text:
# Numbered subsection within unnumbered section

How to add numbered subsections with unnumbered subsection in article template? I want the following format

Unnumbered section
A First numbered subsection
B Second numbered subsection
so on.....

• this \section*{test} \subsection{test A} \subsection{test B} ? – touhami Aug 26 '15 at 7:08
• Maybe Section numbering without numbers can help you. – moewe Aug 26 '15 at 7:09
• What is the number format of numbered sections, numbered subsections and numbered subsections within unnumbered sections? – Heiko Oberdiek Aug 26 '15 at 7:31
• @touhami: that is not working. It will produce test 0.1 test A 0.2 test B But I want test A test#1 B test#2 A and B are the subsection numbering. – Soumyajit Roy Aug 26 '15 at 9:39
• @HeikoOberdiek: A section, A.1 subsection#1, A.2 subsection#2..... But I want---section, A subsection#1, B subsection#2,....... The template is article. – Soumyajit Roy Aug 26 '15 at 9:44

The star form of \section generates a section without number. Since the subsection number usually is build with the section number, \thesubsection needs to be redefined:

\documentclass{article}
\renewcommand*{\thesubsection}{\Alph{subsection}}

\begin{document}
\section*{Unnumbered section}
\subsection{First numbered subsection}
\subsection{Second numbered subsection}
\end{document}

• Thanks a lot. It has worked. However, if I want a specific unnumbered section to be having subsection numbering A, B,C, like that but not for the other numbered section, how should I modify the code? 1. Section 1 1.1 Subsection 1 1.2 Subsection 2........ 2. Section 2 2.1 Subsection 1 2.2 Subsection 2.... .................................... .................................... Unnumbered Section A Subsection 1 B Subsection 2.... Template: Article – Soumyajit Roy Aug 28 '15 at 13:13
• The above issue has been solved... – Soumyajit Roy Aug 28 '15 at 13:19


text:
# Question #363c9

Jan 29, 2016

$Y \propto X$ with a proportionality constant of 2 means $Y = 2 X$

#### Explanation:

We can write the statement "Y is proportional to the value of X ", which means that if X is double, Y is also doubled etc, as follows:

$Y \propto X$
Y is proportional to X

To convert a proportionality relationship to an equation, we need some constant or coefficient, which we would write as:

$Y = k X$, where $k$ is the proportionality constant.

In this case, $k$ is 2, so

$Y = 2 X$

format: plain


text:
# Smog

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]
Gallery [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.
(PD) Photo: Dr. Edwin P. Ewing, Jr. / Centers for Disease Control and Prevention (CDC)
Smog over New York skyline in 1988

Smog is the hazy, unhealthy polluted air that accumulates over cities and other regions under certain conditions.

Modern smog is derived primarily from precursor chemical pollutants, emitted to the atmosphere from vehicular internal combustion engines and industrial plants, that react in the atmosphere with sunlight to produce secondary chemical pollutants that also combine with the precursor emissions to form the components of what is called photochemical smog.

## Origin of the term "smog"

The word smog is a blend of the words smoke and fog. Coinage of the term is generally attributed to Dr. Henry Antoine des Voeux in his 1905 paper, "Fog and Smoke" for a meeting of the Public Health Congress in London. The July 26, 1905 edition of the Daily Graphic, a London newspaper, quoted des Voeux: "... he said it required no science to see that there was something produced in great cities which was not found in the country, and that was smoky fog, or what was known as 'smog'."[1] The next day, the Globe newspaper remarked that "Dr. des Voeux did a public service in coining a new word for the London fog.">[1]

## Photochemical smog

(CC) Diagram: Milton Beychok
Atmospheric chemistry involved in smog formation[2][3][4][5]

Originally, Dr. des Voeux's term smog referred to a mixture of smoke, sulfur dioxide (SO2) and fog that was once prevalent in London when coal with a high sulfur content was widely used throughout the city as heating fuel. The dark, sulfurous London smog caused reduced visibility, respiratory problems and had a noticeable deleterious affect on human health. The Great Smog of 1952 darkened the streets of London and killed approximately 4,000 people in a 4-day period (another 8,000 died from its effects in the following months)[6] The type of smog experienced in London many decades ago is no longer encountered since other fuels have largely replaced the wide-spread use of high-sulfur coal for heating.

Photochemical smog is chemically quite different than the old London-type smog and has a long history. In 1542, when exploring what is now Southern California, Juan Rodriguez Cabrillo named San Pedro Bay the "Bay of Smokes" because of the thick haze that covered the area. Complaints of eye irritation from the polluted air in Los Angeles date back to the late 1860s. In the 1940s, photochemical smog first became apparent in Los Angeles and other large cities on sunny days, although none of those cities had any significant use of coal for heating fuel or for industrial activities.[2][3][4][7]

Photochemical smog is a daytime phenomenon that occurs mostly during warm, sunny summertime days and is characterized by a brown haze that reduces visibility and contains oxidants, such as ozone (O3), that cause respiratory problems, eye irritation and damage to plants. It is a mixture of variable amounts of ozone, reactive hydrocarbons, nitrogen oxide (NO), nitrogen dioxide (NO2), aldehydes, peroxyacetyl nitrate (PAN), particulate matter and other components. By contrast, the London-type smog (also referred to as "classic smog") occurs mostly during cold winter days and consists primarily of a mixture of sulfur dioxide and particulate matter (i.e., "soot") usually derived from burning coal.[8]

### Photochemical smog precursor pollutants

The chemistry involved in the formation of smog is highly complex and involves many different reactions (see the adjacent diagram). The three major ingredients required to form photochemical smog are solar energy (i.e., sunlight), reactive hydrocarbons and nitrogen oxide, the latter two being referred to as the precursor pollutants. As shown in the diagram, the two precursors enter the atmosphere as gases emitted for the most part from vehicular internal combustion engines fuelled by gasoline and diesel oil. To a much lesser extent, nitrogen oxide is also emitted from industrial combustion sources, and reactive hydrocarbons are emitted by evaporation from the handling and storage of volatile hydrocarbons[9] such as gasoline, solvents, some pesticides and some paints. Biogenic sources such as pine trees and certain other trees are also a source, though of relatively minor importance, of reactive hydrocarbon emissions such as isoprene and α-pinene.

### Simplified chemistry of photochemical smog formation

In the simplified schematic overview depicted in the adjacent diagram, the major chemical reactions that contribute to the formation of photochemical smog include the following:[2][3][4][5][10]

• The gaseous precursor nitrogen oxide, emitted primarily from vehicular internal combustion engines, is oxidized to produce gaseous nitrogen dioxide.
nitrogen oxide + oxygen $\rightarrow$ nitrogen dioxide
2NO(g) + O2(g) $\mathrm{\rightarrow}$ 2NO2(g)
• The gaseous nitrogen dioxide is broken down by solar energy from sunlight to produce gaseous nitrogen oxide and atomic oxygen (O, an oxygen radical). This process is referred to as photolysis and hence the term "photochemical smog".
nitrogen dioxide +solar energy $\rightarrow$ nitrogen oxide + atomic oxygen
NO2(g) + solar energy $\rightarrow$ NO (g) + O
• The atomic oxygen reacts with gaseous atmospheric oxygen (O2)[11] to form gaseous ozone (O3).
atomic oxygen + atmospheric oxygen $\rightarrow$ ozone
O + O2(g) $\rightarrow$ O3(g)
• The gaseous ozone also oxidizes gaseous nitrogen oxide to form gaseous nitrogen dioxide and gaseous oxygen.
ozone + nitrogen oxide $\rightarrow$ nitrogen dioxide + oxygen
O3(g) + NO(g) $\rightarrow$ NO2(g) + O2(g)
• The gaseous precursor reactive hydrocarbons (RH), emitted primarily from vehicular internal combustion engines, reacts with atomic oxygen, atmospheric oxygen and ozone to produce various highly reactive hydrocarbon free radicals (RO2).
• The hydrocarbon free radicals then react with other species such as nitrogen dioxide to form peroxyacetyl nitrate (PAN),[12] aldehydes and other smog components.
Nitrogen dioxide + oxygen + hydrocarbon free radicals $\rightarrow$ peroxyacetyl nitrate
2NO2(g) + O2(g) + 2RO2 $\rightarrow$ 2CH3CO–OO–NO2(g)

The various products produced or formed from the precursor pollutants are referred to as secondary pollutants. All of the above chemical reactions (as well as others) occur more or less simultaneously during sunny, summertime days in most large cities with a great number of automobiles and other vehicles.

## Health effects

Smog in São Paulo, Brazil in 2006.
United States' AQI[13]
Air Quality
Index
(AQI)
Air Quality
Category
Color
Code
0 – 50 Good
51 – 100 Moderate
101 – 150 Unhealthy for
Sensitive Groups

151 – 200 Unhealthy
201 – 300 Very Unhealthy
301 – 500 Hazardous

Photochemical smog constitutes a serious health problem in most large cities. It is especially harmful for senior citizens, children, and people with heart and lung conditions such as emphysema, bronchitis, and asthma. It can inflame breathing passages, decrease the lungs' working capacity, cause shortness of breath, pain when inhaling deeply, wheezing, and coughing. It can cause eye and nose irritation and it dries out the protective membranes of the nose and throat. In general, it interferes with the body's ability to fight infection, thereby increasing susceptibility to illness.[14][15]

The adjacent table explains the Air Quality Index (AQI) ranges used by the U.S. Environmental Protection Agency (U.S. EPA) and their corresponding health effect categories and color codes. The U.S. EPA's AQI is also known as the Pollution Standards Index (PSI). A national map of the United States containing daily AQI forecasts across the nation, using that same color-coding, is published online jointly by the U.S. EPA and the National Oceanic and Atmospheric Administration (NOAA).[16]

If multiple pollutants are measured at a monitoring site, then the largest or "dominant" AQI value is reported for the location.

The U.S. EPA has developed conversion calculators, available online,[17][18] for the conversion of AQI values to concentration values and for the reverse conversion of concentrations to AQI values.

The South Coast Air Quality Management District (SCAQMD) of Southern California has published an excellent, comprehensive discussion of the undesirable health effects caused by air pollution. It also includes an extensive listing and discussion of the many medical health effect studies of pollution performed during the past decades starting around 1950.[15]

(CC) Photo: Brian Jeffrey Beggerly
Smog over Beijing in 2005.

## Areas affected

Smog can form in almost any location where the level of air pollutants is high, and ii is more likely to occur during periods of warm, sunny weather when the upper air is warm enough to inhibit vertical circulation. It often persists for extended time periods over densely populated cities, such as Athens, Beijing, Berlin, Cairo, Hong Kong, Kuala Lumpur, London, Los Angeles, Milan, Moscow, New York, Paris, Rome, São Paulo, Seoul and Shanghai. Smog is especially prevalent in geologic basins encircled by hills or mountains.

### England

In 1306, concerns over air pollution were sufficient for Edward I to briefly ban coal fires in London.[19] In 1661, John Evelyn, the noted English diarist of his day, published Fumifugium in which he suggested the widespread planting of fragrant trees, shrubs and flowers around London so as to counteract the foul effects of the smoke from coal-burning.[20] Two years later, in 1663, a satirical poem known as the Ballad of Gresham College commemorates Evelyn's book by this verse:[21]

"He shewes that 't is the seacoale smoake,
That allways London doth Inviron,
Which doth our Lungs and Spiritts choake,
Our hanging spoyle, and rust our Iron."
(CC) Photo: Jim Budd
Smog over London in 2007

In the 1800s, more than a million London residents were burning coal, and winter "fogs" became more than a nuisance. An 1873 coal-smoke saturated fog, thicker and more persistent than natural fog, hovered over the city for days. As now known from subsequent epidemiological findings, the fog caused 268 deaths from bronchitis. Another fog in 1879 lasted from November to March, four long months darkened skies and gloom. Severe episodes of smog continued even into the 20th centuries and were nicknamed "pea-soupers".[22]

As discussed earlier above, the Great Smog of 1952 killed approximately 4,000 people in the short time of 4 days and a further 8,000 died from its effects in the following months. That finally prompted real legislative reform. In 1956, the Clean Air Act 1956, sponsored by the Ministry of Housing and Local Government in England and the Department of Health in Scotland, was enacted by the Parliament of the United Kingdom and introduced smokeless zones in London. It was in effect until 1964 and consequently reduced the sulfur dioxide levels in London's air to the point that made the intense and persistent London smog a thing of the past. It was after this that the great clean-up of London began and buildings recovered their original clean stone façades which, during two centuries, had gradually blackened.

However, as clearly shown in the adjacent photograph, photochemical smog still does occur in modern London.

### European Union

(CC) Photo: Andrea Squatrito
Smog in Milan in 2006.

The European Union (EU) is composed of 27 member states[23] and was established by the Treaty of Maastricht in 1993. Its executive functions are performed by the European Commission and to a limited extent by the European Council.

The large cities within the EU, such as Athens, Berlin, Milan, Paris, Rome and others, all experience episodes of photochemical smog in varying degrees of severity and especially so in the summer months. For example, see the adjacent photograph of smog in Milan.

The EU uses the ground-level atmospheric ozone concentration as an indicator of air quality and has adopted these limiting ozone values:[24][25]

Target value (TV): 120 µg/m3 for 1 hour averaging time for 8-hour average, daily maximum. Not to be exceeded more than 25 days per year. Defined as the target value for the protection of human health.

Information threshold (IT): 180 µg/m3 for 1 hour averaging time. Defined as the level beyond which there is a risk to human health from brief exposure for particularly sensitive sections of the population. Any exceedance of this threshold should be reported by the member state in which it occurs to the European Commission.

Alert threshold (IT): 240 µg/m3 for 1 hour averaging time. Defined as the level beyond which there is a risk to human health from brief exposure for the general population. If exceeded in a member state, national authorities of the member state are required to warn the public and give advice.

According to a recent report by the European Environmental Agency (EEA),[26] the overall area of the EU that experiences exceedances of the ozone target value has declined by 60% during the period from 2006 − 2009. The executive summary of that report states: In contrast to previous summers, in 2009 there were no pan-European multi-day episodes. Summer 2009 was characterized by ozone episodes of two to five days followed by spells with few exceedances. However, the executive summary also states that a significant area of the EU still experienced exceedance of the ozone target value.

### Russia

(CC) Photo: Yulia Smirnova
Smog in Moscow in 2010

In early August 2010, the city of Moscow was covered by a heavy, dark smog caused by extensive wildfires across much of western Russia., turning the city's buildings into blurs, grounding air flights, and forcing the city's residents and tourists to wear surgical masks.[27] At the beginning of August, forest fires in Russia exceeded 100, 000 hectares which by August 21 had decreased to 7,000 hectares according to the Emergencies Ministry.[28]

The head of the Moscow's health department said that the abnormal heat and smog coming from fires had seriously exacerbated the environmental conditions in Moscow and “The death rate has doubled in the city. The regular daily death rate in Moscow is 360 – 380. We had about 700 deaths today." [28]

Some airborne pollutants were four times higher than normal, the worst concentrations seen to date according to city health officials. The concentrations appeared likely to get even worse as the state news agency, ITAR - Tass, reported that the smoke was thickening.[27]

### Ruhr Valley Incident of 1985

The Ruhr district, located in the Rhine river valley of western Germany, is an urban area in the state of North Rhine-Westphalia]. With a population of about 7 million (2008) and an area of 4,435 square kilometers (1,713 square miles), it is the largest urban area in Germany. Historically, it has been a heavily industrialized area that experienced decades of polluted air containing smoke and high amounts of sulfur dioxide — the key ingredients of London-type smog.

In January of 1985, the Ruhr district experienced an extended episode of London-type smog that lasted 5 days with the 24-hour averages of sulfur dioxide in the air as high as 0.8 mg/m3 and particulate matter as high as 0.6 mg/m3, both primarily derived from the extensive burning of coal by the area's steel mills and other industries. In the 5 days of the episode, the daily death rate increased by 8% and the hospital admissions for respiratory and cardiovascular problems increased by 15%.[29][30]

During the smog episode, a smog alarm was called, personal automobile driving was banned and many schools were closed.[31]

### Iran

In December 2005, schools and public offices had to close in Tehran, Iran, and 1600 people were taken to hospital, in a severe smog blamed largely on automobile exhausts.[32]

### Mexico

(CC) Photo: Lauren Brown
Mexico City smog in 2006

Mexico City is located in the tropical latitudes on a high plain at an altitude of 2,200 meters, and is surrounded on three sides by mountains that reach elevations of 5,000 meters. The major sources of air pollutants within Mexico City's urban area include exhaust from 3.5 million vehicles, thousands of industries, and mineral dust. The ancient lakebed valley in which the city is situated became a major source of dust when it was drained in the 16th century.[33] The area is known as the Valley of Mexico, sometimes called the Bowl of Mexico.

Due to its location in a highland bowl, cold air sinks down onto the urban area of Mexico City, trapping industrial and vehicle pollution underneath and, as a result, the city has one of the world's worst air pollution problems. Within one generation, the city has changed from being known for some of the cleanest air of the world into one with some of the worst pollution, with pollutants like nitrogen dioxide measured at twice or thrice the level of international standards.[34] Since solar radiation does not vary significantly with season at tropical latitudes, photochemical smog is present over the city much of the year.[33]

### Southeast Asia

(CC) Photo: Thomas Galvez
Haze and smoke in Kuala Lumpur, Malaysia in 2006

Southeast Asia consists of a mainland section and a maritime section. The mainland section includes Mayanmar,[35] Cambodia, Laos, Thailand, Vietnam and Peninsular Malaysia and the maritime section includes Brunei, East Malaysia, East Timor, Indonesia, Philippines, and Singapore.[36] The large cities in Southeast Asia have been plagued by smoky haze and photochemical smog ever since the late 1990s. The primary cause of the smoky haze have been the extensive forest fires occurring in Indonesia.[37] Farmers and plantation owners are said to be responsible for the fires, which they use to clear tracts of land for further plantings. Other contributing factor are that El Niño conditions have created some very dry seasons with very little rainfall and the peat swamps that underlie the Indonesian forests and add fuel to the fires.

One of the recent major occurrences of smoky haze in Malaysia, Singapore and other parts of Southeast Asia occurred in October 2006 and was caused by smoke from forest fires in Indonesia being blown across the Straits of Malacca by south-westerly winds.

Since December 1997, member countries of the Association of Southeast Asia Nations (ASEAN) have been undertaking joint efforts in monitoring, preventing and mitigating transboundary haze pollution resulting from land and forest fires, guided by the Regional Haze Action Plan (RHAP) that was endorsed by the ASEAN Environment Ministers. In addition, the ASEAN Agreement on Transboundary Haze Pollution (or ASEAN Haze Agreement) was adopted in June2002 and comprehensively addresses all aspects of fire and haze including prevention, emphasizing the underlying causes, monitoring, and mitigation.[38]

In addition to the smoky haze caused by the forest fires, they have resulted in large amounts of the photochemical smog precursor, nitrogen oxide, entering the atmosphere. That, coupled with the great numbers of automobiles and other vehicles used in the large cities of Southeast Asia, has resulted in those cities being subjected to photochemical smog as well as hazy smoke.

### United States

The U.S. EPA has designated over 300 U.S. counties in the United States to be "non-attainment areas" for one or more air pollutants, meaning that they have failed to attain and maintain the National Ambient Air Quality Standards as required by the Clean Air Act enacted by the U.S. Congress . The non-attainment areas are largely clustered around large metropolitan areas, with the largest contiguous non-attainment zones being in California and the Northeast, as can be seen in the map below.[39] Various U.S. and Canadian government agencies collaborate to produce real-time air quality maps and forecasts.[40]

(PD) Image: U.S. Environmental Protection Agency

#### Los Angeles and the San Joaquin Valley

(PD) Photo: Jon Sullivan, www.pdphoto.org
Los Angeles smog in 2003.

Located in low basins surrounded by mountains, Los Angeles and the San Joaquin Valley are known for their photochemical smog. The millions of vehicles in these basins, plus the added effects of the port complex in Los Angeles, result in the accumulation of photochemical smog precursors and subsequently to the formation of photochemical smog. Strict regulations by the California Air Resources Board and other California government agencies overseeing this problem have reduced the number of smog alerts from several hundred annually to just a few. However, these geographically predisposed smog areas still have air pollution levels that are a pressing issue for the more than 25 million people who live there.

(CC) Photo: Rick Kimpel
Denver smog in 2006.

#### Denver

Denver, the "Mile High City" suffers from frequent smogs that are made worse by its location in the Rocky Mountains.

Because of Denver's high altitude, it has frequent temperature inversions when warm polluted air is trapped underneath a layer of cold air and cannot rise and disperse [41] more widely. As a result, photochemical smog may envelope the city for many days at a time (see adjacent photo).[42]

In the past, Denver regularly failed to meet the federal National Ambient Air Quality Standards for air quality on dozens of days each year and the city has worked diligently to solve its air pollution problem. In 2007, the U.S. EPA noted that Denver was ahead of its schedule for reducing smog. Yet, Denver still exceeds the federal standards for several days a year.[42]

#### Major incidents in the United States

• 1948, October, Donora, Pennsylvania: 20 killed by smog, 600 hospitalized, thousands more stricken.[43]
• 1953, November, New York: Smog kills between 170 and 260 people.[43]
• 1954, October, Los Angeles: Heavy smog shuts down schools and industry for most of the month.[43]
• 1963, New York: Smog caused 200 deaths[44]
• 1966, New York: Smog caused 168 deaths[44]

## References

1. Beatrice Trum Hunter (2004). Air and Your Health. Basic Health Publications, page 82. ISBN 1-59120-057-1.
2. Stanley E. Manahan (2006). Environmental Science and Technology: A Sustainable Approach to Green Science and Technology, 2nd Edition. CRC Press. ISBN 0-8493-9512-7.
3. G.S. Sodhi (2005). Fundamental Concepts of Environmental Chemistry, 2nd Edition. Alpha Science International. ISBN 1-84265-218-8.
4. The Chemistry of Photochemical Smog From the website of AUS-e-TUTE, an Australian interactive learning resource.
5. Photochemical Smog From the website of Middle Tennessee State University (MTSU)
6. Michelle L. Bell, Devra L. Davis and Tony Fletcher (January 2004). "A Retrospective Assessment of Mortality from the London Smog Episode of 1959: The Role of Influenza and Pollution". Environmental Health Perspectives 112 (1): 6–8.
7. Smog From the website of Pollution Issues
8. R.B. Singh (2006). Sustainable Urban Development. Concept Publishing, pp. 59-60. ISBN 81-8069-295-7.
9. Note: Also referred to as volatile organic compounds (VOC)
10. Photochemical smog - what it means for us The Environmental Protection Authority of South Australia
11. Note: May also be referred to as molecular oxygen
12. Formula of peroxyacetyl nitrate: C2H3NO5
13. Air Quality Index (AQI) - A Guide to Air Quality and Your Health From the AIRNow web site jointly maintained by the U.S. EPA, National Oceanic and Atmospheric Administration, National Park Service, tribal, state, and local agencies
14. Smog - Who does it hurt? From the AIRNow web site jointly maintained by the U.S. EPA, National Oceanic and Atmospheric Administration, National Park Service, tribal, state, and local agencies.
15. Smog and Health From the website of the South Coast Air Quality Management District (SCAQMD).
16. Today's National Air Quality Forecast
17. AQI Calculator: AQI to Concentration
18. AQI Calculator: Concentration to AQI
19. Barbara Freese (2003). Coal: A Human History, Ist Edition. Perseus Publishing, p.1. ISBN 0-7382-0400-5.
20. John Evelyn (1661). Fumifugium; or the Inconvenience of the Air and Smoke of London Dissipated, Ist Edition. Gabriel Bedel and Thomas Collins, London.  The full text is available online at The Internet Archive
21. Fog and Filthy Air From the website of the Meteo Group in the United Kingdom]].
22. London's Historic "Pea-Soupers"] by David Urbinato.1994, (from the website of the U.S. EPA).
23. Note: The EU member states are: Austria, Belgium, Bulgaria, [[Cyprus, the Czech Republic, Denmark, Estonia, Finland, France, Germany, [[Greece], Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, and the United Kingdom
24. Air pollution by ozone across Europe during summer 2009 Table 1.1 of European Environmental Agency (EEA) Technical report No 2/2009, Overview of exceedances of European Commission(EC) ozone threshold values for April–September 2009
25. Note: Directive 2002/3/EC of 2002. Although Directive 2008/50/EC, which came into force in June 2010, repealed the 2002 Directive, it did not change the Target, Information, and Alert thresholds defined in the 2002 Directive.
26. Air pollution by ozone across Europe during summer 2009 Table 2.5 of European Environmental Agency (EEA) Technical report No 2/2009, Overview of exceedances of European Commission(EC) ozone threshold values for April–September 2009
27. Fires lay ghostly shroud of smoke on Moscow, Jim Heintz, Associated Press, August 6, 2010
28. News report of ITAR-Tass The state news agency of Russia
29. What can we learn today from the Central European smog episode of 1985 (and earlier episodes)? From the website of PubMed.gov.
30. Health Effects During a Smog Episode in West Germany in 1985 E. Wichmann et al, 1988
31. 1985, Environment, Phase III smog alarm, produced by the Landeszentrale für politische Bildung — State Center for Political Education, North Rhine – Westphalia (NRW). Click on 1985 in the right hand column.
32. Hundreds treated over Tehran smog, BBC News, December 10, 2005.
33. A Hazy Day in Mexico City From the website of NASA.
34. Air pollution in Mexico City] by Maricela Yip and Pierre Madl, December 14, 2000, University of Salzburg, Austria
35. Also known as Burma. However, Myanmar is the country's name as recognized by the United Nations.
36. World Macro Regions and Components] From the United Nations website.
37. Mark E. Harrison, Susan E. Page and Suwido H. Limin (August 2009). "The global impact of Indonesian forest fires". Biologist 56 (3): 156-163.
38. Combating Haze in ASEAN: Frequently Asked Questions From the website of the ASEAN.
39. EPA Green Book, Counties Designated Non-attainment and Maintenance
40. Airnow.gov
41. Milton R. Beychok (2005). Fundamentals Of Stack Gas Dispersion, 4th Edition. Author-published. ISBN 0-9644588-0-2.
42. Stanley D. Bruun, Maureen Hays-Mitchell, Donald J. Ziegler and Ellen R. White (2008). Cities of the World: World Regional Urban Development, 4th Edition. Rowman & Littlefield Publishers, p. 99. ISBN 0-7425-5597-6.
43. Environmental History Timeline: 1940-1960 From the website of Radford University in Virginia.
44. Arnold W. Reitze Jr. (2001). Air Pollution Control Law: Compliance and Enforcement, 1st Edition, pp. 14-15. ISBN 1-58576-027-7.


text:
Mathematical Biosciences and Engineering

2018, Issue 1: 125-140. doi: 10.3934/mbe.2018005

Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis

• Received: 30 October 2016 Accepted: 20 January 2017 Published: 01 February 2018
• MSC : Primary: 92D30; Secondary: 34C99

• Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic $\mathcal{R}_0$, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ ≥ 0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most). The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction. The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

Citation: Andrea Pugliese, Abba B. Gumel, Fabio A. Milner, Jorge X. Velasco-Hernandez. Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis[J]. Mathematical Biosciences and Engineering, 2018, 15(1): 125-140. doi: 10.3934/mbe.2018005

Related Papers:

• Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic $\mathcal{R}_0$, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ ≥ 0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most). The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction. The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

 [1] [ C. L. Althaus,N. Low, How Relevant Is Sexual Transmission of Zika Virus?, PLOS Med., 13 (2016): 1-3. [2] [ M.-C. Boily,R. F. Baggaley,L. Wang,B. Masse,R. G. White,R. J. Hayes,M. Alary, Heterosexual risk of HIV-1 infection per sexual act: Systematic review and meta-analysis of observational studies, Lancet Infect. Dis., 9 (2009): 118-129. [3] [ R. B. Brooks,M. P. Carlos,R. A. Myers,M. Grace White,T. Bobo-Lenoci,D. Aplan,D. Blythe,K. A. Feldman, Likely Sexual Transmission of Zika Virus from a Man with No Symptoms of Infection --Maryland, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016): 915-916. [4] [ CDC, Hepatitis A Questions and Answers for the Public, URL https://www.cdc.gov/hepatitis/hav/afaq.htm. [5] [ F. C. Coelho,B. Durovni,V. Saraceni,C. Lemos,C. T. Codeco,S. Camargo,L. M. de Carvalho,L. Bastos,D. Arduini,D. A. M. Villela,M. Armstrong, Higher incidence of Zika in adult women than adult men in Rio de Janeiro suggests a significant contribution of sexual transmission from men to women, Int. J. Infect. Dis., 51 (2016): 128-132. [6] [ W. D. Davidson,S Slavinski,K. Komoto,J. Rakeman, Suspected Female-to-Male Sexual Transmission of Zika Virus-New York City, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016): 716-717. [7] [ O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, New York: John Wiley and Sons, 2000. [8] [ M. R. Duffy,T.-H. Chen,W. T. Hancock,A. M. Powers,J. L. Kool,R. S. Lanciotti,M. Pretrick,M. Marfel,S. Holzbauer,C. Dubray,L. Guillaumot,A. Griggs,M. Bel,A. J. Lambert,J. Laven,O. Kosoy,A. Panella,B. J. Biggerstaff,M. Fischer,E. B. Hayes, Zika virus outbreak on Yap Island, Federated States of Micronesia., N. Engl. J. Med., 360 (2009): 2536-2543. [9] [ N. M. Ferguson, Z. M. Cucunubá, I. Dorigatti, G. L. Nedjati-Gilani, C. A. Donnelly, M.- G. Basáñez, P. Nouvellet and J. Lessler, Countering the Zika epidemic in Latin America, Science (80-. )., 353 (2016), 353-354, URL http://www.sciencemag.org/cgi/content/full/science.aag0219/DC1. [10] [ K. Fonseca, B. Meatherall, D. Zarra, M. Drebot, J. MacDonald, K. Pabbaraju, S. Wong, P.Webster, R. Lindsay and R. Tellier, First case of zika virus infection in a returning canadian traveler, Am. J. Trop. Med. Hyg., 91 (2014), 1035-1038, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4228871/. [11] [ P. J. Fos, null, Epidemiology Foundations: The Science of Public Health, Jossey-Bass, San Francisco, 2011. [12] [ E. B. Hayes, Zika Virus Outside Africa, Emerg. Infect. Dis., 15 (2009), 1347-1350, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/. [13] [ W. O. Kermack,A. G. McKendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. London A, 115 (1927): 700-721. [14] [ A. J. Kucharski,S. Funk,R. M. Eggo,H.-P. Mallet,W. J. Edmunds,E. J. Nilles, Transmission dynamics of zika virus in island populations: A modelling analysis of the 2013-14 french polynesia outbreak, PLoS Negl. Trop. Dis, 10 (2016): e0004726. [15] [ W. D. Z. Lopes, J. D. Rodriguez, F. A. Souza, T. R. dos Santos, R. S. dos Santos, W. M. Rosanese, W. R. Z. Lopes, C. A. Sakamoto and A. J. da Costa, Sexual transmission of Toxoplasma gondii in sheep, Vet. Parasitol., 195 (2013), 47-56, URL http://linkinghub.elsevier.com/retrieve/pii/S0304401713000083. [16] [ G. MacDonald, The analysis of equilibrium in malaria, Trop Dis Bull, 49 (1952): 813-829. [17] [ J. M. Mansuy,C. Pasquier,M. Daudin,S. Chapuy-Regaud,N. Moinard,C. Chevreau,J. Izopet,C. Mengelle,L. Bujan, Zika virus in semen of a patient returning from a non-epidemic area, Lancet Infect. Dis., 16 (2016): 894-895. [18] [ J. C. Miller, Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes, Infect. Dis. Model., 2 (2017), 35-55, URL http://linkinghub.elsevier.com/retrieve/pii/S2468042716300203. [19] [ R. Pellissier,A. Rousselot, Enquete serologique sur l'incidence des virus neurol ropes chez quelques singes de l'Afrique Equatoriale Francaise (French) [A Serological Investigation of the Incidence of Neurotropic Viruses in Certain Monkeys of French Equatorial Africa], Bull. Société Pathol. Exot., 47 (1954): 228-231. [20] [ A. J. Rodriguez-Morales, A. C. Bandeira and C. Franco-Paredes, The expanding spectrum of modes of transmission of Zika virus: A global concern, Ann. Clin. Microbiol. Antimicrob. , 15 (2016), 13, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4776405/. [21] [ R. Ross, The Prevention of Malaria, Churchill, London, 1911. [22] [ Q. Zhang, K. Sun, M. Chinazzi, A. Pastore-Piontti, N. E. Dean, D. P. Rojas, S. Merler, D. Mistry, P. Poletti, L. Rossi, M. Bray, M. E. Halloran, I. M. Longini and A. Vespignani, Spread of Zika virus in the Americas, Proc. Natl. Acad. Sci. (2017).
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沈阳化工大学材料科学与工程学院 沈阳 110142

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# In a stepped-up transformer, why does input power equal to output power even though voltage is increased?

I just learned about the basics of transformers. I am aware that according to the law of energy conservation, power input should be equal to power output. However, when I think about this with ohm's law, in the secondary coil, power remains the same but voltage is increased implies resistance is somehow increased. But what causes this "increase in resistance"? I looked through some of the previous posts on this topic but I think all of them don't explicitly explain this increase in resistance. Maybe there is some more advanced stuff that is not covered in our high-school syllabus, but I want to know more. Thanks in advance!

• Forget the output circuit - how the transformer may couple to that is irrelevant. Just looking at the transformer, one coil puts power/energy into a magnetic field, the other coil takes it out. By changing the ratio of windings, the relative voltage changes, so the current into a matched load must change to keep the power constant. If the load is not matched, all bets are off on actually getting the input power into the output load. Aug 4 at 15:59
• @Jon Yes, I understood what you meant but the point of my problem is how does the load magically adjusted itself so that the power matches. Aug 4 at 16:06
• Why does you floor lamp only take 1 amp off a 15 amp circuit? The transformer can provide the full power at the step-up(down) voltage, but that doesn't mean some random load will take it all. Aug 4 at 16:08
• Ok, but I want to know will the $R$ in the secondary coil change when the turn ratio of the transformer changes? If yes, why? Aug 4 at 16:14
• The resistance of the secondary coil is whatever the resistance of that wire is - it does not know anything about the turn ratio. If the resistance of the transformer itself has a major influence on the operation of the transformer than something is wrong. If you put an infinite resistance load on the output, you would measure full voltage and no current. If you put a (near)-zero resistance load on the output you would get lots of current (limited perhaps by the transformer wiring) and collapse the voltage output. If you put a matched load on you can get optimal power transfer. Aug 4 at 16:18

You have $$P_1=P_2$$, or $$V_1^2/R_1=V_2^2/R_2$$, where $$R_1$$ is the apparent resistance of the primary circuit. If you change $$R_2$$ you are changing both, the power and the apparent resistance $$R_1$$ ("apparent" because is not linked to any actual resistance), but not the voltages.

• Why changing $R_2$ will affect $R_1$? Aren't the primary and secondary circuits in a transformer separate, where the only linkage between them is the magnetic flux? Aug 4 at 17:02
• Yes, as I said, it an apparent resistance. But you can forget about it, the basic answer is that the power changes in both sides, that is, as in any electrical circuit, if you fix the input V then the power changes with R Aug 4 at 17:04

but I want to know will the 𝑅 in the secondary coil change when the turn ratio of the transformer changes? If yes, why?

The $$R$$ of the secondary coil will not change because the $$R$$ for the primary and secondary coils is zero for an ideal transformer in which power in equals power out. If they had any resistance, some energy would dissipated as heat and therefore not be available to a load connected to the secondary.

but then according to $$P=\frac{V^2}{R}$$, when the power is step-up(or down), the power will change. What is wrong with this argument?

If the secondary voltage is greater than the primary voltage, then the primary current has to be greater than the secondary current since

$$\frac{V_{p}}{V_{s}}=\frac{I_{s}}{I_{p}}$$

In other words, the power will change for both the primary and the secondary

Hope this helps.

• but then according to $P=\frac{V^2}{R}$, when the power is step-up(or down), the power will change. What is wrong with this argument? Aug 4 at 16:28
• Did you mean $$P=\frac{V^2}{R}$$? Aug 4 at 16:30
• And when you say $R$ do you mean the resistance connected to the secondary winding? Aug 4 at 16:34
• Yes, that's right Aug 4 at 16:35
• Are you saying the output power of the secondary is greater than the input power to the primary? I'm not sure what you are arguing. Aug 4 at 16:49


text:
Introduction

This worksheet covers the Wilcoxon rank-sum test, which is an alternative to the between-subjects t-test, and the Kruskal-Wallis H test, which is an alternative to the between-subjects one-factor ANOVA.

The Wilcoxon rank-sum and Kruskal-Wallis H tests are both non-parametric tests. This means they make fewer assumptions about your data than do standard parametric tests (such as t-tests, and ANOVA). Specifically, where your sample size is small, parametric tests assume that your population is approximately normally distributed. However, it’s important to realise that as your sample size increases, parametric tests make fewer and fewer assumptions about your population distribution, due to the central limit theorem. It’s also important to realise that parametric tests have greater statistical power than non-parametric tests, so we should use parametric tests when their assumptions are met.

Putting this all together, there’s a relatively small set of situations where it makes sense to use a non-parametric test such as the Wilcoxon rank-sum or Kruskal-Wallis H. These are when:

1. Your sample size is small (N < 30 per group) and,
2. you do not know whether the population distribution is approximately normal and
3. you have reason to expect the effect size will be large (d > 1)

For example, you wouldn’t use an non-parametric test on a small sample of IQ scores, because IQ is known to be normally distributed. The point about effect size follows from the other two - if your sample size is small, you will only be able to detect large effects - see the statistical power worksheet.

Where’s the Bayes Factor? The Wilcoxon rank-sum and Kruskal-Wallis are traditional tests, in the sense that they give us a p-value rather than a Bayes Factor. As we have previously covered, p-values are widely misinterpreted by psychologists, and can never provide evidence for the null hypothesis. For this reason, we have generally advised that you instead use the Bayesian equivalents of these tests - such as Bayesian t-tests, Bayesian ANOVA, and Bayesian chi-square. However, it is not straight forward to calculate Bayesian equivalents of the Mann-Whitney and Kruskal-Wallis tests in R at the moment. So, in this case, we’ll stick to the traditional tests.

Getting started

To prepare for this worksheet:

1. Open the rminr-data project we used previously.

2. Open the Files tab. You should see a folder called going-further. This folder should contain the files picture-naming-long.csv and music-emotion-preproc.csv.

3. If you don’t see the folder or the files, it means you created your project before the data required for this worksheet was added to the rminr-data git repository. You fix this by asking git to “pull” the repository. Select the Git tab, which is located in the row of tabs which includes the Environment tab. Click the Pull button with a downward pointing arrow. A window will open showing the files which have been pulled from the repository. Close the Git pull window.

4. Create a script named non-parametric.R in the rminr-data folder (the folder above going-further). Add the code to this script as you work through each section of the worksheet.

Wilcoxon rank-sum test

We’ll demonstrate the Wilcoxon rank-sum test using data from an experiment which evaluated children’s language development using the Words in Game (WinG) test. WinG consists of a set of picture cards that are used in four tests: noun comprehension, noun production, predicate comprehension, and predicate production. The Italian and English versions of the WinG cards use different pictures to depict the associated concepts. The experiment tested whether English-speaking children aged approximately 30 months produce different responses for the two sets of cards.

rm(list = ls()) # clear the environment
library(tidyverse)
wing_preproc <- read_csv('going-further/picture-naming-long.csv')

Explanation of commands:

These commands should be familiar from previous worksheets. Line 1 clears the environment. Line 2 loads the tidyverse package. Line 3 reads the data.

The first few lines of wing_preproc look like this:

1 female english nc 12
1 female english np 4
1 female english pc NA
1 female english pp NA
2 male italian nc 18
2 male italian np 12
2 male italian pc 17
2 male italian pp 9
3 female english nc 18
3 female english np 13
3 female english pc 17
3 female english pp 9

The first three columns are the participant ID number, gender of the participant, and the type of card presented. The fourth column is the test (e.g. “nc” = “Noun comprehension”). The final column is the number of correct responses. Some data is missing - indicated as “NA”.

In the next section, we are going to compare English and Italian on the noun comprehension task. So, we filter the data (which contains all four tasks) to include just that task. We also remove any missing data.

# Filter
nc_include <- wing_preproc %>% filter(task == 'nc') %>% drop_na()

Explanation of command: The filter command should be familiar from many previous worksheets. drop_na() removes any row that contains an NA in it.

Calculating descriptive statistics

When we report non-parametric tests, we normally report the median (rather than the mean) as our descriptive statistic. This makes practical sense, because the mean can be misleading when the distribution is skewed and, if we have chosen to do a non-parametric test, we are (presumably) uncertain whether the distribution is skewed or not.

# summary statistics
nc_include %>%
group_by(cards) %>%
summarise(median = median(correct))
summarise() ungrouping output (override with .groups argument)
# A tibble: 2 x 2
cards   median
<chr>    <dbl>
1 english     19
2 italian     17

Explanation of commands: These commands should be familiar from several previous worksheets. We group the data by cards, and use summarise, to calculate the median() score for each group.

Calculating the Wilcoxon rank-sum test

As we said earlier in this worksheet, the Wilcoxon rank sum test is a non-parametric equivalent of a between-subjects t-test. It works by ranking all of the scores in the two groups, adding the ranks in each group, and comparing these “summed ranks” to determine if they differ.

We’ll run a Wilcoxon rank-sum test to see if there were any significant differences between scores for the Italian and English cards on the noun comprehension (nc) task.

# Wilcoxon rank-sum test (Mann-Whitney U) for noun comprehension
wilcox.test(correct ~ cards, nc_include)
Warning in wilcox.test.default(x = c(12, 18, 17, 17, 20, 19, 19, 20, 19), :
cannot compute exact p-value with ties

Wilcoxon rank sum test with continuity correction

data:  correct by cards
W = 58, p-value = 0.125
alternative hypothesis: true location shift is not equal to 0

Explanation of commands:

The command wilcox.test(correct ~ cards, nc_include) runs the test to compare the correct scores for the Italian and English cards.

Explanation of output:

The phrase with continuity correction is a technical detail that can be safely ignored. If you’re curious, you can read more in the help file, by typing ?wilcoxon.test into the console.

The phrase correct by cards reminds you that you compared the values in correct between the levels in the cards factor. A p value of less than .05 is generally considered by psychologists to be evidence that the two groups are different.

The phrase alternative hypothesis: true location shift is not equal to 0 can be safely ignored, as it’s just another (rather obscure) way of saying that you are testing whether the two groups are different.

The warning cannot compute exact p-value with ties lets you know that the method used to calculate p will not be exact, because some items in the Italian and English scores had identical rankings. It is possible to calculate the exact p value using the pwilcox command, but that’s beyond what we’ll cover in this worksheet. If your p-value is sufficiently close to .05 that it would matter if the estimate is a bit off, then a better solution would be to attempt to replicate your finding in a second study with a larger sample.

We now have all of the information we need to report the results. For the noun comprehension task, there was no significant difference in accuracy between the Italian (Mdn = 17) and English (Mdn = 19) cards W = 58, p = 0.13.

In some journal articles, you may see a non-parametric test called a “Mann-Whitney U”. This is exactly the same test as computed in the Wilcoxon rank-sum test above, just with a different name (and represented by a U rather than a W).

Exercise 1

Calculate summary statistics and the Wilxocon rank-sum for the noun production task. Your results should look like this:

summarise() ungrouping output (override with .groups argument)
# A tibble: 2 x 2
cards   median
<chr>    <dbl>
1 english     13
2 italian     11
Warning in wilcox.test.default(x = c(4, 13, 15, 10, 14, 12, 10, 16, 13), :
cannot compute exact p-value with ties

Wilcoxon rank sum test with continuity correction

data:  correct by cards
W = 47.5, p-value = 0.5619
alternative hypothesis: true location shift is not equal to 0

Copy the R code you used for this exercise into PsycEL.

Kruskal-Wallis H test

The Kruskal-Wallis H test is a non-parametric equivalent of a one-way between subjects ANOVA. It extends the Mann-Whitney test to situations where there are more than two groups. Like the Mann-Whitney test, the Kruskal-Wallis test works on ranked data.

We’ll demonstrate the Kruskal-Wallis test using data from a study which compared emotion regulation strategies between fans of mainstream (control group), goth, metal and emo music. Participants were measured using the Emotion Regulation Strategies for Artistic Creative Activities Scale (ERS-ACA), an 18 item inventory, with each item scored from 1 (‘strongly disagree’) to 5 (‘strongly agree’). The ERS-ACA gives an overall measure of the strategy people use to regulate their emotions when they engage in artistic, creative activities, and scores on three strategy sub-scales; avoidance, approach and self-development.

# Kruskal-Wallis H test
ers_l <- read_csv('going-further/music-emotion-preproc.csv')
Parsed with column specification:
cols(
subj = col_double(),
subculture = col_character(),
ers = col_character(),
score = col_double()
)

Explanation of command:

This data has already undergone some preprocessing and is in long format. The first few lines of ers_l look like this:

subj subculture ers score
17 Goth avoidance 1.286
17 Goth approach 4.167
17 Goth development 4.2
17 Goth total 3.056
18 Metal avoidance 4
18 Metal approach 3.5
18 Metal development 3.4
18 Metal total 3.667

Descriptive statistics

We’ll start by calculating the medians for the ‘approach’ subscale.

approach <- ers_l %>% filter(ers == 'approach') %>% drop_na()
approach %>%
group_by(subculture) %>%
summarise(median = median(score))
summarise() ungrouping output (override with .groups argument)
# A tibble: 4 x 2
subculture median
<chr>       <dbl>
1 Emo          3.83
2 Goth         3.67
3 Mainstream   3.67
4 Metal        3.83

Explanation of commands:

Line 1 filters the data to only include measurements for the ‘approach’ subscale and removes missing data. Lines 2-5 are very similar to the summary statistics we generated for the Mann-Whitney test. In this case we group by music subculture.

Explanation of output:

The differences in medians between groups look quite small.

Calculating the Kruskal-Wallis

We can now run the Kruskal-Wallis test:

kruskal.test(score ~ subculture, data = approach)

Kruskal-Wallis rank sum test

data:  score by subculture
Kruskal-Wallis chi-squared = 5.2313, df = 3, p-value = 0.1556

Explanation of commands:

The command kruskal.test(score ~ subculture, data = approach) runs the test to compare the ERS-ACA score scores for the four groups in subculture.

Explanation of output:

The string score by subculture reminds you that you compared the values in score between the levels in the subculture factor. The Kruskal-Wallis H statistic is 5.2313. R describes it as chi-squared because it is possible to estimate the relevant p-value using a chi-square distribution with the degrees of freedom (df) for that distribution set to one less than the number of groups, in this case df = 3. The p value tells us whether there was a significant difference between the four medians. It does not tell us which pairs of groups differ significantly from each other (for that, using a Wilcoxon rank-sum).

The results of this test are as follows:

There was no significant difference in approach style between the mainstream (Mdn = 3.67), goth (Mdn = 3.67), metal (Mdn = 3.83) and emo (Mdn = 3.83) groups, H = 5.23, p = 0.16.

Exercise 2

Calculate summary statistics and Kruskal-Wallis H for the self-development emotional response subscale of the ERS-ACA. Your results should look like this:

summarise() ungrouping output (override with .groups argument)
# A tibble: 4 x 2
subculture median
<chr>       <dbl>
1 Emo           3.8
2 Goth          4
3 Mainstream    3.6
4 Metal         3.8

Kruskal-Wallis rank sum test

data:  score by subculture
Kruskal-Wallis chi-squared = 8.5011, df = 3, p-value = 0.03671

Copy the R code you used for this exercise into PsycEL.


text:
1887

### Abstract

The high consume of gas in Argentina has demanded the increase of gas production and the developing of the tight gas resources. Here, we will describe the most characteristics tight gas reservoirs of the Neuquén Basin: the Mulichinco, Las Lajas and PreCuyano formations. Mulichincos, with permeability as low as 0.001 mD, is in early development stage and has several pilot wells planned for 2007 to better define its potential; Las Lajas, focus of the Argentinean Tight Gas main project, produces only if hydraulic fractured, and needs new technologies to better define its sands thicknesses and trap system; Precuyano`s, with reservoirs of volcanic origin, presents a highly structural complexity and requires different fracture approach for each pay zone.

/content/papers/10.3997/2214-4609.201405160
2007-06-10
2021-01-15


text:
# Associations to the word «Laws»

## Wiktionary

LAW, noun. (uncountable) The body of rules and standards issued by a government, or to be applied by courts and similar authorities.
LAW, noun. A particular such rule.
LAW, noun. (more generally) A written or understood rule that concerns behaviours and their consequences. Laws are usually associated with mores.
LAW, noun. (scientific) (strictly) A well-established, observed physical characteristic or behavior of nature. The word is used to simply identify "what happens," without implying any explanatory mechanism or causation. Compare to theory.
LAW, noun. (mathematics) A statement that is true under specified conditions.
LAW, noun. A category of English "common law" petitions that request monetary relief, as opposed to relief in forms other than a monetary judgment; compare to "equity".
LAW, noun. (cricket) One of the official rules of cricket as codified by the MCC.
LAW, noun. (slang) (uncountable) () The police.
LAW, noun. (fantasy) One of the two metaphysical forces of the world in some fantasy settings, as opposed to chaos.
LAW, noun. An oath, as in the presence of a court. See wager of law.
LAW, noun. (obsolete) a tumulus of stones
LAW, noun. (Scottish and northern dialectal, archaic) a hill
LAW, interjection. (dated) An exclamation of mild surprise; lawks.
LAW, proper noun. A patronymic surname.
LAW, proper noun. A diminutive of Lawrence.
LAW, proper noun. A topographic surname, perhaps originally meaning someone who lives near a burial mound.
LAW, proper noun. (Scotland) a conical hill
LAW, proper noun. (Judaism) the Torah
LAW, proper noun. (Christianity) a generic term which can refer to the Divine commandments (primarily the Decalogue), the Old Testament in general or, most specifically, the Torah.
LAW AND ORDER, noun. (dated) (17th century) The principles under which the world and its components operate.
LAW AND ORDER, noun. The strict enforcement of law, statutes, and social conventions.
LAW BINDING, noun. A plain style of leather binding, used for law books.
LAW CLERK, noun. (legal) a clerk who assists a judge with legal proceedings. Depending on the court, such positions may be professional careers or may be temporary positions for graduating law students.
LAW ENFORCEMENT, noun. Insuring obedience to law.
LAW ENFORCEMENT, noun. The various government agencies involved in the prevention of crime and the apprehension of criminals.
LAW FIRM, noun. (legal) A business formed by one or more lawyers to engage in the practice of law.
LAW FIRMS, noun. Plural of law firm
LAW FRENCH, proper noun. (legal) The body of now archaic legal terms used in the United Kingdom descended from the Anglo-Norman dialect of Old French.
LAW LATIN, noun. Alternative term for legal Latin.
LAW LATIN, noun. (1811) (jocular) (obsolete) Barbarous Latin.
LAW LEATHER, noun. (historical) A specific kind and grade of sheepskin leather traditionally used for binding statute books and other official documents.
LAW OF ATTRACTION, proper noun. (new age) The belief that thinking about something attracts it and causes it to manifest in your life.
LAW OF AVERAGES, noun. (statistics) The statistical tendency toward a fixed proportion in the results when an experiment is repeated a large number of times; the law of large numbers.
LAW OF AVERAGES, noun. (informal) An imaginary or perceived "law" of probabilities which is wrongly used to predict results in the short-term.
LAW OF CONSERVATION OF ENERGY, noun. (physics) The law stating that the total amount of energy in any isolated system remains constant, and cannot be created or destroyed, although it may change forms.
LAW OF CONSERVATION OF NINJUTSU, proper noun. (narratology) (fiction) (informal) (humorous) In a fictional fight, the phenomenon of the team with fewer people often being apparently more powerful than the other team.
LAW OF COSINES, noun. (trigonometry) A statement that relates the lengths of the sides of a triangle to the cosine of one of its angles.
LAW OF DEMETER, proper noun. (software) (object-oriented) A design guideline for developing particularly object-oriented programs that mandates loose coupling between objects.
LAW OF DIMINISHING MARGINAL UTILITY, noun. (economics) In economics, the theory there is a general decrease in the utility of a product, as more units of it are consumed.
LAW OF DIMINISHING RETURNS, noun. The tendency for a continuing effort toward a particular goal to decline in effectiveness after a certain amount of success has been achieved.
LAW OF DIMINISHING RETURNS, noun. (economics) A relationship between input and output, such that adding units of any one input (labour, capital etc.) to fixed amounts of the others will yield successively smaller increments of output.
LAW OF DOUBLE NEGATION, noun. (logic) The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically: $$\neg \neg A \to A$$.
LAW OF EXCLUDED MIDDLE, noun. (logic) A principle of classical logic which states all statements must be either true or false, i.e. in symbols: $$P \vee \neg P$$.
LAW OF HOBSON-JOBSON, noun. The rule that words or phrases borrowed between languages will be modified in their pronunciation as necessary to conform to the set of sounds used by the borrowing language.
LAW OF IDENTITY, noun. (logic) The rule of classical logic, that anything equals itself. Expressed by the symbolic equation A=A.
LAW OF JANTE, proper noun. A supposed negative pattern of group behavior in Scandinavian communities toward people who achieve individual success.
LAW OF LARGE NUMBERS, noun. (statistics) The statistical tendency toward a fixed ratio in the results when an experiment is repeated a large number of times; law of averages.
LAW OF NATURE, noun. A physical law (a scientific generalization based upon empirical observation)
LAW OF NATURE, noun. A natural law (any of a number of doctrines in moral, political and legal theory)
LAW OF NONCONTRADICTION, noun. (logic) The principle that no statement may be simultaneously true and false at the same time and in the same sense.
LAW OF NONTRIVIALITY, noun. (logic) The rule that states that not all propositions are true, the opposite of trivialism.
LAW OF SINES, noun. (trigonometry) A statement that relates the lengths of the sides of a triangle to the sines of its angles.
LAW OF THE EXCLUDED MIDDLE, noun. Alternative form of law of excluded middle
LAW OF THE JUNGLE, noun. (idiomatic) The survival of the fittest, strongest or most cunning
LAW OF THE LAND, noun. (legal) A particular law or the complete set of laws currently in effect within a jurisdiction, especially with emphasis on the official and authoritative nature of such law.
LAW OF THE TONGUE, noun. Appropriate control of one's words.
LAW OF THE TONGUE, noun. (Can we verify([1]) this sense?) An implicit agreement between whalers of Eden, Australia, and local orcas, in which the orcas would invite whalers on a whale hunt, in exchange for dibs on the catch, primarily the whale's tongue.
LAW OF UNINTENDED CONSEQUENCES, noun. An adage which proposes that every act has unintended consequences.
LAW REVIEW, noun. (legal): A scholarly journal focusing on legal issues, normally published by an organization of students at a law school or through a bar association.
LAW REVIEW, noun. (legal): An article published in such a journal.
LAW REVIEW, noun. (legal): The student organization responsible for publishing such a scholarly journal.
LAW REVIEWS, noun. Plural of law review
LAW SCHOOL, noun. A post-graduate academic program in which students are prepared for the practice of law
LAW SCHOOL, noun. A building which houses such an academic program
LAW SCHOOLS, noun. Plural of law school
LAW STUDENT, noun. (legal) a student who studies the law. Someone enrolled in a school for legal education.
LAW SUIT, noun. Alternative form of lawsuit

## Dictionary definition

LAW, noun. The collection of rules imposed by authority; "civilization presupposes respect for the law"; "the great problem for jurisprudence to allow freedom while enforcing order".
LAW, noun. Legal document setting forth rules governing a particular kind of activity; "there is a law against kidnapping".
LAW, noun. A rule or body of rules of conduct inherent in human nature and essential to or binding upon human society.
LAW, noun. A generalization that describes recurring facts or events in nature; "the laws of thermodynamics".
LAW, noun. The branch of philosophy concerned with the law and the principles that lead courts to make the decisions they do.
LAW, noun. The learned profession that is mastered by graduate study in a law school and that is responsible for the judicial system; "he studied law at Yale".
LAW, noun. The force of policemen and officers; "the law came looking for him".

## Wise words

A word carries far, very far, deals destruction through time as the bullets go flying through space.


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Like a lot of other R users I’ve felt the need for sending email from R. I haven’t surveyed CRAN for such a package but looked for the possibility of sending command line email in Windows.

Found a nice application called sendEmail that can be found here

Below are code snippets in R that will allow you to make use of this application.

First, download the application from the link provided above. Get the TLS supported variant. Unzip and add to your PATH variable, the location of the sendEmail executable file.

Create a text file with some sample content, place it in C:/ and name it “filetest.txt”. This is the file used as a sample attachment in the code below.

I am assuming you’re using Gmail and have a username: [email protected] with password: xyz.

In the sendEmail folder, README.txt provides the necessary information of the command line parameters (see section ‘Usage Overview’). The code snippet below simply creates the relevant parameter-value combination(s) and uses system() to issue a system call.

# Create the required command line parameters for sendEmail to work
paramsList <- list()
paramsList$fromAddress <- c("-f",'[email protected]') paramsList$toAddress <- c("-t",'[email protected]')
paramsList$emailSubject <- c("-u","Test Email from R") paramsList$listemailMessage <- c("-m",paste("Sent at ",format(Sys.time(),"%Y-%d-%m:%H-%M-%S"),sep=" "))
paramsList$serverAndPort <- c("-s","smtp.gmail.com:587") paramsList$fileAttachPath <- c("-a","C:/filetest.txt")
paramsList$accUsername <- c("-xu","[email protected]") paramsList$accPassword <- c("-xp","xyz")

# Add double quotes to the parameter values
paramsList1 <- lapply(paramsList,function(x){x[2] <- dQuote(x[2]);paste(x,collapse = " ")})

# Combine to create one single function call
suffixCall <- paste(do.call("c",paramsList1),collapse = " ")
commandCall <- paste("sendEmail",suffixCall,sep = " ")
# Issue the command via system() – sending it to CMD
returnVal <- system(commandCall,intern=T,wait=T)
print(returnVal)

You could use the R2HTML package in R to send model summary report in HTML like below:

# ———
library(R2HTML)
data(iris)
# From the help files of R2HTML package
tmpfic <- HTMLInitFile(outdir="C:/",filename="regressionTest")
HTML(as.title(“Regression Example”),file=tmpfic)
HTML(summary(lm(Sepal.Length~Sepal.Width+ Petal.Length + Petal.Width,data=iris)),file=tmpfic)
HTMLEndFile(tmpfic)
browseURL(tmpfic) # View the file in your local browser

# Update the paramsList appropriately
paramsList\$fileAttachPath <- c("-a",tmpfic)

# Recreate the new command and execute
paramsList1 <- lapply(paramsList,function(x){x[2] <- dQuote(x[2]);paste(x,collapse = " ")})
suffixCall <- paste(do.call("c",paramsList1),collapse = " ")
commandCall <- paste("sendEmail",suffixCall,sep = " ")
returnVal <- system(commandCall,intern=T,wait=T)
print(returnVal)

The solution presented here is not a pure R implementation and could be considered more of a hack.


text:
## Redefine Statistical Significance Part II: Caught in a Bad Romance?

In the previous post we discussed the paper “Redefine Statistical Significance”. The key point of that paper was that p-values near .05 provide (at best) only weak evidence against the null hypothesis. This contradicts current practice, where p-values slightly lower than .05 bring forth an epistemic jamboree, one where researchers merrily draw bold conclusions such as “we reject the null…


text:
# What is the vertex form of  y= (2x-3)(7x-12)+17x^2-13x?

Jan 9, 2017

Vertex form of equation is $y = 31 {\left(x - \frac{29}{31}\right)}^{2} + \frac{275}{31}$

#### Explanation:

Vertex form of equation is $y = a {\left(x - h\right)}^{2} + k$

As we have $y = \left(2 x - 3\right) \left(7 x - 12\right) + 17 {x}^{2} - 13 x$

$= 2 x \times 7 x - 2 x \times 12 - 3 \times 7 x - 3 \times \left(- 12\right) + 17 {x}^{2} - 13 x$

$= 14 {x}^{2} - 24 x - 21 x + 36 + 17 {x}^{2} - 13 x$

$= 14 {x}^{2} - 24 x - 21 x + 36 + 17 {x}^{2} - 13 x$

$= 31 {x}^{2} - 58 x + 36$

$= 31 \left({x}^{2} - \frac{58}{31} x\right) + 36$

$= 31 \left({x}^{2} - 2 \times \frac{29}{31} x + {\left(\frac{29}{31}\right)}^{2}\right) + 36 - 31 \times {\left(\frac{29}{31}\right)}^{2}$

$= 31 {\left(x - \frac{29}{31}\right)}^{2} + 36 - \frac{841}{31}$

$= 31 {\left(x - \frac{29}{31}\right)}^{2} + \frac{275}{31}$
graph{(2x-3)(7x-12)+17x^2-13x [-5, 5, -2.88, 37.12]}


text:
# Nice ques to solve

1. Aug 18, 2011

### Vishalrox

Nice ques to solve !!!

Given a triangle ABC find a point M on its circumcircle such that MA=MB+MC....It is easy for an equilateral triangle... I got it... but I couldn't get how to find for any arbitrary triangle....

2. Aug 18, 2011

### gsal

Re: Nice ques to solve !!!

Hhhmmm...in other words, you have 3 points on a circle and you need to find a 4th one that meets such condition.

I would first find out the angle that the radius makes to each point A,B,C; let's call these angles THA, THB, THC (TH for theta).

Then, let's have a point M and its corresponding THM.

Let's call the center of the circle O.

By taking the center of the circle, the point M and one point (A,B,C) at a time, we have 3 isosceles triangles where the base-lines (chords) of those triangles are those lines you want to match the equation MA=MB+MC. Also, it is true that MA/2 = MB/2 + MC/2.

Those base-lines are chords whose lengths are 2Sin(TH/2), where TH is the angle the two sides make at the center of the circle (radius=1).

For triangle AOM, we have one TH, THAOM = THA - THM

and

THBOM = THB - THM
THCOM = THC - THM

And so, we want:

Sin(THAOM/2) = Sin(THBOM/2) + Sin(THCOM/2)

I think it now gets a bit hairy to isolate THM from here; so, you could simply apply brut force and substitute values between 0 and 2Pi and that's it.

3. Aug 18, 2011

### Yuqing

Re: Nice ques to solve !!!

The three sides of the triangle as well as the chords MA, MB and MC will form the sides and diagonals of a cyclic quadrilateral. Now if you apply Ptolemy's theorem to the quadrilateral, you should be able to obtain a condition relating the three sides. With that condition, you should be able to derive an expression for the length of either MA, MB or MC and the position of M can be found with that.

Last edited: Aug 18, 2011
4. Aug 18, 2011

### DaveC426913

Re: Nice ques to solve !!!

Guys? Help him to the answer. Spoonfeeding is not allowed on PF.

5. Aug 18, 2011

### Yuqing

Re: Nice ques to solve !!!

Oh my apologies, since this wasn't posted in the homework forums, I though he just wanted some form of solution. I'll edit my answer to provide hints instead.

6. Aug 19, 2011

### superg33k

Re: Nice ques to solve !!!

Now we just do THAOM/2=THA/2-THM/2
Expand out the sines using sin(A-B)=sinAcosB-sinBcosA
Pull out the sin(THM/2) to one side and the cos(THM/2) to the other.
Divide for tan(THM/2)

7. Aug 21, 2011

### Vishalrox

Re: Nice ques to solve !!!

i got it mysely...just say whether its r8.....Barycentric coordinates as a function of the side lengths. The circumcenter has trilinear coordinates (cos α, cos β, cos γ) where α, β, γ are the angles of the triangle. The circumcenter M has coordinates:

M = (a^2(-a^2 + b^2 + c^2), b^2(a^2 - b^2 + c^2), c^2(a^2 + b^2 - c^2))

where a, b, c are edge lengths (BC, CA, AB respectively) of the triangle.

and i found the three co-ordinates....now we can find the answer easily....!!!...is it correct.....


text:
Definition 29.49.1. Let $X$, $Y$ be schemes.

1. Let $f : U \to Y$, $g : V \to Y$ be morphisms of schemes defined on dense open subsets $U$, $V$ of $X$. We say that $f$ is equivalent to $g$ if $f|_ W = g|_ W$ for some $W \subset U \cap V$ dense open in $X$.

2. A rational map from $X$ to $Y$ is an equivalence class for the equivalence relation defined in (1).

3. If $X$, $Y$ are schemes over a base scheme $S$ we say that a rational map from $X$ to $Y$ is an $S$-rational map from $X$ to $Y$ if there exists a representative $f : U \to Y$ of the equivalence class which is an $S$-morphism.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01RS. Beware of the difference between the letter 'O' and the digit '0'.


text:
# Thread: Techniques of integration (5)

1. ## Techniques of integration (5)

$\int \left(x \ln x - \frac{(n-1)!}{x^n} \right)e^x \ dx = ?$

2. Originally Posted by NonCommAlg
$\int \left(x \ln x - \frac{(n-1)!}{x^n} \right)e^x \ dx = ?$
I bet there is a better way , but I thought this was fun.

First lets break this up a bit

$
\int x\ln(x)e^xdx-(n-1)!\int x^{-n}e^xdx
$

Lets focus on the 2nd integral to start

$(n-1)!\int x^{-n}e^xdx$

so if we integrate by parts with $u=e^x \implies du =e^xdx$ and $dv=x^{-n} \implies v=-\frac{1}{n-1}x^{-n+1}$

$(n-1)!\int x^{-n}e^xdx=(n-1)!\left(-\frac{1}{n-1}x^{-n+1}e^x+\frac{1}{n-1}\int x^{-n+1}e^xdx \right)=$
$-(n-2)!x^{-n+1}e^{x}+(n-2)!\int x^{-n+1}e^x$

If we integrate by parts again we get

$-(n-2)!x^{-n+1}e^{x}-(n-3)!x^{-n+2}e^{x}+(n+3)!\int x^{-n+2}e^x$

Doing this n-1 times we get

$(n-1)!\int x^{-n}e^xdx=-e^{x}\sum_{i=1}^{n-1}[n-(1+i)]!x^{-n+i}+\int \frac{1}{x}e^{x}dx$

Now lets look at the last bit

$\int \frac{1}{x}e^{x}dx$ if we integrate this by parts we get

$\int \frac{1}{x}e^{x}dx=e^{x}\ln(x)-\int e^{x}\ln(x)dx$

Now lets integrate the last integral by parts again with

$u=e^{x} \implies du=e^{x},dv=\ln(x ) \implies v=x\ln(x)-x$

$\int \frac{1}{x}e^{x}dx=e^{x}\ln(x)-[x\ln(x)-x]e^{x}+\int [x\ln(x)-x]e^xdx$

$\int \frac{1}{x}e^{x}dx=e^{x}\ln(x)-[x\ln(x)-x]e^{x}+\int x\ln(x)e^{x}dx-\int xe^xdx$

So finally we get

$(n-1)!\int x^{-n}e^xdx=-e^{x}\sum_{i=1}^{n-1}[n-(1+i)]!x^{-n+i}+e^{x}\ln(x)-[x\ln(x)-x]e^{x}+$
$\int x\ln(x)e^{x}dx-\int xe^xdx$

Now subbing this back into the very first line we get

$e^{x}\sum_{i=1}^{n-1}[n-(1+i)]!x^{-n+i}+e^{x}\ln(x)+[x\ln(x)-x]e^{x}+\int xe^xdx$

evaluateing the last integral and simplifying gives

$e^{x}\sum_{i=1}^{n-1}[n-(1+i)]!x^{-n+i}+e^{x}\ln(x)+xe^{x}\ln(x)-e^x$

3. Hello,

Here is what I tried...

$I_n(x)=\int\left(x\ln(x)-\frac{(n-1)!}{x^n}\right)e^x ~dx$

By differentiating this (under the integral sign), we can get :
$I_n'(x)=I_n(x)+\int\left(1+\ln(x)+\frac{n!}{x^{n+1 }}\right)e^x~dx$

$I_n'(x)=I_n(x)-I_{n+1}(x)+\int\left(1+\ln(x)+x\ln(x)\right)e^x~dx$

But the latter is exactly $x\ln(x)e^x+C$

And we also have $I_n'(x)=\left(x\ln(x)-\frac{(n-1)!}{x^n}\right)e^x$

So finally :
$I_{n+1}(x)-I_n(x)=\frac{(n-1)!}{x^n}\cdot e^x+C$

And :
$I_n(x)=e^x \sum_{k=2}^n \frac{(k-2)!}{x^{k-1}}+I_2+D$

That's where I've gotten to. I didn't have much time this morning. And since I'm not at home, I don't have a paper in front of me and cannot continue immediately... I think the problem will be to compute $I_2$... The sum is just the (n-1) (or (n-2)) th derivative of x/(x-1)=1+1/(x-1), which can easily be found.

I'll think about it later. See ya

4. Originally Posted by Moo
Hello,

Here is what I tried...

$I_n(x)=\int\left(x\ln(x)-\frac{(n-1)!}{x^n}\right)e^x ~dx$

By differentiating this (under the integral sign), we can get :
$I_n'(x)=I_n(x)+\int\left(1+\ln(x)+\frac{n!}{x^{n+1 }}\right)e^x~dx$
here Moo actually used by parts: $x \ln x - \frac{(n-1)!}{x^n} = u, \ e^x dx = dv$ and the fact that $uv=I_n'(x).$

$I_n'(x)=I_n(x)-I_{n+1}(x)+\int\left(1+\ln(x)+x\ln(x)\right)e^x~dx$

But the latter is exactly $x\ln(x)e^x+C$

And we also have $I_n'(x)=\left(x\ln(x)-\frac{(n-1)!}{x^n}\right)e^x$

So finally :
$I_{n+1}(x)-I_n(x)=\frac{(n-1)!}{x^n}\cdot e^x+C$

And :
$I_n(x)=e^x \sum_{k=2}^n \frac{(k-2)!}{x^{k-1}}+I_2+D$

That's where I've gotten to. I didn't have much time this morning. And since I'm not at home, I don't have a paper in front of me and cannot continue immediately... I think the problem will be to compute $I_2$... The sum is just the (n-1) (or (n-2)) th derivative of x/(x-1)=1+1/(x-1), which can easily be found.

I'll think about it later. See ya
this is a nice approach. you probably don't need to think about finding $I_2$ because $\emptyset$'s solution, which is exactly how i created the problem, has already answered your question.


text:
clubSandwich provides several cluster-robust variance estimators (i.e., sandwich estimators) for ordinary and weighted least squares linear regression models, two-stage least squares regression models, and generalized linear models. Several adjustments are incorporated to improve small-sample performance. The package includes functions for estimating the variance-covariance matrix and for testing single- and multiple-contrast hypotheses based on Wald test statistics. Tests of single regression coefficients use Satterthwaite or saddlepoint corrections. Tests of multiple-contrast hypotheses use an approximation to Hotelling’s T-squared distribution. Methods are provided for a variety of fitted models, including lm, mlm, glm, ivreg (from package AER), plm (from package plm), gls and lme (from nlme), lmer (from lme4), robu (from robumeta), and rma.uni and rma.mv (from metafor).

# Installing clubSandwich

The package is available on the Comprehensive R Archive Network. To install it, type

install.packages("clubSandwich")

To install the latest development version directly from Github, type:

install.packages("remotes")
remotes::install_github("jepusto/clubSandwich")

Once installed, have a look at the available vignettes by typing:

browseVignettes(package="clubSandwich")


text:
# Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds

Let $$(M,g)$$ be a homogeneous Riemannian manifold. That is, the isometry group $$Iso(M,g)$$ acts transitively on $$M$$. Let $$\pi_1(M)$$ be the fundamental group of $$M$$. Then $$\pi_1(M)$$ has finite commutator subgroup.

I am looking for a converse of sorts. Let $$M$$ be a compact manifold. Suppose that $$M$$ is smooth homogeneous. In other words, there exists some finite dimensional Lie group $$G$$ not necessarily compact acting smoothly but not necessarily by isometries and transitively on $$M$$. Furthermore suppose that $$\pi_1(M)$$ has finite commutator subgroup. Then can we conclude that there exists some homogeneous metric $$g$$ on $$M$$ (i.e. the isometry group $$Iso(M,g)$$ acts transitively on $$M$$)?

EDIT:

A counter example cannot be simply connected (or even have finite fundamental group) because then it would be the case for the smooth transitive action of the noncompact group $$G$$ that the action of the maximal compact subgroup is still transitive [Corollary 3, Montgomery, Simply Connected Homogeneous Spaces]. So a counterexample must have infinite fundamental group. In particular that means for any counter example the universal cover cannot be compact

https://math.stackexchange.com/questions/1848792/fundamental-group-of-a-compact-space-with-compact-universal-covering-space

Also no aspherical manifold can be a counterexample. To see why, suppose that $$M$$ is a smooth homogeneous aspherical closed manifold whose fundamental group has finite commutator subgroup. An aspherical manifold has torsion free fundamental group. So if the commutator subgroup is finite then the fact that the fundamental group is torsion free implies that the commutator subgroup vanishes and the fundamental group is abelian. Every finitely generated torsion free abelian group is isomorphic to $$\mathbb{Z}^n$$, the fundamental group of the torus. A smooth homogeneous aspherical closed manifold with abelian (even nilpotent, even virtually nilpotent) fundamental group is determined up to diffeomorphism by its fundamental group see

https://math.stackexchange.com/questions/4386449/transitive-lie-group-actions-and-exotic-smooth-tori/4386739#4386739

So $$M$$ must be diffeomorphic to the standard torus $$T^n$$, which admits a flat metric. In conclusion, any smooth homogeneous aspherical closed manifold whose fundamental group has finite commutator subgroup is diffeomorphic to the standard torus and thus admits a Riemannian homogeneous metric (the flat metric).

Also, all counterexamples have to be in at least dimension $$4$$.

In $$n \leq 2$$ every compact smooth homogeneous space is Riemannian homogeneous except the Klein bottle which is aspherical so cannot be a counterexample.

For $$n=3$$ most compact smooth homogeneous spaces are aspherical. The ones that aren't are all Riemannian homogeneous with the exception of $$\mathbb{R}P^3 \# \mathbb{R}P^3$$ which has infinite fundamental group (the free product $$C_2 * C_2$$) so it is ruled out by the fundamental group condition since the abelianization is finite $$C_2 \oplus C_2$$ so the commutator subgroup of $$\pi_1$$ is not finite.

EDIT 2: Since abelian groups have trivial commutator subgroup then certainly the commutator subgroup is finite and thus the simplest possible fundamental group for a counterexample is $$\pi_1(M) \cong\mathbb{Z}$$ So the simplest counterexample would be a compact (non-aspherical) 4 manifold with fundamental group $$\pi_1(M) \cong\mathbb{Z}$$ that admits a transitive action by a noncompact group $$G$$ but does not admit any Riemannian homogeneous metric. Note that this is exactly what Robin Goodfellow attempted in his answer. The issue is that his manifold is not even smooth homogeneous to begin with, therefore not a counterexample.

• I think you should add the word 'compact' to the question itself because, as it stands, it's easy to overlook the one place that specifies that you are asking only about compact manifolds. In fact, I was misled this way when I gave a (now deleted) answer that didn't take this condition into account because it wasn't in the question. Jan 23 at 15:27
• Your argument in the case of the tori is incomplete as it does not rule out exotic tori which exist in high dimensions. but it's easy to see that if a compact Riemannian homogeneous space $M$ is aspherical then $Iso(M)$ is also aspherical which does imply that $Iso_0(M)$ is a torus and so is $M$. Feb 19 at 16:42
• @IanGershonTeixeira yes, this is easy. there are several ways to see it. The simplest is that any $M=G/H$ where $G$ is compact admits a metric of nonnegative sectional curvature (coming from a biinvariant metric on $G$). It's a standard consequence of Cheeger-Gromoll splitting theorem that any nonnegatively curved metric on a topological torus is flat. you can also give a lie group argument based on structure theory of compact Lie groups but the above suffices. Feb 19 at 22:51
• to elaborate further Cheeger and Gromoll proved that the universal cover of any closed nonnegatively curved $M$ is isometric to $\mathbb R^k\times C$ where $C$ is simply connected and closed. If $M$ is aspherical then $C$ must be a point which means that $M$ is flat. Feb 19 at 23:00
• @IanGershonTeixeira sorry, i was going by your main question where you assume that the metric is Riemannian homogeneous. if you only assume that it's smooth homogeneous then I don't immediately see how to show that any homogeneous torus must be standard although I am sure this is true. You might want to ask a separate question about it. Feb 19 at 23:20

The isometry group of a compact Riemannian manifold is always compact, so the answer to this question doesn’t really depend on the fundamental group at all: if $$G$$ is noncompact, then it doesn’t act isometrically on a compact $$M$$ for any choice of metric, but if $$G$$ is compact, then we can get an invariant Riemannian metric using the usual averaging trick.
Assuming you want a transitive isometric action of some Lie group other than $$G$$ itself, this is a bit trickier, though it does still come down to finding a compact Lie group acting transitively on $$M$$ (for the same reasons given above). I imagined that there would be many solvmanifold counterexamples, but all the ones I can think of do not satisfy your fundamental group condition.
One possible counterexample would be the mapping torus for an orientation-reversing isometry on an odd-dimensional sphere $$S^{2n+1}$$ with the round metric. This is homogeneous under the action of $$\mathrm{O}(2n+2)\times\mathbb{R}$$ and has fundamental group $$\mathbb{Z}$$, but any transitive isometric Lie group action on it must descend from a transitive isometric action on the universal cover $$S^{2n+1}\times\mathbb{R}$$, and—assuming things work similar to the case in dimension 3, though I don’t have my copy of Thurston’s geometry book on hand to recall the argument—the maximal Lie group acting effectively and transitively on $$S^{2n+1}\times\mathbb{R}$$ with compact stabilizers is $$\mathrm{O}(2n+2)\times\mathbb{R}\rtimes\mathbb{Z}/2\mathbb{Z}$$; for odd-dimensional spheres, we don’t have central orientation-reversing isometries, so the largest normalizer (in $$\mathrm{O}(2n+2)$$) of a subgroup generated by such an isometry is probably small enough that the action of $$\mathrm{O}(2n+2)\times\mathbb{R}\rtimes\mathbb{Z}/2\mathbb{Z}$$ can’t descend to a transitive action on the mapping torus.
• Note that in OP there is no requirement that $G$ is related to $Isom(M,g)$ in any way (except for both acting on the same manifold). Dec 28, 2021 at 14:44
• Thanks MoisheKohan! In particular the spirit of my question is "If a non compact G acts transitively then when can we arrange for the maximal compact subgroup K of G to act transitively by isometries"? But I kept the wording vague because the real line acts transitive isometries on the flat circle but the maximal compact is trivial. I expect the answer is "yes when G acts transitively and the fundamental group of M has finite commutator subgroup then there is always a transitive isometric action by $K \times T^n$ where $K$ is maximal semisimple compact, $n$ dimension maximal abelian Dec 28, 2021 at 14:53
• Oh wow the mapping torus for an orientation reversing isometry of the three sphere! This is just the sort of interesting counterexample I was looking for. I'll definitely have to look in my copy of Thurston's geometry book. I guess this would be in the section on $S^2 \times \mathbb{R}$ geometry? If you feel confident about the details I'm happy to accept your answer. If not we can wait until maybe you get a chance to look more at Thurston or someone else like @MoisheKohan comes back and say the details look good. Dec 31, 2021 at 16:25
• I think your statement "the mapping torus for an orientation-reversing isometry on an odd-dimensional sphere $S^{2n+1}$ with the round metric... is homogeneous under the action of $\mathrm{O}(2n+2)\times\mathbb{R}$" is wrong and therefore your suggestion is not a valid counterexample. Note that the natural action of $\mathrm{O}(2n+2)\times\mathbb{R}$ on the universal cover $S^{2n+1} \times \mathbb{R}$ does not descend to an action on the mapping torus see math.stackexchange.com/questions/4360976/… Jan 23 at 4:40
• If you have another action of $O(2n+2)×\mathbb{R}$ on the mapping torus $S^{2n+1} \rtimes S^1$ in mind I would love to hear about it. If not and you just made a mistake when you claimed the existence of such a homogeneous action then I would encourage you to edit your answer accordingly. For what it's worth even spheres don't work as a counterexample either because, as you anticipated by specializing to the case of a non central isometry, for even spheres the natural action factors through the action of a compact group and thus the mapping torus is Riemannian homogeneous. Jan 23 at 4:46


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Asymptotic behaviour of solutions of periodic competition diffusion system. (English) Zbl 0686.35060
In this paper T-periodic solutions of the system $$u\sb t=k\sb 1\Delta u+u(a-bu-cv);\quad v\sb t=k\sb 2\Delta v+v(d-eu-fv)$$ in $\Omega$ $\times (-\infty,\infty)$ are studied which satisfy a Neumann boundary condition. It is assumed that the coefficients are T-periodic and depend also on the space variable. Existence, uniqueness and stability results are established. The techniques rely on monotonicity methods and on the generalized maximum principle. This paper extends previous work of the second author.
Reviewer: C.Bandle

##### MSC:
 35K57 Reaction-diffusion equations 35B35 Stability of solutions of PDE 35B50 Maximum principles (PDE)
##### Keywords:
Neumann boundary condition; Existence; uniqueness
Full Text:
##### References:
 [1] Ahmad, S.: Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition model. J. math. Analysis applic. 127, 377-387 (1987) · Zbl 0648.34037 [2] Alvarez, C.; Lazer, A. C.: An application of topological degree to the periodic competing species problem. J. austral. Math. soc. Ser. B 28, 202-219 (1986) · Zbl 0625.92018 [3] Amann, H.: Periodic solutions of semilinear parabolic equations. Nonlinear analysis, 1-29 (1978) · Zbl 0464.35050 [4] Brown, P. N.: Decay to uniform states in ecological interactions. SIAM J. Appl. math. 38, 22-37 (1980) · Zbl 0511.92019 [5] Cosner, C.; Lazer, A. C.: Stable coexistence states in the Lotka--Volterra competition model with diffusion. SIAM J. Appl. math. 44, 1112-1132 (1984) · Zbl 0562.92012 [6] Gopolosamy, K.: Exchange of equilibria in two species Lotka--Volterra competition models. J. austral. Math. soc. Ser. B 24, 160-170 (1982) · Zbl 0498.92016 [7] Ladysenskaja, O.; Solonnikov, V.; Ural’ceva, N.: Linear and quasilinear equations of parabolic type.. (1968) [8] Pao, C. V.: Coexistence and stability of a competition--diffusion system in population dynamics. J. math. Analysis applic. 83, 54-76 (1981) · Zbl 0479.92013 [9] Sattinger, D.: Topics in stability and bifurcation theory. Lecture notes in mathematics 309 (1973) · Zbl 0248.35003 [10] Smoller, J.: Shock waves and reaction--diffusion equations. (1983) · Zbl 0508.35002 [11] Zhou, L.; Pao, C. V.: Asymptotic behaviour of a competition--diffusion system in population dynamics. Nonlinear analysis 6, 1163-1184 (1982) · Zbl 0522.92017 [12] De Montoni, P.; Schaffino, A.: Bifurcation of periodic solutions of periodic evolution equations in a cone. J. diff. Eqns 45, 408-430 (1982)


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# Mag_Reynolds¶

plasmapy.formulary.dimensionless.Mag_Reynolds(U: Unit("m / s"), L: Unit("m"), sigma: Unit("S / m")) -> Unit(dimensionless)

The Magnetic Reynolds number is a dimensionless quantity that estimates the relative contributions of advection and induction to magnetic diffusion in a conducting medium.

$Rm = \frac{U L}{\eta}$

where $$\eta = \frac{1}{\mu_0 \sigma}$$ and $$\mu_0$$ is the permeability of free space.

Parameters: U (Quantity) – The velocity scale of the plasma. L (Quantity) – The length scale of the plasma. sigma (Quantity) – The conductivity of the plasma. ~astropy.units.UnitsWarning – If units are not provided, SI units are assumed. TypeError – The U is not a Quantity and cannot be converted into a ~astropy.units.Quantity. UnitConversionError – If the U is not in appropriate units.

Examples

>>> import astropy.units as u
>>> sigma = 5.96e7 * u.S / u.m
>>> U = 10 * u.m / u.s
>>> L = 1 * u.cm
>>> Mag_Reynolds(U, L, sigma)
<Quantity 7.48955689>
>>> rho = 1e-8 * u.S / u.m
>>> U = 0.1 * u.m / u.s
>>> L = 0.05 * u.m
>>> Mag_Reynolds(U, L, sigma)
<Quantity 0.37447784>

Returns: Rm – Dimensionless quantity. Quantity