Geodetic datum - Revision history

178.239.73.50: Added conversion for miles.

2014-11-27T14:23:35Z

Added conversion for miles.

← Older revision		Revision as of 16:23, 27 November 2014
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	~~56 year old Bed~~ and ~~Breakfast Operator Bud from Picton~~, ~~usually spends time with interests which includes football~~, ~~property developers~~ in ~~singapore~~ and ~~textiles~~. ~~Lately took some time to~~ take a ~~trip to Gusuku Sites and Related~~ [http://www.videome.org/profile/DeannaLDKi properties for sale in singapore] of the ~~Kingdom~~ of ~~Ryukyu~~.		The Astaka in Iskandar Malaysia is the ultra luxurious personal condominium to be launched in Johor Bahru to this point. It's made up of two residential towers of sixty four and 70 storeys each. Standing at 301 meters tall, all of the condominium items will take pleasure in unparalleled seaview and in addition that of Singapore island. In addition to residential units, The Astaka can even have business and retail space that may consolidate the project's position as the monetary centre within the heart of Johor Bahru.<br><br>Ultimately, we can not assume that the real estate market in Iskandar and Singapore behaves in the same manner. In addition, certain components, resembling political and foreign money dangers, that we frequently take without any consideration once we buy or promote a Singapore property, actually change into essential considerations on the subject of shopping for [http://www.videome.org/profile/DeannaLDKi properties for sale in singapore] throughout the causeway. Hence, it will be significant for all investors to do their research earlier than they plunge in. Also, builders of future EC websites from the Authorities Land Gross sales programme will probably be allowed to launch items for sale only 15 months from the date of award of the sites, or after the bodily completion of the foundation works, whichever is earlier. ORCHARD HIGHWAY – Essentially the most treasured residential area in Singapore. Semi-Detached Home<br><br>Consisting of one other five condominiums - The Marina Collection, Turquoise, Seascape , The Pinnacle Collection, Seven Palms and two islands, most builders purchased land in Sentosa South Cove for much greater prices than the sooner Sentosa North Cove. Sentosa South Cove spans from the japanese edge of the island to the southern tip the place the beaches are discovered – and that is what makes it so enticing. In Core Central Area , prices of non-landed personal residential properties rose by zero.7% q-q. four,296 models of uncompleted non-public residential properties had been offered in This fall 2012, 25.8% down on the previous quarter. Sales of accomplished personal residential properties declined by fifty five.5% to fifty seven models throughout the identical interval. Attending to Tanjong Katong Secondary School<br><br>Regent Quarter Residences - a collection of 34 one and two bed room apartments and one penthouse for sale at the coronary heart of King's Cross. Just 4 minutes' stroll from King's Cross St Pancras underground station, Regent Quarter lies at the heart of the thrilling King's Cross regeneration space. This is a computation of your total monthly debt obligations (e.g. current home mortgage, new house loan that you are applying for, automotive loan, overdraft amenities, other credit facilities and recurrent debt obligations) to your whole month-to-month revenue. Month-to-month reimbursement instalments for all property loans and different debt obligations are to not exceed a TDSR of 60%. This is to encourage prudent borrowing by households. Near nature & higher class landed enclave<br><br>1998-2000 block with 4-Room Model A (100 sqm), 5-Room Improved (120 sqm) and Govt Residence (one hundred forty sqm), some blocks have 4-Room Mannequin A2 (85 sqm) Due of enormous quantity of unsold flats of bigger sizes, HDB stopped building Govt Residences, last one being topped in 2004, additionally very few 5-Room had been topped in 2007-2009 (Executive Maisonettes not built since 2000). Overall, all rooms became smaller, since Executive are not construct, most 5-room got again the study space. The ratios were revised and raised in Grasp Plan 1998, as we speak most public housing are in 2.eight – four.2 vary, low-density condos are low as 1.4 and office buildings in Central Business District attain a plot ratio of 12 My Featured Properties of the Day! Click it! View it! Metropolis Gate Apartment<br><br>22.To support right-sizing, we're constructing extra studio residences for seniors in varied towns. I do know many are ready to right-size, offered they don't have to depart their neighbourhood. We're introducing a new Studio Condo Precedence Scheme (SAPS) and we are going to reserve half of the provision for seniors who apply for one close to their current flat, near their current neighborhood or close to the place their youngsters live. It is a further enhancement to the present Ageing-in-Place Precedence Scheme and the Married Little one Priority Scheme which award priority by way of giving the seniors extra poll possibilities. The new Studio Apartment Precedence Scheme will exchange these two precedence schemes. This will probably be applied from the Could BTO train.

en>Vsmith at 19:48, 9 February 2014

2014-02-09T19:48:20Z

← Older revision		Revision as of 21:48, 9 February 2014
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	~~In [[algebraic topology]], '''universal coefficient theorems''' establish relationships between~~		56 year old Bed and Breakfast Operator Bud from Picton, usually spends time with interests which includes football, property developers in singapore and textiles. Lately took some time to take a trip to Gusuku Sites and Related [http://www.videome.org/profile/DeannaLDKi properties for sale in singapore] of the Kingdom of Ryukyu.
	~~homology~~ and ~~cohomology theories. For instance~~, ~~the ''integral [[homology theory]]'' of a [[topological space]] ''X'', and its ''homology~~ with ~~coefficients'' in any [[abelian group]] ''A'' are related as follows: the integral homology groups~~

	~~:''H<sub>i</sub>''(''X'', '''Z''')~~

	~~completely determine the groups~~

	~~:''H<sub>i</sub>''(''X''~~, ~~''A'')~~

	Here ''H<sub>i</sub>'' might be the [[simplicial homology]] or more general [[singular homology]] theory: the result itself is a pure piece of [[homological algebra]] about [[chain complex]]es of [[free abelian group]]s. The form of the result is that other coefficients ''A'' may be used, at the cost of using a [[Tor functor]].

	~~For example it is common to take ''A'' to be '''Z'''/2'''Z''', so that coefficients are modulo 2. This becomes straightforward~~ in ~~the absence of 2-[[torsion_(algebra)\|torsion]] in the homology. Quite generally, the result indicates the relationship that holds between the [[Betti number]]s ''b<sub>i</sub>'' of ''X''~~ and ~~the Betti numbers ''b''<sub>''i'',''F''</sub> with coefficients in a [[field (mathematics)\|field]] ''F''~~. ~~These can differ, but only when the [[characteristic (algebra)\|characteristic]] of ''F'' is a [[prime number]] ''p'' for which there is~~ some ~~''p''-torsion in the homology.~~

	~~==Statement of the homology case ==~~
	~~Consider the [[tensor product of modules]] ''H<sub>i</sub>''(''X'', '''Z''') ⊗ ''A''.The theorem states that there is an [[injective]] [[group homomorphism]] ι from this group~~ to ~~''H<sub>i</sub>''(''X, A''), which has [[cokernel]] Tor(''H''<sub>''i''-1</sub>(''X'', '''Z'''), ''A''). In other words, there is~~ a ~~[[natural (category theory)\|natural]] [[short exact sequence]]~~

	~~:<math> 0 \rightarrow H_i(X, \mathbf{Z})\otimes A\rightarrow H_i(X,A)\rightarrow\mbox{Tor}(H_{i-1}(X, \mathbf{Z}),A)\rightarrow 0.</math>~~

	~~Furthermore, this is a [[splitting lemma\|split sequence]] (but the splitting is ''not'' natural).~~

	~~The [[Tor functor\|Tor group]] on the right can be thought of as the obstruction~~ to ~~ι being an isomorphism.~~

	~~==Universal coefficient theorem for cohomology==~~
	~~Let ''G'' be a module over a principal ideal domain ''R'' (e.g.,<math>\mathbf{Z}</math> or a field.)~~

	~~There is also a '''universal coefficient theorem for~~ [~~[cohomology]]''' involving the [[Ext functor]], which asserts that there is a natural short exact sequence~~
	:~~<math> 0 \rightarrow \operatorname{Ext}(\operatorname{H}_{i-1}(X; R), G) \rightarrow \operatorname{H}^i(X; G) \overset{h}\rightarrow \operatorname{Hom}_R(H_i(X; R), G)\rightarrow 0.</math>~~
	~~As in the homology case, the sequence splits, though not naturally.~~

	~~In fact, suppose <math>\operatorname{H}_i(X;G) = \operatorname{ker}\operatorname{\partial}_i \otimes G~~ / ~~\operatorname{im}\operatorname{\partial}_{i+1} \otimes G<~~/~~math> and <math>\operatorname{H}^*(X; G)</math> is defined as <math>\operatorname{ker}(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G))</math>~~. ~~Then ''h'' above is the canonical map: <math>h([f])([x]) = f(x)~~.~~</math> An alternative point-of-view can be based on representing cohomology via [[Eilenberg-MacLane_ space]] where the map ''h'' takes a homotopy class of maps from <math>X<~~/~~math> to <math>K(G,i)<~~/~~math> to the corresponding homomorphism induced~~ in ~~homology. Thus, the Eilenberg-MacLane space is a ''weak right [[adjoint~~]~~]'' to the homology [[functor]]. <ref>{{Harv\|Kainen\|1971}}</ref>~~

	~~== Example: ''mod'' 2 cohomology~~ of the ~~real projective space==~~
	~~Let ''X'' = '''P'''<sup>''n''</sup>('''R'''), the [[real projective space]]. We compute the singular cohomology~~ of ~~''X'' with coefficients in ''R'' := '''Z'''<sub>2</sub>~~.

	~~Knowing that the integer homology is given by:~~

	~~:<math>H_i(X; \mathbf{Z}) =~~
	~~\begin{cases}~~
	~~\mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\~~
	~~\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\~~
	~~0 & \mbox{else.}~~
	~~\end{cases}</math>~~

	~~We have Ext(''R'', ''R'') = ''R'', Ext('''Z''', ''R'')= 0, so that the above exact sequences yield~~
	~~:<math>\forall i = 0 \ldots n , \ H^i (X; R) = R</math>.~~

	~~In fact the total [[cohomology ring]] structure is~~
	~~:<math>H^*(X; R) = R [w] / \langle w^{n+1} \rangle </math>.~~

	~~==Corollaries==~~
	A special case of the theorem is computing integral cohomology. For a finite CW complex ''X'', <math>H_i(X; \mathbf{Z})</math> is finitely generated, and so we have the following [[Fundamental_theorem_of_finitely_generated_abelian_groups#Classification\|decomposition]].

	~~:<math> H_i(X; \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)}\oplus T_i .</math>~~

	~~Where <math> \beta_i(X) </math> are the [[betti numbers]] of ''X''. One may check that~~

	:<math> \mbox{Hom}(H_i(X),\mathbf{Z}) \cong \mbox{Hom}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Hom}(T_i, \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} </math>, and <math> \mbox{Ext}(H_i(X),\mathbf{Z}) \cong \mbox{Ext}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Ext}(T_i, \mathbf{Z}) \cong T_i. </math>

	~~This gives the following statement for integral cohomology:~~

	~~:<math> H^i(X;\mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} \oplus T_{i-1}. </math>~~

	For ''X'' an [[orientability\|orientable]], [[closed manifold\|closed]], and [[connected space\|connected]] ''n''-[[manifold]], this corollary coupled with [[Poincaré duality]] gives that <math>\beta_i(X)=\beta_{n-i}(X) </math>.

	~~{{reflist}}~~

	~~==References==~~
	*[[Allen Hatcher]], ''Algebraic Topology'', Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the [http://www.math.cornell.edu/~hatcher/AT/ATpage.html author's homepage].

	* {{cite journal
	~~\| last = Kainen~~
	~~\| first = P. C.~~
	~~\| authorlink = Paul Chester Kainen~~
	~~\| coauthors =~~
	~~\| title = Weak Adjoint Functors~~
	~~\| journal = Mathematische Zeitschrift~~
	~~\| volume = 122~~
	~~\| issue =~~
	~~\| pages = 1–9~~
	~~\| publisher =~~
	~~\| year = 1971~~
	~~\| pmid =~~
	~~\| pmc =~~
	~~\| doi =~~
	}}

	~~[[Category:Homological algebra]]~~
	~~[[Category:Theorems in algebraic topology]]~~

en>Mario1952: {{Geodesy}}

2014-01-30T14:55:23Z

New page

In [[algebraic topology]], '''universal coefficient theorems''' establish relationships between
homology and cohomology theories. For instance, the ''integral [[homology theory]]'' of a [[topological space]] ''X'', and its ''homology with coefficients'' in any [[abelian group]] ''A'' are related as follows: the integral homology groups

:''Hi''(''X'', '''Z''')

completely determine the groups

:''Hi''(''X'', ''A'')

Here ''Hi'' might be the [[simplicial homology]] or more general [[singular homology]] theory: the result itself is a pure piece of [[homological algebra]] about [[chain complex]]es of [[free abelian group]]s. The form of the result is that other coefficients ''A'' may be used, at the cost of using a [[Tor functor]].

For example it is common to take ''A'' to be '''Z'''/2'''Z''', so that coefficients are modulo 2. This becomes straightforward in the absence of 2-[[torsion_(algebra)|torsion]] in the homology. Quite generally, the result indicates the relationship that holds between the [[Betti number]]s ''bi'' of ''X'' and the Betti numbers ''b''''i'',''F'' with coefficients in a [[field (mathematics)|field]] ''F''. These can differ, but only when the [[characteristic (algebra)|characteristic]] of ''F'' is a [[prime number]] ''p'' for which there is some ''p''-torsion in the homology.

==Statement of the homology case ==
Consider the [[tensor product of modules]] ''Hi''(''X'', '''Z''') ⊗ ''A''.The theorem states that there is an [[injective]] [[group homomorphism]] ι from this group to ''Hi''(''X, A''), which has [[cokernel]] Tor(''H''''i''-1(''X'', '''Z'''), ''A''). In other words, there is a [[natural (category theory)|natural]] [[short exact sequence]]

:<math> 0 \rightarrow H_i(X, \mathbf{Z})\otimes A\rightarrow H_i(X,A)\rightarrow\mbox{Tor}(H_{i-1}(X, \mathbf{Z}),A)\rightarrow 0.</math>

Furthermore, this is a [[splitting lemma|split sequence]] (but the splitting is ''not'' natural).

The [[Tor functor|Tor group]] on the right can be thought of as the obstruction to ι being an isomorphism.

==Universal coefficient theorem for cohomology==
Let ''G'' be a module over a principal ideal domain ''R'' (e.g.,<math>\mathbf{Z}</math> or a field.)

There is also a '''universal coefficient theorem for [[cohomology]]''' involving the [[Ext functor]], which asserts that there is a natural short exact sequence
:<math> 0 \rightarrow \operatorname{Ext}(\operatorname{H}_{i-1}(X; R), G) \rightarrow \operatorname{H}^i(X; G) \overset{h}\rightarrow \operatorname{Hom}_R(H_i(X; R), G)\rightarrow 0.</math>
As in the homology case, the sequence splits, though not naturally.

In fact, suppose <math>\operatorname{H}_i(X;G) = \operatorname{ker}\operatorname{\partial}_i \otimes G / \operatorname{im}\operatorname{\partial}_{i+1} \otimes G</math> and <math>\operatorname{H}^*(X; G)</math> is defined as <math>\operatorname{ker}(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G))</math>. Then ''h'' above is the canonical map: <math>h([f])([x]) = f(x).</math> An alternative point-of-view can be based on representing cohomology via [[Eilenberg-MacLane_ space]] where the map ''h'' takes a homotopy class of maps from <math>X</math> to <math>K(G,i)</math> to the corresponding homomorphism induced in homology. Thus, the Eilenberg-MacLane space is a ''weak right [[adjoint]]'' to the homology [[functor]]. <ref>{{Harv|Kainen|1971}}</ref>

== Example: ''mod'' 2 cohomology of the real projective space==
Let ''X'' = '''P'''''n''('''R'''), the [[real projective space]]. We compute the singular cohomology of ''X'' with coefficients in ''R'' := '''Z'''2.

Knowing that the integer homology is given by:

:<math>H_i(X; \mathbf{Z}) =
\begin{cases}
\mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\
\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\
0 & \mbox{else.}
\end{cases}</math>

We have Ext(''R'', ''R'') = ''R'', Ext('''Z''', ''R'')= 0, so that the above exact sequences yield
:<math>\forall i = 0 \ldots n , \ H^i (X; R) = R</math>.

In fact the total [[cohomology ring]] structure is
:<math>H^*(X; R) = R [w] / \langle w^{n+1} \rangle </math>.

==Corollaries==
A special case of the theorem is computing integral cohomology. For a finite CW complex ''X'', <math>H_i(X; \mathbf{Z})</math> is finitely generated, and so we have the following [[Fundamental_theorem_of_finitely_generated_abelian_groups#Classification|decomposition]].

:<math> H_i(X; \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)}\oplus T_i .</math>

Where <math> \beta_i(X) </math> are the [[betti numbers]] of ''X''. One may check that

:<math> \mbox{Hom}(H_i(X),\mathbf{Z}) \cong \mbox{Hom}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Hom}(T_i, \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} </math>, and <math> \mbox{Ext}(H_i(X),\mathbf{Z}) \cong \mbox{Ext}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Ext}(T_i, \mathbf{Z}) \cong T_i. </math>

This gives the following statement for integral cohomology:

:<math> H^i(X;\mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} \oplus T_{i-1}. </math>

For ''X'' an [[orientability|orientable]], [[closed manifold|closed]], and [[connected space|connected]] ''n''-[[manifold]], this corollary coupled with [[Poincaré duality]] gives that <math>\beta_i(X)=\beta_{n-i}(X) </math>.

{{reflist}}

==References==
*[[Allen Hatcher]], ''Algebraic Topology'', Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the [http://www.math.cornell.edu/~hatcher/AT/ATpage.html author's homepage].

* {{cite journal
| last = Kainen
| first = P. C.
| authorlink = Paul Chester Kainen
| coauthors =
| title = Weak Adjoint Functors
| journal = Mathematische Zeitschrift
| volume = 122
| issue =
| pages = 1–9
| publisher =
| year = 1971
| pmid =
| pmc =
| doi =
}}

[[Category:Homological algebra]]
[[Category:Theorems in algebraic topology]]