid the risks of exceeding thresholds of global warming.

In the face of these constraints, this paper introduces an alternative approach. It starts with policymakers selecting a net-zero CO2 emissions target informed by the best available science and economics. Then, near-term to net zero (NT2NZ) CO2 prices are combined with a broader policy strategy to achieve an emissions pathway consistent with the net-zero target in the near term, when the projections of energy–economic models are most useful.

## Helping policymakers set CO2 prices

In textbooks, optimal CO2 prices are identified with perfect precision. Net benefits to society are largest if the government taxes an activity that creates a negative externality (such as CO2 emissions) at the rate that equalizes the marginal social benefits and marginal social costs of emissions reductions3.

Economists have long recognized that, in the real world, approaches for developing optimal policies can be constrained by various uncertainties and measurement difficulties4,5, including imprecision, ambiguity, intractability and indeterminacy6. Therefore, in addition to maximizing net benefits, analysts should strive to identify CO2 prices using approaches with the following attributes:

• Credible precision. The approach produces a range of CO2 prices that provides policymakers with valuable information to incorporate into policy design decisions. For example, if adding complexity to a model injects disproportionate uncertainty, it can be more informative to use a simpler framework7.

• Transparency. The approach enables policymakers to understand the causes of relatively high or low estimates within the range of CO2 prices identified.

• Consistency with policy objectives. The approach produces CO2 prices that fit the objectives of the policy. The most common rationale for a CO2 price is to reduce emissions as part of a broader global response to the risks of climate change.

Policymakers setting CO2 prices are also concerned with additional factors that are outside our scope, such as public health benefits from reducing air pollution, energy security, competitiveness, the expected actions of other jurisdictions and (perhaps most importantly) political viability.

## Challenges with setting CO2 prices using SCC estimates

The SCC, commonly used as the optimal CO2 price that maximizes net benefits to society8, is estimated with projections of the following parameters: global emissions over the next few centuries, the effects of emissions on temperatures and other climate impacts, and the impacts of climate change on the economy and human welfare, using economic methods that aggregate centuries of impacts into a single value representing the net benefits of emissions reductions9.

Unfortunately, the degree of uncertainty in SCC estimates spans virtually any conceivable stringency level for a CO2 pricing policy. Meta-analyses find recent SCC estimates that range from under US$0 per ton of CO2 to over US$2,000 per ton (excluding outliers still leaves a range of hundreds of dollars)8,10.

SCC estimates will continue to improve9,11,12, but methodological advancements are unlikely to narrow the range of SCC estimates much. After all, large uncertainties come from parameters that are inherently uncertain, such as the appropriate discount rates9, risk aversion levels13, issues around inequality14 and attempts to assign monetary values to non-economic climate damages15. In addition, methodological improvements often involve incorporating new uncertain elements that were omitted from previous estimates, which can widen the range of SCC estimates16,17.

The advantage of the SCC approach is that it attempts to perfectly maximize net benefits to society; however, the SCC approach provides limited practical value to policymakers setting specific CO2 prices due to its difficulty satisfying the desired attributes identified in the previous section. First, the range of SCC estimates is too wide to credibly support the use of any single CO2 price. (In contrast, the US government developed SCC estimates for use in the separate context of benefit–cost analysis, where a wide range can be incorporated9.) Second, the differences in SCC estimates hinge partly on assumptions that are not usually transparent to policymakers (such as the value placed on future generations by discount rates). Third, the SCC approach is disconnected from constraining global warming beyond specific levels—the goal of most policymakers setting CO2 prices. For example, William Nordhaus’s 2018 Nobel Prize Lecture shows an optimal pathway of 4 °C of warming by the mid-2100s as the implication of his SCC estimates18, an outcome far outside the bounds of the Paris Agreement’s goals (which other analyses have found would pass a cost–benefit test19). Helping policymakers set CO2 prices in practice therefore necessitates an approach that balances benefits and costs only imperfectly, such as the NT2NZ approach described in the next section.

The classic alternate approach in the face of this sort of uncertainty is a cap-and-trade programme, which involves mandating emissions targets. We assume that policymakers have other reasons for selecting a price instrument, such as the interactions with complementary policies, a desire for certainty in business planning, concerns about market manipulation or political economy considerations20.

## The NT2NZ approach

NT2NZ CO2 prices are designed to accommodate uncertainties and measurement difficulties and to align with real-world policy objectives. They are estimated using the following four steps:

### Step 1: select a net-zero CO2 emissions date

While the international climate change negotiations have focused primarily on temperature targets, policymakers are increasingly shifting to net-zero emissions targets for both substantive and political reasons21,22,23,24. A global net-zero CO2 emissions target has a science-based rationale: surface temperatures will continue to increase until the sources and sequestration of CO2 are equal, at which point temperatures will roughly stabilize25,26 (reductions in non-CO2 GHGs and land-use change emissions are required for full stabilization24, but we focus on CO2 emissions, which are the bulk of what would be covered by a CO2 price). Views differ on what threshold of global warming should not be exceeded, but failing to achieve net-zero CO2 emissions implies ever-increasing temperatures, which will eventually reach unacceptably high levels regardless of one’s threshold. Net-zero targets also naturally scale to any jurisdiction, because achieving global net zero requires, on average, all jurisdictions to achieve net zero.

Policymakers must balance a range of factors to set net-zero targets, including the risks of even-higher temperature changes and the additional costs of decarbonizing faster. International climate agreements recognize that countries have the responsibility to decarbonize at different paces27, which means that jurisdictions will set different net-zero target dates. Like estimating an SCC, setting a net-zero target involves judgements about concern for future generations, willingness to tolerate risks and aversion to inequality (among other factors). Under the NT2NZ approach, these trade-offs are made by the governing officials selecting the target.

### Step 2: select an emissions pathway to the net-zero target

An infinite number of pathways are conceivable between current emissions levels and a future net-zero target. Some frameworks emphasize the benefits of reducing near-term disruption and enabling innovation to bring down technology costs, which argue for a slower initial rate of reductions28. Other frameworks emphasize the importance of near-term deployments in reducing costs and avoiding technology lock-in29,30,31, as well as benefits from reducing cumulative emissions and respecting intergenerational equity.

Policymakers, weighing these considerations as well as technical and political constraints, may choose a straight-line pathway to net zero for simplicity and transparency, or a different trajectory that fits the circumstances of the jurisdiction (such as a developing country with a peak-and-decline pathway)32.

### Step 3: estimate CO2 prices consistent with the emissions pathway in the near term

Energy–economic models can be used to estimate the CO2 prices required to reduce emissions on a desired pathway under a given set of assumptions about future technologies, prices and behaviour33. Unlike the SCC approach, energy–economic models enable analysts to combine CO2 prices with other policy measures to overcome multiple market barriers to emissions reductions.

While net-zero emissions is the long-term goal, the NT2NZ approach focuses on the near term (the next decade, for example). Models that simulate economic and energy systems are built using historical data on production, consumption and market dynamics, which may be a reasonable assumption in the near term. After all, most energy technologies and consumer behaviours evolve relatively slowly. But such models become less useful as the time horizon of the exercise lengthens34. Changes in technologies, preferences and policies will inevitably impact energy systems in unexpected ways in a rapidly decarbonizing country—just as advancements in solar energy technologies and the shale revolution in the United States were almost entirely unforeseen decades ago35.

Focusing on the near term means that CO2 price estimates should not be unduly influenced by assumptions about the highly uncertain long-term evolution of technologies and behaviour. Analysts often use models with ‘foresight’, which means that decisions are contingent on assumed future changes to the energy system. For example, if the costs of breakthrough low-carbon technologies are assumed to remain prohibitively high, models with foresight may suggest that the cost-effective pathway involves higher near-term CO2 prices, so that additional emissions reductions can be achieved in sectors where stock turnover is slow. In contrast, if the costs of breakthrough technologies are assumed to fall precipitously, the same model may suggest a cost-effective pathway that involves relatively less near-term mitigation (see Supplementary Information, Appendix 2).

### Step 4: periodically update Steps 1–3

Knowledge about climate science and the costs of mitigation technologies will continue to change rapidly, especially given the substantial social and economic shifts associated with decarbonization. This calls for an adaptive management strategy36 whereby the analysis described above is repeated periodically. Using emissions outcomes and other relevant metrics (such as deployment rates of low-carbon infrastructure and progress in hard-to-decarbonize sectors30), both the CO2 price pathway and the broader climate policy strategy can be revised and extended, capturing the most up-to-date information.

Adaptive management can enable jurisdictions to stay close to the desired emissions pathway without making policy details contingent on assumptions about highly uncertain long-term variables. Various mechanisms for periodically updating policies have been proposed in recent years37, including CO2 prices that are contingent on emissions outcomes (that is, if emissions exceed target levels, price increases accelerate in future years).

## Illustrative NT2NZ CO2 prices for the United States

We demonstrate the approach described above to produce illustrative NT2NZ CO2 prices for the United States. For this analysis we use the 50-state version of the Global Change Assessment Model (GCAM-USA), an integrated assessment model of energy–economy–environment systems, and data available as of 2019 (Methods). Real-world policy design should be informed by multiple analytic tools using the most up-to-date available information32.

We begin with three straight-line annual emissions pathways from current (2020) levels to net-zero CO2 emissions targets in 2060, 2050 and 2040 (Fig. 1a) to reflect a range of emissions pathways discussed in recent years by US policymakers. In the absence of a consensus favouring larger or smaller near-term emissions reductions38,39, an illustrative straight-line emissions pathway for the United States may be appealing due to its simplicity and transparency. These pathways correspond to 2030 CO2 emissions of 35%, 42% and 57% below 2005 levels, respectively. Consistent with both economic theory and policy practice, we assume that the CO2 price is surrounded by complementary policies that address separate market failures40: efficiency policies, air pollution regulations and early-stage support for the deployment of low-carbon technologies (such as electric vehicles).

For our benchmark scenario, we find NT2NZ CO2 prices in 2025 of US$32, US$52 and US$93 per metric ton (in 2018 dollars) for consistency with net-zero targets in 2060, 2050 and 2040, respectively. The NT2NZ CO2 prices in 2030 are roughly two times larger (Fig. 1b), reflecting a much higher annual growth rate than typical CO2 price estimates based on the SCC or rising at the rate of interest (see Supplementary Information, Appendix 2). For each emissions pathway, we show a range of NT2NZ CO2 prices from sensitivity scenarios that intend to capture uncertainty in influential model inputs (Methods). Figure 2 shows that compared with our benchmark scenario (with a 2050 target), more stringent and successful complementary policies (that is, air quality regulations that lead to higher coal retirements, more aggressive energy efficiency measures and more aggressive early-stage deployment support for certain low-carbon technologies) lowered the CO2 prices by US$10–US$20 per ton. The prices rise by about the same amount with less aggressive complementary policies. Changing the future oil price trajectory from a pathway to either US$45 or US$176 per barrel in 2030 leads to a swing in CO2 prices of US$40 per ton in 2030. Figure 2 also displays the impacts on NT2NZ CO2 prices from changing inputs related to natural gas prices, innovation and economic growth. Appendix 1 in the Supplementary Information provides additional results.

For a 2050 target, the range of NT2NZ CO2 prices is largely consistent with the range of CO2 prices in legislation proposed to the US Congress in 2019 (Fig. 1). However, the prices are on the lower end of the range of CO2 prices that global energy–economic models have identified as consistent with constraining average global temperature increases to 1.5 and 2 degrees Celsius41 for at least three reasons. First, NT2NZ CO2 prices will differ by jurisdiction, and the United States has a large amount of coal-fired electricity generation that can be replaced at a relatively low cost. Second, while many studies assume that the CO2 price is implemented without other policies, we assume that multiple policies are implemented to address multiple market barriers to emissions reductions. Third, the actors within GCAM do not have foresight, so their energy consumption is based on their myopic vision of current market conditions and not based on long-term projections of technological progress (see Supplementary Information, Appendix 2 and Extended Data Fig. 1).

## Discussion and policy implications

Economists have long referred to the SCC as the optimal CO2 price. The use of the SCC as a CO2 price has become commonplace in recent years, including in federal carbon tax proposals and state-level subsidies for clean electricity generation in the United States (see Supplementary Information, Appendix 3). Using the NT2NZ approach instead offers several advantages.

First, CO2 prices can be estimated with more precision. All of the uncertainties in the CO2 prices estimated using the NT2NZ approach—such as near-term clean energy innovation and fossil fuel prices—are also uncertainties using the SCC approach. But the NT2NZ approach avoids much larger uncertainties, including assigning monetary values to climate change damages. The NT2NZ approach focuses on the most important and (relatively) better understood aspects of the problem.

Second, while any approach can be transparent, the NT2NZ approach focuses on changes to the energy system in the near future, which should enable policymakers to better understand the rationale for selecting CO2 prices of varying levels. An NT2NZ analysis enables external stakeholders to assess whether a jurisdiction’s policies are consistent with its targets and how the outcomes are influenced by key assumptions about technology costs and energy prices (Fig. 2). While the full range of our estimates is ~US\$40 per ton for any given target, policymakers are not likely to regard all sensitivities as equally likely and can choose within that range accordingly. By comparison, the complexity of modelling the SCC can make it more difficult to communicate why estimates differ (often by hundreds of dollars per ton under different plausible assumptions about discounting, adaptation and so on).

Third, the NT2NZ approach is more consistent with the objectives of most policymakers implementing CO2 prices. While economists have traditionally recommended a harmonized global CO2 price to maximize efficiency, national annual emissions targets have long been the lingua franca of international negotiations on climate change. The illustrative example in the previous section shows how US policymakers could use the NT2NZ approach to help design a federal carbon price as an element of its 2030 Nationally Determined Contribution to the Paris Agreement. Other countries, groups of countries or subnational jurisdictions could do the same. The NT2NZ approach also enables the CO2 price to be one piece of a broader policy strategy to address multiple market failures around climate, which better aligns with economic theory compared with a CO2-price-alone approach40. It also better aligns with the real-world practice of combining CO2 prices with a range of other (often sectoral)42 policies, recognizing that preparation for deep decarbonization31 requires adaptive management using an ecosystem of policies. For example, passenger vehicle decarbonization might include goals and policy measures to encourage electric vehicles, charging infrastructure, fuel economy and modal shifts away from single-occupancy vehicle travel43.

Pairing a long-term emissions target with a set of iterative near-term policies is not novel. The United Kingdom, for example, has adopted a national target of net-zero GHG emissions by 2050 and sets five-year carbon budgets to act as stepping-stones44. Indeed, the Paris Agreement encourages this framework by calling on nations to produce long-term development strategies for low GHG emissions and near-term Nationally Determined Contributions that are updated every five years26.

The NT2NZ approach does not attempt to set CO2 prices by perfectly balancing costs and benefits, so it does not satisfy the definition of an ‘optimal CO2 price’ found in economics textbooks. Instead, it enables policymakers to consider both qualitative and quantitative information about climate science and economics when selecting a net-zero target. Models can then estimate the CO2 prices for cost-effective reductions in the near term, when their projections are most useful, with the understanding that policy details can be updated as uncertainties are resolved.

We describe our results as illustrative because no single model should inform policy setting. The NT2NZ approach should be implemented across a suite of models to further characterize uncertainties, identify results that are robust across methods34 and explore how prices are best combined with other policies. Future work should also examine the roles of the domestic land-use sink and non-CO2 GHG emissions.

## Methods

### Overview of the model

To numerically estimate CO2 prices using the NT2NZ approach, we use GCAM-USA. GCAM is an economy-wide global integrated assessment model representing the energy and land sectors linked with a climate model; GCAM is used to explore the interactions of emissions-reducing investments and activities across the US and global economies. We begin with the same version of the model and the standard assumption set used for the US Mid-century Strategy (MCS)45. The technical assumptions used in the US MCS are documented in the MCS technical appendix46. We have also made updates and changes reflecting the developments in the energy markets and policies as of late 2019, when this analysis was conducted (before the COVID-19 crisis). These are described in detail below. More detail on GCAM can be found in the Supplementary Information.

### Key updated assumptions

Most of the assumptions follow the benchmark scenario described in the MCS technical appendix. We updated key assumptions to align the model with the changing market dynamics in the past several years, using the sources in Supplementary Table 3. Given the uncertain and influential nature of these assumptions, our sensitivity scenarios are designed by selecting low, benchmark and high assumptions for each.

#### Innovation and low-carbon energy technologies

To reflect the rapidly falling costs of renewable energy, we have updated the renewable energy costs to those from the National Renewable Energy Laboratory’s (NREL) Annual Technology Baseline47. To reflect uncertainties in the evolution of future energy markets as well as concerted research, development and deployment policies to further reduce low-carbon energy costs, we conduct a sensitivity analysis using the high, medium and low estimates of solar, wind and nuclear costs. The detailed cost assumptions are summarized in Supplementary Table 4.

#### Oil and gas prices

To reflect future uncertainty in oil and gas prices, we have updated the oil and gas price trajectories consistent with the Energy Information Administration’s (EIA) oil and gas price trajectories from Annual Energy Outlook 2019 (ref. 48). Each five-year price point in GCAM is based on a three-year running average of prices to avoid short-term fluctuations. We conduct a sensitivity analysis using the high, medium and low estimates of the oil and gas price trajectories. The detailed assumptions are summarized in Supplementary Table 5. We note that oil prices have fallen considerably since our analysis was performed, and we probably will not know how projections for 2025 or 2030 will be influenced until after the immediate COVID-19 crisis subsides.

#### Population and GDP

We conducted a sensitivity analysis using high, medium and low estimates of population and GDP. Medium population projections are based on US Census Bureau National Population Projections Tables49. The high population growth scenario assumes 0.1% per year additional growth, and the low population growth scenario assumes 0.1% per year less growth. The GDP variations are based on Annual Energy Outlook 2019, which provides high, medium and low variants of GDP growth projections. The COVID-19 pandemic will considerably depress GDP growth in the near term and has the potential to do so through 2025—future analysis with updated projections from the EIA (or another source) should take this effect into account. The detailed population and GDP data used are tabulated in Supplementary Table 6.

#### Complementary policies

We developed three complementary policy scenarios (detailed below), both to reflect policies needed to overcome market barriers left unaddressed by CO2 prices and to reflect the uncertain future of sectoral policies that influence CO2 emissions.

### Coal retirements

Coal-fired power plants release not only CO2 emissions but also various other air pollutants, such as particulate matter and ozone. Policymakers have long recognized the need for regulations to protect constituents from the harmful impacts of these pollutants. Coal-fired power plants have been rapidly retiring in the United States due to changing market dynamics and concerted efforts to reduce emissions of CO2 and other air pollutants. To capture the potential impact of future environmental regulations on coal generation, we developed high, medium and low coal-retirement scenarios. These coal-retirement pathways assume no CO2 emission mitigation policy; they thus serve as a starting point for our CO2 price scenarios. The low trajectory tracks the EIA Annual Energy Outlook 2018 coal-retirement reference case projections48. The medium (benchmark) trajectory roughly tracks the US Environmental Protection Agency Integrated Planning Model May 2019 reference case50 (which also assumes no new policy). The high trajectory tracks the Integrated Planning Model reference case to 2021 and then assumes a trajectory that is consistent with the Enhanced Engagement scenario in the America’s Pledge Report51 (this post-2021 rate is consistent with a capacity retirement rate between the medium and average retirement rates since 2012). The impact of these coal-retirement trajectories on generation is shown in Table M5. In addition to scheduled retirements, GCAM allows the power plants to prematurely retire when they are no longer profitable in the market. Sufficiently high CO2 prices can therefore force coal power plants out of the market (Supplementary Fig. 3).

### Energy demand

A CO2 price alone is often insufficient to encourage consumers to take advantage of all cost-effective opportunities to reduce energy usage due to market barriers including informational failures and consumer short-sightedness. To reflect uncertainty in the future growth of energy demand, including policies to reduce the rate of energy demand growth, we developed high, medium and low scenarios for future demand growth in key energy sectors using assumptions from the Smart Growth scenario of the MCS report45. Specifically, our low and medium energy demand scenarios assume less vehicle travel and greater efficiency in the buildings sector than does the Benchmark scenario. The detailed energy demand differences are shown in Supplementary Table 7.

### Early-stage deployment support for low-carbon technologies

In addition to the lack of a CO2 price, numerous market barriers stand in the way of a large-scale shift to products that do not directly burn fossil fuels (such as electric vehicles and electric heat pumps). To reflect the highly uncertain future of electrification, including policies designed to support electrification, we develop high, medium and low sensitivity scenarios for electrification in buildings and transportation, drawn from the NREL Electrification Futures Study52. The electrification rates are shown in Supplementary Table 7.

### Land-sector sink assumptions

We develop three net-zero emissions targets (2040, 2050 and 2060) (Fig. 1). In our analysis, net CO2 emissions are constrained to linearly decrease to zero in the specified target years. In 2030, this formulation corresponds to an emissions range of 3,400 MtCO2 to 2,200 MtCO2 (35–57% below the 2005 emission level). For simplicity, we assume the CO2 emissions from land-use change are constant at current levels: 714 MtCO2 (ref. 53). The US MCS has considered multiple possible pathways for CO2 emissions from land-use change, but given the large fluctuations in year-by-year estimates for land-use change emissions, we did not find other estimates to be of additional value that warrants the additional complexity. A decrease in the land sink over time would require additional emissions reductions from fossil fuel combustion beyond what we model here.

### Reporting Summary

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

## Data availability

All data generated or analysed during this study are from publicly available sources and are either included in this article (and its Supplementary Information files) or available from the corresponding author on reasonable request. Source data are provided with this paper.

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United States Mid-century Strategy for Deep Decarbonization (White House, 2016); https://unfccc.int/files/focus/long-term_strategies/application/pdf/mid_century_strategy_report-final_red.pdf

46. 46.

United States Mid-century Strategy for Deep Decarbonization—Technical Appendix (White House, 2016); https://unfccc.int/files/focus/long-term_strategies/application/pdf/us_mcs_documentation_and_output.pdf

47. 47.

Annual Technology Baseline (ATB) from the National Renewable Energy Laboratory (NREL) (NREL, 2019); https://atb.nrel.gov/

48. 48.

Annual Energy Outlook 2018 (EIA, 2018); https://www.eia.gov/outlooks/aeo/

49. 49.

2017 National Population Projections Datasets: Projections for the United States: 2017 to 2060 (US Census Bureau, 2017); https://www.census.gov/data/datasets/2017/demo/popproj/2017-popproj.html

50. 50.

Incremental Documentation for IPM Platform v6 May 2019 Reference Case (US EPA, 2019); https://www.epa.gov/airmarkets/incremental-documentation-ipm-platform-v6-may-2019-reference-case

51. 51.

Fulfilling America’s Pledge: How States, Cities, and Business Are Leading the United States to a Low-Carbon Future (America’s Pledge Initiative on Climate, 2018); https://www.americaspledgeonclimate.com/fulfilling-americas-pledge/

52. 52.

Jadun, P. et al. Electrification Futures Study: End-Use Electric Technology Cost and Performance Projections through 2050 (NREL, 2017); https://www.nrel.gov/docs/fy18osti/70485.pdf

53. 53.

Inventory of U.S. Greenhouse Gas Emissions and Sinks: 1990–2017 (US EPA, 2019).

## Acknowledgements

We thank K. Gillingham, G. Heal, A. Fawcett, J. Larsen, S. Sayre, J. Goffman and P. Kelleher for comments on an early version of this manuscript. This work was made possible by support from the Center on Global Energy Policy at the School of International and Public Affairs of Columbia University.

## Author information

Authors

### Contributions

N.K., A.R.B., H.M. and P.M. contributed to the design of the research. W.K. and H.M. conducted the modelling. All authors contributed to the analysis of the results. N.K., A.R.B., H.M. and P.M. wrote the manuscript.

### Corresponding author

Correspondence to Noah Kaufman.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Climate Change thanks Emily Wimberger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

## Extended data

### Extended Data Fig. 1 SI Figure 4. NT2NZ versus CO2 Prices Rising at the Interest Rate.

a, Linear emissions reduction pathways to net zero targets (dark lines) vs emissions reduction pathways when the price is constrained to grow at the interest rate (as a rough proxy for perfect foresight). b, CO2 prices for the linear pathways (closed symbols) and interest rate-constrained pathways (open symbols) in the year 2025 and 2030 for net zero targets in 2040 (squares), 2050 (circles), and 2060 (triangles), respectively.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–5, discussion and Tables 1–7.

## Source data

### Source Data Fig. 1

Numerical data used to generate graphs from GCAM output.

### Source Data Fig. 2

Numerical data used to generate graphs from GCAM output.

### Source Data Extended Data Fig. 1

Numerical data used to generate graphs from GCAM output.

## Rights and permissions

Reprints and Permissions

Kaufman, N., Barron, A.R., Krawczyk, W. et al. A near-term to net zero alternative to the social cost of carbon for setting carbon prices. Nat. Clim. Chang. 10, 1010–1014 (2020). https://doi.org/10.1038/s41558-020-0880-3

• Accepted:

• Published:

• Issue Date:

• ### A price on warming with a supply chain directed market

• John F. Raffensperger

Discover Sustainability (2021)


text:
# Weight of light and Gravity Speed

My math teacher was talking to me about how NASA weighed light. If it doesn't have any mass, how is this possible? Also, he was saying that they are beginning to study a speed faster than the speed of light. Called Gravity Speed. Which has no lag, and is instantaneous. Do any of you know about these?

Related Other Physics Topics News on Phys.org
Originally posted by Guard
My math teacher was talking to me about how NASA weighed light. If it doesn't have any mass, how is this possible? Also, he was saying that they are beginning to study a speed faster than the speed of light. Called Gravity Speed. Which has no lag, and is instantaneous. Do any of you know about these?

And your math teacher told you the derivative of x is 0.

Integral
Staff Emeritus
Gold Member
Here is a concept, read some of the other threads on the forum.

Gravity is not instantaneous, it propagates at c.

Originally posted by Guard
My math teacher was talking to me about how NASA weighed light. If it doesn't have any mass, how is this possible? Also, he was saying that they are beginning to study a speed faster than the speed of light. Called Gravity Speed. Which has no lag, and is instantaneous. Do any of you know about these?
Perhaps you misunderstood him. Maybe he was talking about the gravitational aspects of light? Do you recall exactly what he meant by that? Perhaps be meant the same thing Alan Guth did when he wrote the following. From his class notes on The Early Universe
We are perhaps not used to thinking of electromagnetic radiation as having mass, but it is well-known that radiation has energy density. If the energy density is denoted by u, then special relativity implies that the electromagnetic radiation has a mass density $$\rho$$ given by

$$\rho = u/c^{2}$$

To my knowledge nobody has ever actually "weighed" electrogmagnetic radiation in any way, but the theoretical evidence in favor of Eq. (7.3) is overwhelming - light does have mass. (Nonetheless, the photon has zero rest mass, ...)
The full page can be read at
http://www.geocities.com/physics_world/guth.gif

format: plain


text:
# 3-D seismic acquisition geometry design and analysis: Investigation of the requirements to include illumination from all multiples

A Kumar

Research output: ThesisDissertation (TU Delft, preparation external)

## Abstract

A seismic survey should be designed such that imaging of the acquired data leads to a sufficiently accurate subsurface image. For that purpose, methods for acquisition geometry analysis and design are available. These methods are used to judge whether an acquisition geometry is suited for the specified objectives. Conventional 2D and 3D acquisition geometry analysis methods are largely based on common-midpoint processing that assumes a horizontally layered earth model. Consequently, the influence of the inhomogeneity of the subsurface on the data quality and image quality is not taken into account. However, in practice it has been shown that the data and image quality can suffer significantly from complex geology. Therefore, the inhomogeneity of the subsurface must be taken into account in the acquisition analysis methods. This can be done by the use of a macro model of the subsurface. Additionally, recent developments in seismic imaging and reservoir characterization use the multiple-reflections in the data to extend the illumination in areas that cannot be reached by the primary-reflections. The use of multiples yields a better vertical resolution as well as to suppress migration artefacts caused by crosstalk of multiple-reflections. The seismic value chain suggests that the new developments in seismic imaging and reservoir characterization should lead to new opportunities in the seismic acquisition. Therefore, the goal of this research is the development of a method that meets all the above mentioned requirements. The method presented in this thesis is based on the previously developed $\it{focal}$ $\it{beam}$ $\it{analysis}$ concept. This concept emphasizes the separate analysis of the source geometry and the detector geometry, leading to two outputs: the focal source beam and the focal detector beam. This gives the opportunity to separately judge and adjust the configuration of the sources and the configuration of the detectors. These beams are inspected in the space-frequency domain and in the Radon-frequency domain. In the spatial domain, it is visible whether the wavefield has properly been focused into a point. In the Radon domain, it is visible how the angle-dependent amplitudes are affected by the acquisition geometry and overburden structures. This approach provides thorough understanding of the cause of image deficiencies. The source and detector beams can be multiplied to compute a migrated image. The multiplication can be carried out in two domains to assess the different quality parameters of a seismic image: • Multiplication of the source and detector beam in the space-frequency domain yields the resolution function, which represents the image of a unit point diffractor. • Multiplication in the Radon-frequency domain yields the AVP-function, which is the angle-dependent image of one reflection point on an angle-independent reflector. However, so far this method assumed primary reflections only as signal, leaving out multiple reflections as noise. In this thesis, I discussed that multiples can be considered as signal in the seismic imaging if they are handled correctly. Therefore, I extended this focal beam method to the multiples as well. In the extended focal beam method, the full wavefield propagation between a subsurface point and the acquisition surface is simulated using a recursive full wavefield modeling engine. It uses a macro velocity model for the wavefield extrapolation from one depth level to another depth level and a reflectivity model to include all the reflection and transmission properties related to the same depth level. Subsequently, the data are focused to the target point using the source and the detector geometry. This full wavefield data is a complex wavefield which includes the effects of all multiples (i.e., propagation as well as reflection and transmission effects). Therefore, the focusing is achieved by a minimization scheme. I have demonstrated with the help of some examples, that the gap in the Radon transformed focal source beam due to sparse sampling issues can be filled with the use of all multiples. It means more angle-dependent information can be obtained. Imaging of subsalt sediments is challenging in practice, because of the high velocity contrasts and irregular shapes. My analysis shows that in such situation, image quality varies strongly with the position of the target point with respect to salt. In that case, even if one has a perfect source distribution, primary illumination may be limited due to geology. The image quality cannot be improved any further by adjusting the acquisition geometry in this case. I show that utilizing all multiples may be the part of the solution. The illumination from below is important in such cases. The method presented in this thesis offers opportunities for the investigation of the added value of the surface-related and the internal multiples. To summarize, this new method meets all the following criteria: • Dealing with the complexity of the subsurface model. • Considering all the multiple reflections as useful information. • Illumination from below in the complex sub-salt scenarios. • Development of geophysical-based infill specifications to assess the impact of coverage holes on data quality
Original language English Doctor of Philosophy Delft University of Technology Wapenaar, C.P.A., SupervisorBlacquière, G., Advisor 9 Dec 2015 Delft A. Kumar 978-94-6203-962-9 https://doi.org/10.4233/uuid:392ff377-0133-473c-9796-32ce61535aca Published - 2015

## Keywords

• seismic acquisition
• imaging
• illumination
• multiples


text:
## BdMO National 2012: Primary 10

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
Zzzz
Posts: 172
Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

### BdMO National 2012: Primary 10

Problem 10:
Tusher chose some consecutive numbers starting from $1$. He noticed that the least common multiple of those numbers is divisible by $100$. What is the minimum number of numbers he chose?
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)

nafistiham
Posts: 829
Joined: Mon Oct 17, 2011 3:56 pm
Location: 24.758613,90.400161
Contact:

### Re: BdMO National 2012: Primary 10

$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.


text:
lilypond-user
[Top][All Lists]

## Re: troubles with breaking lines

 From: Carl Sorensen Subject: Re: troubles with breaking lines Date: Sat, 30 Apr 2011 08:25:31 -0600

On 4/30/11 7:41 AM, "Federico Bruni" <address@hidden> wrote:

> Hi,
>
> I came across a problem I'm not able to troubleshoot.
> I cut down the music I'm writing to the attached minimal example.
>
> I'm letting lilypond do the line breaking and what it does? It breaks
> the line after the first measure, then it puts all the other measures in
> one line.  And I can't see any warning in the log output.
>
> If I try to put some manual breaks I get this warning:
>
> warning: forced break was overridden by some other event, should you be
> using bar checks?
>
> I'm using v2.13.61 recently compiled from source (because I wanted to
> test the new chord glissando feature... BTW, kudos to the developers!)

This is a bug in \glissando.

If you look carefully, you'll notice that there is no glissando in measure 2
of the Staff between the c and the cis, although it's there in the TabStaff.

If you comment out the \glissando in that measure, then everything works
properly (except the glissando is missing, of course).

I've created issue 1639 to cover this.

Thanks for the report.

Carl

reply via email to


text:
# How to check if a 3D point is in a planar polygon?

Following up on ndroock1's question, I naively tried to apply the solutions to a 3D point and polygon and they didn't work. For example, functions involving ArcTan that are used in kguler's answer don't work with three arguments, Mac's answer doesn't consider the 3rd dimension at all, the undocumented function GraphicsMeshPointWindingNumber doesn't work in 3D, and complex numbers only map to 2D planes. And so on.

So, is there any way to check if a 3D point is in a 3D planar polygon?

-
Why not just apply a linear transformation sending the polygon's plane to the xy plane and proceed from there? – whuber Sep 7 '12 at 15:07
@whuber I was thinking about that, but I'm not that familiar with FindGeometricTransform. – Eli Lansey Sep 7 '12 at 15:09
@EliLansey, you could compute the normal n to the plane by taking the Cross product of two vectors in the plane, and then use RotationMatrix[{n,{0,0,1}}] to get a rotation matrix which will rotate everything into the xy plane. – Simon Woods Sep 7 '12 at 15:22
@Simon, for polygons with inexact coordinates, it'd be safer to use Newell's algorithm to compute the normal... – J. M. Sep 7 '12 at 15:31
@SimonWoods That's a great idea. Want to write it up as an answer? – Eli Lansey Sep 7 '12 at 15:32
show 1 more comment

Just work in local coordinates for the polygon's plane. This requires finding a change of basis matrix and then applying it (which is just matrix multiplication).

One way to obtain a basis is to use any three points on the polygon, assuming they are not collinear. Thus:

basis[{x_, y_, z_, ___}] := Orthogonalize[{y - x, z - x}] // Transpose;

(Orthogonalize is not really necessary but it assures the transformation preserves distances and angles for applications that might need that.)

Right-multiplication by basis[...] converts 3D coordinates to 2D coordinates (ignoring any component of the 3D coordinate orthogonal to the polygon's plane). Then apply whatever 2D algorithm you prefer.

### Example

Let's generate a random 3D polygon and another random point:

{vertices, p} = Through[{Most, Last}[RandomReal[NormalDistribution[0, 1], {4, 3}]]];

To illustrate the use of basis, here are the 3D and local 2D renderings of this configuration:

Graphics3D[{Polygon[vertices], PointSize[0.02], Darker[Red], Point[p]}]
With[{a = basis[vertices]},
Graphics[{Lighter[Gray], Polygon[ vertices . a], Darker[Red], PointSize[0.02],  Point[p . a]}]]

One check that the point actually is in the plane of the polygon is achieved this way:

inPlane[p_, basis_, origin_] := Abs[Det[Append[Transpose[basis], p - origin]] ] < 10.^(-12)

The origin must be a point known to be in the plane (such as one of the polygon's vertices). E.g.,

inPlane[#, basis[vertices], First[vertices]] &  /@ Append[vertices, p]

{True, True, True, False}

verifies that the triangle does lie in its plane and, as it happens, this particular point does not.

-
Can you give more detail about checking if the point is in the plane? – Eli Lansey Sep 7 '12 at 15:34
@Eli: equation 18 here might give you a clue on how to do the checking. (I used it in the coplanarQ[] function I mentioned in another comment.) – J. M. Sep 7 '12 at 15:38
One way to safen the implementation of basis[] would be to take the Mean[] of the polygon's points and use that as the x... on another note, for your inPlane[], might it be better to do the test as Chop[Det[(* stuff *)]] == 0? – J. M. Sep 7 '12 at 15:50
@J.M. Right; Chop works well. I considered using the centroid, but it potentially has similar numerical problems for many polygons. If you're worried about this, perhaps the most efficient approach is to choose, say, five or six vertices at random and for the basis calculation use three that form a sufficiently large angle between them. A less efficient but very stable and trustworthy method is to take the first two principal components in an SVD of the vertices(considered as a $3$ by $n$ matrix) relative to their centroid: that is, a centered PCA. – whuber Sep 7 '12 at 16:02
Hah, nice! I forgot that PCA can do the job here, even though it is not the most efficient way to go about things... but safe nevertheless. – J. M. Sep 7 '12 at 16:12

I use the algorithm described here for convex 3D polygons. Basically, if a point is inside a polygon, the sum of the angles between the point and each pair of vertices should be $2\pi$, otherwise it's outside the polygon. The angle between two vectors is given by $$\theta=\arccos\left[{\vec a\cdot\vec b\over \|\vec a\|\|\vec b\|}\right].$$ So,

inPolyQ[poly_,pt_]:=2.π==Total[ArcCos[
Dot@@@#/Times@@@Map[Norm,#,{2}]&@Transpose@{#,RotateRight[#]}
]]&@(#-pt&/@poly)

This only works for a convex polygon, however. Additionally, this doesn't check if the polygon is, in fact, planar.

Edit As per J.M.'s comment, this can be simplified using the VectorAngle function:

inPolyQ[poly_,pt_]:=2.π==Total[
VectorAngle@@@Transpose@{#,RotateRight[#]}
]&@(#-pt&/@poly)

-
You know there's a VectorAngle[] function, don't you? – J. M. Sep 7 '12 at 14:58
@J.M. Doh! Well, that'll make this simpler-looking... – Eli Lansey Sep 7 '12 at 15:01
Here is a not-very-efficient way to check if a set of given points are all coplanar: coplanarQ[pts_?MatrixQ] := If[Length[pts] < 4, True, And @@ ((Chop[Det[PadRight[#, {4, 4}, 1]]] == 0) & /@ Subsets[pts, {4}])]. – J. M. Sep 7 '12 at 15:10
An alternative implementation for inPolyQ[]: inPolyQ[pt_?VectorQ, poly_?MatrixQ] := Chop[Total[VectorAngle @@@ Partition[(# - pt) & /@ poly, 2, 1, {1, 1 - 2 Boole[TrueQ[First[poly] == Last[poly]]]}]] - 2 π] == 0. – J. M. Sep 7 '12 at 15:25
@J.M. What are the benefits of your approach? – Eli Lansey Sep 7 '12 at 15:32


text:
# Monte Carlo Parameters

## Index

param

Description

param.type

The type of parameter mapping.

param.move_range

The parameter move range.

param.func

The callable for performing the Monte Carlo moves.

param.kwargs

A dictionary with keyword arguments for param.func.

param.validation.allow_non_existent

Whether to allow parameters, that are explicitly specified, for absent atoms.

param.validation.charge_tolerance

Check whether the net charge of the system is integer within a given tolerance.

param.validation.enforce_constraints

Whether to enforce the constraints for the initial user-specified parameters.

param.block.param

The name of the forcefield parameter.

param.block.unit

The unit in which the forcefield parameters are expressed.

param.block.constraints

A string or list of strings with parameter constraints.

param.block.guess

Estimate all non-specified forcefield parameters.

param.block.frozen

A sub-block with to-be frozen parameters.

psf

Description

psf.str_file

The path+filename to one or more stream file.

psf.rtf_file

The path+filename to one or more MATCH-produced rtf file.

psf.psf_file

The path+filename to one or more psf files.

psf.ligand_atoms

All atoms within a ligand.

pes

Description

pes.block.func

The callable for performing the Monte Carlo moves.

pes.block.ref

A list of reference values for when func operates on qmflows.Result objects.

pes.block.kwargs

A dictionary with keyword arguments for pes.block.func.

pes.block.err_func

A function for computing the auxilary error of the specified PES descriptor.

pes_validation

Description

pes_validation.block.func

The callable for performing the Monte Carlo validation.

pes_validation.block.ref

A list of reference values for when func operates on qmflows.Result objects.

pes_validation.block.kwargs

A dictionary with keyword arguments for pes_validation.block.func.

job

Description

job.type

The type of package manager.

job.molecule

One or more .xyz files with reference (QM) potential energy surfaces.

job.lattice

One or more CP2K .cell files with the lattice vectors of each mol in job.molecule.

job.block.type

An instance of a QMFlows Package.

job.block.settings

The job settings as used by job.block.type.

job.block.template

A settings template for updating job.block.settings.

monte_carlo

Description

monte_carlo.type

The type of Monte Carlo procedure.

monte_carlo.iter_len

The total number of ARMC iterations $$\kappa \omega$$.

monte_carlo.sub_iter_len

The length of each ARMC subiteration $$\omega$$.

monte_carlo.logfile

The name of the ARMC logfile.

monte_carlo.hdf5_file

The name of the ARMC .hdf5 file.

monte_carlo.path

The path to the ARMC working directory.

monte_carlo.folder

The name of the ARMC working directory.

monte_carlo.keep_files

Whether to keep all raw output files or not.

phi

Description

phi.type

The type of phi updater.

phi.gamma

The constant $$\gamma$$.

phi.a_target

The target acceptance rate $$\alpha_{t}$$.

phi.phi

The initial value of the variable $$\phi$$.

phi.func

The callable for updating phi.

phi.kwargs

A dictionary with keyword arguments for phi.func.

## param

All forcefield-parameter related options.

This settings block accepts an arbitrary number of sub-blocks.

Examples

param:
type: FOX.armc.ParamMapping
move_range:
start: 0.005
stop: 0.1
step: 0.005
ratio: null
func: numpy.multiply
kwargs: {}
validation:
allow_non_existent: False
charge_tolerance: 0.01
enforce_constraints: False

charge:
param: charge
constraints:
- '0.5 < Cd < 1.5'
- '-0.5 > Se > -1.5'
Cd: 0.9768
Se: -0.9768
O_1: -0.47041
frozen:
C_1: 0.4524
lennard_jones:
-   unit: kjmol
param: epsilon
Cd Cd: 0.3101
Se Se: 0.4266
Cd Se: 1.5225
frozen:
guess: uff
-   unit: nm
param: sigma
Cd Cd: 0.1234
Se Se: 0.4852
Cd Se: 0.2940
frozen:
guess: uff

param.type
Parameter

The type of parameter mapping.

Used for storing and moving user-specified forcefield values.

FOX.armc.ParamMapping

A ParamMappingABC subclass.

param.move_range
Parameter
• Type - array-like or dict

• Default Value - {"start": 0.005, "stop": 0.1, "step": 0.005, "ratio": None}

The parameter move range.

This value accepts one of the following two types of inputs:

1. A list of allowed moves (e.g. [0.9, 0.95, 1.05, 1.0]).

2. A dictionary with the "start", "stop" and "step" keys.

For example, the list in 1. can be reproduced with {"start": 0.05, "stop": 0.1, "step": 0.05, "ratio": None}.

When running the ARMC parallel procedure (monte_carlo.type = FOX.armc.ARMCPT) option 1. should be supplied as a nested list (e.g. [[0.9, 0.95, 1.05, 1.0], [0.8, 0.9, 1.1, 1.2]]) and option 2. requires the additional "ratio" keyword (e.g. [1, 2]).

param.func
Parameter
• Default Value - "numpy.multiply"

The callable for performing the Monte Carlo moves.

The passed callable should be able to take two NumPy arrays as a arguments and return a new one.

numpy.multiply()

Multiply arguments element-wise.

param.kwargs
Parameter
• Default Value - {}

A dictionary with keyword arguments for param.func.

param.validation.allow_non_existent
Parameter

Whether to allow parameters, that are explicitly specified, for absent atoms.

This check is performed once, before the start of the ARMC procedure.

param.validation.charge_tolerance
Parameter

Check whether the net charge of the system is integer within a given tolerance.

This check is performed once, before the start of the ARMC procedure. Setting this parameter to inf disables the check.

param.validation.enforce_constraints
Parameter

Whether to enforce the constraints for the initial user-specified parameters.

This option checks if the initially supplied parameters are compatible with all the supplied constraints; an error will be raised if this is not the case. Note that the constraints will always be enforced once the actual ARMC procedure starts.

param.block.param
Parameter

The name of the forcefield parameter.

Important

Note that this option has no default value; one must be provided by the user.

param.block.unit
Parameter

The unit in which the forcefield parameters are expressed.

See the CP2K manual for a comprehensive list of all available units.

param.block.constraints
Parameter

A string or list of strings with parameter constraints. Accepted types of constraints are minima/maxima (e.g. 2 > Cd > 0) and fixed parameter ratios (e.g. Cd == -1 * Se). The special $LIGAND alias can be used for representing all atoms within a single ligand. For example, $LIGAND is equivalent to 2 * O + C + H in the case of formate.

param.block.guess
Parameter

Estimate all non-specified forcefield parameters.

If specified, expects a dictionary with the "mode" key, e.g. {"mode": "uff"} or {"mode": "rdf"}.

param.block.frozen
Parameter

A sub-block with to-be frozen parameters.

Parameters specified herein will be treated as constants rather than variables. Accepts forcefield parameters (e.g. "Cd Cd" = 1.0) and, optionally, the guess key.

## psf

Settings related to the construction of protein structure files (.psf).

Note that the psf.str_file, psf.rtf_file and psf.psf_file options are all mutually exclusive; only one should be specified. Furthermore, this block is completelly optional.

Examples

psf:
rtf_file: ligand.rtf
ligand_atoms: [C, O, H]

psf.str_file
Parameter

The path+filename to one or more stream files.

Used for assigning atom types and charges to ligands.

psf.rtf_file
Parameter

The path+filename to one or more MATCH-produced rtf files.

Used for assigning atom types and charges to ligands.

psf.psf_file
Parameter

The path+filename to one or more psf files.

Used for assigning atom types and charges to ligands.

psf.ligand_atoms
Parameter

A list with all atoms within the organic ligands.

Used for defining residues.

## pes

Settings to the construction of potentialy energy surface (PES) descriptors.

This settings block accepts an arbitrary number of sub-blocks, each containg the func and, optionally, kwargs keys.

Examples

pes:
rdf:
func: FOX.MultiMolecule.init_rdf
kwargs:
atom_subset: [Cd, Se, O]
kwargs:
atom_subset: [Cd, Se]
energy:
func: FOX.properties.get_attr  # i.e. qmflows.Result(...).energy
ref: [-17.0429775897]
kwargs:
name: energy
hirshfeld_charges:
func: FOX.properties.call_method  # i.e. qmflows.Result(...).get_hirshfeld_charges()
ref:
- [-0.1116, 0.1930, -0.1680, -0.2606, 0.1702, 0.0598, 0.0575, 0.0598]
kwargs:
name: get_hirshfeld_charges

pes.block.func
Parameter

A callable for constructing a PES descriptor.

The callable should return an array-like object and, as sole positional argument, take either a FOX.MultiMolecule or qmflows.Results instance. In the latter case one must supply a list of reference PES-descriptor-values to pes.block.ref.

Important

Note that this option has no default value; one must be provided by the user.

FOX.MultiMolecule.init_rdf()

Initialize the calculation of radial distribution functions (RDFs).

FOX.MultiMolecule.init_adf()

Initialize the calculation of angular distribution functions (ADFs).

pes.block.ref
Parameter

A list of reference values for when func operates on qmflows.Result objects.

If not None, a list of array_like objects must be supplied here, one equal in length to the number of supplied molecules (see job.molecule).

pes.block.kwargs
Parameter
• Default Value - {}

A dictionary with keyword arguments for func.

pes.block.err_func
Parameter

A function for computing the auxilary error of the specified PES descriptor. The callable should be able to take two array-like objects as arguments and return a scalar.

FOX.armc.mse_normalized() & FOX.armc.mse_normalized_v2()

Return a normalized mean square error (MSE) over the flattened input.

FOX.armc.mse_normalized_weighted() & FOX.armc.mse_normalized_weighted_v2()

Return a normalized mean square error (MSE) over the flattened subarrays of the input.

FOX.armc.mse_normalized_max()

Return a maximum normalized mean square error (MSE) over the flattened subarrays of the input.

## pes_validation

Settings to the construction of potentialy energy surface (PES) validators.

Functions identically w.r.t. to the pes block, the exception being that PES descriptors calculated herein are do not affect the error; they are only calculated for the purpose of validation.

This settings block accepts an arbitrary number of sub-blocks, each containg the func and, optionally, kwargs keys.

Examples

pes_validation:
kwargs:
atom_subset: [Cd, Se]
mol_subset: !!python/object/apply:builtins.slice  # i.e. slice(None, None, 10)
- null
- null
- 10

pes_validation.block.func
Parameter

A callable for constructing a PES validators.

The callable should return an array-like object and, as sole positional argument, take either a FOX.MultiMolecule or qmflows.Results instance. In the latter case one must supply a list of reference PES-descriptor-values to pes_validation.block.ref.

The structure of this block is identintical to its counterpart in pes.block.func.

Important

Note that this option has no default value; one must be provided by the user.

FOX.MultiMolecule.init_rdf()

Initialize the calculation of radial distribution functions (RDFs).

FOX.MultiMolecule.init_adf()

Initialize the calculation of angular distribution functions (ADFs).

pes_validation.block.ref
Parameter

A list of reference values for when func operates on qmflows.Result objects.

If not None, a list of array_like objects must be supplied here, one equal in length to the number of supplied molecules (see job.molecule).

pes_validation.block.kwargs
Parameter
• Default Value - {}

A dictionary with keyword arguments for func.

The structure of this block is identintical to its counterpart in pes.block.kwargs.

## job

Settings related to the running of the various molecular mechanics jobs.

In addition to having two constant keys (type and molecule) this block accepts an arbitrary number of sub-blocks representing quantum and/or classical mechanical jobs. In the example above there are two of such sub-blocks: geometry_opt and md. The first step consists of a geometry optimization while the second one runs the actual molecular dynamics calculation. Note that these jobs are executed in the order as provided by the user-input.

Examples

job:
type: FOX.armc.PackageManager
molecule: .../mol.xyz

geometry_opt:
type: qmflows.cp2k_mm
settings:
prm: .../ligand.prm
cell_parameters: [50, 50, 50]
template: qmflows.templates.geometry.specific.cp2k_mm
md:
type: qmflows.cp2k_mm
settings:
prm: .../ligand.prm
cell_parameters: [50, 50, 50]
template: qmflows.templates.md.specific.cp2k_mm

job.type
Parameter

The type of Auto-FOX package manager.

Used for managing and running the actual jobs.

FOX.armc.PackageManager

A PackageManagerABC subclass.

job.molecule
Parameter

One or more .xyz files with reference (QM) potential energy surfaces.

Important

Note that this option has no default value; one must be provided by the user.

job.lattice
Parameter

One or more CP2K .cell files with the lattice vectors of each mol in job.molecule.

This option should be specified is one is performing calculations on periodic systems.

job.block.type
Parameter
• Type - str or qmflows.packages.Package instance

• Default Value - "qmflows.cp2k_mm"

An instance of a QMFlows Package.

qmflows.cp2k_mm

An instance of CP2KMM.

job.block.settings
Parameter
• Default Value - {}

The job settings as used by type.

In the case of PES-averaged ARMC one can supply a list of dictionaries, each one representing the settings for its counterpart in job.molecule.

If a template is specified then this block may or may not be redundant, depending on its completeness.

job.block.template
Parameter

A Settings template for updating settings.

The template can be provided either as a dictionary or, alternativelly, an import path pointing to a pre-existing dictionary. For example, "qmflows.templates.md.specific.cp2k_mm" is equivalent to import qmflows; template = qmflows.templates.md.specific.cp2k_mm.

qmflows.templates.md

Templates for molecular dynamics (MD) calculations.

qmflows.templates.geometry

Templates for geometry optimization calculations.

## monte_carlo

Settings related to the Monte Carlo procedure itself.

Examples

monte_carlo:
type: FOX.armc.ARMC
iter_len: 50000
sub_iter_len: 10
logfile: armc.log
hdf5_file: armc.hdf5
path: .
folder: MM_MD_workdir
keep_files: False

monte_carlo.type
Parameter

The type of Monte Carlo procedure.

FOX.armc.ARMC

The Addaptive Rate Monte Carlo class.

FOX.armc.ARMCPT

An ARMC subclass implementing a parallel tempering procedure.

monte_carlo.iter_len
Parameter

The total number of ARMC iterations $$\kappa \omega$$.

monte_carlo.sub_iter_len
Parameter

The length of each ARMC subiteration $$\omega$$.

monte_carlo.logfile
Parameter

The name of the ARMC logfile.

monte_carlo.hdf5_file
Parameter

The name of the ARMC .hdf5 file.

monte_carlo.path
Parameter

The path to the ARMC working directory.

monte_carlo.folder
Parameter

The name of the ARMC working directory.

monte_carlo.keep_files
Parameter

Whether to keep all raw output files or not.

## phi

Settings related to the ARMC $$\phi$$ parameter.

Examples

phi:
type: FOX.armc.PhiUpdater
gamma: 2.0
a_target: 0.25
phi: 1.0
kwargs: {}

phi.type
Parameter

The type of phi updater.

The phi updater is used for storing, keeping track of and updating $$\phi$$.

FOX.armc.PhiUpdater

A class for applying and updating $$\phi$$.

phi.gamma
Parameter
• Default Value - 2.0

The constant $$\gamma$$.

See (4). Note that a list must be supplied when running the ARMC parallel tempering procedure (monte_carlo.type = FOX.armc.ARMCPT)

phi.a_target
Parameter
• Default Value - 0.25

The target acceptance rate $$\alpha_{t}$$.

See (4). Note that a list must be supplied when running the ARMC parallel tempering procedure (monte_carlo.type = FOX.armc.ARMCPT)

phi.phi
Parameter
• Default Value - 1.0

The initial value of the variable phi.

See (3) and (4). Note that a list must be supplied when running the ARMC parallel tempering procedure (monte_carlo.type = FOX.armc.ARMCPT)

phi.func
Parameter
• Default Value - "numpy.add"

The callable for updating phi.

The passed callable should be able to take two floats as arguments and return a new float.

numpy.add()
A dictionary with further keyword arguments for phi.func.


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text:
# feets: feATURE eXTRACTOR FOR tIME sERIES¶

In time-domain astronomy, data gathered from the telescopes is usually represented in the form of light-curves. These are time series that show the brightness variation of an object through a period of time (for a visual representation see video below). Based on the variability characteristics of the light-curves, celestial objects can be classified into different groups (quasars, long period variables, eclipsing binaries, etc.) and consequently be studied in depth independentely. In order to characterize this variability, some of the existing methods use machine learning algorithms that build their decision on the light-curves features. Features, the topic of the following work, are numerical descriptors that aim to characterize and distinguish the different variability classes. They can go from basic statistical measures such as the mean or the standard deviation, to complex time-series characteristics such as the autocorrelation function. In this package we present a library with a compilation of some of the existing light-curve features. The main goal is to create a collaborative and open tool where every user can characterize or analyze an astronomical photometric database while also contributing to the library by adding new features. However, it is important to highlight that this library is not restricted to the astronomical field and could also be applied to any kind of time series. Our vision is to be capable of analyzing and comparing light-curves from all the available astronomical catalogs in a standard and universal way. This would facilitate and make more efficient tasks as modelling, classification, data cleaning, outlier detection and data analysis in general. Consequently, when studying light-curves, astronomers and data analysts would be on the same wavelength and would not have the necessity to find a way of comparing or matching different features. In order to achieve this goal, the library should be run in every existent survey (MACHO, EROS, OGLE, Catalina, Pan-STARRS, etc) and future surveys (LSST) and the results should be ideally shared in the same open way as this library.

## Help & discussion mailing list¶

You can contact me at: jbc.develop@gmail.com (if you have a support question, try the mailing list first)

## Code Repository & Issues¶

https://github.com/carpyncho/feets

feets is under The MIT License

A short and simple permissive license with conditions only requiring preservation of copyright and license notices. Licensed works, modifications, and larger works may be distributed under different terms and without source code.

## Citation¶

If you use feets in a scientific publication, we would appreciate citations to the following paper:

Cabral, J. B., Sánchez, B., Ramos, F., Gurovich, S., Granitto, P., & Vanderplas, J. (2018). From FATS to feets: Further improvements to an astronomical feature extraction tool based on machine learning. Astronomy and Computing.

Bibtex entry:

@article{cabral2018fats,
title={
From FATS to feets: Further improvements to
an astronomical feature extraction tool based on
machine learning},
author={
Cabral, JB and S{\'a}nchez, B and Ramos, F and
Gurovich, S and Granitto, P and Vanderplas, J},
journal={Astronomy and Computing},
year={2018},
publisher={Elsevier}
}


text:
# Area of a surface confined by a sphere

Let $$S$$ be a hypersurface enclosed inside the unit sphere in $$R^n$$. We may assume that every ray $$\{t x: t \geq 0 \}$$ intersects $$S$$ at most once.

Under what extra condition is $${\rm Area}(S) \leq {\rm Area}(S^{n-1})$$ ?

(I am mostly interested in the 2-dimensional case.)

Thanks.

• Should be able to use Radon transforms. Try the paper of Alvarez Paiva and Fernandez. – Ben McKay Feb 15 at 15:58
• I think you mean the unit ball, don't you? – Pietro Majer Feb 15 at 16:01
• If you regard the hypersurface $S$ as a radial graph over a domain $D\subseteq S^{n-1}$, i.e, $$S = \{ r(p)\,p\ |\ p\in D\,\}$$ for some function $r:D\to[0,1]$, then the standard formula for the $(n{-}1)$-content of $S$ is $$A_{n-1}(S) = \int_D r^{n-2}(r^2+|\nabla r|^2)^{1/2} \,\mathrm{d}V_{n-1},$$ so it's a question of what you want to assume about the function $r$. – Robert Bryant Feb 15 at 18:22
• Thanks very much. It seems like a useful tool. I am trying to find a reference for this formula. (For instance, do you mean $D$ is somehow centered at the origin? I also wonder how to interpret the gradient of $r$ exactly, as it is defined on $D$.) BTW, i refined the question in: mathoverflow.net/questions/352836/… – Thomas Feb 16 at 4:21
• $D$ is a domain (i.e., open set) in $S^{n-1}$. (I imagine that you want $D$ to be all of $S^{n-1}$, but you didn't say that.) I'm not sure what you mean by 'centered at the origin'. If $D$ is a hemisphere in $S^{n-1}$, is it 'centered at the origin'? Gradient has its usual meaning for a function defined on a hypersurface: Extend the function to be constant in the normal direction and compute the gradient as usual in $n$-space. I saw the other question (and, really, you should have just modified or added to this one, they are so close). What do you mean by 'elliptic'? – Robert Bryant Feb 16 at 10:05


text:
# A capacitor made of two parallel plates, each of area ‘A’ and separation 'd’ is charged by an external dc source. Show that during charging, the displacement current inside the capacitor is the same as the current charging the capacitor.

Total current

$i=i_{c}+i_{d}$

Where, $i_{c}$ is conduction current and

$i_{d}$ is displacement current

Outside the capacitor, the displacement current; $i_{d}=0$ so, $i=i_{c}$

Inside the capacitor, the conduction current;       $i_{c}=0$ So, $i=i_{d}$

Hence,

the displacement current is the same as the current charging the capacitor.

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text:
# Freddy Cachazo

Gluskin Sheff Freeman Dyson Chair in Theoretical Physics at Perimeter Institute

Areas of Research:
Phone: x8501

## Research Interests

In recent years it has become clear that the scattering matrix of massless particles is a very special object. Massless particles are the backbone of our best theory of nature; the Standard Model of particle physics and gravity. In 2003, Witten succeeded in combining twistor theory and string theory to produce a new formulation for the scattering matrix of gauge bosons - massless particles of helicity plus or minus one. Since then finding a similar formulation for gravitons - particles with helicity plus or minus two - has been a very important open problem. In 2012, in collaboration with Y. Geyer (a PSI student), I found one such formulation. Not long after this first formulation was found, I found a second one with D. Skinner (a PI postdoctoral fellow). All these constructions are based on four dimensional kinematics. A natural question is whether similar constructions can be found in any dimension. Such a general formulation could lead to a deeper understanding of field theory in general. In 2013 and in collaboration with E. Yuan (a PI graduate student) and S. He (a PI postdoctoral fellow), I found such a construction. This is now known as the CHY construction and it gives tree-level scattering matrices of a variety of theories, including gluons and gravitons, in any dimensions as integrals over the space of n-punctured spheres. This new formulation has become an active area of research. In 2014 our group found CHY representations for theories such as Dirac-Born-Infeld and Galileons. Very recently we have found extensions of the theory of pions using soft limits which were previously hidden and become manifest using their CHY formulation.

## Positions Held

• September 2009 - June 2010 School of Natural Sciences. Institute for Advanced Study. Princeton NJ. Member.
• 2005-present. Perimeter Institute. Faculty Member
• 2002-2005 School of Natural Sciences. Institute for Advanced Study. Princeton NJ. Postdoctoral Member.

## Awards

• 2016 CAP-CRM Prize in Theoretical and Mathematical Physics.
• 2014 New Horizons Prize (FPP).
• 2012 Herzberg Medal (CAP).
• 2011 Rutherford Medal, Royal Society of Canada (RSC).
• 2009 Gribov Medal, European Physical Society (EPS)
• Early Reseach Award of Ontario (ERA).

## Recent Publications

• F. Cachazo, P. Cha and S. Mizera, Extensions of Theories from Soft Limits, JHEP 1606, 170 (2016) [arXiv: 1604.03893 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, JHEP 1608, 008 (2016) [arXiv: 1512.05001 [hep-th]].
• N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 1506, 179 (2015) [arXiv: 1412.8475 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 1507, 149 (2015) [arXiv: 1412.3479 [hep-th]].
• N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett. 113, no. 26, 261603 (2014) [arXiv: 1410.0354 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 1501 (2015) 121 [arXiv: 1409.8256 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 1407, 033 (2014) [arXiv: 1309.0885 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113, no. 17, 171601 (2014) [arXiv: 1307.2199 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90, no. 6, 065001 (2014) [arXiv: 1306.6575 [hep-th]].
• F. Cachazo, S. He and E. Y. Yuan, Scattering in Three Dimensions from Rational Maps, JHEP 1310, 141 (2013) [arXiv: 1306.2962 [hep-th]].
• F. Cachazo and D. Skinner, Gravity from Rational Curves, Phys. Rev. Lett. 110, 161301 (2013) [arXiv: 1207.0741 [hep-th]].
• F. Cachazo, L. Mason and D. Skinner, Gravity in Twistor Space and its Grassmannian Formulation, SIGMA 10, 051 (2014) [arXiv: 1207.4712 [hep-th]].
• F. Cachazo and G. Zhang, Minimal Basis in Four Dimensions and Scalar Blocks, arXiv: 1601.06305 [hep-th].
• F. Cachazo and H. Gomez, Computation of Contour Integrals on {\cal M}_{0,n}, arXiv: 1505.03571 [hep-th].
• F. Cachazo, S. He and E. Y. Yuan, New Double Soft Emission Theorems, arXiv: 1503.04816 [hep-th].
• F. Cachazo and E. Y. Yuan, Are Soft Theorems Renormalized?, arXiv: 1405.3413 [hep-th].
• F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv: 1404.4091 [hep-th].
• F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles in Arbitrary Dimension, arXiv: 1307.2199 [hep-th].
• F. Cachazo, S. He and E. Y. Yuan, Scattering in Three Dimensions from Rational Maps, arXiv: 1306.2962 [hep-th].
• F. Cachazo, S. He and E. Y. Yuan, Scattering Equations and KLT Orthogonality, arXiv: 1306.6575 [hep-th].
• F. Cachazo, Resultants and Gravity Amplitudes, arXiv: 1301.3970 [hep-th].
• Grassmannian Geometry of Scattering Amplitudes, N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, J. Trnka, Cambridge University Press, 2016

## Seminars

• Plenary talk on S-Matrix Theory", Canadian Association of Physicist Congress, University of Ottawa, Canada, June 2016.
• Plenary talk on Unification and Extension", Theory CANADA 11, Carleton University, Ottawa, Canada, June 2016.
• Plenary talk on "Developments in String and Field Theory" at the European Physical Society 2015 meeting in Vienna, Austria, July 2015.
• Plenary talk at PASCOS 2015, Abdus Salam ICTP, Trieste, Italy, June 2015
• Plenary talk at Amplitudes 2015, EHT, Zurich, Switzerland, June 2015
• Five lectures at TASI 2015 "New Frontiers in Fields and Strings", University of Colorado, Colorado, USA, June 2015
• Colloquium. Department of Physics, University of Toronto, Toronto, Canada, March 2015
• Series of three lectures given at the Spring School on Superstring Theory and Related Topics. Abdus Salam ICTP, Trieste, Italy, March 2015
• Plenary talk at STRINGS 2014 in Princeton USA.
• Plenary talk at Quantum Gravity in Paris 2014. Paris. March 2014.
• Academics Program Department Presentation to the Scientific Advisory Committee, October 2015.
• PIRSA:15060045, Welcome to Convergence - Opening Panel: Physics Today, 2015-06-22, Convergence
• PIRSA:15050055, Scattering Amplitudes and Riemann Surfaces, 2015-05-27, Geometry and Physics 2015
• PIRSA:15050063, Scattering Amplitudes and Riemann Surfaces, 2015-05-26, Geometry and Physics 2015
• PIRSA:15050047, Scattering Amplitudes and Riemann Surfaces, 2015-05-25, Geometry and Physics 2015
• PIRSA:14100026, 14/15 PSI - Conformal Field Theory - Lecture 14, 2014-10-24, 14/15 PSI - Conformal Field Theory
• PIRSA:14100025, 14/15 PSI - Conformal Field Theory - Lecture 13 , 2014-10-23, 14/15 PSI - Conformal Field Theory
• PIRSA:14100024, 14/15 PSI - Conformal Field Theory - Lecture 12 , 2014-10-22, 14/15 PSI - Conformal Field Theory
• PIRSA:14100023, 14/15 PSI - Conformal Field Theory - Lecture 11 , 2014-10-21, 14/15 PSI - Conformal Field Theory
• PIRSA:14100022, 14/15 PSI - Conformal Field Theory - Lecture 10, 2014-10-20, 14/15 PSI - Conformal Field Theory
• Quantum Gravity Group Meeting. Resultants and Gravity Amplitudes.