formulasearchengine - User contributions [en]

Birthday problem

2015-01-12T16:33:05Z

137.122.114.44: /* An upper bound */

There are many proven ways to treat hemorrhoids that are effective for extended expression hemorrhoid relief. Because, whilst hemorrhoids, like other illnesses have a genetic component - if your mother or daddy had them you're more probably to get them - they equally are influenced by lifestyle. Some of the points which lead to the occurrence of hemorrhoids are chronic irregularity, sitting for extended periods of time, and a sofa potato lifestyle. Therefore, hemorrhoids can be healed or at least place into noticeable remission by utilizing certain good sense life-style techniques. I will focus found on the simplest of these techniques to implement. I believe which you need to have tried many kinds of [http://hemorrhoidtreatmentfix.com/hemorrhoid-symptoms hemorrhoids symptoms]. In this particular article, you are capable to discover how each type of treatment is employed, so you can do the right treatment with a symptom. There are actually several treatments which is employed for hemorrhoid. The first and the many popular is the cream plus ointment. These are to be rubbed onto the affected piece of the anus. It assists to soothe the absolutely inflamed blood vessels plus a momentary relief is accomplished. There is a relaxation of the tissues of the rectal column thus far the hemorrhoid is not thus much bulged. If there is a bulge nevertheless, the pain relief could not do thus much to aid. Lunch plus Dinner. Gradually add real fruits plus veggies to a meals, and substitute whole grains for white flour and pasta for an extra fiber punch. So, he HAD heard about a hemorrhoids house remedy or 2, he mentioned, yet couldn't truly remember any details about them. I told him I needed time to consider the next step plus got from there and into the bright sunshine as quickly because I may. Surgery for hemorrhoids definitely wasn't my initially choice. Ice is one of the simplest nevertheless the most effective hemorrhoid treatments we can utilize to minimize swelling, swelling, bleeding and pain. Wrap it in chipped form inside a piece of cheese fabric plus apply it onto the hemorrhoid itself. These hemorrhoid treatments because mentioned above are considered to be the top methods which have been utilized by countless people. However, should you have tried these techniques and they cannot aid you, then you have to see the doctor, which is surprisingly possible which you'll be recommended to test a surgical solution. Although surgery is considered to be an efficient solution, nevertheless there are some risks involved. Besides, it's truly expensive and it takes longer to heal as well.

Chain-growth polymerization

2014-08-27T19:44:34Z

137.122.205.68: /* Comparison with other polymerization methods */

However even in case you are not a PR you might be entitled to purchase non-landed property as Singapore is a really liberal market in comparison with other nations within the area. Non-landed refers to properties similar to apartments and condominiums Singapore's change from British colony in direction of a developed financial system has been characterized by population growth fuelling the larger demand for property and this in turn has given property builders the means to make cash. Singapore can be a strong hub for trading and finance, which has driven property values even larger. Most derived their wealth from property The Company is geographically diversified in Asia, with Singapore and China as its core markets in addition to Vietnam and Indonesia as its development markets. It focuses on a two-pronged technique of property development for sale and property fund administration. CITYSCAPE @ FARRER PARK PERSONAL CONDOMINIUM APARTMENT MERGUI ROAD, SINGAPORE (DISTRICT 08) Eligibility to Purchase Personal Property Condominium The client pays for his unit progressively, relying on progress of building. This is known as the Regular Progressive Fee Schedule, and is also mandated by regulation. The venture must complete ie. get its Temporary Occupation Permit (popularly shortened to PRIME), and subsequent Certificates of Statutory Completion (CSC), inside an accepted timeframe. Capital Land - It deals in places of work and residential properties and leisure complexes. Developments permitted as a condominium growth below he Planning Act a) For individual borrowers who haven't any outstanding housing loans, the LTV limit will probably be 80%, or 60% if the loan tenure exceeds 30 years or the loan period extends beyond the borrower's retirement age of sixty five. Can sell before HIGH is Issued The property tax on owner-occupied properties is charged at 4 per cent of the annual value. This concession is applicable to just one property at anybody time If the property is not proprietor-occupied, the tax fee is 10 per cent of the property's annual value. A staff of pros to help shoppers to look for property funding alternatives in China and Singapore. Offering management and letting of properties. Commercial real estate requirements for corporations. Relocation, growth, renewal, lease negotiations or new workplace set-up both in Singapore and abroad. Wing Tai Holdings Limited is one among Singapore's main property developer and way of life company widely recognised for high quality in its property developments. Diversified Property Builders? Singapore-based mostly Park Lodge Group is certainly one of Asia Pacific's fastest expanding hospitality brands. The privately-held entity owns and operates properties in Singapore, Hong Kong, China and Japan. Ship routine queries to numerous authorities departments to ensure that are not any adversarial regulatory notices or authorities schemes that can affect the property you want to purchase. We expect these factors to be key options of the native actual estate market in the coming 12 months." ONE OTHER major project that may hit the market next yr is South Beach, a blended-use growth situated between Raffles Lodge and Suntec Metropolis and subsequent to the Esplanade MRT station Wee Cho Yaw, chairman, United Abroad Bank Group Matching Cityscape HIGH date and school start date. Property Services Kengfu Growth, part of the Lian Huat Group, has numerous curiosity in actual estate growth, resort administration and possession, development, trading and property funding. Its actual estate actions in Singapore contain growth of business, residential and conservation initiatives. MCL Land is a number one property group listed on the Singapore Alternate. A member of the Jardine Matheson Group under Hongkong Land Holdings, MCL Land has an extended track record of constructing high quality properties in Singapore and Malaysia during the last 40 years Properties in Singapore are bought both on a freehold or leasehold tenure. When service is carried out (i.e. when sale is accomplished or when tenancy contract for leasing is entered into). There isn't any capital acquire tax in Singapore Subsequent yr there can be quite a no. of new challenge launches that can sooth the market. Whether to purchase now or later may also depends on which mission / district you are taking a look at and the transactions happening for that space. I think the subprime crisis in the U.S., the ensuing credit crunch and a u.s. recession will dent the property market in Singapore for a stable two-three years. After that, it should resume its upward development and hit ranges near what we see in Hong Kong, New York and London. I might wait. Properties that don't fall inside the definition of residential properties stated above are non-residential properties JOOL SUITES PRIVATE CONDOMINIUM [http://providers.myvaxines.com/groups/singapores-main-property-portal-for-new-condominium-launches-singaporepropertyforsale-information/ condo prices in singapore] SING JOO WALK, FARRER PARK (DISTRICT 08) IP Real Estate Investments Pte Ltd CapitaLand Restricted Property growth in Singapore; contains one of Asia-Pacific's largest purpose-built conference areas, office towers with spectacular views of the harbour, Singapore's largest mall, the world's largest fountain and so forth Main property builders with operations in Singapore, Malaysia, Hong Kong & China; the group additionally has interests in hospitality (investment & management), garments (manufacture, retail, distribution), food retailing, & mobile phone networks Property improvement company that is part of the Frasers Centrepoint Limited group ("FCL", previously referred to as Centrepoint Properties Ltd); has built greater than 9,000 high quality houses Property tycoon Ng Teng Fong's passing yesterday marks the tip of an period of bigger-than-life property titans. Hayden Properties

Jamshīd al-Kāshī

2014-02-18T16:42:16Z

137.122.85.125:

The title of the writer is Jayson. I am really fond of handwriting but I can't make it my profession really. For a whilst I've been in Alaska but I will have to transfer in a year or two. Credit authorising is how he makes money. Have a look at my site - [http://kjhkkb.net/xe/notice/374835 psychic love readings]

Partial residual plot

2013-11-06T19:34:43Z

137.122.64.24: /* See also */ WP:ALSO

'''Essential manifold''' a special type of closed manifolds.
The notion was first introduced explicitly by [[Mikhail Gromov (mathematician)|Mikhail Gromov]].<ref>Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147.</ref>

==Definition==

A closed [[manifold]] ''M'' is called essential if its [[fundamental class]] [''M''] defines a nonzero element in the [[homology (mathematics)|homology]] of its [[fundamental group]] ''π'', or more precisely in the homology of the corresponding [[Eilenberg–MacLane space]] ''K''(''π'', 1), via the natural homomorphism
:<math>H_n(M)\to H_n(K(\pi,1))</math>,
where ''n'' is the dimension of ''M''. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.

==Examples==

*All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere ''S2''.
*Real projective space ''RPn'' is essential since the inclusion
*:<math>\mathbb{RP}^n \to \mathbb{RP}^{\infty}</math>
:is injective in homology, where
::<math>\mathbb{RP}^{\infty} = K(\mathbb{Z}_2, 1)</math>
:is the Eilenberg-MacLane space of the finite cyclic group of order 2.
*All compact [[aspherical manifold]]s are essential;
**In particular all compact [[hyperbolic manifold]]s are essential.
*All [[lens space]]s are essential.

==Properties==

*[[Connected sum]] of essential manifolds is essential.

==References==
{{Reflist}}

==See also==

*[[Gromov's systolic inequality for essential manifolds]]
*[[Systolic geometry]]

{{Systolic geometry navbox}}

[[Category:Algebraic topology]]
[[Category:Riemannian geometry]]
[[Category:Differential geometry]]
[[Category:Systolic geometry]]
[[Category:Manifolds]]

Product operator formalism

2013-09-03T16:15:17Z

137.122.61.28: I have added that this method is a simplification of the complete theory, the density matrix, which was not clear in the previous article.

{{multiple issues
| notability=May 2012
| orphan=May 2012
| essay=May 2012
| more footnotes=May 2012
| refimprove=May 2012
}}
[[File:Energy and fluid flow in a solar turbine power plant.jpg|400px|thumb|Figure 1. Energy and fluid flow in a solar turbine power plant]]
A '''solar turbine power plant''' uses the energy in solar radiation captured by so-called [[Solar thermal collector|solar collector]]s. Solar power is a renewable source of energy. The solar radiant energy reaching the earth's surface is around 1.783*1014 KJ or 1.353kJ/s per square meter. Solar power plants, such as the [[Blythe Solar Power Project]], operate mainly on closed power cycles: [[Rankine cycle]]s (for low temperature ranges) and [[Brayton cycle]]s (for high temperature ranges). Solar plants provide energy ranging from a few kilowatts to a few megawatts. The constraints associated with solar plants are size, space, high capital cost, and the inevitable fluctuations in the daily supply of solar radiant energy.

==Efficiency==
===Concentration ratio===
[[File:Variation of Receiver temperature to Concentration Ratio.jpg|300px|thumb|Figure 2. Receiver temperature as related to concentration ratio]]
The concentration ratio is the ratio of the area of the concentrator to the area of the receiver surface. The amount of solar energy incident on the concentrator is directed towards the receiver, so the ratio is a measure of the energy concentrated towards the receiver.<ref name = sukhatme>{{cite book|title=Solar Energy: Principles of Thermal Collection and Storage|first=Suhas P.|last=Sukhatme|first2=J. K.|last2=Nayak|edition=3|isbn=978-0070142961|year=2008|publisher=McGraw-Hill}}</ref>{{rp|210}}

Higher concentration ratio values can be attained by using large apertures and small receiver. Receiver temperature increases with the increase in concentration ratio, as shown in Figure 2. Concentration ratios vary from 1.5 to 3000 depending on the type of collector, i.e., whether it is a medium- or high-temperature collector. This is an important parameter in determining the efficiency of a solar plant.

<math>{CR}={{aperture\,area\,of\,concentrator}\over{receiver\,surface\,area}} = {Ac\over Ar}</math>

===Optical efficiency===
[[File:Variation of Collector efficiency (%) with Temperature ratio.jpg|300px|thumb|Figure 3. Variation of collector efficiency (%) with temperature ratio]]
The optical efficiency of a solar collector relates the percentage of the solar rays penetrating the transparent cover of the collector (transmission) and the percentage being absorbed.

<math>{\eta_0}={{heat\,energy\,received\,by\,the\,receiver}\over{incident\,radiation\,on\,the\,collector}} = {Qr\over Qc}</math>

Where <math>{Q_c} = {I_c} . {A_c}</math>

<math>{I_c} =</math> incident solar radiation

Therefore, <math>{Q_r} = {\eta_o}.. {I_c} . {A_c}</math>

===Collector efficiency===
There are three types of solar collectors: low (100°C),<ref name=Yahya>{{cite book|title=Turbines, Compressors and Fans|first=S. M.|last=Yahya|edition=3|isbn=978-0070597709|year=2005|publisher=[[Tata McGraw-Hill]]}}</ref>{{rp|699}} medium (300-400°C),<ref name=Yahya />{{rp|701}} and high (400-700°C)<ref name=Yahya />{{rp|706}} temperature collectors. Each type has its own efficiency.

<math>{\eta_c} = {{useful\,heat\,received\,by\,the\,coolant}\over{incident\,radiation\,on\,the\,collector}} = {Qu\over Qc}</math>

<math>{Q_u}={Q_r}-{L}={Q_r}-{Losses}</math>

The losses can be expressed by the overall co-efficient <math>{U}</math> based on receiver area

<math>{L} = {U}.{Ar}.({Tr}-{Ta})</math>

<math>{Q_u}={\eta_0}.{Ic}.{Ac}-{U}.{Ar}.({Tr}-{Ta})</math>

<math>{\eta_c}={\eta_0}-({1\over{cr}}).{{U\,Ta\,}\over{Ic}}.({{Tr}\over{Ta}}-{1})</math>

<math>{\eta_c}=</math> f(CR,TR)

where TR is the receiver temperature ratio. TR increases with the concentration ratio, as shown in Figure 2. Collector efficiency decreases with the temperature ratio, as shown in Figure 3.

==Solar receiver==
[[File:A Heliostat and External receiver.jpg|300px|thumb|Figure 4. A heliostat and external receiver]]
[[File:A cavity of solar radiation receiver.jpg|300px|thumb|Figure 5. A heliostat and external receiver]]
[[File:A tubular collector receiver.jpg|300px|thumb|Figure 6. A tubular collector receiver]]
The receiver absorbs heat transmitted by the collector. Sometimes the receiver is an integral part of the system, for example, in solar ponds and flat plate collectors. Receivers may be stationary or portable.

There are three types of receivers: external, tubular, and cavity.

===External receivers===
A working fluid is provided on the external surface of a vertical body (Figure 4).

Major losses are due to:
* Non-focusing
* [[Thermal conduction|Conduction]], [[convection]], and [[radiation]]
* [[Mirror|Reflection]]

Concentration ratio and fluid temperature attained are 1000 and 500°C, respectively.<ref name=Yahya />{{rp|709}}

===Cavity receivers===
Heat flux enters through the apertures as shown in Figure 5; concentrators transmit the heat flux to the surface of coolant tubes through the apertures. Heat energy is transferred to other parts (where the direct beam is unable to reach) through [[Total internal reflection|internal reflection]]. Overall size is large due to the number of coolant tubes.

===Tubular receivers===
This consists of a row of coaxial tubes. The outer tube receives the radiation, whereas the working fluid enters through the inner tube and leaves through the annular space between the tubes (Figure 6). Concentration ratio and maximum fluid temperature attained are around 1.5 and 200°C, respectively.<ref name=Yahya />{{rp|710}}

==Receiver system==
[[File:Distributed Receiver System.jpg|thumb|Figure 7. Distributed receiver system]]

===Distributed receiver system===
In this system the three collectors (as shown in Figure 7) collect the heat flux and transfer it to receiver from where the coolant takes this energy to the heat exchanger (Path A). The coolant at times serves the purpose of working fluid, as depicted by Path B.

===Central receiver system===
In this system, the solar collectors transmit the heat flux to a receiver which is large in size. External and cavity types of receivers can be employed for this purpose. Example: [[heliostat]]s.

==Net efficiency==
[[File:Variation of Collector efficiencies with Receiver temperature.jpg|300px|thumb|Figure 8. Variation of collector efficiencies with receiver temperature]]
The collector efficiency (ɳc) decreases with increases in receiver temperature. The thermal efficiency (ɳth) increases with increases in the inlet temperature of the working fluid. Therefore, overall plant efficiency (ɳn) varies, as shown in Figure 8. The net efficiency is between 15% and 20%.<ref name=Yahya />{{rp|725}}. The curve is flat at maximum efficiency.

==Solar energy storage==
Since solar radiation is not always available, it becomes necessary to store the energy in some form. [[Solar thermal energy]] storage can be done in:

# Solids: Some rocks absorb heat. The amount of energy stored depends on the mass of the solid material, its specific heat, and the allowable temperature rise.
# Liquids: If the heat is stored below the boiling point of fluids at ambient pressure then some fluids can be used as heat storage media. Some liquids which can be used for this purpose are [[Sodium#Heat transfer|Sodium]], [[Methylcyclopentadienyl manganese tricarbonyl|Hitec]], [[Polychlorinated biphenyl|Therminol]], and oils.
# Latent heat of fusion: In this type of system a solid is heated until it melts. Thus heat is stored in the body at constant temperature in the form of latent heat. Examples LiF (latent heat = 1050 kJ/Kg melting point = 848°C) and LiOH (latent heat = 1080 kJ/Kg melting point =471°C).<ref name=Yahya />{{rp|717}}
# A combination of any of the above approaches can also be used to store solar energy.

==Solar turbines==
The performance of the coolants, working fluid, and steam or gas turbines together determine the efficiency of the plant.

===Coolants and working fluids===
A coolant absorbs energy in the receiver and transfers the energy to the working fluid in the heat exchanger. Example water/steam, liquid metals, molten salts, gases and oils.

Water can be used as coolants in low and medium temperature solar power plants.

The maximum temperature deployed in an oil type of coolant is 250°C.<ref name=Yahya />{{rp|717}} Oil can be dangerous because it is inflammable. It is also relatively costly.

Gases that can be used as coolants are air, helium, argon, and carbon dioxide. They can be used for high temperature ranges (Tmax=800°C).<ref name=Yahya />{{rp|717}}

Molten salts are also used for high temperature regions. They have high specific heat.

Molten metals (sodium or aluminium) can also be used as coolants. Since their density is high they require a smaller receiver.

steam, freon, or helium are some of gases used as working fluids.

===Steam turbines===
[[Steam turbine]]s operate on a [[Rankine cycle]].<ref name = cengel>{{cite book|title=Thermodynamics: An Engineering Approach |first=Yunus|last=Çengel|last2=Boles|first2=Michael|edition=7|year=2011|publisher=[[McGraw-Hill]]|isbn=978-0073529325}}</ref> Steam can be generated by a receiver directly from the solar heat flux, which eliminates the need for a heat exchanger. However, some plants deploy molten salts to attain higher temperatures, which eliminates the need for steam boilers. Values of pressure and temperature in solar plants are 50-100 bar and 400-500°C, respectively. Both impulse and reaction stages can be used. For small values of power, impulse stages are preferable.

===Gas turbine===
[[Gas turbine]]s operate on a [[Brayton cycle]], that is, with inlet temperatures around 500-800°C.<ref name = cengel /> A conventional gas turbine power plant uses a combustion chamber, but here the combustion chamber is the receiver/heat exchanger. However, using a gas solar turbine power plants with stored thermal energy is quite difficult because the power plant operates at a high temperature range and it is quite difficult to store heat energy at this high temperature. Gas turbines use fewer stages, do not require feed water heaters or condensers and have a low cooling requirement.

==Advantages and disadvantages==

===Advantages===
* Being a renewable form of energy, the fuel is free and surplus
* No fuel storage, processing or handling equipment is required
* Being an alternative form of energy saves of oil/petrol/diesel resources
* Less environmental pollution
* Can be operated in remote areas or places which are unfit for habitation

===Disadvantages===
* Reliable power generation (dependence on weather)
* Large amount of area required for its establishment
* High capital cost
* Overall efficiency is low

==See also==
* [[Concentrated solar power]]
* [[List of concentrating solar thermal power companies]]
* [[List of solar thermal power stations]]

== References ==
{{reflist}}

==Further reading==
* {{cite book|series=Solar Thermal Central Receiver Systems|volume=3|title=Performance Evaluation Standards for Solar Central Receivers|editor-last=Carasso|editor-first=Meir|editor2-last=Becker|editor2-first=Manfred|isbn=9783540532705|year=1990|publisher=[[Springer Science+Business Media|Springer-Verlag]]}}
* {{cite book|title=Solar Power Plants: Fundamentals, Technology, Systems, Economics|first=Carl-Jochen|last=Winter|first2=Rudolf L.|last2=Sizmann|isbn=978-0387188973|year=1991|publisher=Springer-Verlag}}
* {{cite journal|url=http://www.wired.com/science/planetearth/news/2005/11/69528|title=Huge Solar Plants Bloom in Desert|work=[[Wired (magazine)|Wired]]|first=Will|last=Wade|date=15 November 2005|accessdate=20 May 2012}}

{{Solar energy}}
{{Renewable energy by country}}

[[Category:Solar power stations]]
[[Category:Solar thermal energy]]

Algebraic statistics

2013-03-27T19:04:22Z

137.122.49.102: /* External links */ rm redundant red link. If there was a wiki article, it would be in a See also section

In [[mathematics]] the '''cotangent complex''' is roughly a universal linearization of a [[morphism]] of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. [[Luc Illusie]], [[Daniel Quillen]], and M. André independently came up with a definition that works in all cases.

==Motivation==
Suppose that ''X'' and ''Y'' are [[algebraic variety|algebraic varieties]] and that {{nowrap|''f'' : ''X'' → ''Y''}} is a morphism between them. The cotangent complex of ''f'' is a more universal version of the relative [[Kähler differentials]] Ω''X''/''Y''. The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If ''Z'' is another variety, and if {{nowrap|''g'' : ''Y'' → ''Z''}} is another morphism, then there is an exact sequence
:<math>f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0.</math>
In some sense, therefore, relative Kähler differentials are a [[right exact functor]]. (Literally this is not true, however, because the category of algebraic varieties is not an [[abelian category]], and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the [[Lichtenbaum–Schlessinger functor]]s ''T''''i'' and [[imperfection module]]s. Most of these were motivated by [[deformation theory]].

This sequence is exact on the left if the morphism ''f'' is smooth. If Ω admitted a first [[derived functor]], then exactness on the left would imply that the [[connecting homomorphism]] vanished, and this would certainly be true if the first derived functor of ''f'', whatever it was, vanished. Therefore a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials.

Another natural exact sequence related to Kähler differentials is the [[conormal exact sequence]]. If ''f'' is a closed immersion with ideal sheaf ''I'', then there is an exact sequence
:<math>I/I^2 \to f^*\Omega_{Y/Z} \to \Omega_{X/Z} \to 0.</math>
This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of ''f'', and the relative differentials Ω''X''/''Y'' have vanished because a closed immersion is [[formally unramified]]. If ''f'' is the inclusion of a smooth subvariety, then this sequence is a short exact sequence.<ref>{{Harvard citations|last = Grothendieck|year = 1967|loc = Proposition 17.2.5|nb = yes}}</ref> This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term.

==Early work on cotangent complexes==
The cotangent complex dates back at least to SGA 6 VIII 2, where [[Pierre Berthelot]] gave a definition when ''f'' is a ''smoothable'' morphism, meaning there is a scheme ''V'' and morphisms {{nowrap|''i'' : ''X'' → ''V''}} and {{nowrap|''h'' : ''V'' → ''Y''}} such that {{nowrap|''f'' {{=}} ''hi''}}, ''i'' is a closed immersion, and ''h'' is a smooth morphism. (For example, all projective morphisms are smoothable, since ''V'' can be taken to be a projective bundle over ''Y''.) In this case, he defines the cotangent complex of ''f'' as an object in the [[derived category]] of [[coherent sheaf|coherent sheaves]] ''X'' as follows:
*<math>L^{X/Y}_0 = i^*\Omega_{V/Y},</math>
*If ''J'' is the ideal of ''X'' in ''V'', then <math>L^{X/Y}_1 = J/J^2 = i^*J</math>,
*<math>L^{X/Y}_i = 0</math> for all other ''i'',
*The differential <math>L^{X/Y}_1 \to L^{X/Y}_0</math> is the pullback along ''i'' of the inclusion of ''J'' in the structure sheaf <math>\mathcal{O}_V</math> of ''V'' followed by the universal derivation <math>d : \mathcal{O}_V \to \Omega_{V/Y}</math>.
*All other differentials are zero.
Berthelot proves that this definition is independent of the choice of ''V''<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.2|nb = yes}}</ref> and that for a smoothable complete intersection morphism, this complex is perfect.<ref>{{Harvard citations|last = Berthelot|year = 1966|loc = VIII Proposition 2.4|nb = yes}}</ref> Furthermore, he proves that if {{nowrap|''g'' : ''Y'' → ''Z''}} is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an [[exact triangle]]
:<math>\mathbf{L}f^*L^{Y/Z}_\bullet \to L^{X/Z}_\bullet \to L^{X/Y}_\bullet \to \mathbf{L}f^*L^{Y/Z}_\bullet[1].</math>

==The definition of the cotangent complex==
The correct definition of the cotangent complex begins in the [[homotopic algebra|homotopical setting]]. Quillen and André worked with the [[simplicial set#Simplicial objects|simplicial]] commutative rings, while Illusie worked with simplicial ringed [[topos|topoi]]. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that ''A'' and ''B'' are [[simplicial ring]]s and that ''B'' is an ''A''-algebra. Choose a resolution {{nowrap|''r'' : ''P''• → ''B''}} of ''B'' by simplicial free ''A''-algebras. Applying the Kähler differential functor to ''P''• produces a simplicial ''B''-module. The total complex of this simplicial object is the '''cotangent complex''' ''L''''B''/''A''. The morphism ''r'' induces a morphism from the cotangent complex to Ω''B''/''A'' called the '''augmentation map'''. In the homotopy category of simplicial ''A''-algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler differential functor.

Given a commutative square as follows:
:[[File:Commutative square.svg]]
there is a morphism of cotangent complexes {{nowrap|''L''''B''/''A'' ⊗''B'' ''D'' → ''L''''D''/''C''}} which respects the augmentation maps. This map is constructed by choosing a free simplicial ''C''-algebra resolution of ''D'', say {{nowrap|''s'' : ''Q''• → ''D''}}. Because ''P''• is a free object, the composite ''hr'' can be lifted to a morphism {{nowrap|''P''• → ''Q''•}}. Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms {{nowrap|''A'' → ''B'' → ''C''}}, this produces the sequence
:<math>L^{B/A} \otimes_B C \to L^{C/A} \to L^{C/B}.</math>
There is a connecting homomorphism <math>L^{C/B} \to (L^{B/A} \otimes_B C)[1]</math> which turns this sequence into an exact triangle.

The cotangent complex can also be defined in any combinatorial [[model category]] ''M''. Suppose that <math>f\colon A\rightarrow B</math> is a morphism in ''M''. The cotangent complex <math>L^f</math> (or <math>L^{B/A}</math>) is an object in the category of spectra in <math>M_{B//B}</math>. A pair of composable morphisms <math>A\xrightarrow{f} B\xrightarrow{g} C</math> induces an exact triangle in the homotopy category, <math>L^{B/A}\otimes_BC\rightarrow L^{C/A}\rightarrow L^{C/B}\rightarrow (L^{B/A}\otimes_BC)[1]</math>.

==Properties of the cotangent complex==
===Flat base change===
Suppose that ''B'' and ''C'' are ''A''-algebras such that {{nowrap|Tor''A''''q''(''B'', ''C'') {{=}} 0}} for all {{nowrap|''q'' > 0}}. Then there are quasi-isomorphisms<ref>{{Harvard citations|last = Quillen|year = 1970|loc = Theorem 5.3|nb = yes}}</ref>
:<math>L^{B \otimes_A C/C} \cong B \otimes_A L^{C/A},</math>
:<math>L^{B \otimes_A C/A} \cong (L^{B/A} \otimes_A C) \oplus (B \otimes_A L^{C/A}).</math>
If ''C'' is a flat ''A''-algebra, then the condition that {{nowrap|Tor''A''''q''(''B'', ''C'')}} vanishes for {{nowrap|''q'' > 0}} is automatic. The first formula then proves that the construction of the cotangent complex is local on the base in the [[flat topology]].

===Vanishing properties===
Let {{nowrap|''f'' : ''A'' → ''B''}}. Then:<ref>{{Harvard citations|last = Quillen|year = 1970|loc = Theorem 5.4|nb = yes}}</ref><ref>{{Harvard citations|last = Quillen|year = 1970|loc = Corollary 6.14|nb = yes}}</ref>
*If ''B'' is a [[localization of a ring|localization]] of ''A'', then {{nowrap|''L''''B''/''A'' {{=}} 0}}.
*If ''f'' is an [[étale morphism]], then {{nowrap|''L''''B''/''A'' {{=}} 0}}.
*If ''f'' is a [[smooth morphism]], then {{nowrap|''L''''B''/''A''}} is quasi-isomorphic to Ω''B''/''A''. In particular, it has [[projective dimension]] zero.
*If ''f'' is a [[local complete intersection morphism]], then {{nowrap|''L''''B''/''A''}} has projective dimension at most one.
*If ''A'' is Noetherian, {{nowrap|''B'' {{=}} ''A''/''I''}}, and ''I'' is generated by a regular sequence, then <math>I/I^2</math> is a [[projective module]] and ''L''''B''/''A'' is quasi-isomorphic to <math>I/I^2[1]</math>.

==Examples==
*Let ''X'' be smooth over ''S''. Then the cotangent complex is Ω''X''/''S''. In Berthelot's framework, this is clear by taking {{nowrap|''V'' {{=}} ''X''}}. In general, étale locally on ''S'', ''X'' is a finite dimensional affine space and the morphism from ''X'' to ''S'' is projection, so we may reduce to the situation where {{nowrap|''S'' {{=}} Spec ''A''}} and {{nowrap|''X'' {{=}} Spec ''A''[''x''1, ..., ''x''''n'']}}. We can take the resolution of {{nowrap|''A''[''x''1, ..., ''x''''n'']}} to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.

*Let ''X'' and ''Y'' be smooth over ''S'', and assume that {{nowrap|''i'' : ''X'' → ''Y''}} is a closed embedding. Using the exact triangle corresponding to the morphisms {{nowrap|''X'' → ''Y'' → ''S''}}, we may determine the cotangent complex ''L''''X''/''Y''. To do this, note that by the previous example, the cotangent complexes ''L''''X''/''S'' and ''L''''Y''/''S'' consist of the Kähler differentials Ω''X''/''S'' and Ω''Y''/''S'' in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that ''L''''X''/''Y'' is nonzero only in the first degree, and in that degree, it is the kernel of the map {{nowrap|''i''*Ω''Y''/''S'' → Ω''X''/''S''}}. This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, ''L''''X''/''Y'' is the conormal bundle of ''X'' in ''Y''.

==See also==
*[[André–Quillen cohomology]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1=André | first1=M. | title=Homologie des Algèbres Commutatives | series=Grundlehren der mathematischen Wissenschaften | volume=206 | publisher=[[Springer-Verlag]] | year=1974}}
*{{Citation
| last = Berthelot
| first = Pierre
| authorlink = Pierre Berthelot (mathematician)
| coauthors = [[Alexandre Grothendieck]], [[Luc Illusie]], eds.
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
| year = 1971
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = xii+700
| nopp = true
}}
*{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1967__32_ | year=1967 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=32 | pages=5–361 | doi=10.1007/BF02732123}}
*{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | title=Catégories cofibrées additives et complexe cotangent relatif | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=French | series=Lecture Notes in Mathematics '''79''' | isbn=978-3-540-04248-8 | date=01/07/1969 }}
*{{Citation | last1=Harrison | first1=D. K. | title=Commutative algebras and cohomology | journal=Transactions of the American Mathematical Society | volume=104 | year=1962 | pages=191–204 | doi=10.2307/1993575 | jstor=1993575 | issue=2 | publisher=American Mathematical Society}}
*{{Citation | last1=Illusie | first1=Luc | author1-link=Luc Illusie | title=Complexe Cotangent et Déformations I | origyear=1971 | publisher=[[Springer-Verlag]] | location=Berlin, New York | language=French | series=Lecture Notes in Mathematics '''239''' | isbn=978-3-540-05686-7 | year=2009}}
*{{Citation | last1=Lichtenbaum | last2=Schlessinger | title=The cotangent complex of a morphism | journal=Transactions of the American Mathematical society | issue=128 | year=1967 | pages=41–70}}
*{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=On the (co-)homology of commutative rings | series=Proc. Symp. Pure Mat. | volume=XVII | publisher=[[American Mathematical Society]] | year=1970}}

{{DEFAULTSORT:Cotangent Complex}}
[[Category:Algebraic geometry]]
[[Category:Category theory]]
[[Category:Homotopy theory]]

Group of rational points on the unit circle

2012-08-21T16:00:32Z

137.122.114.44: /* Other ways to describe the group */ G is not described with blackboard bold elsewhere

The first Worldwide Chefs knives were designed in 1985 by Komin Yamada in Japan. The most critical feature of any knife is its edge, and the Global edge is truly its signature. The straight edge outcomes in a substantially sharper knife which stays sharper longer. The edge on a Global knife is so big and prominent that it is effortlessly noticed with the naked eye and extends a quarter of an inch or far more up from the tip of the knife. Years ago, cooks only had a couple of knife designs from which they could pick. Wipe clean and the knife is ready to use. Some individuals are searching for Wustof knives so please forgive me when I purposely misspell Wustof so men and women can discover the fabulous Wusthof Knives Set that I cherish. I asked for only A single actually good Pricey Knife for Christmas. Trying to do what he believed was even better, he bought me a entire set of Low-cost knives. It so occurred that a lady helped me that had the knife that I bought. Any of the above brands will give you a lifetime of great functionality with minimal care. A buyer wanting a set of really great to great good quality knives will do nicely to stay away from something endorsed by a Television chef. Although I am positive that there is a set of endorsed knives on the industry that are worth their salt I have not identified them yet, and have given up the hunt. Her 40 years of campaigns changed Thanksgiving for the subsequent 150 years. Many have a tendency to opt for non-electric models, probably due to the fact the act of knife sharpening down the ages has constantly been a manual process. Even Chicago Cutlery 15 Pc Review so, the Presto removes the need to have for you to do something other than feed the blade by means of both sides in order to hone the knife back to its original state. The ideal electric knife sharpeners never come along usually. The Fiskars 7861 is the finest axe and knife sharpener in its class. The Spyderco Tri-Angle is the best contemporary version of a whetstone knife sharpener. As long as the knife rides against the knuckle, you often know where the sharp blade is. It's the non-knife hand that dictates the increment of the cut. If you treasured this article so you would like to acquire more info relating to chicago cutlery metropolitan 15 reviews ([http://www.thebestkitchenknivesreviews.com/chicago-cutlery-reviews-best-chicago-cutlery-knives/ http://www.thebestkitchenknivesreviews.com/chicago-cutlery-reviews-best-chicago-cutlery-knives/]) generously visit the web page. Moving the left hand in greater increments when cutting vegetables gives bigger pieces. It is the hand that's NOT holding the knife that is the brains of the operation. Start out applying your chef's knife and you will delight in visually attractive food and a talent to final a lifetime. Knife to be utilised by a Chef in the kitchen. You can go with the old German names, and knives by Wustoff or Henkel's are pretty great, or you can go with some of the new and lighter weight Japanese models, but get a knife that feels superior in your hand, and enjoy a lifetime of better cooking. Keeping your blade razor sharp does not sound safer, but it genuinely is and with a sharp knife you can operate with gentle precision, and get the job carried out a lot more quickly. Knife manage and shape must suit user. Anticipate to pay $20 for the least-pricey knife of serviceable high quality. Chicago Cutlery Design Pro Review Chinese cleavers are becoming additional popular in Western kitchens as a substitute for the chef's knife. Carbon steel-Most knife blades used to be created of carbon steel. The second consideration when deciding on a knife is to find out from what material it is made. Stamped-A stamped blade is made by die stamping, or cutting the standard knife shape from a sheet of metal. A sharp knife cuts superior, much easier and safer.

History of elementary algebra

2012-08-16T17:23:43Z

137.122.114.44: /* Bloom of Thymaridas */ corrected indices

Hi there, I am Andrew Berryhill. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time lately. Since he was eighteen he's been operating as an information officer but he plans on changing it. My wife and I live in Mississippi but now I'm considering other choices. Feel [http://chungmuroresidence.com/xe/reservation_branch2/152663 free online tarot card readings] to surf to my website :: online [http://cspl.postech.ac.kr/zboard/Membersonly/144571 real psychic] ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 simply click for source])

Octal game

2012-07-10T20:06:22Z

137.122.114.44: /* Computation records */ 63 > 1

CMS provides the best platform to create websites that fulfill all the specifications of SEO. It is very easy to customize plugins according to the needs of a particular business. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. If you adored this article and you would like to receive more info relating to [http://htxurl.info/wordpressdropboxbackup626162 wordpress backup] nicely visit our web site. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. This plugin is a must have for anyone who is serious about using Word - Press. They provide many such popular products which you can buy for your baby. That's a total of 180$ for each Wordpress theme if you sell 3 links and keep the designer link for your own website, or 240$ if you sell all links. You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Setting Up Your Business Online Using Free Wordpress Websites. User friendly features and flexibility that Word - Press has to offer is second to none. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It supports backup scheduling and allows you to either download the backup file or email it to you. Make sure you have the latest versions of all your plugins are updated. Millions of individuals and organizations are now successfully using this tool throughout the world. I don't want that for my visitors and I'm quite sure they don't either. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. You should stay away from plugins that are full of flaws and bugs. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins.

Trapezoidal rule (differential equations)

2012-05-18T04:39:32Z

137.122.49.83: /* Motivation */

Hello, I'm Timmy, a 18 year old from Marlboro, United States. My hobbies include (but are not limited to) Vehicle restoration, Hiking and watching Psych.