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# Emergy

(Redirected from Emergy synthesis)

Emergy is a type of energy (exergy) that is consumed in direct and indirect transformations needed to make a product or service.[1] Emergy is a measure of quality differences between different forms of energy. Emergy is an expression of all the energy used in the work processes that generate a product or service in units of one type of energy. Emergy is measured in units of emjoules, a unit referring to the available energy consumed in transformations. Emergy accounts for different forms of energy and resources (e.g. sunlight, water, fossil fuels, minerals, etc.) Each form is generated by transformation processes in nature and each has a different ability to support work in natural and in human systems. The recognition of these quality differences is a key concept.

## History

The theoretical and conceptual basis for the emergy methodology is grounded in thermodynamics[citation needed], general system theory[2] and systems ecology.[3] Evolution of the theory by Howard T. Odum over the first thirty years is reviewed in Environmental Accounting[1] and in the volume edited by C.A.S. Hall titled Maximum Power.[4]

### Background

Beginning in the 1950s, Odum analyzed energy flow in ecosystems (e.g. Silver Springs, Florida;[5] Enewetak atoll in the south Pacific;[6] Galveston Bay, Texas[7] and Puerto Rican rainforests,[8] amongst others) where energies in various forms at various scales were observed. His analysis of energy flow in ecosystems, and the differences in the potential energy of sunlight, fresh water currents, wind and ocean currents led him to make the suggestion that when two or more different energy sources drive a system, they cannot be added without first converting them to a common measure that accounts for their differences in energy quality. This led him to introduce the concept of "energy of one kind" as a common denominator with the name "energy cost".[9] He then expanded the analysis to model food production in the 1960s,[9] and in the 1970s to fossil fuels.[10][11]

Odum's first formal statement of what would later be termed emergy was in 1973:

Energy is measured by calories, btu’s, kilowatthours, and other intraconvertable units, but energy has a scale of quality which is not indicated by these measures. The ability to do work for man depends on the energy quality and quantity and this is measurable by the amount of energy of a lower quality grade required to develop the higher grade. The scale of energy goes from dilute sunlight up to plant matter, to coal, from coal to oil, to electricity and up to the high quality efforts of computer and human information processing.[12]

In 1975, he introduced a table of “Energy Quality Factors”, kilocalories of sunlight energy required to make a kilocalorie of a higher quality energy,[13] the first mention of the energy hierarchy principle which states that “energy quality is measured by the energy used in the transformations” from one type of energy to the next.

These energy quality factors, were placed on a fossil-fuel basis and called "Fossil Fuel Work Equivalents" (FFWE), and the quality of energies were measured based on a fossil fuel standard with rough equivalents of 1 kilocalorie of fossil fuel equal to 2000 kilocalories of sunlight. "Energy quality ratios" were computed by evaluating the quantity of energy in a transformation process to make a new form and were then used to convert different forms of energy to a common form, in this case fossil fuel equivalents. FFWE's were replaced with Coal equivalents (CE) and by 1977, the system of evaluating quality was placed on a solar basis and termed solar equivalents (SE).[14]

### Embodied energy

The term "embodied energy" was used for a time in the early 1980s to refer to energy quality differences in terms of their costs of generation, and a ratio called a “quality factor” for the calories (or joules) of one kind of energy required to make those of another.[15] However, since the term embodied energy was used by other groups who were evaluating the fossil fuel energy required to generate products and were not including all energies or using the concept to imply quality, embodied energy was dropped in favor of “embodied solar calories”, and the quality factors became known as "transformation ratios".

### Introduction of the term "emergy"

Use of the term "embodied energy" for this concept was modified in 1986 when David Scienceman, a visiting scholar at the University of Florida from Australia, suggested the term “emergy” and "emjoule" or "emcalorie" as the unit of measure to distinguish emergy units from units of available energy.[16] The term transformation ratio was shortened to transformity in about the same time. It is important to note that throughout this twenty years the baseline or the basis for evaluating forms of energy and resources shifted from organic matter, to fossil fuels and finally to solar energy.

After 1986, the emergy methodology continued to develop as the community of scientists expanded and as new applied research into combined systems of humans and nature presented new conceptual and theoretical questions. The maturing of the emergy methodology resulted in more rigorous definitions of terms and nomenclature and refinement of the methods of calculating transformities. The International Society for the Advancement of Emergy Research and a biennial International Conference at the University of Florida support this research.

### Chronology

Table 1: Development of emergy, transformity and conversion ratios.
Years Baseline Unit Emergy Values Units Reference
1967–1971 Organic matter the baseline. All energies of higher quality (wood, peat, coal, oil, living biomass, etc.) expressed in units of organic matter. Sunlight equivalent to organic matter = 1000 solar kilocalories per kilocalories of organic matter. g dry wt O.M.; kcal, conversion from OM to kcal = 5kcal/g dry wt. [9][17]
1973–1980 Fossil fuels and then coal the baseline. Energy of lower quality (sunlight, plants, wood, etc.) were expressed in units of fossil fuels and later in units of coal equivalents. Direct sunlight equivalents of fossil fuels = 2000 solar kilocalories per fossil fuel kilocalorie Fossil fuel work equivalents (FFWE) and later, coal equivalents (CE) [10][11]
1980–1982 Global solar energy the baseline. All energies of higher quality (wind, rain, wave, organic matter, wood, fossil fuels, etc.) expressed in units of solar energy 6800 global solar Calories per Calorie of available energy in coal Global solar calories (GSE). [3][18]
1983–1986 Recognized that solar energy, deep heat, and tidal momentum were basis for global processes. Total annual global sources equal to the sum of these (9.44 E24 solar joules/yr) Embodied solar joules per joule of fossil fuels = 40,000 seJ/J Embodied solar equivalents (SEJ) and later called "emergy" with nomenclature (seJ) [19]
1987–2000 Further refinements of total energy driving global processes, Embodied solar energy renamed to EMERGY Solar Emergy per Joule of coal energy ~ 40,000 solar emjoules/ Joule (seJ/J) named Transformity seJ/J = Transformity; seJ/g = Specific emergy [1]
2000–present Emergy driving the biosphere reevaluated as 15.83 E24 seJ/yr raising all previously calculated transformities by the ratio of 15.83/9.44 = 1.68 Solar emergy per Joule of coal energy ~ 6.7 E 4 seJ/J seJ/J = Transformity; seJ/g = Specific emergy [20]

## Definitions and examples

Emergy— amount of energy of one form that is used in transformations directly and indirectly to make a product or service. The unit of emergy is the emjoule or emergy joule. Using emergy, sunlight, fuel, electricity, and human service can be put on a common basis by expressing each of them in the emjoules of solar energy that is required to produce them. If solar emergy is the baseline, then the results are solar emjoules (abbreviated seJ). Although other baselines have been used, such as coal emjoules or electrical emjoules, in most cases emergy data are given in solar emjoules.

Unit Emergy Values (UEVs) — the emergy required to generate one unit of output. Types of UEVs:

Transformity — emergy input per unit of available energy output. For example, if 10,000 solar emjoules are required to generate a joule of wood, then the solar transformity of that wood is 10,000 solar emjoules per joule (abbreviated seJ/J). The solar transformity of the sunlight absorbed by the earth is 1.0 by definition.
Specific emergy — emergy per unit mass output. Specific emergy is usually expressed as solar emergy per gram (seJ/g). Because energy is required to concentrate materials, the unit emergy value of any substance increases with concentration. Elements and compounds not abundant in nature therefore have higher emergy/mass ratios when found in concentrated form since more environmental work is required to concentrate them, both spatially and chemically.
Emergy per unit money — the emergy supporting the generation of one unit of economic product (expressed in monetary terms). It is used to convert money into emergy units. Since money is paid for goods and services, but not to the environment, the contribution to a process represented by monetary payments is the emergy that money purchases. The amount of resources that money buys depends on the amount of emergy supporting the economy and the amount of money circulating. An average emergy/money ratio in solar emjoules/$can be calculated by dividing the total emergy use of a state or nation by its gross economic product. It varies by country and has been shown to decrease each year, which is one index of inflation. This emergy/money ratio is useful for evaluating service inputs given in money units where an average wage rate is appropriate. Emergy per unit labor — the emergy supporting one unit of direct labor applied to a process. Workers apply their efforts to a process and in so doing they indirectly invest in it the emergy that made their labor possible (food, training, transport, etc). This emergy intensity is generally expressed as emergy per time (seJ/yr; seJ/hr), but emergy per money earned (seJ/$) is also used. Indirect labor required to make and supply the inputs to a process is generally measured with the dollar cost of services, so that its emergy intensity is calculated as seJ/$. Empower — a flow of emergy (i.e., emergy per unit time). Table 2. Nomenclature Term Definition Abbreviation Units Extensive Properties Emergy The amount of available energy of one type (usually solar) that is directly or indirectly required to generate a given output flow or storage of energy or matter. Em seJ (solar equivalent Joules) Emergy Flow Any flow of emergy associated with inflowing energy or materials to a system/process. R=renewable flows; N= nonrenewable flows; F= imported flows; S= services seJ*time−1 Gross Emergy Product Total emergy annually used to drive a national or regional economy GEP seJ*yr−1 Product-related Intensive Properties Transformity Emergy investment per unit process output of available energy Τr seJ*J−1 Specific Emergy Emergy investment per unit process output of dry mass SpEm seJ*g−1 Emergy Intensity of currency Emergy investment per unit of GDP generated in a country, region or process EIC seJ*curency−1 Space-related Intensive Properties Emergy Density Emergy stored in a volume unit of a given material EmD seJ*volume−3 Time-related Intensive Properties Empower Emergy flow (released, used) per unit time EmP seJ*time−1 Empower Intensity Areal Empower (emergy released per unit time and area) EmPI seJ*time−1*area−1 Empower Density Emergy released per unit time by a unit volume (e.g. a power plant or engine) EmPd seJ*time−1*volume−3 Selected Performance Indicators Emergy released (used) Total emergy investment in a process (measure of a process footprint) U= N+R+F+S (see Fig.1) seJ Emergy Yield Ratio Total emergy released (used up) per unit of emergy invested EYR= U/(F+S) (see Fig.1) Environmental Loading Ratio Total nonrenewable and imported emergy released per unit of local renewable resource ELR= (N+F+S)/R (see Fig.1) Emergy Sustainability Index Emergy yield per unit of environmental loading ESI= EYR/ELR (see Fig.1) Renewability Percentage of total emergy released (used) that is renewable. %REN= R/U (see Fig.1) Emergy Investment Ratio Emergy investment needed to exploit one unit of local (renewable and nonrenewable) resource. EIR= (F+S)/(R+N) (see Fig.1) ## Accounting method Emergy accounting converts the thermodynamic basis of all forms of energy, resources and human services into equivalents of a single form of energy, usually solar. To evaluate a system, a system diagram organizes the evaluation and account for energy inputs and outflows. A table of the flows of resources, labor and energy is constructed from the diagram and all flows are evaluated. The final step involves interpreting the results.[1] ### Purpose In some cases, an evaluation is done to determine the fit of a development proposal within its environment. It also allows comparison of alternatives. Another purpose is to seek the best use of resources to maximize economic vitality. ### Systems diagram Figure 1: Energy system diagram of a city in its support region System diagrams show the inputs that are evaluated and summed to obtain the emergy of a flow. A diagram of a city and its regional support area is shown in Figure 1.[21] ### Evaluation table A table (see example below) of resource flows, labor and energy is constructed from the diagram. Raw data on inflows that cross the boundary are converted into emergy units, and then summed to obtain total emergy supporting the system. Energy flows per unit time (usually per year) are presented in the table as separate line items. Table 3. Example emergy evaluation table Note Item(name) Data(flow/time) Units UEV (seJ/unit) Solar Emergy (seJ/time) 1. First item xxx.x J/yr xxx.x Em1 2. Second item xxx.x g/yr xxx.x Em2 -- n. nth item xxx.x J/yr xxx.x Emn O. Output xxx.x J/yr or g/yr xxx.x ${\displaystyle \sum _{n}^{1}Em_{i}}$ Legend • Column #1 is the line item number, which is also the number of the footnote found below the table where raw data sources are cited and calculations are shown. • Column # 2 is the name of the item, which is also shown on the aggregated diagram. • Column # 3 is the raw data in joules, grams, dollars or other units. • Column # 4 shows the units for each raw data item. • Column # 5 is the unit emergy value, expressed in solar emergy joules per unit. Sometimes, inputs areexpressed in grams, hours, or dollars, therefore an appropriate UEV is used (sej/hr; sej/g; sej/$).
• Column # 6 is the solar emergy of a given flow, calculated as the raw input times the UEV (Column 3 times Column 5).

All tables are followed by footnotes that show citations for data and calculations.

### Calculating unit values

The table allows a unit emergy value to be calculated. The final, output row (row “O” in the example table above) is evaluated first in units of energy or mass. Then the input emergy is summed and the unit emergy value is calculated by dividing the emergy by the units of the output.

### Performance indicators

Figure 2: Systems diagram showing flows used in the performance indicator ratios

Figure 2 shows non-renewable environmental contributions (N) as an emergy storage of materials, renewable environmental inputs (R), and inputs from the economy as purchased (F) goods and services. Purchased inputs are needed for the process to take place and include human service and purchased non-renewable energy and material brought in from elsewhere (fuels, minerals, electricity, machinery, fertilizer, etc.). Several ratios, or indices are given in Figure 2 that assess the global performance of a process.

• Emergy Yield Ratio (EYR) — Emergy released (used up) per unit invested. The ratio is a measure of how much an investment enables a process to exploit local resources.
• Environmental Loading Ratio (ELR) — The ratio of nonrenewable and imported emergy use to renewable emergy use. It is an indicator of the pressure a transformation process exerts on the environment and can be considered a measure of ecosystem stress due to a production (transformation activity.
• Emergy Sustainability Index (ESI) — The ratio of EYR to ELR. It measures the contribution of a resource or process to the economy per unit of environmental loading.
• Aerial Empower Intensity — The ratio of emergy use in the economy of a region to its area. Renewable and nonrenewable emergy density are calculated separately by dividing the total renewable emergy by area and the total nonrenewable emergy by area, respectively.

Other ratios are useful depending on the type and scale of the system under evaluation.

• Percent Renewable Emergy (%Ren) — The ratio of renewable emergy to total emergy use. In the long run, only processes with high %Ren are sustainable.
• Emprice. The emprice of a commodity is the emergy one receives for the money spent in sej/\$.
• Emergy Exchange Ratio (EER) — The ratio of emergy exchanged in a trade or purchase (what is received to what is given). The ratio is always expressed relative to a trading partner and is a measure of the relative trade advantage of one partner over the other.
• Emergy per capita — The ratio of emergy use of a region or nation to the population. Emergy per capita can be used as a measure of potential, average standard of living of the population.

## Uses

The recognition of the relevance of energy to the growth and dynamics of complex systems has resulted in increased emphasis on environmental evaluation methods that can account for and interpret the effects of matter and energy flows at all scales in systems of humanity and nature. The following table lists some general areas in which the emergy methodology has been employed.

 Emergy and ecosystems Self-organization (Odum, 1986; Odum, 1988) Aquatic and marine ecosystems (Odum et al., 1978a; Odum and Arding, 1991; Brandt-Williams, 1999) Food webs and hierarchies (Odum et al. 1999; Brown and Bardi, 2001) Ecosystem health (Brown and Ulgiati, 2004) Forest ecosystems (Doherty et al., 1995; Lu et al. 2006) Complexity (Odum, 1987a; Odum, 1994; Brown and Cohen, 2008) Biodiversity (Brown et al. 2006) Emergy and Information Diversity and information (Keitt, 1991; Odum, 1996, Jorgensen et al., 2004) Culture, Education, University (Odum and Odum, 1980; Odum et al., 1995; Odum et al., 1978b) Emergy and Agriculture Food production, agriculture (Odum, 1984; Ulgiati et al. 1993; Martin et al. 2006; Cuadra and Rydberg, 2006; de Barros et al. 2009; Cavalett and Ortega, 2009) Livestock production (Rótolo et al.2007) Agriculture and society (Rydberg and Haden, 2006; Cuadra and Björklund, 2007; Lu, and Campbell, 2009) Soil erosion (Lefroy and Rydberg, 2003; Cohen et al. 2006) Emergy and energy sources and carriers Fossil fuels (Odum et a.l 1976; Brown et al., 1993; Odum, 1996; Bargigli et al., 2004; Bastianoni et al. 2005; Bastianoni et al. 2009) Renewable and nonrenewable electricity (Odum et al. 1983; Brown and Ulgiati, 2001; Ulgiati and Brown, 2001; Peng et al. 2008) Hydroelectric dams (Brown and McClanahan, 1992) Biofuels (Odum, 1980a; Odum and Odum, 1984; Carraretto et al., 2004; Dong et al. 2008; Felix and Tilley, 2009; Franzese et al., 2009) Hydrogen (Barbir, 1992) Emergy and the Economy National and international analyses (Odum, 1987b; Brown, 2003; Cialani et al. 2003; Ferreyra and Brown. 2007; Lomas et al., 2008; Jiang et al.,2008) Trade (Odum, 1984a; Brown, 2003) Environmental accounting (Odum, 1996) Development policies (Odum, 1980b) Sustainability (Odum, 1973; Odum, 1976a; Brown and Ulgiati, 1999; Odum and Odum, 2002; Brown et al. 2009) Tourism (Lei and Wang, 2008a; Lei et al., 2011; Vassallo et al., 2009) Gambling industry (Lei et al., 2011) Emergy and cities Spatial organization and urban development (Odum et al., 1995b; Huang, 1998; Huang and Chen, 2005; Lei et al.,2008; Ascione, et. al 2009) Urban metabolism (Huang et al.,2006; Zhang et al., 2009) Transportation modes (Federici, et al. 2003; Federici et al., 2008; Federici et al., 2009; Almeida et al., 2010 ) Emergy and landscapes Spatial empower, Land development indicators (Brown and Vivas, 2004; Reiss and Brown, 2007) Emergy in landforms (Kangas, 2002) Watersheds (Agostinho et al., 2010) Emergy and ecological engineering Restoration models (Prado-Jartar and Brown, 1996) Reclamation projects (Brown, 2005; Lei and Wang, 2008b; Lu et al., 2009 ) Artificial Ecosystems: wetlands, pond (Odum, 1985) Waste treatment (Kent et al. 2000; Grönlund, et al. 2004; Giberna et al. 2004; Lei and Wang, 2008c) Emergy, material flows and recycling Mining and minerals processing (Odum, 1996; Pulselli et al.2008) Industrial production, ecodesign (Zhang et al. 2009; Almeida et al., 2009) Recycling pattern in human-dominated ecosystems (Brown and Buranakarn, 2003) Emergy and thermodynamics Efficiency and Power (Odum and Pinkerton, 1955; Odum, 1995) Maximum Empower Principle (Odum, 1975; Odum, 1983; Cai e al., 2004) Pulsing paradigm (Odum, 1982; Odum, W.P. et al., 1995) Thermodynamic principles (Giannantoni, 2002, 2003) Emergy and systems modeling Energy systems language and modeling (Odum, 1971; Odum, 1972) National sustainability (Brown et al. 2009; Lei and Zhou, 2012) Sensitivity analysis, uncertainty (Laganis and Debeljak, 2006; Ingwersen, 2010) Emergy and policy Tools for decision makers (Giannetti et al., 2006; Almeida, et al. 2007; Giannetti et al., 2010) Conservation and economic value (Lu et al.2007) References for each of the citations in this table are given in a separate list at the end of this article

## Controversies

The concept of emergy has been controversial within academe including ecology, thermodynamics and economy.[22][23][24][25][26][27] Emergy theory has been criticized for allegedly offering an energy theory of value to replace other theories of value.[citation needed] The stated goal of emergy evaluations is to provide an "ecocentric" valuation of systems, processes. Thus it does not purport to replace economic values but to provide additional information, from a different point of view.[citation needed]

The idea that a calorie of sunlight is not equivalent to a calorie of fossil fuel or electricity strikes many as absurd, based on the 1st Law definition of energy units as measures of heat (i.e. Joule's mechanical equivalent of heat).[28] Others have rejected the concept as impractical since from their perspective it is impossible to objectively quantify the amount of sunlight that is required to produce a quantity of oil. In combining systems of humanity and nature and evaluating environmental input to economies, mainstream economists criticize the emergy methodology for disregarding market values.[citation needed]

## Notes

1. ^ a b c d Odum, Howard T. (1996). Environmental Accounting: Emergy and Environmental Decision Making. Wiley. p. 370. ISBN 978-0-471-11442-0.
2. ^ von Bertalanffy. L. 1968. General System Theory. George Braziller Publ. New York 295 p.
3. ^ a b Odum, H.T. 1983. Systems Ecology: An Introduction. John Wiley, NY. 644 p.
4. ^ Odum, H.T., 1995. Self organization and maximum power. Chapter 28, pp. 311-364 in Maximum Power, Ed. by C.A.S. Hall, University Press of Colorado, Niwot.
5. ^ Odum, H.T. 1957. Trophic structure and productivity of Silver Springs, Florida. Ecol. Monogr. 27:55-112.
6. ^ Odum, H.T. and E.P. Odum. 1955. Trophic structure and productivity of a windward coral reef at Eniwetok Atoll, Marshall Islands. Ecol. Monogr. 25:291-320.
7. ^ Odum, H.T. and C.M. Hoskin. 1958. Comparative studies of the metabolism of Texas Bays. Pubi. Inst. Mar. Sci., Univ. Tex. 5:16-46.
8. ^ Odum, H.T. and R.F. Pigeon, eds. 1970. A Tropical Rain Forest. Division of Technical Information, U.S. Atomic Energy Commission. 1600 pp.
9. ^ a b c Odum, H.T. 1967. Energetics of food production. In: The World Food Problem, Report of the President's Science Advisory Committee, Panel on World Food Supply, Vol. 3. The Whitehouse. pp. 55-94.
10. ^ a b Odum, H.T. et al. 1976. Net Energy Analysis of Alternatives for the United States. In U.S. Energy Policy: Trends and Goals, Part V - Middle and Long-term Energy Policies and Alternatives. 94th Congress 2nd Session Committee Print. Prepared for the Subcommittee on Energy and Power of the Committee on Interstate and Foreign Commerce of the U.S. House of Representatives, 66-723, U.S. Govt. Printing Office, Wash, DC. pp. 254-304.
11. ^ a b Odum, H.T. and E.C. Odum. 1976. Energy Basis for Man and Nature. McGraw-Hill, NY. 297 pp
12. ^ Odum, H.T. 1973. Energy, ecology and economics. Royal Swedish Academy of Science. AMBIO 2(6):220-227.
13. ^ Odum, H.T. 1976. 'Energy quality and carrying capacity of the earth. Response at Prize Ceremony, Institute de la Vie, Paris. Tropical Ecology 16(l):1-8.
14. ^ Odum, H.T. 1977. Energy analysis, energy quality and environment. In Energy Analysis: A New Public Policy Tool, M.W. Gilliland, ed. American Association for the Advancement of Science, Selected Symposium No. 9, Wash. DC. Westview Press. pp. 55-87.
15. ^ Odum, E.C., and Odum, H.T., 1980. Energy systems and environmental education. Pp. 213-231 in: Environmental Education- Principles, Methods and Applications, Ed. by T.S. Bakshi and Z. Naveh. Plenum Press, New York.
16. ^ Scienceman, D.M., 1987. "Energy and Emergy," in G. Pillet and T. Murota (eds), Environmental Economics: The Analysis of a Major Interface, R. Leimgruber, Geneva, pp. 257–276. (CFW-86-26)
17. ^ Odum, H.T. 1971. Environment, Power and Society. John Wiley, NY. 336 pp.
18. ^ Odum, H.T., M.J. Lavine, F.C. Wang, M.A. Miller, J.F. Alexander, Jr. and T. Butler. 1983. A Manual for Using Energy Analysis for Plant Siting with an Appendix on Energy Analysis of Environmental Values. Final report to the Nuclear Regulatory Commission, NUREG/CR-2443 FINB-6155. Energy Analysis Workshop, Center for Wetlands, University of Florida, Gainesville. 221 pp.
19. ^ Odum, H.T. and E.C. Odum, eds. 1983. Energy Analysis Overview of Nations. Working Paper WP-83-82. International Institute for Applied Systems Analysis, Laxenburg, Austria. 469 pp.
20. ^ Odum, H.T., M.T. Brown and S.B. Williams. 2000. Handbook of Emergy Evaluation: A Compendium of Data for Emergy Computation Issued in a Series of Folios. Folio #1 - Introduction and Global Budget. Center for Environmental Policy, Environmental Engineering Sciences, Univ. of Florida, Gainesville, 16 pp. Available on line at: http://emergysystems.org/folios.php.
21. ^ Many example diagrams can be found at EmergySystems.org).
22. ^ Ayres, R.U., 1998. Ecology vs. Economics: Confusing Production and Consumption. Center of the Management of Environmental Resources, INSEAD, Fontainebleau, France.
23. ^ Cleveland, C.J., Kaufmann, R.K., Stern, D.I., 2000. Aggregation and the role of energy in the economy. Ecol. Econ. 32, 301–317.
24. ^ Hau JL, Bakshi BR. 2004. Promise and problems of emergy analysis. Ecological Modelling 178:215–225.
25. ^ Mansson, B.A., McGlade, J.M., 1993. Ecology, thermodynamics and H.T. Odum’s conjectures. Oecologia 93, 582–596.
26. ^ Silvert W. 1982. The theory of power and efficiency in ecology. Ecological Modelling 15:159–164.
27. ^ Spreng, D.T., 1988. Net-Energy Analysis and the Energy Requirements of Energy Systems. Praeger Publishers, New York, 289 pp.
28. ^ Sciubba, E., 2010. On the Second-Law inconsistency of Emergy Analysis. Energy 35, 3696-3706.
References for Table 4
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Almeida, C.M.V.B., D. Borges Jr., S.H. Bonilla, B.F. Giannetti 2010. Identifying improvements in water management of bus-washing stations in Brazil Resources, Conservation and Recycling, In Press, Corrected Proof, Available online 13 February 2010
Almeida, C.M.V.B., F.A. Barrella, B.F. Giannetti. 2007. Emergetic ternary diagrams: five examples for application in environmental accounting for decision-making. Journal of Cleaner Production, Volume 15, Issue 1, 2007, Pages 63-74
Ascione, M., L. Campanella, F. Cherubini, and S. Ulgiati. 2009. Environmental driving forces of urban growth and development: An emergy-based assessment of the city of Rome, Italy. Landscape and Urban Planning, Volume 93, Issues 3-4, 15 December 2009, Pages 238-249
Barbir, F., 1992. Analysis and Modeling of Environmental and Economic Impacts of the Solar Hydrogen Energy System. Ph.D. Dissertation, Dept. of Mechanical Engineering, University of Miami, Florida, 176 pp.
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Brown, M.T. and T. McClanahan 1996. Emergy Analysis Perspectives for Thailand and Mekong River Dam Proposals. Ecological Modeling 91:pp105-130
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Brown, M.T., and Ulgiati, S., 1999. Emergy Evaluation of the Biosphere and Natural Capital. Ambio, 28(6): 486-493.
Brown, M.T., and Ulgiati, S., 2002. The Role of Environmental Services in Electricity Production Processes. Journal of Cleaner Production, 10: 321-334.
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Brown, M.T., Woithe, R.D., Montague, C.L., Odum, H.T., and Odum, E.C., 1993. Emergy Analysis Perspectives of the Exxon Valdez Oil Spill in Prince William Sound, Alaska. Final Report to the Cousteau Society. Center for Wetlands, University of Florida, Gainesville, FL, 114 pp.
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Carraretto, C., A. Macor, A. Mirandola, A. Stoppato, S. Tonon. 2004. Biodiesel as alternative fuel: Experimental analysis and energetic evaluations. Energy, Volume 29, Issues 12-15, October–December 2004, Pages 2195-2211
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Cuadra, M., T. Rydberg. 2006. Emergy evaluation on the production, processing and export of coffee in Nicaragua. Ecological Modelling, Volume 196, Issues 3-4, 25 July 2006, Pages 421-433
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Doherty, S.J., Odum, H.T., and Nilsson, P.O., 1995. Systems Analysis of the Solar Emergy Basis for Forest Alternatives in Sweden. Final Report to the Swedish State Power Board, College of Forestry, Garpenberg, Sweden, 112 pp.
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Federici, M., S. Ulgiati, R. Basosi. 2008. A thermodynamic, environmental and material flow analysis of the Italian highway and railway transport systems. Energy, Volume 33, Issue 5, May 2008, Pages 760-775
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Huang, S-L., C-W. Chen. 2005. Theory of urban energetics and mechanisms of urban development. Ecological Modelling, Volume 189, Issues 1-2, 25 November 2005, Pages 49-71
Huang, S-L., C-L. Lee, C-W. Chen. 2006. Socioeconomic metabolism in Taiwan: Emergy synthesis versus material flow analysis. Resources, Conservation and Recycling, Volume 48, Issue 2, 15 August 2006, Pages 166-196
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Jorgensen, S. E., H. T. Odum, M. T. Brown. 2004. Emergy and exergy stored in genetic information. Ecological Modelling, Volume 178, Issues 1-2, 15 October 2004, Pages 11-16
Kangas, P.C., 2002. Emergy of Landforms. Folio No. 5 of Handbook of Emergy Evaluation. The Center for Environmental Policy, University of Florida, Gainesville 93 p. (http://www.emergysystems.org/downloads/Folios/Folio_5.pdf)
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Laganis, J., M. Debeljak. 2006. Sensitivity analysis of the emergy flows at the solar salt production process in Slovenia. Ecological Modelling, Volume 194, Issues 1-3, 25 March 2006, Pages 287-295
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Lei K., Z. Wang. 2008c. Municipal Wastes and Their Solar Transformities: Emergy Synthesis for Macao. Waste management, Volume 28, Issue 12, pages 2522-2531
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Lei K., S. Zhou, D. Hu, Z. Wang. 2010. Ecological energy accounting for the gambling sector: A case study in Macao. Ecological complexity, Volume 7, pages 149-155
Lei K., S. Zhou, D. Hu, Z. Guo, A. Cao. 2011. Emergy analysis for tourism systems: Principles and a case study for Macao. Ecological complexity, Volume 8, 192-200
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Lomas, P.L., S. Álvarez, M. Rodríguez, C. Montes. 2008. Environmental accounting as a management tool in the Mediterranean context: The Spanish economy during the last 20 years. Journal of Environmental Management, Volume 88, Issue 2, July 2008, Pages 326-347
Lu, H-F., W-L.Kang, D.E. Campbell, H. Ren, Y-W. Tan, R-X. Feng, J-T. Luo, F-P. Chen. 2009. Emergy and economic evaluations of four fruit production systems on reclaimed wetlands surrounding the Pearl River Estuary, China. Ecological Engineering, Volume 35, Issue 12, December 2009, Pages 1743-1757
Lu, H. D.E. Campbell, Z. Li, H. Ren. 2006.Emergy synthesis of an agro-forest restoration system in lower subtropical China. Ecological Engineering, Volume 27, Issue 3, 2 October 2006, Pages 175-192
Lu, H., D. Campbell, J. Chen, P. Qin, H. Ren . 2007. Conservation and economic viability of nature reserves: An emergy evaluation of the Yancheng Biosphere Reserve. Biological Conservation, Volume 139, Issues 3-4, October 2007, Pages 415-438
Lu, H., D. E. Campbell. 2009. Ecological and economic dynamics of the Shunde agricultural system under China's small city development strategy. Journal of Environmental Management, Volume 90, Issue 8, June 2009, Pages 2589-2600
Martin, J.F., S.A.W. Diemont, E. Powell, M. Stanton, S. Levy-Tacher. 2006. Emergy evaluation of the performance and sustainability of three agricultural systems with different scales and management. Agriculture, Ecosystems & Environment, Volume 115, Issues 1-4, July 2006, Pages 128-140
Odum H.T. and E.C. Odum , 2001. A Prosperous Way Down: Principles and Policies. University Press of Colorado.
Odum H.T. and Pinkerton R.C., 1955. Time's speed regulator: the optimum efficiency for maximum power output in physical and biological systems. American Scientist, 43: 331-343.
Odum H.T., 1983. Maximum power and efficiency: a rebuttal. Ecological Modelling, 20: 71-82.
Odum H.T., 1988. Self organization, transformity and information. Science, 242: 1132-1139.
Odum H.T., 1996. Environmental Accounting. Emergy and Environmental Decision Making. John Wiley & Sons, N.Y.
Odum, E.C., and Odum, H.T., 1980. Energy systems and environmental education. Pp. 213-231 in: Environmental :Education- Principles, Methods and Applications, Ed. by T.S. Bakshi and Z. Naveh. Plenum Press, New York.
Odum, E.C., and Odum, H.T., 1984. System of ethanol production from sugarcane in Brazil. Ciencia e Cultura, 37(11): 1849-1855.
Odum, E.C., Odum, H.T., and Peterson, N.S., 1995a. Using simulation to introduce systems approach in education. Chapter 31, pp. 346-352, in Maximum Power, ed. by C.A.S. Hall, University Press of Colorado, Niwot.
Odum, H. T., Brown, M. T., Whitefield, L. S., Woithe, R., and Doherty, S., 1995b. Zonal Organization of Cities and Environment: A Study of Energy System Basis for Urban Society. A Report to the Chiang Ching-Kuo Foundation for International Scholarly Exchange, Center for Environmental Policy, University of Florida, Gainesville, FL.
Odum, H.T, M.T. Brown, and S. Ulgiati. 1999. Ecosystems as Energetic Systems. pp.281-302 in S.E. Jorgensen and F. Muller (eds) Handbook of Ecosystem Theories. CRC Press, New York
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Odum, H.T. 1971b. An energy circuit language for ecological and social systems: its physical basis. Pp. 139-211, in Systems Analysis and Simulation in Ecology, Vol. 2, Ed. by B. Patten. Academic Press, New York.
Odum, H.T. 1972b. Chemical cycles with energy circuit models. Pp. 223-257, in Changing Chemistry of the Ocean, ed. by D. Dryssen and D. Jagner. Nobel Symposium 20. Wiley, New York.
Odum, H.T. 1973. Energy, ecology and economics. Royal Swedish Academy of Science. AMBIO 2(6):220-227.
Odum, H.T. 1976a. 'Energy quality and carrying capacity of the earth. Response at Prize Ceremony, Institute de la Vie, Paris. Tropical Ecology 16(l):1-8.
Odum, H.T. 1987a. Living with complexity. Pp. 19-85 in The Crafoord Prize in the Biosciences, 1987, Lectures. Royal Swedish Academy of Sciences, Stockholm, Sweden. 87 pp
Odum, H.T. 1987b. Models for national, international, and global systems policy. Chapter 13, pp. 203-251, in Economic-Ecological Modeling, ed. by L.C. Braat and W.F.J. Van Lierop. Elsevier Science Publishing, New York, 329 pp.
Odum, H.T. et al. 1976. Net energy Analysis of Alternatives for the United States. In U.S. Energy Policy: Trends and Goals. Part V - Middle and Long-term Energy Policies and Alternatives. 94th Congress 2nd Session Committee Print. Prepared for the Subcommittee on Energy and Power of the Committee on Interstate and Foreign Commerce of the U.S. House of Representatives, 66-723, U.S. Govt. Printing Office, Wash, DC. pp. 254-304.
Odum, H.T., 1975. Combining energy laws and corollaries of the maximum power principle with visual system mathematics. Pp. 239-263, in Ecosystems: Analysis and Prediction, ed. by Simon Levin. Proceedings of the conference on ecosystems at Alta, Utah. SIAM Institute for Mathematics and Society, Philadelphia.
Odum, H.T., 1980a. Biomass and Florida’s future. Pp. 58-67 in: A Hearing before the Subcommittee on Energy Development and Applications of the Committee on Science and Technology of the U.S. House of Representatives, 96th Congress. Government Printing Office, Washington, D.C.
Odum, H.T., 1980b. Principle of environmental energy matching for estimating potential economic value: a rebuttal. Coastal Zone Management Journal, 5(3): 239-243.
Odum, H.T., 1982. Pulsing, power and hierarchy. Pp. 33-59, in Energetics and Systems, ed. by W.J. Mitsch, R.K. Ragade, R. W. Bosserman, and J.A. Dillon Jr., Ann Arbor Science, Ann Arbor, Michigan.
Odum, H.T., 1984a. Energy analysis of the environmental role in agriculture. Pp. 24-51, in Energy and Agriculture, ed. by G. Stanhill. Springer Verlag, Berlin. 192 pp.
Odum, H.T., 1985. Water conservation and wetland values. Pp. 98-111, in Ecological Considerations in Wetlands Treatment of Municipal Wastewaters, ed. by P.J. Godfrey, E.R. Kaynor, S. Pelezrski, and J. Benforado. Van Nostrand Reinhold, New York. 473 pp.
Odum, H.T., 1986. Enmergy in ecosystems. In Environmental Monographs and Symposia, N. Polunin, ed. John Wiley, NY. pp. 337-369.
Odum, H.T., 1994. Ecological and General Systems: An Introduction to Systems Ecology. University Press of Colorado, Niwot. 644 pp. Revised edition of Systems Ecology, 1983, Wiley.
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Odum, H.T., 2000. Handbook of Emergy Evaluation: A Compendium of Data for Emergy Computation Issued in a Series of Folios. Folio #2 – Emergy of Global processes. Center for Environmental Policy, Environmental (http://www.emergysystems.org/downloads/Folios/Folio_2.pdf)
Odum, H.T., and Arding, J.E., 1991. Emergy analysis of shrimp mariculture in Ecuador. Report to Coastal Studies Institute, University of Rhode Island, Narragansett. Center for Wetlands, University of Florida, Gainesville, pp. 87.
Odum, H.T., Gayle, T., Brown, M.T., and Waldman, J., 1978b. Energy analysis of the University of Florida. Center for Wetlands, University of Florida, Gainesville. Unpublished manuscript.
Odum, H.T., Kemp, W., Sell, M., Boynton W., and Lehman, M., 1978a. Energy Analysis and the coupling of man and estuaries. Environmental Management, 1: 297-315.
Odum, H.T., Lavine, M.J., Wang, F.C., Miller, M.A., Alexander, J.F., and Butler, T., 1983. Manual for using energy analysis for plant siting. Report to the Nuclear Regulatory Commission, Washington, DC. Report No. NUREG/CR-2443. National Technical Information Service, Springfield, Va. Pp. 242.
Odum, H.T., M.T. Brown and S.B. Williams. 2000. Handbook of Emergy Evaluation: A Compendium of Data for Emergy Computation Issued in a Series of Folios. Folio #1 - Introduction and Global Budget. Center for Environmental Policy, Environmental . (http://www.emergysystems.org/downloads/Folios/Folio_1.pdf)
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Pizzigallo, A.C.I., C. Granai, S. Borsa. 2008. The joint use of LCA and emergy evaluation for the analysis of two Italian wine farms. Journal of Environmental Management, Volume 86, Issue 2, January 2008, Pages 396-406
Prado-Jatar, M.A., and Brown, M.T., 1997. Interface ecosystems with an oil spill in a Venezuelan tropical savannah. Ecological Engineering, 8: 49-78.
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Rydberg, T., A.C. Haden. 2006. Emergy evaluations of Denmark and Danish agriculture: Assessing the influence of changing resource availability on the organization of agriculture and society. Agriculture, Ecosystems & Environment, Volume 117, Issues 2-3, November 2006, Pages 145-158
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## Get a free weekly practice problem!

Keep that axe sharp.

### You only have free questions left (including this one).

But it doesn't have to end here! Sign up for the 7-day coding interview crash course and you'll get a free Interview Cake problem every week.

Write a function fib that takes an integer n and returns the nth Fibonacci number.

Let's say our Fibonacci series is 0-indexed and starts with 0. So:

fib(0); // => 0 fib(1); // => 1 fib(2); // => 1 fib(3); // => 2 fib(4); // => 3 ...

Our solution runs in n time.

There's a clever, more mathy solution that runs in time, but we'll leave that one as a bonus.

If you wrote a recursive function, think carefully about what it does. It might do repeat work, like computing fib(2) multiple times!

We can do this in space. If you wrote a recursive function, there might be a hidden space cost in the call stack!

The nth Fibonacci number is defined in terms of the two previous Fibonacci numbers, so this seems to lend itself to recursion.

fib($n) = fib($n - 1) + fib($n - 2); Can you write up a recursive solution? As with any recursive function, we just need a base case and a recursive case: 1. Base case: n is 0 or 1. Return n. 2. Recursive case: Return fib($n - 1) + fib($n - 2). function fib($n) { if ($n === 0 ||$n === 1) { return $n; } return fib($n - 1) + fib($n - 2); } Okay, this'll work! What's our time complexity? It's not super obvious. We might guess n, but that's not quite right. Can you see why? Each call to fib makes two more calls. Let's look at a specific example. Let's say n=5. If we call fib(5), how many calls do we make in total? Try drawing it out as a tree where each call has two child calls, unless it's a base case. Here's what the tree looks like: We can notice this is a binary tree whose height is n, which means the total number of nodes is . So our total runtime is . That's an "exponential time cost," since the n is in an exponent. Exponential costs are terrible. This is way worse than or even . Our recurrence tree above essentially gets twice as big each time we add 1 to n. So as n gets really big, our runtime quickly spirals out of control. The craziness of our time cost comes from the fact that we're doing so much repeat work. How can we avoid doing this repeat work? We can memoize! Let's wrap fib in a class with an instance variable where we store the answer for any n that we compute: class Fibber { private$memo = []; public function fib($n) { // edge case: negative index if ($n < 0) { throw new Exception('Index was negative. No such thing as a negative index in a series.'); } // base case: 0 or 1 if ($n === 0 ||$n === 1) { return $n; } // see if we've already calculated this if (isset($this->memo[$n])) { return$this->memo[$n]; }$result = $this->fib($n - 1) + $this->fib($n - 2); // memoize $this->memo[$n] = $result; return$result; } }

What's our time cost now?

Our recurrence tree will look like this:

The computer will build up a call stack with fib(5), fib(4), fib(3), fib(2), fib(1). Then we'll start returning, and on the way back up our tree we'll be able to compute each node's 2nd call to fib in constant time by just looking in the memo. n time in total.

What about space? $memo takes up n space. Plus we're still building up a call stack that'll occupy n space. Can we avoid one or both of these space expenses? Look again at that tree. Notice that to calculate fib(5) we worked "down" to fib(4), fib(3), fib(2), etc. What if instead we started with fib(0) and fib(1) and worked "up" to n? We use a bottom-up approach, starting with the 0th Fibonacci number and iteratively computing subsequent numbers until we get to n. function fib($n) { // edge cases: if ($n < 0) { throw new Exception('Index was negative. No such thing as a negative index in a series.'); } if ($n === 0 || $n === 1) { return$n; } // we'll be building the fibonacci series from the bottom up // so we'll need to track the previous 2 numbers at each step // 0th fibonacci $prevPrev = 0; // 1st fibonacci$prev = 1; // Declare current $current = 0; for ($i = 1; $i <$n; $i++) { // Iteration 1: current = 2nd fibonacci // Iteration 2: current = 3rd fibonacci // Iteration 3: current = 4th fibonacci // To get nth fibonacci ... do n-1 iterations.$current = $prev +$prevPrev; $prevPrev =$prev; $prev =$current; } return \$current; }

time and space.

• If you're good with matrix multiplication you can bring the time cost down even further, to . Can you figure out how?

This one's a good illustration of the tradeoff we sometimes have between code cleanliness and efficiency.

We could use a cute, recursive function to solve the problem. But that would cost time as opposed to n time in our final bottom-up solution. Massive difference!

In general, whenever you have a recursive solution to a problem, think about what's actually happening on the call stack. An iterative solution might be more efficient.

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## LaTeX forum ⇒ Calendars and Miscellaneous ⇒ template for a notation tabel Topic is solved

Calendars, invoices, tables, memos, contracts, dictionaries, code snippets
Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm

### template for a notation tabel

Hello,

does anybody have a template for a notation tabel? (see pictures)

Thanks
Last edited by Elmaltus on Fri Jan 18, 2019 4:21 pm, edited 1 time in total.

Stefan Kottwitz
Posts: 9367
Joined: Mon Mar 10, 2008 9:44 pm
Welcome to the forum!

Here is a quick start:

1. \documentclass{article}
2. \usepackage{tabularx}
3. \usepackage{booktabs}
4. \usepackage[labelfont=bf,format=plain,justification=raggedright,singlelinecheck=false]{caption}
5. \begin{document}
6. \begin{table}
7. \caption{Notation for\ldots}
8. \begin{tabularx}{\textwidth}{@{}XX@{}}
9. \toprule
10. Indices \\
11. $K$ & Number of ... \\
12. $T$ & Number of ... \\
13. Parameters \\
14. c & something \\
15. \bottomrule
16. \end{tabularx}
17. \end{table}
18. \end{document}

Stefan

Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm
Thank you, that helped a lot!
Could you please explain to me how I can adjust the width of each collums? For example the first collum should be 20% of the textwidth...

Stefan Kottwitz
Posts: 9367
Joined: Mon Mar 10, 2008 9:44 pm

1. \begin{tabularx}{\textwidth}{@{}p{0.2\textwidth}X@{}}

p{0.2\textwidth} is standard tabular syntax and specifies a column with 20% of the text width.

X is tabularx syntax and specifies a column that fills all available space up to the max we set above (\textwidth).

Stefan

Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm
Thanks now i got it

Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm
Hello,
my notationtable is nearly done but now I don't know how I can fix the following problems:
I just want to move all variable descriptions to the left while the topics (like Parameters: or Decision variables:) can fit a whole row. (see screenshot) And why is there so much Space between Random Variables and BL_kt?

Thank you!
1. \begin{table}[h]
2. \caption{Notation}
3. \begin{tabularx}{\textwidth}{p{0.22\textwidth}X}
4. \toprule
5. {\ul Indices: } \\
6. $K$ & number of products (k \in\{1,...,K\}) \\
7. $T$ & number of periods (t \in\{1,...,T\})\\
8.
9. {\ul Parameters: } \\
10. \delta_k & target δ-service level for product k \\
11. \epsilon & \\
12. c_t & production capacity in period t \\
13. c^r_t & Remanufacturing capacity in period t \\
14. hc_k & holding cost of product k per unit and period \\
15. hc_k & holding cost of product k per unit and period \\
16. oc & overtime costs per unit \\
17. M_{kt} & bignumber for product k in period t\\
18. pc_{k} & production cost of product k per unit\\
19. pc^r_{k} & remanufacturing cost of product k per unit\\
20. sc_{k} & setup cost of product k\\
21. sc^r_{k} & setup cost of returned product k\\
22. tp_{k} & production time for one unit of product k\\
23. tp^r_{k} & remanufacturing time for one unit of product k\\
24. ts_{k} & setup time of product k\\
25. ts^r_{k} & setup time of returned product k\\
26. {\ul Random variables: } \\
27. BL_{kt} & backlog of product k at the end of period t\\
28. D_{kt} & external demand of product k in period t \\
29. I_{kt} & net inventory of product k at the end of period t \\
30. I^r_{kt} & net inventory of returns of product k at the end of period t \\
31. IP_{kt} & physical inventory of product k at the end of period t\\
32. IP^r_{kt} & physical inventory of recoverables at the end of period t\\
33. R_{kt} & returns of product k in period t\\
34. SF^r_{kt} & Shortfall of recoverables of product k in period t\\
35. {\ul Decision variables: } \\
36. q_{kt} & production quantity of product k in period t\\
37. q^r_{kt} & remanufacturing quantity of product k in period t \\
38. o_{t} & amount of overtime for production in period t \\
39. o^r_{t} & amount of overtime for remanufacturing in period t\\
40. % \gamma_{kt} & binary setup indicator for production of product k in period t\\
41. % \gamma^r_{kt} & binary setup variable for remanufactured product k in period t\\
42. \bottomrule
43. \end{tabularx}
44. \end{table}
Attachments
Unbenannt.PNG (62.54 KiB) Viewed 231 times

Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm
Okay I fixed the layout problems myself. But I don't know why the table moves to the end of my document when I add two more rows. (see pictures) There should be enough space for two more rows but it still is not on the right spot.
Any Ideas what I can do?
Thanks!

1. \begin{table}[h]
2. \caption{Notation des SCLSP-RM\textsuperscript{gen} Modells nach Sahling et all.(2016)}
3. \begin{tabularx}{\textwidth}{p{0.08\textwidth}X}
4. \toprule
5.
6. \multicolumn{2}{l}{{\ul Indices:}} \\
7. $K$ & number of products (k \in\{1,...,K\}) \\
8. $T$ & number of periods (t \in\{1,...,T\})\\
9.
10. \multicolumn{2}{l}{{\ul Parameters:}} \\
11. \delta_k & target δ-service level for product k \\
12. \epsilon & \\
13. c_t & production capacity in period t \\
14. c^r_t & Remanufacturing capacity in period t \\
15. hc_k & holding cost of product k per unit and period \\
16. hc_k & holding cost of product k per unit and period \\
17. oc & overtime costs per unit \\
18. M_{kt} & bignumber for product k in period t\\
19. pc_{k} & production cost of product k per unit\\
20. pc^r_{k} & remanufacturing cost of product k per unit\\
21. sc_{k} & setup cost of product k\\
22. sc^r_{k} & setup cost of returned product k\\
23. tp_{k} & production time for one unit of product k\\
24. tp^r_{k} & remanufacturing time for one unit of product k\\
25. ts_{k} & setup time of product k\\
26. ts^r_{k} & setup time of returned product k\\
27.
28. \multicolumn{2}{l}{{\ul Random variables:}} \\
29. BL_{kt} & backlog of product k at the end of period t\\
30. D_{kt} & external demand of product k in period t \\
31. I_{kt} & net inventory of product k at the end of period t \\
32. I^r_{kt} & net inventory of returns of product k at the end of period t \\
33. IP_{kt} & physical inventory of product k at the end of period t\\
34. IP^r_{kt} & physical inventory of recoverables at the end of period t\\
35. R_{kt} & returns of product k in period t\\
36. SF^r_{kt} & Shortfall of recoverables of product k in period t\\
37.
38. \multicolumn{2}{l}{{\ul Decision variables:}} \\
39. q_{kt} & production quantity of product k in period t\\
40. q^r_{kt} & remanufacturing quantity of product k in period t \\
41. o_{t} & amount of overtime for production in period t \\
42. o^r_{t} & amount of overtime for remanufacturing in period t\\
43. % \gamma_{kt} & binary setup indicator for production of product k in period t\\
44. % \gamma^r_{kt} & binary setup variable for remanufactured product k in period t\\
45. \bottomrule
46. \end{tabularx}
47. \end{table}
Attachments
Unbenannt.PNG (54.64 KiB) Viewed 219 times
2.PNG (54.42 KiB) Viewed 219 times

Elmaltus
Posts: 6
Joined: Thu Jan 17, 2019 1:48 pm
Okay I fixed the layout problems myself. But I don't know why the table moves to the end of my document when I add two more rows. (see pictures) There should be enough space for two more rows but it still is not on the right spot.
Any Ideas what I can do?
Thanks!

1. \begin{table}[h]
2. \caption{Notation des SCLSP-RM\textsuperscript{gen} Modells nach Sahling et all.(2016)}
3. \begin{tabularx}{\textwidth}{p{0.08\textwidth}X}
4. \toprule
5.
6. \multicolumn{2}{l}{{\ul Indices:}} \\
7. $K$ & number of products (k \in\{1,...,K\}) \\
8. $T$ & number of periods (t \in\{1,...,T\})\\
9.
10. \multicolumn{2}{l}{{\ul Parameters:}} \\
11. \delta_k & target δ-service level for product k \\
12. \epsilon & \\
13. c_t & production capacity in period t \\
14. c^r_t & Remanufacturing capacity in period t \\
15. hc_k & holding cost of product k per unit and period \\
16. hc_k & holding cost of product k per unit and period \\
17. oc & overtime costs per unit \\
18. M_{kt} & bignumber for product k in period t\\
19. pc_{k} & production cost of product k per unit\\
20. pc^r_{k} & remanufacturing cost of product k per unit\\
21. sc_{k} & setup cost of product k\\
22. sc^r_{k} & setup cost of returned product k\\
23. tp_{k} & production time for one unit of product k\\
24. tp^r_{k} & remanufacturing time for one unit of product k\\
25. ts_{k} & setup time of product k\\
26. ts^r_{k} & setup time of returned product k\\
27.
28. \multicolumn{2}{l}{{\ul Random variables:}} \\
29. BL_{kt} & backlog of product k at the end of period t\\
30. D_{kt} & external demand of product k in period t \\
31. I_{kt} & net inventory of product k at the end of period t \\
32. I^r_{kt} & net inventory of returns of product k at the end of period t \\
33. IP_{kt} & physical inventory of product k at the end of period t\\
34. IP^r_{kt} & physical inventory of recoverables at the end of period t\\
35. R_{kt} & returns of product k in period t\\
36. SF^r_{kt} & Shortfall of recoverables of product k in period t\\
37.
38. \multicolumn{2}{l}{{\ul Decision variables:}} \\
39. q_{kt} & production quantity of product k in period t\\
40. q^r_{kt} & remanufacturing quantity of product k in period t \\
41. o_{t} & amount of overtime for production in period t \\
42. o^r_{t} & amount of overtime for remanufacturing in period t\\
43. % \gamma_{kt} & binary setup indicator for production of product k in period t\\
44. % \gamma^r_{kt} & binary setup variable for remanufactured product k in period t\\
45. \bottomrule
46. \end{tabularx}
47. \end{table}


text:
Image Details

Choose export citation format:

High-contrast Imaging with Fizeau Interferometry: the Case of Altair

• Authors: E. Spalding, K. M. Morzinski, P. Hinz, J. Males, M. Meyer, S. P. Quanz, J. Leisenring, J. Power

E. Spalding et al 2022 The Astronomical Journal 163 .

• Provider: AAS Journals

Caption: Figure 11.

An example of KS statistic interpolations in the λ/B EE regime, for the “S” half-strips (i.e., those pointed most closely to the direction south after derotation). Dots show the grid of tested companion contrasts (0 < Δm < 10) and distances from the host star. The bold curve is at a KS value of 0.2716, which represents the boundary between companions that are from a different parent population, to a confidence of 95% (see Appendix A.3). The gray region is the region where the injected companions are 1 FWHM or less from the edge of the strip. Since the PSF of the fake companion is truncated when it is this close to the edge, the KS statistic will decrease as an edge artifact.


text:
# My triangle needs more nodes

Consider the standard equilateral triangle, with nodes labeled using barycentric coordinates:

We can turn this 3 node triangle into a 6 node triangle by adding a new line of 3 vertices (one more than was present on a side of the original 3 node triangle), remove any internal edges (but not internal nodes) and re-normalize the coordinates:

Repeating the process to go from a 6 node triangle to a 10 node triangle, add a line of 4 vertices (again, one more than was present on a side of the original 6 node triangle), remove any internal edges (but not internal nodes) and re-normalize the coordinates:

This process can be repeated indefinitely. The goal of this challenge is given an integer N representing how many times this process has been performed, output all the nodes for the associated triangle in barycentric coordinates.

# Input

Your program/function should take as input a single non-negative integer N representing how many times this process has been applied. Note that for N=0, you should output the original triangle with 3 nodes.

The input may come from any source (function parameter, stdio, etc.).

# Output

Your program/function should output all the nodes in normalized barycentric coordinates. The order of the nodes does not matter. A number can be specified as a fraction (fraction reduction not required) or a floating point number. You may also output "scaled" vectors to specify a node. For example, all 3 of the following outputs are equivalent and allowed:

0.5,0.5,0

1/2,2/4,0

[1,1,0]/2

If using floating point output, your output should be accurate to within 1%. The output may be to any sink desired (stdio, return value, return parameter, etc.). Note that even though the barycentric coordinates are uniquely determined by only 2 numbers per node, you should output all 3 numbers per node.

# Examples

Example cases are formatted as:

N
x0,y0,z0
x1,y1,z1
x2,y2,z2
...

where the first line is the input N, and all following lines form a node x,y,z which should be in the output exactly once. All numbers are given as approximate floating point numbers.

0
1,0,0
0,1,0
0,0,1

1
1,0,0
0,1,0
0,0,1
0.5,0,0.5
0.5,0.5,0
0,0.5,0.5

2
1,0,0
0,1,0
0,0,1
0.667,0,0.333
0.667,0.333,0
0.333,0,0.667
0.333,0.333,0.333
0.333,0.667,0
0,0.333,0.667
0,0.667,0.333

3
1,0,0
0.75,0,0.25
0.75,0.25,0
0.5,0,0.5
0.5,0.25,0.25
0.5,0.5,0
0.25,0,0.75
0.25,0.25,0.5
0.25,0.5,0.25
0.25,0.75,0
0,0,1
0,0.25,0.75
0,0.5,0.5
0,0.75,0.25
0,1,0

# Scoring

This is code golf; shortest code in bytes wins. Standard loopholes apply. You may use any built-ins desired.

• You say "If using floating point output". What alternatives are there? Fractions? If so, do they have to be reduced? How about scaled vectors like [1,2,3]/6? – Peter Taylor Aug 23 '16 at 9:54
• Yes, all of those alternatives are allowed. I'll update the problem statement. – helloworld922 Aug 23 '16 at 9:55

## CJam (22 bytes)

{):X),3m*{:+X=},Xdff/}

This is an anonymous block (function) which takes N on the stack and leaves an array of arrays of doubles on the stack. Online demo

### Dissection

{         e# Define a block
):X     e# Let X=N+1 be the number of segments per edge
),3m*   e# Generate all triplets of integers in [0, X] (inclusive)
{:+X=}, e# Filter to those triplets which sum to X
Xdff/   e# Normalise
}

f n|m<-n+1=[map(/m)[x,y,m-x-y]|x<-[0..m],y<-[0..m-x]]

# Python 3, 87 bytes

This is actually supposed to be a comment to the solution by TheBikingViking but I don't have enough reputation for comments.

One can save a few bytes by only iterating over the variables i,j and using the fact that with the third one they add up to n+1.

def f(n):d=n+1;r=range(n+2);print([[i/d,j/d,(d-i-j)/d]for i in r for j in r if d>=i+j])

## Mathematica,  44  43 bytes

Select[Range[0,x=#+1]~Tuples~3/x,Tr@#==1&]&

This is an unnamed function taking a single integer argument. Output is a list of lists of exact (reduced) fractions.

Generates all 3-tuples of multiples of 1/(N+1) between 0 and 1, inclusive, and then selects those whose sum is 1 (as required by barycentric coordinates).

# 05AB1E, 10 bytes

ÌL<¤/3ãDOÏ

Explanation

ÌL<          # range(1,n+2)-1
¤/        # divide all by last element (n+1)
3ã      # cartesian product repeat (generate all possible triples)
DO    # make a copy and sum the triples
Ï   # keep triples with sum 1

Try it online

• Since ¤ consumes the array, why does / divide the array by that? Does it "remember" that last popped value and use it if needed? – Luis Mendo Aug 23 '16 at 13:40
• @LuisMendo: ¤ is one of the few commands that does not pop and consume from the stack. It pushes the last element of the list while leaving the list on the stack. – Emigna Aug 23 '16 at 13:47
• – Luis Mendo Aug 23 '16 at 14:04
• @LuisMendo 05AB1E only outputs the top of the stack at the end of a program. – Emigna Aug 23 '16 at 14:20
• Oh, of course! Thanks for the explanations – Luis Mendo Aug 23 '16 at 15:08

# MATL, 17 bytes

2+:qGQ/3Z^t!s1=Y)

Try it online!

### Explanation

The approach is the same as in other answers:

1. Generate the array [0, 1/(n+1), 2/(n+1), ..., 1], where n is the input;
2. Generate all 3-tuples with those values;
3. Keep only those whose sum is 1.

More specifically:

2+     % Take input and add 2: produces n+2
:q     % Range [0 1 ... n+1]
GQ/    % Divide by n+1 element-wise: gives [0, 1/(n+1), 2/(n+1)..., 1]
3Z^    % Cartesian power with exponent 3. Gives (n+1)^3 × 3 array. Each row is a 3-tuple
t      % Duplicate
!s     % Sum of each row
1=     % Logical index of entries that equal 1
Y)     % Use that index to select rows of the 2D array of 3-tuples

## Jellyfish, 37 33 bytes

Thanks to Zgarb for saving 4 bytes.

p
*%
# S

=E   S
/
1+r#>>i
3

Try it online!

Like my Mathematica and Peter's CJam answers, this generates a set of candidate tuples and then selects only those that sum to 1. I'm not entirely happy with the layout yet, and I wonder whether I can save some bytes with hooks or forks, but I'll have to look into that later.

# Perl 6: 50 40 bytes

{grep *.sum==1,[X] (0,1/($_+1)...1)xx 3}  Returns a sequence of 3-element lists of (exact) rational numbers. Explanation: • $_
Implicitly declared parameter of the lambda.
• 0, 1/(\$_ + 1) ... 1
Uses the sequence operator ... to construct the arithmetic sequence that corresponds to the possible coordinate values.
• [X] EXPR xx 3
Takes the Cartesian product of three copies of EXPR, i.e. generates all possible 3-tuples.
• grep *.sum == 1, EXPR
Filter tuples with a sum of 1.

# Ruby, 62

I'd be surprised if this can't be improved on:

->x{0.step(1,i=1.0/(x+1)){|a|0.step(1-a,i){|b|p [a,b,1-a-b]}}}

Taking the advice latent in the puzzle, this calculates the second node options based on the first, and the third node by subtracting the first two.

# Brachylog, 24 bytes

+:1f
yg:2j:eaL+?/g:Lz:*a

Try it online!

# Python 3, 106 bytes

def f(n):r=range(n+2);print([x for x in[[i/-~n,j/-~n,k/-~n]for i in r for j in r for k in r]if sum(x)==1])

A function they takes input via argument and prints a list of lists of floats to STDOUT.

Python is not good at Cartesian products...

How it works

def f(n):                         Function with input iteration number n
r=range(n+2)                      Define r as the range [0, n+1]
for i in r for j in r for k in r  Length 3 Cartesian product of r
[i/-~n,j/-~n,k/-~n]               Divide each element of each list in the product
by n+1
[x for x in ... if sum(x)==1]     Filter by summation to 1
print(...)                           Print to STDOUT

Try it on Ideone

# Actually, 15 bytes

This uses an algorithm similar to the one in TheBikingViking's Python answer. Golfing suggestions welcome. Try it online!

u;ur♀/3@∙Σ1=░

Ungolfed:

u;                Increment implicit input and duplicate.
ur              Range [0..n+1]
♀/            Divide everything in range by (input+1).
3@∙         Take the Cartesian product of 3 copies of [range divided by input+1]
Σ1=    Create function that takes a list checks if sum(list) == 1.
░   Push values of the Cartesian product where f returns a truthy value.

# Ruby, 77 74 bytes

Another answer using the algorithm in TheBikingViking's Python answer. Golfing suggestions welcome.

->n{a=[*0.step(1,i=1.0/(n+1))];a.product(a,a).reject{|j|j.reduce(&:+)!=1}}

Another 74-byte algorithm based on Not that Charles's Ruby answer.

->x{y=x+1;z=1.0/y;[*0..y].map{|a|[*0..y-a].map{|b|p [a*z,b*z,(y-a-b)*z]}}}

## JavaScript (Firefox 30-57), 88 81 bytes

n=>[for(x of a=[...Array(++n+1).keys()])for(y of a)if(x+y<=n)[x/n,y/n,(n-x-y)/n]]

Returns an array of arrays of floating-point numbers. Edit: Saved 7 bytes by computing the third coordinate directly. I tried eliminating the if by calculating the range of y directly but it cost an extra byte:

n=>[for(x of a=[...Array(++n+1).keys()])for(y of a.slice(x))[x/n,(y-x)/n,(n-y)/n]]

• At the end, you wrote [x/n,y/n/z/n], did you forget a comma? – kamoroso94 Aug 24 '16 at 3:44
• @kamoroso94 You're right, I typoed the last comma, thanks for letting me know. – Neil Aug 24 '16 at 7:28


text:
# ICHEP 2016 Chicago

3-10 August 2016
Chicago IL USA
US/Central timezone
There is a live webcast for this event.

## Measurement of the Jet Mass Scale and Resolution for Large Radius Jets at sqrt(s) = 8 TeV using the ATLAS Detector

Aug 8, 2016, 6:30 PM
2h
Riverwalk A/B

### Riverwalk A/B

Poster Strong Interactions and Hadron Physics

### Speaker

Ruchika Nayyar (University of Arizona (US))

### Description

A measurement of the jet mass scale and jet mass resolution uncertainty for large radius jets using the full sqrt(s) = 8 TeV dataset from the ATLAS experiment is presented. Large radius jets are calibrated so that on average the reconstructed jet transverse momentum is the same as the corresponding particlelevel jet transverse momentum in simulation. The ratio of the reconstructed jet mass to the particlelevel jet mass is the jet mass response. The mean response is the jet mass scale and the standard deviation of the jet mass response distribution is the jet mass resolution. In this study the uncertainty on these quantities is measured by tting the W boson resonant peak in the large radius jet mass spectrum from lepton plus jets tt events in both data and Monte Carlo. Large radius jets with pT > 200 GeV and |η_jj| < 2:0 are used. Two different fitting procedures are used and give comparable results. For the more precise method, the ratio between the data and the Monte Carlo simulation is 1.001±0.003(stat)± 0.024(syst) for the jet mass scale and 0.96± 0:05(stat) ±0.10(syst) for the jet mass resolution.

### Primary author

ATLAS Collaboration (CERN)

### Presentation materials

There are no materials yet.


text:
emacs-orgmode
[Top][All Lists]

## Re: [Orgmode] latex export and beamer columns

 From: Thomas S. Dye Subject: Re: [Orgmode] latex export and beamer columns Date: Sat, 21 Nov 2009 06:33:52 -1000

On Nov 21, 2009, at 2:49 AM, Eric S Fraga wrote:

Recently, in September, there was a discussion about exporting org
files to latex using the beamer class and generating slides with
multiple columns:

http://article.gmane.org/gmane.emacs.orgmode/17546

No solution was presented at that time but I think I've managed to
create one that is quite reasonable.  I define a beamer entry in
org-export-latex-classes which has the following definitions for the
different headings in an org-mode file:

--8<---------------cut here---------------start------------->8---
("\\section{%s}" . "\\section*{%s}")
("\\begin{frame}\\frametitle{%s}" "\\end{frame}"
"\\begin{frame}\\frametitle{%s}" "\\end{frame}")
("\\begin{columns} \% %s" "\\end{columns}"
"\\begin{columns} \% %s" "\\end{columns}")
("\\begin{column}{%s\\textwidth}" "\\end{column}"
"\\begin{column}{%s\\textwidth}" "\\end{column}")
--8<---------------cut here---------------end--------------->8---

Then, if I have a 3rd level heading (whose actual text is ignored),
multiple columns mode is entered and the level 4 headings indicate the
widths (relative to \textwidth although obviously one could generalise
this) of each subsequent column.

*Note*: for this to work, you have to specify H:4 in the document
OPTIONS or else the 4th level headings are turned into items in an
itemized list.

So, for instance, this will generate a slide with two columns of text
(noting that I use odd levels only for my headings):

--8<---------------cut here---------------start------------->8---
# -*- coding: utf-8; -*-
#+TITLE:     Beamer column test
#+AUTHOR:    Eric
#+OPTIONS:   H:4 num:t toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t

#+OPTIONS: TeX:t LaTeX:t skip:nil d:nil todo:t pri:nil tags:not-in- toc
#+LaTeX_CLASS: beamer

*** An interesting slide
***** columns (this text is ignored)
******* 0.3
- Some text on the left of the slide
- make sure we have at least two bullet points
- this is a narrow column
******* 0.7
- text on the right side of the slide
- and more
- and even more
- and this is a much wider column
--8<---------------cut here---------------end--------------->8---

I can now create the slides for presentations and lectures completely
within org-mode so I'm a very happy camper!

eric

_______________________________________________
Emacs-orgmode mailing list
Remember: use Reply All' to send replies to the list.
http://lists.gnu.org/mailman/listinfo/emacs-orgmode

Aloha Eric,

Thanks very much for this approach to creating beamer slides with columns. The earlier discussion didn't come up with a solution at least partly because I was clinging to the notion that low-level headings, below the levels specified by the H option, should be used for itemized lists. Once this mistaken assumption is abandoned, your solution becomes clear.

Nice!

All the best,
Tom

`


text:
#### Authors

Pierre Garrigues, Laurent Ghaoui

#### Abstract

It has been shown that the problem of $\ell_1$-penalized least-square regression commonly referred to as the Lasso or Basis Pursuit DeNoising leads to solutions that are sparse and therefore achieves model selection. We propose in this paper an algorithm to solve the Lasso with online observations. We introduce an optimization problem that allows us to compute an homotopy from the current solution to the solution after observing a new data point. We compare our method to Lars and present an application to compressed sensing with sequential observations. Our approach can also be easily extended to compute an homotopy from the current solution to the solution after removing a data point, which leads to an efficient algorithm for leave-one-out cross-validation.


text:
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# Pre-requisites

You should know what a Bayes Factor is and what a p value is.

# Overview

Sellke et al (Thomas Sellke 2001) study the following question (paraphrased and shortened here).

Consider the situation in which experimental drugs D1; D2; D3; etc are to be tested. Each test will be thought of as completely independent; we simply have a series of tests so that we can explore the frequentist properties of p values. In each test, the following hypotheses are to be tested: $H_0 : D_i \text{ has negligible effect}$ versus $H_1 : D_i \text{ has a non-negligible effect}.$

Suppose that one of these tests results in a p value $$\approx p$$. The question we consider is: How strong is the evidence that the drug in question has a non-negligible effect?

The answer to this question has to depend on the distribution of effects under $$H_1$$. However, Sellke et al derive a bound for the Bayes Factor. Specifically they show that, provided $$p<1/e$$, the BF in favor of $$H_1$$ is not larger than $1/B(p) = -[e p \log(p)]^{-1}.$ (Note: the inverse comes from the fact that here we consider the BF in favor of $$H_1$$, whereas Sellke et al consider the BF in favor of H_0).

Here we illustrate this result using Bayes Theorem to do calculations under a simple scenario.

# Details

Let $$\theta_i$$ denote the effect of drug $$D_i$$. We will translate the null $$H_0$$ above to mean $$\theta_i=0$$. We will also make an assumption that the effects of the non-null drugs are normally distributed $$N(0,\sigma^2)$$, where the value of $$\sigma$$ determines how different the typical effect is from 0.

Thus we have: $H_{0i}: \theta_i = 0$ $H_{1i}: \theta_i \sim N(0,\sigma^2)$.

In addition we will assume that we have data (e.g. the results of a drug trial) that give us imperfect information about $$\theta$$. Specifically we assume $$X_i | \theta_i \sim N(\theta_i,1)$$. This implies that: $X_i | H_{0i} \sim N(0,1)$ $X_i | H_{1i} \sim N(0,1+\sigma^2)$

Consequently the Bayes Factor (BF) comparing $$H_1$$ vs $$H_0$$ can be computed as follows:

BF= function(x,s){dnorm(x,0,sqrt(s^2+1))/dnorm(x,0,1)}

Of course the BF depends both on the data $$x$$ and the choice of $$\sigma$$ (here s in the code).

We can plot this BF for $$x=1.96$$ (which is the value for which $$p=0.05$$):

s = seq(0,10,length=100)
plot(s,BF(1.96,s),xlab="sigma",ylab="BF at p=0.05",type="l",ylim=c(0,4))
BFbound=function(p){1/(-exp(1)*p*log(p))}
abline(h=BFbound(0.05),col=2)

Version Author Date
c3b365a John Blischak 2017-01-02

Here the horizontal line shows the bound on the Bayes Factor computed by Sellke et al.

And here is the same plot for $$x=2.58$$ ($$p=0.01$$):

plot(s,BF(2.58,s),xlab="sigma",ylab="BF at p=0.01",type="l",ylim=c(0,10))
abline(h=BFbound(0.01),col=2)

Version Author Date
c3b365a John Blischak 2017-01-02

Note some key features:

• In both cases the BF is below the bound given by Sellke et al, as expected.
• As $$\sigma$$ increases from 0 the BF is initially 1, rises to a maximum, and then gradually decays. This behavior, which occurs for any $$x$$, perhaps helps provide intuition into why it is even possible to derive an upper bound.
• In the limit as $$\sigma \rightarrow \infty$$ it is easy to show that (for any $$x$$) the BF $$\rightarrow 0$$. This is an example of “Bartlett’s paradox”, and illustrates why you should not just use a “very flat” prior for $$\theta$$ under $$H_1$$: the Bayes Factor will depend on how flat you mean by “very flat”, and in the limit will always favor $$H_0$$.

## References

Thomas Sellke, James O. Berger, M. J. Bayarri. 2001. “Calibration of $$p$$ Values for Testing Precise Null Hypothesis.” The American Statistician 55 (1).

sessionInfo()
R version 3.5.2 (2018-12-20)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.1

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
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attached base packages:
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loaded via a namespace (and not attached):
[1] workflowr_1.2.0 Rcpp_1.0.0      digest_0.6.18   rprojroot_1.3-2
[5] backports_1.1.3 git2r_0.24.0    magrittr_1.5    evaluate_0.12
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[17] yaml_2.2.0      compiler_3.5.2  htmltools_0.3.6 knitr_1.21

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text:
# Differential equations - exact differential equations

I am self studying ode from boyce diprima book and while doing exercises on chapter 2.6.

I couldnt undestand how question 17 is derived I checked the solution manual and I found the same answer which I have already found by myself but what the the question asks is different.

The integral on solution manual is not definite and the question asks me to show how a definite integral is derived and I am not sure exactly how that definite integral is derived

M(s,y0) : this expression on question makes me suspicious

I uploaded the question to scribd because it is too long to write together with the theorem they refer to.

http://www.scribd.com/fullscreen/77494675?access_key=key-1fqwhkmcex23ufgyz0sh

can anyone show me how the definite integral, question asks, can be derived ? I also also put the solution manuals answer to pdf.

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I'm pretty shaky with DEs myself, and the notation always confuses the heck out of me, but nobody else has answered this yet, so I thought I'd take a stab at it. Hopefully my answer will at least give you something to work with (or encourage better answers):

The $\int_{x_0}^{x} M(s, y_0) \, ds$ is probably just changing the name of the first variable inside $M$'s parenthesis. It is used to get get the result in the correct letter, $x$, after integrating.

Note also that the book probably uses the notation that $x$ is a variable, and $x_0$ is a fixed unknown (a.k.a. constant).

Now let me change notations here for a minute. Have you seen where sometimes they represent a function with a lowercase letter and the function's integral with an uppercase letter (e.g., $M'(x, y) = m(x, y)$)? I'm going to use that notation and try to find $\int_{x_0}^{x} m(s, y_0) \, ds$.

If the integral of $m(s, y_0)$ is $M(s, y_0)$ (that is, $\int m(s, y_0) \, ds = M(s, y_0)$), then

$$\int_{x_0}^{x} m(s, y_0) \, ds = \left. M(s, y_0) \right|_{x_0}^{x} = M(x, y_0) - M(x_0, y_0)$$

Since the last term, $M(x_0, y_0)$, is a number (aka constant), taking the derivative of this with respect to (wrt) $x$ yields $m(x, y_0)$.

Next, in the textbook's answer they used the definite integral of $N$ to get the value for $\psi$. That is if the partial derivative of $\psi$ wrt $y$ equals $N$ (e.g. $\frac{\partial \psi}{\partial y} = N$), then

$$\psi(x, y) = \int_{y_0}^{y} \frac{\partial \psi}{\partial y} \, dy = \int_{y_0}^{y} N(x, t) \, dt = \int N(x, y) \, dy - \int N(x, y_0) \, dy$$

The first term on the right hand side (rhs), $\int N(x, y) \, dy$, is written that way because we don't know the integral of $N$ (it's the capital of capital $n$, or a really big $N$ :/ ). It's not an indefinite integral in the usual sense, it's just a notation to indicate "the function that you'd get when you integrate $N(x, t)$ with respect to $t$ and then evaluate that function at $t = y$".

The second term, $\int N(x, y_0) \, dy$, is obtained the same way and it means "the function you get when you integrate $N(x, t)$ wrt $t$ and then evaluate the function at $t = y_0$". It is a function of $x$, and I think they just renamed it $h$, so that $h(x) = \int N(x, y_0)$.

Next, they took the partial derivative of $\psi$ wrt $x$ and set it equal to $M$ and solved for $h'(x)$. I don't understand why they reversed the integral and partial derivative signs, but I guess that's allowed, and I don't understand the rest of the book's answer from this point on. But I hope I at least clarified why the answer looked like it was using indefinite integrals. Maybe someone can now chip in with an explanation of the rest of the text's answer.

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Welcome to math.SE. You can typeset mathematics using MathJax by enclosing LaTeX code in $or$\$. –  Zhen Lin Jan 8 '12 at 3:06