text:
Talk:Great Fire of London

British English
I should hope this article IS written in British standard English - it is a British topic, after all. However, there are nevertheless some creeping Americanisms within it. For example, we don't use the word 'specialty' in the UK.

Capitalization of "City"
I noticed this throughout the article- is there a special reason for the word "City" always being capitalized? Is it part of London's official name? Nothing urgent- I'm just curious. --Wafulz 02:28, 28 November 2006 (UTC)
 * The "city of London" and "City of London" aren't the same thing. It's pretty crazy, but take a look at London, Greater London and especially City of London for the gory details. --Spangineerws  (háblame)  02:43, 28 November 2006 (UTC)
 * Thank you, Spangineer, for that clarification! It is really nice when people bother to explain some local naming detail that can make the reader really confused. I wonder if there is a good, non-intrusive way of working that fact into the text of the article itself? Maybe the introduction? All in all, an absolutely beautiful article, especially considering the grim story it has to tell. Big thanks to all who did this! --Ronja Addams-Moring 07:59, 1 December 2006 (UTC)
 * Sorry, I managed to miss the expression "the City proper—the area bounded by the City wall and the river Thames" - that makes it quite clear. --Ronja Addams-Moring 16:32, 1 December 2006 (UTC)
 * It's not 'crazy', as you put it. It is a semantic and ultimately legal distinction. 82.40.43.135 (talk) 10:32, 10 March 2022 (UTC)

Capping "City" still doesn't make a lot of sense in spite of this answer. "The city" would serve as well in most cases, and WP avoids unnecessary capitalization, per MOS:CAPS. Dicklyon (talk) 07:25, 7 December 2021 (UTC)
 * I tend to agree, but I'd like to discuss this before making the change. No idea why this discussion was abandoned 15 years ago, but this article has more problems right now. I'll put "unnecessary capitalization" on the list. Renerpho (talk) 16:32, 8 December 2021 (UTC)
 * I downcased to "the city wall"; in "the City proper", I suppose it means specifically "the City of London itself", so I can accept that. In many cases, where "City" if just short for "City of London" specifically, I'm OK with it, but we should look for and fix any more generic uses. Dicklyon (talk) 17:28, 8 December 2021 (UTC)
 * Thanks, . The relevant section of the City of London article, Early modern period, is confusing in that regard, as it sometimes writes "City", sometimes "city"; and so does the rest of that article. For example, the lead says "The city is now only a tiny part of the metropolis of London [...]; however, the City of London is not a London borough", which leaves me scratching my head. Renerpho (talk) 17:43, 8 December 2021 (UTC)
 * Can you speak English? 'Capping' means putting a limit on something, usually measurables like expenditure.  What you mean is 'Capitalizing'. 82.40.43.135 (talk) 10:38, 10 March 2022 (UTC)

Panorama used in article
I've recently been reading about some of the pre-1666 panorama artworks of London, some mentioned at Panorama of London. The one used in this article is the Visscher panorama from around 1600. There is a later panorama, the Hollar panorama of 1647, which is at the image file: Long view of London From Bankside (no article yet, but it would be possible to have one). The Hollar panorama is described here in A Descriptive Catalogue of the Etched Work of Wenceslaus Hollar 1607-1677 (2002) by Richard Pennington. Following the description of the original state, Pennington goes on to describe some of the later states, including some versions that show details from after the fire. Pennington also quotes from another work (Printed maps of London circa 1553-1850 by Darlington and Howgego) that says the Hollar panorama is "the most important ... and the most accurate of the pre-Fire panoramas". Given the differences between the Hollar panorama and the Visscher panorama (particularly the use of a single perspective in the Hollar panorama), maybe a crop of the Hollar panorama could be used here instead, or at least referred to for a more accurate representation? It might also help to make clear (I'd not fully realised this before) that the Visscher panorama is from several different viewpoints, and doesn't quite show what the actual view looked like (i.e. the bend in the river is 'flattened' somewhat in the Visscher panorama). I suppose what is needed is to find someone able to crop the massive file we have of the Hollar panorama, and see if the result is worth using or not. Carcharoth (talk) 11:16, 14 September 2013 (UTC)

Why is this article protected?
It appears this article has been protected for several years. If Wikipedia policy has changed, please indicate, but my understanding is articles should only be protected from editing if there is consistent and ongoing vandalism, edit warring, etc. Closest I can see from looking at the history is some editor claimed to have never heard of Pepys' Diary, though that may have been in jest. 68.146.52.234 (talk) 20:01, 1 November 2014 (UTC)

kkjxjx# — Preceding unsigned comment added by 90.209.3.97 (talk) 16:18, 29 April 2015 (UTC)

There's a comma after "however" missing here: "However an enormous stroke of bad luck meant". — Preceding unsigned comment added by 89.133.180.198 (talk) 09:12, 14 August 2015 (UTC)

Close to not meeting featured article criteria
The article has multiple issues at the moment, including a lack of citations, and some outdated information, contradicting research published in 2016. I'll give it a few weeks before opening a featured article review. Renerpho (talk) 20:40, 25 September 2020 (UTC)
 * I've addressed these issues; just waiting on a book to come in for some additional touchups. Nikkimaria (talk) 20:19, 26 September 2020 (UTC)

 * Hi, I was just checking this over during FA sweeps and it seems to me that the issues might be deeper than just a few touch-ups. It seems that the Field book (London, Londoners and the Great Fire of 1666: Disaster and Recovery) is credited by reviews by bringing a new perspective to this topic by examining it as social history. (There is also other new research, eg:) The citation format is also inconsistent and I wonder how diaries and other primary sources from 1666 can be considered "high-quality RS" according to the FA criteria. Renerpho might be on the right track when calling for a FAR. (t · c)  buidhe  04:39, 29 November 2021 (UTC)
 * Hello and, and thanks. I also think that these problems are still considerable (despite the efforts by Nikkimaria back in September 2020), and go beyond what a single editor can be expected to handle. More help is needed. Hopefully we can get the article back into shape with an FAR. Renerpho (talk) 05:30, 29 November 2021 (UTC)

Featured article review
The article has been nominated for FAR. Please comment here and help improving the article. Renerpho (talk) 06:13, 29 November 2021 (UTC)

Images
I'd like to work on fixing some of the image issues (fixed pixels, etc), but first, a question about what images to include. Could we stop fretting about image sizes, when we shouldn’t be using fixed pixel sizes anyway? I was planning to address all of the images, using the more correct upright= parameter, once we finish with the text. All of the images will need to be adjusted, and that will not be done by using fixed pixel sizes. Sandy Georgia (Talk)  03:07, 26 December 2021 (UTC)
 * Bish originally had an image of John Evelyn and an image of James II. Do you have a sense of why they were removed, or if we have images of them that meet policy?
 * Also, regarding the suggestion in this section; I'm not able to make much of those images, wonder what others think.
 * Archived, Sandy Georgia (Talk)  17:04, 11 December 2021 (UTC)
 * And should this section be archived, or is there something there to work on? Sandy Georgia  (Talk)  04:17, 11 December 2021 (UTC)
 * At the moment we have IMO too many images for the available content. I would be inclined to deal with the content first and then consider if there is room to include additional images. Nikkimaria (talk) 15:57, 11 December 2021 (UTC)
 * Got it. The images that show the burned areas in pink breach MOS:ACCIM, since they depend on color only. I was thinking of asking Guerillero if he could code the pink areas somehow, if you’re agreeable to that. Sandy Georgia  (Talk)  17:30, 11 December 2021 (UTC)
 * Sounds good. Nikkimaria (talk) 18:39, 11 December 2021 (UTC)

 * Thank you,, for upgrading the maps to account for color blindedness and color issues per MOS:ACCIM; you're the best! Sandy Georgia (Talk)  15:48, 30 December 2021 (UTC)

Comprehensive
Thoughts on whether these is anything to uncover in these sections above, or should they be archived? Sandy Georgia (Talk)  04:20, 11 December 2021 (UTC)
 * Talk:Great Fire of London/Archive 1
 * Talk:Great Fire of London/Archive 1
 * I think they can be archived, since they don't provide any sourcing for proposed additions. Nikkimaria (talk) 15:57, 11 December 2021 (UTC)

 * Five ways the Great Fire changed London, BBC.
 * How London might have looked: five masterplans after the great fire of 1666, The Guardian
 * It appears that the Great Fire of London impacted building codes worldwide:, , , and see NYT report after Grenfell Tower fire. And these from Buidhe (see above):  “There is also other new research, eg:” Sandy Georgia  (Talk)  17:06, 11 December 2021 (UTC)

 * does this help? It seems to have a lot to say, not sure what the source is, iPad typing from the car. Sandy Georgia (Talk)  16:39, 11 December 2021 (UTC)

Aftermath
It appears from available sourcing that the Aftermath section could do with significant expansion and therefore subdivision, but I'm struggling a bit with what that might look like. Here are the pieces currently in the article and those with significant sourcing to support additions that I've found so far: I don't think each of these warrants its own subsection, but does anyone have thoughts? Pinging Sandy, Buidhe, Renerpho. Nikkimaria (talk) 21:27, 11 December 2021 (UTC)
 * blaming foreigners
 * reactions from abroad
 * legal proceedings (Fire Court, resettlement proclamation, committee of inquiry)
 * rebuilding plans
 * impact on building codes
 * development of insurance industry
 * impact on population
 * political impact
 * economic impact
 * cultural responses (including pop culture)
 * monuments
 * plague myth
 * Reactions (blaming foreigners, from abroad, legal); Rebuilding plans (include Wren, Evelyn, monuments, St. Paul’s etc); Impact (population, political, economic, insurance industry, building codes, fire brigades); Cultural responses (myths, pop culture). Perhaps eliminate “Aftermath” and have four new level I sections, as there is so much. But that’s just a very quick glance after a very long day. Sandy Georgia  (Talk)  03:38, 12 December 2021 (UTC)

remark on a minor change
Hi folks,

I removed some instruction from the captions for newspaper articles, specifically "Click on the image to enlarge and read". I imagine I understand why these were placed there--normally when we have images of newspaper articles it is under fair use and navigating to the image won't actually enlarge it. However the instruction itself is a self-reference to avoid (WP:CLICKHERE--we say "the image" and not "here" but I still feel we should avoid it) and the term "click" no longer describes well the action taken to navigate somewhere (a reader may very well come across the article having never used a mouse!). Protonk (talk) 19:33, 2 September 2022 (UTC)

Thomas Dagger
Interesting article on the BBC today about the identity of the person who raised the alarm. Possibly worth adding - or at least considering. Cheers - SchroCat (talk) 11:08, 1 September 2023 (UTC)

 * Definitely worth adding. Additional details in Londonist and The Guardian. Sadly I cannot find a research paper that goes with it. Renerpho (talk) 11:19, 1 September 2023 (UTC)

Layout problems
In my web browser, the first few words of the article are squeezed into a very narrow vertical channel to the left of the first two big pictures, then there is a huge vertical gap below before the text continues below the pictures, at the word ‘London’.

Could someone fix this with CSS or wiki markup wizardry? CecilWard (talk) 20:02, 13 September 2023 (UTC)

 * I don't see this on my screen. Can you tell us more about your setup? What skin are you using? Nikkimaria (talk) 01:29, 14 September 2023 (UTC)

title: Talk:Great Fire of London


text:
Global warming potential

Global warming potential (GWP) is an index to measure how much infrared thermal radiation a greenhouse gas would absorb over a given time frame after it has been added to the atmosphere (or emitted to the atmosphere). The GWP makes different greenhouse gases comparable with regard to their "effectiveness in causing radiative forcing". It is expressed as a multiple of the radiation that would be absorbed by the same mass of added carbon dioxide, which is taken as a reference gas. Therefore, the GWP has a value of 1 for. For other gases it depends on how strongly the gas absorbs infrared thermal radiation, how quickly the gas leaves the atmosphere, and the time frame being considered.

For example, methane has a GWP over 20 years (GWP-20) of 81.2 meaning that, for example, a leak of a tonne of methane is equivalent to emitting 81.2 tonnes of carbon dioxide measured over 20 years. As methane has a much shorter atmospheric lifetime than carbon dioxide, its GWP is much less over longer time periods, with a GWP-100 of 27.9 and a GWP-500 of 7.95.

The carbon dioxide equivalent (e or eq or -e or -eq) can be calculated from the GWP. For any gas, it is the mass of that would warm the earth as much as the mass of that gas. Thus it provides a common scale for measuring the climate effects of different gases. It is calculated as GWP times mass of the other gas.

Definition
The global warming potential (GWP) is defined as an "index measuring the radiative forcing following an emission of a unit mass of a given substance, accumulated over a chosen time horizon, relative to that of the reference substance, carbon dioxide (CO2). The GWP thus represents the combined effect of the differing times these substances remain in the atmosphere and their effectiveness in causing radiative forcing."

In turn, radiative forcing is a scientific concept used to quantify and compare the external drivers of change to Earth's energy balance. Radiative forcing is the change in energy flux in the atmosphere caused by natural or anthropogenic factors of climate change as measured in watts per meter squared.

Values
The global warming potential (GWP) depends on both the efficiency of the molecule as a greenhouse gas and its atmospheric lifetime. GWP is measured relative to the same mass of and evaluated for a specific timescale. Thus, if a gas has a high (positive) radiative forcing but also a short lifetime, it will have a large GWP on a 20-year scale but a small one on a 100-year scale. Conversely, if a molecule has a longer atmospheric lifetime than its GWP will increase when the timescale is considered. Carbon dioxide is defined to have a GWP of 1 over all time periods.

Methane has an atmospheric lifetime of 12 ± 2 years. The 2021 IPCC report lists the GWP as 83 over a time scale of 20 years, 30 over 100 years and 10 over 500 years. A 2014 analysis, however, states that although methane's initial impact is about 100 times greater than that of, because of the shorter atmospheric lifetime, after six or seven decades, the impact of the two gases is about equal, and from then on methane's relative role continues to decline. The decrease in GWP at longer times is because methane decomposes to water and through chemical reactions in the atmosphere. Similarly the third most important GHG, nitrous oxide (N2O), is a common gas emitted through the denitrification part of the nitrogen cycle. It has a lifetime of 109 years and an even higher GWP level running at 273 over 20 and 100 years.

Examples of the atmospheric lifetime and GWP relative to for several greenhouse gases are given in the following table: Estimates of GWP values over 20, 100 and 500 years are periodically compiled and revised in reports from the Intergovernmental Panel on Climate Change. The most recent report is the IPCC Sixth Assessment Report (Working Group I) from 2023.

The IPCC lists many other substances not shown here. Some have high GWP but only a low concentration in the atmosphere.

The values given in the table assume the same mass of compound is analyzed; different ratios will result from the conversion of one substance to another. For instance, burning methane to carbon dioxide would reduce the global warming impact, but by a smaller factor than 25:1 because the mass of methane burned is less than the mass of carbon dioxide released (ratio 1:2.74). For a starting amount of 1 tonne of methane, which has a GWP of 25, after combustion there would be 2.74 tonnes of, each tonne of which has a GWP of 1. This is a net reduction of 22.26 tonnes of GWP, reducing the global warming effect by a ratio of 25:2.74 (approximately 9 times).

Earlier values from 2007
The values provided in the table below are from 2007 when they were published in the IPCC Fourth Assessment Report. These values are still used (as of 2020) for some comparisons.

Importance of time horizon
A substance's GWP depends on the number of years (denoted by a subscript) over which the potential is calculated. A gas which is quickly removed from the atmosphere may initially have a large effect, but for longer time periods, as it has been removed, it becomes less important. Thus methane has a potential of 25 over 100 years (GWP100 = 25) but 86 over 20 years (GWP20 = 86); conversely sulfur hexafluoride has a GWP of 22,800 over 100 years but 16,300 over 20 years (IPCC Third Assessment Report). The GWP value depends on how the gas concentration decays over time in the atmosphere. This is often not precisely known and hence the values should not be considered exact. For this reason when quoting a GWP it is important to give a reference to the calculation.

The GWP for a mixture of gases can be obtained from the mass-fraction-weighted average of the GWPs of the individual gases.

Commonly, a time horizon of 100 years is used by regulators.

Water vapour
Water vapour does contribute to anthropogenic global warming, but as the GWP is defined, it is negligible for H2O: an estimate gives a 100-year GWP between -0.001 and 0.0005.

H2O can function as a greenhouse gas because it has a profound infrared absorption spectrum with more and broader absorption bands than. Its concentration in the atmosphere is limited by air temperature, so that radiative forcing by water vapour increases with global warming (positive feedback). But the GWP definition excludes indirect effects. GWP definition is also based on emissions, and anthropogenic emissions of water vapour (cooling towers, irrigation) are removed via precipitation within weeks, so its GWP is negligible.

Calculation methods
When calculating the GWP of a greenhouse gas, the value depends on the following factors: A high GWP correlates with a large infrared absorption and a long atmospheric lifetime. The dependence of GWP on the wavelength of absorption is more complicated. Even if a gas absorbs radiation efficiently at a certain wavelength, this may not affect its GWP much, if the atmosphere already absorbs most radiation at that wavelength. A gas has the most effect if it absorbs in a "window" of wavelengths where the atmosphere is fairly transparent. The dependence of GWP as a function of wavelength has been found empirically and published as a graph.
 * the absorption of infrared radiation by the given gas
 * the time horizon of interest (integration period)
 * the atmospheric lifetime of the gas

Because the GWP of a greenhouse gas depends directly on its infrared spectrum, the use of infrared spectroscopy to study greenhouse gases is centrally important in the effort to understand the impact of human activities on global climate change.

Just as radiative forcing provides a simplified means of comparing the various factors that are believed to influence the climate system to one another, global warming potentials (GWPs) are one type of simplified index based upon radiative properties that can be used to estimate the potential future impacts of emissions of different gases upon the climate system in a relative sense. GWP is based on a number of factors, including the radiative efficiency (infrared-absorbing ability) of each gas relative to that of carbon dioxide, as well as the decay rate of each gas (the amount removed from the atmosphere over a given number of years) relative to that of carbon dioxide.

The radiative forcing capacity (RF) is the amount of energy per unit area, per unit time, absorbed by the greenhouse gas, that would otherwise be lost to space. It can be expressed by the formula:

$$\mathit{RF} = \sum_{i=1}^{100} \text{abs}_i \cdot F_i / \left(\text{l} \cdot \text{d}\right)$$ where the subscript i represents a wavenumber interval of 10 inverse centimeters. Absi represents the integrated infrared absorbance of the sample in that interval, and Fi represents the RF for that interval.

The Intergovernmental Panel on Climate Change (IPCC) provides the generally accepted values for GWP, which changed slightly between 1996 and 2001, except for methane, which had its GWP almost doubled. An exact definition of how GWP is calculated is to be found in the IPCC's 2001 Third Assessment Report. The GWP is defined as the ratio of the time-integrated radiative forcing from the instantaneous release of 1 kg of a trace substance relative to that of 1 kg of a reference gas:

$$\mathit{GWP} \left(x\right) = \frac{a_x}{a_r} \frac{\int_0^{\mathit{TH}} [x](t)\, dt} {\int_0^{\mathit{TH}} [r](t)\, dt}$$ where TH is the time horizon over which the calculation is considered; ax is the radiative efficiency due to a unit increase in atmospheric abundance of the substance (i.e., Wm−2 kg−1) and [x](t) is the time-dependent decay in abundance of the substance following an instantaneous release of it at time t=0. The denominator contains the corresponding quantities for the reference gas (i.e. ). The radiative efficiencies ax and ar are not necessarily constant over time. While the absorption of infrared radiation by many greenhouse gases varies linearly with their abundance, a few important ones display non-linear behaviour for current and likely future abundances (e.g.,, CH4, and N2O). For those gases, the relative radiative forcing will depend upon abundance and hence upon the future scenario adopted.

Since all GWP calculations are a comparison to which is non-linear, all GWP values are affected. Assuming otherwise as is done above will lead to lower GWPs for other gases than a more detailed approach would. Clarifying this, while increasing has less and less effect on radiative absorption as ppm concentrations rise, more powerful greenhouse gases like methane and nitrous oxide have different thermal absorption frequencies to  that are not filled up (saturated) as much as, so rising ppms of these gases are far more significant.

Carbon dioxide equivalent
Carbon dioxide equivalent (e or eq or -e) of a quantity of gas is calculated from its GWP. For any gas, it is the mass of which would warm the earth as much as the mass of that gas. Thus it provides a common scale for measuring the climate effects of different gases. It is calculated as GWP multiplied by mass of the other gas. For example, if a gas has GWP of 100, two tonnes of the gas have e of 200 tonnes, and 9 tonnes of the gas has e of 900 tonnes.

On a global scale, the warming effects of one or more greenhouse gases in the atmosphere can also be expressed as an equivalent atmospheric concentration of. e can then be the atmospheric concentration of which would warm the earth as much as a particular concentration of some other gas or of all gases and aerosols in the atmosphere. For example, e of 500 parts per million would reflect a mix of atmospheric gases which warm the earth as much as 500 parts per million of would warm it. Calculation of the equivalent atmospheric concentration of of an atmospheric greenhouse gas or aerosol is more complex and involves the atmospheric concentrations of those gases, their GWPs, and the ratios of their molar masses to the molar mass of.

e calculations depend on the time-scale chosen, typically 100 years or 20 years, since gases decay in the atmosphere or are absorbed naturally, at different rates.

The following units are commonly used:
 * By the UN climate change panel (IPCC): billion metric tonnes = n×109 tonnes of equivalent (Gteq)
 * In industry: million metric tonnes of carbon dioxide equivalents (MMTCDE) and MMT eq.
 * For vehicles: grams of carbon dioxide equivalent per mile (ge/mile) or per kilometer (ge/km)

For example, the table above shows GWP for methane over 20 years at 86 and nitrous oxide at 289, so emissions of 1 million tonnes of methane or nitrous oxide are equivalent to emissions of 86 or 289 million tonnes of carbon dioxide, respectively.

Use in Kyoto Protocol and for reporting to UNFCCC
Under the Kyoto Protocol, in 1997 the Conference of the Parties standardized international reporting, by deciding (see decision number 2/CP.3) that the values of GWP calculated for the IPCC Second Assessment Report were to be used for converting the various greenhouse gas emissions into comparable equivalents.

After some intermediate updates, in 2013 this standard was updated by the Warsaw meeting of the UN Framework Convention on Climate Change (UNFCCC, decision number 24/CP.19) to require using a new set of 100-year GWP values. They published these values in Annex III, and they took them from the IPCC Fourth Assessment Report, which had been published in 2007. Those 2007 estimates are still used for international comparisons through 2020, although the latest research on warming effects has found other values, as shown in the tables above.

Though recent reports reflect more scientific accuracy, countries and companies continue to use the IPCC Second Assessment Report (SAR) and IPCC Fourth Assessment Report values for reasons of comparison in their emission reports. The IPCC Fifth Assessment Report has skipped the 500-year values but introduced GWP estimations including the climate-carbon feedback (f) with a large amount of uncertainty.

Other metrics to compare greenhouse gases
The Global Temperature change Potential (GTP) is another way to compare gases. While GWP estimates infrared thermal radiation absorbed, GTP estimates the resulting rise in average surface temperature of the world, over the next 20, 50 or 100 years, caused by a greenhouse gas, relative to the temperature rise which the same mass of would cause. Calculation of GTP requires modelling how the world, especially the oceans, will absorb heat. GTP is published in the same IPCC tables with GWP.

GWP* has been proposed to take better account of short-lived climate pollutants (SLCP) such as methane, relating a change in the rate of emissions of SLCPs to a fixed quantity of. However GWP* has itself been criticised both for its suitability as a metric and for inherent design features which can perpetuate injustices and inequity.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Global warming potential


text:
Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.

Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.

There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies that do not come from topological spaces.

The term "Grothendieck topology" has changed in meaning. In it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.

Overview
André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they define. His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it.

In the early 1960s, Alexander Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry. He used étale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of étale coverings mimicked those of open immersions, and that consequently it was possible to make constructions that imitated the cohomology functor $$H^1$$. Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory that he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use étale coverings instead. Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from.

Motivation
The classical definition of a sheaf begins with a topological space $$X$$. A sheaf associates information to the open sets of $$X$$. This information can be phrased abstractly by letting $$O(X)$$ be the category whose objects are the open subsets $$U$$ of $$X$$ and whose morphisms are the inclusion maps $$V\rightarrow U$$ of open sets $$U$$ and $$V$$ of $$X$$. We will call such maps open immersions, just as in the context of schemes. Then a presheaf on $$X$$ is a contravariant functor from $$O(X)$$ to the category of sets, and a sheaf is a presheaf that satisfies the gluing axiom (here including the separation axiom). The gluing axiom is phrased in terms of pointwise covering, i.e., $$\{U_i\}$$ covers $$U$$ if and only if $$ \bigcup_i U_i = U$$. In this definition, $$U_i$$ is an open subset of $$X$$. Grothendieck topologies replace each $$U_i$$ with an entire family of open subsets; in this example, $$U_i$$ is replaced by the family of all open immersions $$V_{ij} \to U_i$$. Such a collection is called a sieve. Pointwise covering is replaced by the notion of a covering family; in the above example, the set of all $$\{V_{ij} \to U_i\}_j$$ as $$i$$ varies is a covering family of $$U$$. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions that describe other properties of the space $$X$$.

Sieves
In a Grothendieck topology, the notion of a collection of open subsets of U stable under inclusion is replaced by the notion of a sieve. If c is any given object in C, a sieve on c is a subfunctor of the functor Hom(−, c); (this is the Yoneda embedding applied to c). In the case of O(X), a sieve S on an open set U selects a collection of open subsets of U that is stable under inclusion. More precisely, consider that for any open subset V of U, S(V) will be a subset of Hom(V, U), which has only one element, the open immersion V → U. Then V will be considered "selected" by S if and only if S(V) is nonempty. If W is a subset of V, then there is a morphism S(V) → S(W) given by composition with the inclusion W → V. If S(V) is non-empty, it follows that S(W) is also non-empty.

If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by f$^\ast$S. It is defined as the fibered product S ×Hom(−, X) Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, f$^\ast$S(Z) = { g: Z → Y | fg $$\in$$S(Z) }, and f$^\ast$S inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩W}.

Grothendieck topology
A Grothendieck topology J on a category C is a collection, for each object c of C, of distinguished sieves on c, denoted by J(c) and called covering sieves of c. This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve S on an open set U in O(X) will be a covering sieve if and only if the union of all the open sets V for which S(V) is nonempty equals U; in other words, if and only if S gives us a collection of open sets that cover U in the classical sense.

Axioms
The conditions we impose on a Grothendieck topology are:

 * (T 1) (Base change) If S is a covering sieve on X, and f: Y → X is a morphism, then the pullback f$\ast$S is a covering sieve on Y.
 * (T 2) (Local character) Let S be a covering sieve on X, and let T be any sieve on X.  Suppose that for each object Y of C and each arrow f: Y → X in S(Y), the pullback sieve f$\ast$T is a covering sieve on Y.  Then T is a covering sieve on X.
 * (T 3) (Identity) Hom(−, X) is a covering sieve on X for any object X in C.

The base change axiom corresponds to the idea that if {Ui} covers U, then {Ui ∩ V} should cover U ∩ V. The local character axiom corresponds to the idea that if {Ui} covers U and {Vij}j $\in$J i covers Ui for each i, then the collection {Vij} for all i and j should cover U. Lastly, the identity axiom corresponds to the idea that any set is covered by itself via the identity map.

Grothendieck pretopologies
In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category C contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are:

 * (PT 0) (Existence of fibered products) For all objects X of C, and for all morphisms X0 → X that appear in some covering family of X, and for all morphisms Y → X, the fibered product X0 ×X Y exists.
 * (PT 1) (Stability under base change) For all objects X of C, all morphisms Y → X, and all covering families {Xα → X}, the family {Xα ×X Y → Y} is a covering family.
 * (PT 2) (Local character) If {Xα → X} is a covering family, and if for all α, {Xβα → Xα} is a covering family, then the family of composites {Xβα → Xα → X} is a covering family.
 * (PT 3) (Isomorphisms) If f: Y → X is an isomorphism, then {f} is a covering family.

For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology.

For categories with fibered products, there is a converse. Given a collection of arrows {Xα → X}, we construct a sieve S by letting S(Y) be the set of all morphisms Y → X that factor through some arrow Xα → X. This is called the sieve generated by {Xα → X}. Now choose a topology. Say that {Xα → X} is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology.

(PT 3) is sometimes replaced by a weaker axiom:

 * (PT 3') (Identity) If 1X : X → X is the identity arrow, then {1X} is a covering family.

(PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism Y → X is Hom(−, X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent.

Sites and sheaves
Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site.

A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(−, X), F) → Hom(S, F), induced by the inclusion of S into Hom(−, X), is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S. A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on C is the topos defined by the site (C, J).

Using the Yoneda lemma, it is possible to show that a presheaf on the category O(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.

Sheaves on a pretopology have a particularly simple description: For each covering family {Xα → X}, the diagram

 * $$F(X) \rightarrow \prod_{\alpha\in A} F(X_\alpha) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{\alpha,\beta \in A} F(X_\alpha\times_X X_\beta)$$

must be an equalizer. For a separated presheaf, the first arrow need only be injective.

Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.

The discrete and indiscrete topologies
Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, also known as the coarse or chaotic topology, we declare only the sieves of the form Hom(−, X) to be covering sieves. The indiscrete topology is generated by the pretopology that has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf.

The canonical topology
Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−, X), is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms in C. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical. Subcanonical sites are exactly the sites for which every presheaf of the form Hom(−, X) is a sheaf. Most sites encountered in practice are subcanonical.

Small site associated to a topological space
We repeat the example that we began with above. Let X be a topological space. We defined O(X) to be the category whose objects are the open sets of X and whose morphisms are inclusions of open sets. Note that for an open set U and a sieve S on U, the set S(V) contains either zero or one element for every open set V. The covering sieves on an object U of O(X) are those sieves S satisfying the following condition: This notion of cover matches the usual notion in point-set topology.
 * If W is the union of all the sets V such that S(V) is non-empty, then W = U.

This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {Vα $$\sube$$ U} is a covering family if and only if the union $$\cup$$Vα equals U. This site is called the 'small site associated to a topological space X''.

Big site associated to a topological space
Let Spc be the category of all topological spaces. Given any family of functions {uα : Vα → X}, we say that it is a surjective family or that the morphisms uα are jointly surjective if $$\cup$$ uα(Vα) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of whose members are open immersions. Let S be a sieve on Spc. S is a covering sieve for this topology if and only if:
 * For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
 * If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

Fix a topological space X. Consider the comma category Spc/X of topological spaces with a fixed continuous map to X. The topology on Spc induces a topology on Spc/X. The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by Jean Giraud.

The big and small sites of a manifold
Let M be a manifold. M has a category of open sets O(M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product U ×M V is the open set U ∩ V, which is still in O(M). This means that the topology on O(M) is defined by a pretopology, the same pretopology as before.

Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above. Notice that to satisfy (PT 0), we need to check that for any continuous map of manifolds X → Y and any open subset U of Y, the fibered product U ×Y X is in Mfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist in Mfd because the preimage of a smooth map at a critical value need not be a manifold.

Topologies on the category of schemes
The category of schemes, denoted Sch, has a tremendous number of useful topologies. A complete understanding of some questions may require examining a scheme using several different topologies. All of these topologies have associated small and big sites. The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology. The small site over a given scheme is formed by only taking the objects and morphisms that are part of a cover of the given scheme.

The most elementary of these is the Zariski topology. Let X be a scheme. X has an underlying topological space, and this topological space determines a Grothendieck topology. The Zariski topology on Sch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions. The covering sieves S for Zar are characterized by the following two properties: Despite their outward similarities, the topology on Zar is not the restriction of the topology on Spc! This is because there are morphisms of schemes that are topologically open immersions but that are not scheme-theoretic open immersions. For example, let A be a non-reduced ring and let N be its ideal of nilpotents. The quotient map A → A/N induces a map Spec A/N → Spec A, which is the identity on underlying topological spaces. To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do. In fact, this map is a closed immersion.
 * For all Y and every morphism f : Y → X in S(Y), there exists a V and a g : V → X such that g is an open immersion, g is in S(V), and f factors through g.
 * If W is the union of all the sets f(Y), where f : Y → X is in S(Y), then W = X.

The étale topology is finer than the Zariski topology. It was the first Grothendieck topology to be closely studied. Its covering families are jointly surjective families of étale morphisms. It is finer than the Nisnevich topology, but neither finer nor coarser than the cdh and l′ topologies.

There are two flat topologies, the fppf topology and the fpqc topology. fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined to be a family that is a cover on Zariski open subsets. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.

Grothendieck introduced crystalline cohomology to study the p-torsion part of the cohomology of characteristic p varieties. In the crystalline topology, which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together with divided power structures. Crystalline sites are examples of sites with no final object.

Continuous and cocontinuous functors
There are two natural types of functors between sites. They are given by functors that are compatible with the topology in a certain sense.

Continuous functors
If (C, J) and (D, K) are sites and u : C → D is a functor, then u is continuous if for every sheaf F on D with respect to the topology K, the presheaf Fu is a sheaf with respect to the topology J. Continuous functors induce functors between the corresponding topoi by sending a sheaf F to Fu. These functors are called pushforwards. If $$\tilde C$$ and $$\tilde D$$ denote the topoi associated to C and D, then the pushforward functor is $$u_s : \tilde D \to \tilde C$$.

us admits a left adjoint us called the pullback. us need not preserve limits, even finite limits.

In the same way, u sends a sieve on an object X of C to a sieve on the object uX of D. A continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then u is continuous if and only if it sends covering sieves to covering sieves and if and only if it sends covering families to covering families. In general, it is not sufficient for u to send covering sieves to covering sieves (see SGA IV 3, Exemple 1.9.3).

Cocontinuous functors
Again, let (C, J) and (D, K) be sites and v : C → D be a functor. If X is an object of C and R is a sieve on vX, then R can be pulled back to a sieve S as follows: A morphism f : Z → X is in S if and only if v(f) : vZ → vX is in R. This defines a sieve. v is cocontinuous if and only if for every object X of C and every covering sieve R of vX, the pullback S of R is a covering sieve on X.

Composition with v sends a presheaf F on D to a presheaf Fv on C, but if v is cocontinuous, this need not send sheaves to sheaves. However, this functor on presheaf categories, usually denoted $$\hat v^*$$, admits a right adjoint $$\hat v_*$$. Then v is cocontinuous if and only if $$\hat v_*$$ sends sheaves to sheaves, that is, if and only if it restricts to a functor $$v_* : \tilde C \to \tilde D$$. In this case, the composite of $$\hat v^*$$ with the associated sheaf functor is a left adjoint of v* denoted v*. Furthermore, v* preserves finite limits, so the adjoint functors v* and v* determine a geometric morphism of topoi $$\tilde C \to \tilde D$$.

Morphisms of sites
A continuous functor u : C → D is a morphism of sites D → C (not C → D) if us preserves finite limits. In this case, us and us determine a geometric morphism of topoi $$\tilde C \to \tilde D$$. The reasoning behind the convention that a continuous functor C → D is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces. A continuous map of topological spaces X → Y determines a continuous functor O(Y) → O(X). Since the original map on topological spaces is said to send X to Y, the morphism of sites is said to as well.

A particular case of this happens when a continuous functor admits a left adjoint. Suppose that u : C → D and v : D → C are functors with u right adjoint to v. Then u is continuous if and only if v is cocontinuous, and when this happens, us is naturally isomorphic to v* and us is naturally isomorphic to v*. In particular, u is a morphism of sites.

title: Grothendieck topology


text:
Talk:Gomoku

Solution?
Where can the solution be found? Viktor Allis is uncontactable.

Answer: Solutions were found by Computer, so it's hard to show you all positions. But you can trust this because there must be at least 10000 players around the world, who can prove a surewin on board-size 15x15 and bigger. I think there's a surewin on 13x13 too, but nobody would take time to prove it, because it's not interesting. Smaller board sizes would reduce the branch-factor and therefor less complex.

title: Talk:Gomoku


text:
Greens

Greens may refer to:
 * Leaf vegetables such as collard greens, mustard greens, spring greens, winter greens, spinach, etc.

Supranational

 * Green politics
 * Green party, political parties adhering to Green politics
 * Global Greens
 * European Green Party

Established parties

 * Green Party (disambiguation)
 * The Greens (disambiguation)

Other

 * Green Party of the United States
 * Australian Greens
 * Green armies, peasant-based groups participating in the Russian Civil War of 1917–23
 * Green Movement (disambiguation)
 * The Greens, an early 20th-century nationalist and separatist political and military movement in Montenegro
 * Greens, a political faction and associated chariot-racing team in the Byzantine empire; involved in the deadly Nika riots of 532

Places

 * Greens Farms, Connecticut, United States
 * Greens Ledge Light on Long Island Sound, United States
 * Greens Norton village in Northamptonshire, England
 * Greens Pool beach on the south coast of Western Australia
 * Greens Restaurant in San Francisco, California, United States
 * Millennium Greens and Doorstep Greens, locally owned and managed public spaces in England
 * Breckenridge Greens (Edmonton), Potter Greens, Suder Greens - neighbourhoods in Alberta, Canada

In sport

 * Ashland Greens basketball team in Pennsylvania, United States
 * Bentleigh Greens soccer team in a suburb of Melbourne, Victoria, Australia
 * Baywood Greens public golf club in Long Neck, Delaware, United States
 * Greens Worldwide sports management company
 * Greens (golf), the very closely mown areas of a golf course around the holes, maintained by a Greenskeeper

Manufacturing

 * A British brand of railway locomotives, road rollers and other products; see Thomas Green & Son

title: Greens


text:
Ghost in the Shell

Ghost in the Shell is a Japanese cyberpunk media franchise based on the seinen manga series of the same name written and illustrated by Masamune Shirow. The manga, first serialized in 1989 under the subtitle of The Ghost in the Shell, and later published as its own tankōbon volumes by Kodansha, told the story of the fictional counter-cyberterrorist organization Public Security Section 9, led by protagonist Major Motoko Kusanagi, and is set in mid-21st century Japan.

Animation studio Production I.G has produced several anime adaptations of the series. These include the 1995 film of the same name and its 2004 sequel, Ghost in the Shell 2: Innocence; the 2002 television series, Ghost in the Shell: Stand Alone Complex, and its 2020 follow-up, Ghost in the Shell: SAC_2045; and the Ghost in the Shell: Arise original video animation (OVA) series. In addition, an American-produced live-action film was released on March 31, 2017.

Title
The original editor Koichi Yuri says: At first, Ghost in the Shell came from Shirow, but when Yuri asked "something more flashy", Shirow came up with "攻殻機動隊 Koukaku Kidou Tai (Shell Squad)" for Yuri. But Shirow was attached to including "Ghost in the Shell" as well even if in smaller type.

Setting
Primarily set in the mid-twenty-first century in the fictional Japanese city of Niihama, Niihama Prefecture (新浜県新浜市), otherwise known as New Port City (ニューポートシティ), the manga and the many anime adaptations follow the members of Public Security Section 9, a task-force consisting of various professionals skilled at solving and preventing crime, mostly with some sort of police background. Political intrigue and counter-terrorism operations are standard fare for Section 9, but the various actions of corrupt officials, companies, and cyber-criminals in each scenario are unique and require the diverse skills of Section 9's staff to prevent a series of incidents from escalating. In this post-cyberpunk iteration of a possible future, computer technology has advanced to the point that many members of the public possess cyberbrains, technology that allows them to interface their biological brain with various networks. The level of cyberization varies from simple minimal interfaces to almost complete replacement of the brain with cybernetic parts, in cases of severe trauma. This can also be combined with various levels of prostheses, with a fully prosthetic body enabling a person to become a cyborg. The main character of Ghost in the Shell, Major Motoko Kusanagi, is such a cyborg, having had a terrible accident befall her as a child that ultimately required her to use a full-body prosthesis to house her cyberbrain. This high level of cyberization, however, opens the brain up to attacks from highly skilled hackers, with the most dangerous being those who will hack a person to bend to their whims.

Original manga
The original Ghost in the Shell manga ran in Japan from April 1989 to November 1990 in Kodansha's manga anthology Young Magazine, and was released in a tankōbon volume on October 2, 1991. Ghost in the Shell 2: Man-Machine Interface followed in 1997 for 9 issues in Young Magazine, and was collected in the Ghost in the Shell: Solid Box on December 1, 2000. Then a standard version with modifications and new pages was published on June 26, 2001. Four stories from Man-Machine Interface that were not released in tankobon format from previous releases were later collected in Ghost in the Shell 1.5: Human-Error Processor, and published by Kodansha on July 17, 2003. Several art books have also been published for the manga.

Animated films
Two animated films based on the original manga have been released, both directed by Mamoru Oshii and animated by Production I.G. Ghost in the Shell was released in 1995 and follows the "Puppet Master" storyline from the manga. It was re-released in 2008 as Ghost in the Shell 2.0 with new audio and updated 3D computer graphics in certain scenes. Innocence, otherwise known as Ghost in the Shell 2: Innocence, was released in 2004, with its story based on a chapter from the first manga.

Live-action film
In 2008, DreamWorks and producer Steven Spielberg acquired the rights to a live-action film adaptation of the original Ghost in the Shell manga. On January 24, 2014, Rupert Sanders was announced as director, with a screenplay by William Wheeler. In April 2016, the full cast was announced, which included Juliette Binoche, Chin Han, Lasarus Ratuere and Kaori Momoi, and Scarlett Johansson in the lead role; the casting of Johansson drew accusations of whitewashing. Principal photography on the film began on location in Wellington, New Zealand, on February 1, 2016. Filming wrapped in June 2016. Ghost in the Shell premiered in Tokyo on March 16, 2017, and was released in the United States on March 31, 2017, in 2D, 3D and IMAX 3D. It received mixed reviews, with praise for its visuals and Johansson's performance but criticism for its script.

Stand Alone Complex TV series, film and ONA
In 2002, Ghost in the Shell: Stand Alone Complex premiered on Animax, presenting a new telling of Ghost in the Shell independent from the original manga, focusing on Section 9's investigation of the Laughing Man hacker. It was followed in 2004 by a second season titled Ghost in the Shell: S.A.C. 2nd GIG, which focused on the Individual Eleven terrorist group. The primary storylines of both seasons were compressed into OVAs broadcast as Ghost in the Shell: Stand Alone Complex The Laughing Man in 2005 and Ghost in the Shell: Stand Alone Complex Individual Eleven in 2006. Also in 2006, Ghost in the Shell: Stand Alone Complex - Solid State Society, featuring Section 9's confrontation with a hacker known as the Puppeteer, was broadcast, serving as a finale to the anime series. The extensive score for the series and its films was composed by Yoko Kanno.

On April 7, 2017, Kodansha and Production I.G announced that Kenji Kamiyama and Shinji Aramaki would be co-directing a new Kōkaku Kidōtai anime production. On December 7, 2018, it was reported by Netflix that they had acquired the worldwide streaming rights to the original net animation (ONA) anime series, titled Ghost in the Shell: SAC_2045, and that it would premiere on April 23, 2020. The series will be in 3DCG and Sola Digital Arts will be collaborating with Production I.G on the project. It was later revealed that Ilya Kuvshinov will handle character designs. The series had two seasons of 12 episodes each.

In addition to the anime, a series of published books, two separate manga adaptations, and several video games for consoles and mobile phones have been released for Stand Alone Complex.

Arise OVA, TV series and film
In 2013, a new iteration of the series titled Ghost in the Shell: Arise premiered, taking an original look at the Ghost in the Shell world, set before the original manga. It was released as a series of four original video animation (OVA) episodes (with limited theatrical releases) from 2013 to 2014, then recompiled as a 10-episode television series under the title of Kōkaku Kidōtai: Arise - Alternative Architecture. An additional fifth OVA titled Pyrophoric Cult, originally premiering in the Alternative Architecture broadcast as two original episodes, was released on August 26, 2015. Kazuchika Kise served as the chief director of the series, with Tow Ubukata as head writer. Cornelius was brought onto the project to compose the score for the series, with the Major's new voice actress Maaya Sakamoto also providing vocals for certain tracks.

Ghost in the Shell: The New Movie, also known as Ghost in the Shell: Arise − The Movie or New Ghost in the Shell, is a 2015 film directed by Kazuya Nomura that serves as a finale to the Ghost in the Shell: Arise story arc. The film is a continuation to the plot of the Pyrophoric Cult episode of Arise, and ties up loose ends from that arc.

A manga adaptation was serialized in Kodansha's Young Magazine, which started on March 13 and ended on August 26, 2013.

2026 anime
On May 25, 2024, it was announced that a new anime television series adaptation will be produced by Science Saru for a 2026 premiere. Saru will be a production committee with Bandai Namco Filmworks, Kodansha and Production I.G.

Video games
Ghost in the Shell was developed by Exact and released for the PlayStation on July 17, 1997, in Japan by Sony Computer Entertainment. It is a third-person shooter featuring an original storyline where the character plays a rookie member of Section 9. The video game's soundtrack Megatech Body features various techno artists, such as Takkyu Ishino, Scan X and Mijk Van Dijk.

Several video games were also developed to tie into the Stand Alone Complex television series, in addition to a first-person shooter by Nexon and Neople titled Ghost in the Shell: Stand Alone Complex - First Assault Online, released in 2016.

Legacy
Ghost in the Shell influenced some prominent filmmakers. The Wachowskis, creators of The Matrix and its sequels, showed it to producer Joel Silver, saying, "We wanna do that for real." The Matrix series took several concepts from the film, including the Matrix digital rain, which was inspired by the opening credits of Ghost in the Shell, and the way characters access the Matrix through holes in the back of their necks. Other parallels have been drawn to James Cameron's Avatar, Steven Spielberg's A.I. Artificial Intelligence and Jonathan Mostow's Surrogates. James Cameron cited Ghost in the Shell as a source of inspiration, citing it as an influence on Avatar.

Bungie's 2001 third-person action game Oni draws substantial inspiration from Ghost in the Shell setting and characters. Ghost in the Shell also influenced video games such as the Metal Gear Solid series, Deus Ex, and Cyberpunk 2077.

title: Ghost in the Shell


text:
Gauss–Legendre algorithm

The Gauss–Legendre algorithm is an algorithm to compute the digits of $\pi$. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π. However, it has some drawbacks (for example, it is computer memory-intensive) and therefore all record-breaking calculations for many years have used other methods, almost always the Chudnovsky algorithm. For details, see Chronology of computation of π.

The method is based on the individual work of Carl Friedrich Gauss (1777–1855) and Adrien-Marie Legendre (1752–1833) combined with modern algorithms for multiplication and square roots. It repeatedly replaces two numbers by their arithmetic and geometric mean, in order to approximate their arithmetic-geometric mean.

The version presented below is also known as the Gauss–Euler, Brent–Salamin (or Salamin–Brent) algorithm; it was independently discovered in 1975 by Richard Brent and Eugene Salamin. It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with Borwein's algorithm.

Algorithm
a_{n+1} & = \frac{a_n + b_n}{2}, \\ \\                     b_{n+1} & = \sqrt{a_n b_n}, \\ \\                     t_{n+1} & = t_n - p_n(a_{n}-a_{n+1})^2, \\ \\                     p_{n+1} & = 2p_n. \\       \end{align} $$
 * 1) Initial value setting: $$a_0 = 1\qquad b_0 = \frac{1}{\sqrt{2}}\qquad t_0 = \frac{1}{4}\qquad p_0 = 1.$$
 * 2) Repeat the following instructions until the difference of $$a_n$$ and $$b_n$$ is within the desired accuracy: $$ \begin{align}
 * 1) π is then approximated as: $$\pi \approx \frac{(a_{n+1}+b_{n+1})^2}{4t_{n+1}}.$$

The first three iterations give (approximations given up to and including the first incorrect digit):

 * $$3.140\dots$$
 * $$3.14159264\dots$$
 * $$3.1415926535897932382\dots$$

The algorithm has quadratic convergence, which essentially means that the number of correct digits doubles with each iteration of the algorithm.

Limits of the arithmetic–geometric mean
The arithmetic–geometric mean of two numbers, a0 and b0, is found by calculating the limit of the sequences

 * $$\begin{align} a_{n+1} & = \frac{a_n+b_n}{2}, \\[6pt]

b_{n+1} & = \sqrt{a_n b_n}, \end{align} $$

which both converge to the same limit.

If $$a_0=1$$ and $$b_0=\cos\varphi$$ then the limit is ${\pi \over 2K(\sin\varphi)}$ where $$K(k)$$ is the complete elliptic integral of the first kind

 * $$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}.$$

If $$c_0 = \sin\varphi$$, $$c_{i+1} = a_i - a_{i+1}$$, then

 * $$\sum_{i=0}^\infty 2^{i-1} c_i^2 = 1 - {E(\sin\varphi)\over K(\sin\varphi)}$$

where $$E(k)$$ is the complete elliptic integral of the second kind:

 * $$E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\; d\theta$$

Gauss knew of these two results.

Legendre’s identity
Legendre proved the following identity:
 * $$K(\cos \theta) E(\sin \theta ) + K(\sin \theta ) E(\cos \theta) - K(\cos \theta) K(\sin \theta) = {\pi \over 2},$$

for all $$\theta$$.

Elementary proof with integral calculus
The Gauss-Legendre algorithm can be proven to give results converging to π using only integral calculus. This is done here and here.

title: Gauss–Legendre algorithm


text:
Great Internet Mersenne Prime Search

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and its Linux port MPrime. Scott Kurowski wrote the back end PrimeNet server to demonstrate volunteer computing software by Entropia, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. with Kurowski as Executive Vice President and board director. GIMPS is said to be one of the first large scale volunteer computing projects over the Internet for research purposes.

, the project has found a total of seventeen Mersenne primes, fifteen of which were the largest known prime number at their respective times of discovery. The largest known prime is 282,589,933 − 1 (or M82,589,933 for short) and was discovered on December 7, 2018, by Patrick Laroche. On December 4, 2020, the project passed a major milestone after all exponents below 100 million were checked at least once.

From its inception until 2018, the project relied primarily on the Lucas–Lehmer primality test as it is an algorithm that is both specialized for testing Mersenne primes and particularly efficient on binary computer architectures. Before applying it to a given Mersenne number, there was a trial division phase, used to rapidly eliminate many Mersenne numbers with small factors. Pollard's p − 1 algorithm is also used to search for smooth factors.

In 2018, GIMPS adopted a Fermat primality test as an alternative option for primality testing, while keeping the Lucas-Lehmer test as a double-check for Mersenne numbers detected as probable primes by the Fermat test. (While the Lucas-Lehmer test is deterministic and the Fermat test is only probabilistic, the probability of the Fermat test finding a Fermat pseudoprime that is not prime is vastly lower than the error rate of the Lucas-Lehmer test due to computer hardware errors. )

In September 2020,  GIMPS began to support primality proofs based on verifiable delay functions. The proof files are generated while the Fermat primality test is in progress. These proofs, together with an error-checking algorithm devised by Robert Gerbicz, provide a complete confidence in the correctness of the test result and eliminate the need for double checks. First-time Lucas-Lehmer tests were deprecated in April 2021.

GIMPS also has sub-projects to factor known composite Mersenne and Fermat numbers.

History
The project began in early January 1996, with a program that ran on i386 computers. The name for the project was coined by Luke Welsh, one of its earlier searchers and the co-discoverer of the 29th Mersenne prime. Within a few months, several dozen people had joined, and over a thousand by the end of the first year. Joel Armengaud, a participant, discovered the primality of M1,398,269 on November 13, 1996. Since then, GIMPS has discovered a new Mersenne prime every 1 to 2 years on average. However, no new Mersenne prime has been found since 2018, constituting the longest period without a new discovery since the start of the project (over 5 years as of 2024).

Status
, GIMPS has a sustained average aggregate throughput of approximately 4.71 PetaFLOPS (or PFLOPS). In November 2012, GIMPS maintained 95 TFLOPS, theoretically earning the GIMPS virtual computer a rank of 330 among the TOP500 most powerful known computer systems in the world. The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of Hewlett-Packard. As of July 2021 TOP500 results, the current GIMPS numbers would no longer make the list.

Previously, this was approximately 50 TFLOPS in early 2010, 30 TFLOPS in mid-2008, 20 TFLOPS in mid-2006, and 14 TFLOPS in early 2004.

Software license
Although the GIMPS software's source code is publicly available, technically it is not free software, since it has a restriction that users must abide by the project's distribution terms. Specifically, if the software is used to discover a prime number with at least 100,000,000 decimal digits, the user will only win $50,000 of the $150,000 prize offered by the Electronic Frontier Foundation. On the other hand, they will win $3,000 when discovering a smaller prime not qualifying for the prize.

Third-party programs for testing Mersenne numbers, such as Mlucas and Glucas (for non-x86 systems), do not have this restriction.

GIMPS also "reserves the right to change this EULA without notice and with reasonable retroactive effect."

Primes found
All Mersenne primes are of the form, where p is a prime number itself. The smallest Mersenne prime in this table is 21398269 − 1.

The first column is the rank of the Mersenne prime in the (ordered) sequence of all Mersenne primes; GIMPS has found all known Mersenne primes beginning with the 35th.

, 65,723,341 is the largest exponent below which all other prime exponents have been checked twice, so it is not verified whether any undiscovered Mersenne primes exist between the 48th (M57885161) and the 51st (M82589933) on this chart; the ranking is therefore provisional. Furthermore, 114,055,847 is the largest exponent below which all other prime exponents have been tested at least once, so all Mersenne numbers below the 51st (M82589933) have been tested.

The number M82589933 has 24,862,048 decimal digits. To help visualize the size of this number, if it were to be saved to disk, the resulting text file would be nearly 25 megabytes long (most books in plain text format clock in under two megabytes). A standard word processor layout (50 lines per page, 75 digits per line) would require 6,629 pages to display it. If one were to print it out using standard printer paper, single-sided, it would require approximately 14 reams (14 × 500 = 7000 sheets) of paper.

Whenever a possible prime is reported to the server, it is verified first (by one or more independent tests on different machines) before being announced. The importance of this was illustrated in 2003, when a false positive was reported to the server as being a Mersenne prime but verification failed.

The official "discovery date" of a prime is the date that a human first noticed the result for the prime, which may differ from the date that the result was first reported to the server. For example, M74207281 was reported to the server on September 17, 2015, but the report was overlooked until January 7, 2016.

title: Great Internet Mersenne Prime Search


text:
Game.com

The Game.com is a fifth-generation handheld game console released by Tiger Electronics on September 12, 1997. A smaller version, the Game.com Pocket Pro, was released in mid-1999. The first version of the Game.com can be connected to a 14.4 kbit/s modem for Internet connectivity, hence its name referencing the top level domain .com. It was the first video game console to include a touchscreen and the first handheld console to include Internet connectivity.

History
Tiger Electronics had previously introduced its R-Zone game console in 1995 – as a competitor to Nintendo's Virtual Boy – but the system was a failure. Prior to the R-Zone, Tiger had also manufactured handheld games consisting of LCD screens with imprinted graphics.

Original version
By February 1997, Tiger was planning to release a new game console as a direct competitor to Nintendo's Game Boy. Prior to its release, Tiger Electronics stated that the Game.com would "change the gaming world as we know it," while a spokesperson stated that it would be "one of this summer's hits." The Game.com, the only new game console of the year, was on display at the Electronic Entertainment Expo (E3) in May 1997, with sales expected to begin in July. Dennis Lynch of the Chicago Tribune considered the Game.com to be the "most interesting hand-held device" on display at E3, describing it as a "sort of Game Boy for adults".

The Game.com was released in the United States on September 12, 1997, with a retail price of $69.95, while an Internet-access cartridge was scheduled for release in October. Lights Out was included with the console as a pack-in game and Solitaire was built into the handheld itself. The console's release marked Tiger's largest product launch ever. Tiger also launched a website for the system at the domain "game.com". The Game.com was marketed with a television commercial in which a spokesperson insults gamers who ask questions about the console, while stating that it "plays more games than you idiots have brain cells"; GamesRadar stated that the advertisement "probably didn't help matters much". By the end of 1997, the console had been released in the United Kingdom, at a retail price of £79.99.

The Game.com came in a black-and-white color, and featured a design similar to Sega's Game Gear console. The screen is larger than the Game Boy's and has higher resolution. The Game.com included a phone directory, a calculator, and a calendar, and had an older target audience with its PDA features. Tiger designed the console's features to be simple and cheap. The device was powered by four AA batteries, and an optional AC adapter was also available. One of the major peripherals that Tiger produced for the system was the compete.com serial cable, allowing players to connect their consoles to play multiplayer games. The console includes two game cartridge slots. In addition to reducing the need to swap out cartridges, this enabled Game.com games to include online elements, since both a game cartridge and the modem cartridge could be inserted at the same time.

The Game.com was the first video game console to feature a touchscreen and also the first handheld video game console to have Internet connectivity. The Game.com's black-and-white monochrome touchscreen measures approximately one and a half inches by two inches, and is divided into square zones that are imprinted onto the screen itself, to aid players in determining where to apply the stylus. The touchscreen lacks a backlight. The Game.com was also the first handheld gaming console to have internal memory, which is used to save information such as high scores and contact information.

As 1998 opened, the Game.com was considered the only remaining viable competitor for the Game Boy, and Tiger planned to emphasize the Game.com's internet capabilities in marketing, as well as release new games based on major films and Giga Pets.

Game.com Pocket Pro
Because of poor sales with the original Game.com, Tiger developed an updated version known as the Game.com Pocket Pro. The console was shown at the American International Toy Fair in February 1999, and was later shown along with several future games at E3 in May 1999. The Game.com Pocket Pro had been released by June 1999, with a retail price of $29.99. The new console was available in five different colors: green, orange, pink, purple, and teal.

Although it lacked color like its predecessor, the Pocket Pro was reduced in size to be equivalent to the Game Boy Pocket. The screen size was also reduced, and the new console featured only one cartridge slot. Unlike the original Game.com, the Pocket Pro required only two AA batteries. The Game.com Pocket Pro included a phone directory, a calendar and a calculator, but lacked Internet capabilities.

The Game.com Pocket Pro's primary competitor was the Game Boy Color. Despite several games based on popular franchises, the Game.com console line failed to sell in large numbers, and was discontinued in 2000 because of poor sales. The Game.com was a commercial failure, with less than 300,000 units sold, although the idea of a touchscreen would later be used successfully in the Nintendo DS, released in 2004.

Internet features
Accessing the Internet required the use of an Internet cartridge and a dial-up modem, neither of which were included with the console. Email messages could be read and sent on the Game.com using the Internet cartridge, and the Game.com supported text-only web browsing through Internet service providers. Email messages could not be saved to the Game.com's internal memory. In addition to a Game.com-branded 14.4 kbit/s modem, Tiger also offered an Internet service provider through Delphi that was made to work specifically with the Game.com.

Tiger subsequently released the Web Link cartridge, allowing players to connect their system to a desktop computer. Using the Web Link cartridge, players could upload their high scores to the Game.com website for a chance to be listed on a webpage featuring the top high scores. None of the console's games made use of the Internet feature.

Games
Several games were available for the Game.com at the time of its 1997 launch, in comparison to hundreds of games available for the Game Boy. Tiger planned to have a dozen games available by the end of 1997, and hoped to have as many as 50 games available in 1998, with all of them to be produced or adapted internally by Tiger. Some third parties expressed interest in developing for the system, but Tiger decided against signing any initially. Tiger secured licenses for several popular game series, including Duke Nukem, Resident Evil, and Mortal Kombat Trilogy. Game prices initially ranged between $19 and $29. Cartridge size was in the 16 megabit range.

At the time of the Pocket Pro's 1999 release, the Game.com library consisted primarily of games intended for an older audience. Some games that were planned for release in 1999 would be exclusive to Game.com consoles. Game prices at that time ranged from $14 to $30. Twenty games were ultimately released for the Game.com, most of them developed internally by Tiger, in addition to the built-in game Solitaire. • Solitaire (Built in)

• Batman & Robin

• Centipede

• Duke Nukem 3D

• Fighters Megamix

• Frogger

• Henry

• Indy 500

• Jeopardy!

• Lights Out (Pack in)

• The Lost World: Jurassic Park

• Monopoly

• Mortal Kombat Trilogy

• Quiz Wiz: Cyber Trivia

• Resident Evil 2

• Scrabble

• Sonic Jam

• Tiger Casino

• Wheel of Fortune

• Wheel of Fortune 2

• Williams Arcade Classics

Cancelled games
The following is a list of games that were announced in various forms or known to be in development for the console but were never released. • A Bug’s Life

• Castlevania: Symphony of the Night

• Command & Conquer: Red Alert

• Deer Hunter

• Holyfield Champion Boxing (planned to include “Real Feel” force feedback in cart)

• Furbyland

• Giga Pets Deluxe

• Godzilla

• The Legend of the Lost Creator

• Madden 98

• Metal Gear Solid

• Mutoids

• Name That Tune

• NASCAR Racing

• NBA Hangtime

• Shadow Madness

• Sonic 3D Blast

• Small Soldiers

• Turok: Dinosaur Hunter

• WCW Whiplash

Reception
At the time of the Game.com's launch in 1997, Chris Johnston of VideoGameSpot believed that the console would have difficulty competing against the Game Boy. Johnston also believed that text-based Internet and email would attract only limited appeal, stating that such features were outdated. Johnston concluded that the Game.com "is a decent system, but Nintendo is just way too powerful in the industry." Chip and Jonathan Carter wrote that the console did not play action games as well as it did with other games, although they praised the console's various options and wrote, "Graphically, we'd have to say this has the potential to perform better than Game Boy. As for sound, Game.com delivers better than any other hand-held on the market." A team of four Electronic Gaming Monthly editors gave the Game.com scores of 5.5, 4.5, 5.0, and 4.0. They were impressed by the PDA features and touchscreen, but commented that the games library had thus far failed to deliver on the Game.com's great potential. They elaborated that while the non-scrolling games, particularly Wheel of Fortune, were great fun and made good use of the touchscreen, the more conventional action games were disappointing and suffered from prominent screen blurring.

Wisconsin State Journal stated that the Game.com offered "some serious" advantages over the Game Boy, including its touchscreen. It was also stated that in comparison to the Game Boy, the Game.com's 8-bit processor provided "marginal improvements" in the quality of speed and graphics. The newspaper noted that the Game.com had a "tiny, somewhat blurry screen." The Philadelphia Inquirer wrote a negative review of the Game.com, particularly criticizing Internet connectivity issues. Also criticized was the system's lack of a backlit screen, as the use of exterior lighting could cause difficulty in viewing the screen, which was highly reflective.

Steven L. Kent, writing for the Chicago Tribune, wrote that the console had an elegant design, as well as better sound and a higher-definition screen than the Game Boy: "Elegant design, however, has not translated into ideal game play. Though Tiger has produced fighting, racing and shooting games for Game.com, the games have noticeably slow frame rates. The racing game looks like a flickering silent picture show." Cameron Davis of VideoGames.com wrote, "Sure, this is no Game Boy Color-killer, but the Game.Com was never meant to be. To deride it by comparing it with more powerful and established formats would be a bit unfair". Davis also wrote, "The touch screen is pretty sensitive, but it works well - you won't need more than a few seconds to get used to it." However, he criticized the screen's squared zones: "more often than not it proves distracting when you are playing games that don't require it."

GamePro criticized the Pocket Pro's lack of screen color and its difficult controls, but considered its two best qualities to be its cheap price and a game library of titles exclusive to the console. The Philadelphia Inquirer also criticized the Pocket Pro's lack of a color screen, as well as "frustrating" gameplay caused by the "unresponsive" controls, including the stylus. The newspaper stated that, "Even at $29.99, the pocket.pro is no bargain."

Legacy
Brett Alan Weiss of the website AllGame wrote, "The Game.com, the little system that (almost) could, constantly amazes me with the strength and scope of its sound effects. [...] It's astounding what power comes out of such a tiny little speaker." In 2004, Kent included the modem and "some PDA functionality" as the console's strengths, while listing its "Slow processor" and "lackluster library of games" as weaknesses. In 2006, Engadget stated that "You can't fault Tiger Electronics for their ambition," but wrote that the Game.com "didn't do any one thing particularly well", criticizing its text-only Internet access and stating that its "disappointing games were made even worse" by the "outdated" screen.

In 2009, PC World ranked the Game.com at number nine on its list of the 10 worst video game systems ever released, criticizing its Internet aspect, its game library, its low-resolution touchscreen, and its "Silly name that attempted to capitalize on Internet mania." However, PC World positively noted its "primitive" PDA features and its solitaire game, considered by the magazine to be the system's best game. In 2011, Mikel Reparaz of GamesRadar ranked the Game.com at number 3 on a list of 7 failed handheld consoles, writing that while the Game.com had several licensed games, it "doesn't actually mean much when they all look like cruddy, poorly animated Game Boy ports." Raparaz also stated that the Game.com "looked dated even by Game Boy standards," noting that the Game Boy Pocket had a sharper display screen. Reparaz stated that the Game.com's continuation into 2000 was a "pretty significant achievement" considering its competition from the Game Boy Color.

In 2013, Jeff Dunn of GamesRadar criticized the Game.com for its "blurry" and "imprecise" touchscreen, as well as its "limited and unwieldy" Internet and email interfaces. Dunn also criticized the "painful" Internet setup process, and stated that all of the console's available games were "ugly and horrible." Dunn noted, however, that the Game.com's Internet aspect was a "smart" feature. In 2016, Motherboard stated that the Game.com was "perhaps one of the worst consoles of all time," due largely to its low screen quality. In 2018, Nadia Oxford of USgamer noted the Game.com's "paper-thin" library of games and stated that the console "died in record time because it was poorly-made, to say the least."

title: Game.com