formulasearchengine - User contributions [en]

Stochastic dominance

2014-01-05T20:06:20Z

64.121.31.29:

In [[mathematics]], '''Baire functions''' are [[function (mathematics)|function]]s obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by {{harvs|txt|authorlink=René-Louis Baire|first=René-Louis|last=Baire|year=1905}}. A [[Baire set]] is a set whose [[indicator function|characteristic function]] is a Baire function (not necessarily of any particular class, as defined below).

==Classification of Baire functions==
Baire functions of class ''n'', for any countable [[ordinal number]] ''n'', form a vector space of real-valued functions defined on a [[topological space]], as follows.

*The Baire class 0 functions are the [[continuous function]]s.

*The Baire class 1 functions are those functions which are the [[pointwise convergence|pointwise limit]] of a [[sequence]] of Baire class 0 functions.

*In general, the Baire class ''n'' functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than ''n''.

Some authors define the classes slightly differently, by removing all functions of class less than ''n'' from the functions of class ''n''. This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space.

[[Henri Lebesgue]] proved that (for functions on the unit interval) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class.

==Baire class 1==
Examples:
*The [[derivative]] of any differentiable function is of class 1. An example of a differentiable function whose derivative is not continuous (at ''x''=0) is the function equal to <math>x^2\sin(1/x)</math> when ''x''≠0, and 0 when ''x''=0. An infinite sum of similar functions (scaled and displaced by rational numbers) can even give a differentiable function whose derivative is discontinuous on a dense set. However, it necessarily has points of continuity, which follows easily from The Baire Characterisation Theorem (below; take ''K''=''X''='''R''').
*The function equal to 1 if ''x'' is an integer and 0 otherwise. (An infinite number of large discontinuities.)
*The function that is 0 for irrational ''x'' and 1/''q'' for a rational number ''p''/''q'' (in reduced form). (A dense set of discontinuities, namely the set of rational numbers.)

The Baire Characterisation Theorem states that a real valued function f defined on a [[Banach space]] X is a Baire-1 function if and only if for every non-empty closed subset K of X, the restriction of f to K has a point of continuity relative to the topology of K.

By another theorem of Baire, for every Baire-1 function the points of continuity are a [[comeager]] [[Gδ set|''G''δ]]set {{harv|Kechris|1995|loc=Theorem (24.14)}}.

==Baire class 2==
Examples:
*An example of a Baire class two function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers, <math>\chi_\mathbb{Q}</math>, also known as the [[Dirichlet function]]. It is discontinuous everywhere.
*Similarly, the function which is 1 if ''x'' is in the [[Cantor set]] and 0 otherwise. This function is 0 for an uncountable set of ''x'' values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and continuous wherever it equals 0.

==Baire class 3==
Examples:
{{Expand section|date=November 2013}}

==See also==

*[[Baire set]]
*[[Nowhere continuous function]]

==References==

*{{citation|first=R. |last=Baire|title=Leçons sur les fonctions discontinues, professées au collège de France|publisher= Gauthier-Villars |year=1905|url=http://archive.org/detail/leconsdiscontinues00bairrich}}
*{{citation|first=Alexander S.|last=Kechris|author-link=Alexander S. Kechris|title=Classical Descriptive Set Theory|publisher=Springer-Verlag|year=1995}}

==External links==
* [http://eom.springer.de/b/b015030.htm Springer Encyclopaedia of Mathematics article on Baire classes]

[[Category:General topology]]
[[Category:Real analysis]]
[[Category:Types of functions]]

Spherical tokamak

2013-11-04T11:04:57Z

64.121.114.179: /* Disadvantages */

{{distinguish|Meixner polynomials}}
In mathematics, the '''Meixner–Pollaczek polynomials''' are a family of [[orthogonal polynomials]] ''P''{{su|b=''n''|p=(λ)}}(''x'',φ) introduced by {{harvs|txt|authorlink=Josef Meixner|last=Meixner|year=1934}}, which up to elementary changes of variables are the same as the '''Pollaczek polynomials''' ''P''{{su|b=''n''|p=λ}}(''x'',''a'',''b'') rediscovered by {{harvs|txt|authorlink=Felix Pollaczek|last=Pollaczek|year=1949}} in the case λ=1/2, and later generalized by him.

They are defined by
:<math>P_n^{(\lambda)}(x;\phi) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1(-n,\lambda+ix;2\lambda;1-e^{-2i\phi})</math>
:<math>P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1(-n,\lambda+i(a\cos \phi+b)/\sin \phi;2\lambda;1-e^{-2i\phi})</math>

They are orthogonal on the real line with respect to the weight function
:<math> w(x; \lambda, \phi)= |\Gamma(\lambda+ix)|e^{(2\phi-\pi)x}</math>
and the orthogonality is given by
:<math>\int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn}</math>

==See also==

*[[Sieved Pollaczek polynomials]]

==References==

*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
*{{dlmf|id=18.35|title=Pollaczek Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | last1=Meixner | first1=J. | title=Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion | doi=10.1112/jlms/s1-9.1.6 | year=1934 | journal=J. London Math. Soc. | volume=s1-9 | pages=6–13}}
*{{Citation | last1=Pollaczek | first1=Félix | title=Sur une généralisation des polynomes de Legendre | url=http://gallica.bnf.fr/ark:/12148/bpt6k31801/f1363 | mr=0030037 | year=1949 | journal=[[Les Comptes rendus de l'Académie des sciences]] | volume=228 | pages=1363–1365}}

{{DEFAULTSORT:Meixner-Pollaczek polynomials}}
[[Category:Orthogonal polynomials]]

Algebraic geometry of projective spaces

2013-04-16T01:29:32Z

64.121.106.19:

{{Politics of the Philippines}}
[[File:Philippine party-list elections.svg|thumb|Graph showing voter participation in party-list elections.]]
:{{redirects here|Party-list representatives|party-list representatives elsewhere|list MP|the political concept|party-list proportional representation}}
'''Party-list representation in the House of Representatives of the Philippines''' refers to a system in which 20% of the [[House of Representatives of the Philippines|House of Representatives]] is elected. While the House is predominately elected by a [[plurality voting system]], known as a first-past-the-post system, party-list representatives are elected by a type of [[party-list proportional representation]]. Under-represented sectoral groups, such as labor, peasant, urban poor, indigenous cultural communities, women, youth, and other sectors as may be provided by law except the religious sector, may participate in the party-list election. The 1987 [[Constitution of the Philippines]] created the party-list system.

The determination of what parties are allowed to participate—who their nominees should be, how the winners should be determined, and the allocation of seats for the winning parties—has been controversial ever since the party-list election was first contested in 1998 and has resulted in several landmark [[Commission on Elections (Philippines)|Commission on Elections]] (COMELEC) and Supreme Court cases.

Party-list representatives are determined in a party-list election, where the voter votes for the party and not for the nominees ([[closed list]]), the votes are then arranged in descending order, with the parties that won at least 2% of the national vote are given one seat, with additional seats determined by a formula depending on the number of votes garnered, with no party winning more than three seats. If the number of sectoral representatives does not reach 20% of the total number of representatives, parties that haven't won seats are given a seat each until the 57 seats are filled up. A voter therefore has two [[parallel voting|parallel votes]] in House of Representatives elections -- for district representative and for the party-list representative. Neither vote affects the other.

Party-list representation makes use of the tendency for proportional representation systems to favor [[single-issue politics#Single-issue parties|single-issue parties]], and applies that tendency to allow underrepresented sectors to represent themselves in the law-making process.

==Manner of election==

===Constitution===
The [[Constitution of the Philippines|constitution]] mandates that the sectoral representatives shall compose 20% of the House of Representatives. For three consecutive terms after the ratification of the constitution, one-half of the seats allocated to party-list representatives were filled "by selection or election."<ref>{{cite web|url=http://www.comelec.gov.ph/laws/constitution/consti_art6.html |title=CONSTITUTION - Article VI: LEGISLATIVE DEPARTMENT |published=[[Commission on Elections (Philippines)|]] |date=2009-08-10 |accessdate=2010-11-29}}</ref> For the [[Philippine House of Representatives elections, 1987|1987]], [[Philippine House of Representatives elections, 1992|1992]] and [[Philippine House of Representatives elections, 1995|1995]] elections, the president appointed sectoral representatives, subject to the confirmation from the [[Commission on Appointments (Philippines)|Commission on Appointments]], half of whose members are derived from the House of Representatives.

===Party-List System Act===
{|class=wikitable align=right style="text-align:center; font-size:92%; margin: 0 0 0.5em 1em" width=320px cellpadding=20px
|-
!rowspan=2|Election !!rowspan=2|Method !!rowspan=2|Legislative districts !!colspan=2|Sectoral representatives !!rowspan=2|[[Underhang]]
|-
!20% quota !! Seats won
|-
|[[Philippine House of Representatives elections, 2001|2001]] || ''VFP'' || 205 || 51 || 14 || 37
|-
|[[Philippine House of Representatives elections, 2004|2004]] || ''VFP'' || 209 || 52 || 24 || 28
|-
|rowspan=2|[[Philippine House of Representatives elections, 2004|2007]] || ''VFP'' ||rowspan=2|218 ||rowspan=2|54 || 22 || 32
|-
|''BANAT'' || 53 || 1
|-
|[[Philippine House of Representatives elections, 2010|2010]] || ''BANAT'' || 229 || 57 || 56 || 1
|}

On March 3, 1995, Republic Act No. 7941 or the '''Party-List System Act''' was signed into law. It mandated that "the state shall promote [[proportional representation]] in the election of representatives to the House of Representatives through a party-list system". The five political parties with the highest number of members at the start of the [[10th Congress of the Philippines]] were banned from participating. Each voter can vote one party via [[closed list]]; votes are then tallied nationwide as one [[at-large]] district, with the number of sectoral representatives should not surpass 20% of the total number of representatives. The law provided that each party that has 2% of the national vote be entitled one seat each, and an additional seat for every 2% of the vote thereafter until a party has three seats. This means that a party can win the maximum three seats if it surpasses 6% of the national vote.<ref>{{cite web|url=http://www.comelec.gov.ph/laws/republic_acts/ra_7941.html |title=REPUBLIC ACT No. 7941 AN ACT PROVIDING FOR THE ELECTION OF PARTY-LIST REPRESENTATIVES THROUGH THE PARTY-LIST SYSTEM, AND APPROPRIATING FUNDS THEREFOR |published=[[Commission on Elections (Philippines)|]] |date=2009-08-10 |accessdate=2010-11-29}}</ref>

While the law was first used for the [[Philippine House of Representatives elections, 1998|1998 election]], and several parties did meet the 2% quota during the succeeding elections, they did not fill up the required 20% allocation for party-list representatives of the constitution. Furthermore, the votes for parties that had more than 6% of the vote were considered [[wasted vote|wasted]].<ref>{{cite news|url=http://www.mb.com.ph/articles/286667/should-we-retain-partylist-system |title=Should We Retain Party-List System? |first=Florangel |last=Rosario Braid |work=[[Manila Bulletin]] |date=2010-11-09 |accessdate=2010-11-29}}</ref> [[Ateneo de Manila University]] mathematics professor Felix Muga II said that "Any seat allocation formula that imposes a seat-capping mechanism on the party-list proportional representation voting system contradicts the social justice provision of the 1987 Constitution."<ref>{{cite news|url=http://archives.manilatimes.net/national/2009/may/02/yehey/opinion/20090502opi2.html |title=‘Reductio ad absurdum’ |first=Elmer |last=Ordoñez |work=[[The Manila Times]] |date=2009-05-02 |accessdate=2011-04-14}}</ref>

Any vacancy will be filled up by the person next in line in the list; in cases where a seated sectoral representative switches parties, that representative will lose one's seat and the person next in line in the list shall assume the seat.

===''Veterans Federation Party et al. vs. COMELEC''===
{|class=wikitable align=right width=250px
|+Party-list results
|-
![[Philippine House of Representatives elections, 2001#Election for sectoral representatives|2001]]:
|-style="font-size:90%"
|[[File:2001PhilippineHousePartylistElectionChart.PNG|250px]] Note: Majority of the parties were disqualified after the election.
|-
![[Philippine House of Representatives elections, 2004#Election for sectoral representatives|2004]]:
|-
|[[File:2004PhilippineHousePartylistElectionChart.png|250px]]
|-
![[Philippine House of Representatives elections, 2007#Party-list election result|2007]]:
|-
|[[File:2007PhilippineHousePartylistElectionChart.png|250px]]
|-
![[Philippine House of Representatives party-list election, 2010|2010]]:
|-
||[[File:2010PhilippineHousePartylistElectionChart.png|250px]]
|-
!Key:
|-style="font-size:90%"
|
*Inner ring: Proportion of votes, excluding spoiled/invalid votes.
**Gray: Parties that did not win seats.
*Middle ring (2007 only): Proportion of seats won as per ''VFP vs. COMELEC''.
*Outer ring: Proportion of seats won (for 2007, this is the final allocation as per ''BANAT vs. COMELEC'').
**Black: Unfilled seats.
|}

In 2000, the [[Veterans Federation Party]] (VFP), the [[Akbayan! Citizens' Action Party]] and several other parties sued the COMELEC which led a case in the [[Supreme Court of the Philippines|Supreme Court]]; the court ruling changed the way how the seats are allocated for the winning parties. In 1998, only 25 representatives were elected out of 14 winning parties, well short of the then 52 representatives needed to fill up 20% of the House. The so-called "Panganiban formula," named after [[Chief Justice of the Supreme Court of the Philippines|Chief Justice]] [[Artemio Panganiban]], calculates that the number of seats a party will win is dependent on the number of votes of the party with the highest amount of votes.<ref>{{cite news|url=http://newsinfo.inquirer.net/breakingnews/nation/view/20070702-74414/Dilemma_over_partylist_formula_delays_winners%92_proclamation |title=Dilemma over partylist formula delays winners’ proclamation |first=Nikko |last=Dizon |work=[[Philippine Daily Inquirer]] |date=2007-07-02 |accessdate=2010-11-29}}</ref>

The court maintained the four inviolable parameters:
<blockquote>
First, the twenty percent allocation - the combined number of all party-list congressmen shall not exceed twenty percent of the total membership of the House of Representatives, including those elected under the party list.
Second, the two percent threshold - only those parties garnering a minimum of two percent of the total valid votes cast for the party-list system are “qualified” to have a seat in the House of Representatives;
Third, the three-seat limit - each qualified party, regardless of the number of votes it actually obtained, is entitled to a maximum of three seats; that is, one “qualifying” and two additional seats.
Fourth, proportional representation - the additional seats which a qualified party is entitled to shall be computed “in proportion to their total number of votes.”</blockquote>

The court came up with the following procedure on how to determine how many seats a party wins. First, the party with the highest number of votes gets at least one seat. It can win additional seats for every 2% of the national vote until it reaches the three-seat limit.
Therefore:<center>
<math>TP_s = 1~ \mbox{if}~g\ >= 0.02 </math> 
<math>TP_s = 2~ \mbox{if}~g\ >= 0.04 </math> 
<math>TP_s = 3~ \mbox{if}~g\ >= 0.06 </math></center>
where:
:::* ''TPs'' is the number of seats of the top party.
:::* ''g'' is the percentage of votes garnered by the sectoral organization,

For the other parties surpassing the 2% threshold, they all automatically win one seat; additional seats will be won according to the following formula.
<center>:<math>\mathrm{S} = (\frac{\mathrm{PV}}{\mathrm{TP}})\times{TP_s}</math></center>
where:
:::* ''S'' is the number of seats
:::* ''PV'' is the votes for the party
:::* ''TP'' is the votes of the top party.
:::* ''TPs'' is the number of seats of the top party.
The product, disregarding integers, is the number of additional seats for the party.

Prior to the adopting the "Panganiban formula," the court considered applying the [[Largest remainder method|Niemayer formula]] used in the allocation of seats in the German [[Bundestag]]. However, since R.A. 7941 limits the maximum number of seats for each party to three, of the existence of a 2% quota, and that 20% of the seats can be filled up, the court instead devised the formula above to ensure that the 20% allocation for sectoral representatives would not be exceeded, the 2% threshold will be upheld, the three-seat limit enforced and the proportional representation be respected.<ref>{{cite web|url=http://sc.judiciary.gov.ph/jurisprudence/2000/oct2000/136781.htm |title=Veterans Federation Party et al. vs. COMELEC |work=[[Supreme Court of the Philippines]] |accessdate=2010-11-29}}</ref> The formula was first used in determining the result of the [[Philippine general election, 2001|2001]], and was first applied in the [[Philippine House of Representatives elections, 2004|2004 elections]].

The use of this formula by the COMELEC had been labeled by certain groups as to "annihilate independent voices in the House," according to Akbayan representative [[Etta Rosales]].<ref>{{cite news|url=http://newsinfo.inquirer.net/inquirerheadlines/nation/view/20070605-69478/Only_Buhay_may_get_three_seats__ |title=Only Buhay may get three seats |first=Nikko |last=Dizon |work=[[Philippine Daily Inquirer]] |date=2007-06-05 |accessdate=2010-12-30}}</ref> The court upheld this in subsequent cases, such as the ''[[Partido ng Manggagawa]] vs. COMELEC'' and ''[[Citizens' Battle Against Corruption]] vs. COMELEC''.<ref>{{cite news|url=http://newsinfo.inquirer.net/inquirerheadlines/nation/view/20090423-200890/Mechanism-of-proportional-representation |title=Mechanism of proportional representation |work=[[Philippine Daily Inquirer]] |date=2009-04-23 |accessdate=2011-04-14}}</ref>

Panganiban in 2010 remarked in a lecture at the [[Ateneo Law School]] that "It’s very complicated and there must be an easier formula to compute," adding that the party-list law has to be amended by Congress.<ref>{{cite news|url=http://www.gmanews.tv/story/206434/ex-sc-chief-sees-urgent-need-to-amend-partylist-law |title=Ex-SC chief sees urgent need to amend partylist law |first=Anita |last=Legaspi |work=GMANews.tv |date=2010-11-19 |accessdate=2010-11-29}}</ref>

===''BANAT vs. COMELEC''===
In 2007, another party-list group, the Barangay Association for National Advancement and Transparency (BANAT, now [[Barangay Natin!]]) sued the COMELEC for not proclaiming the full number of party-list representatives (they were not among on those who were proclaimed winners). As with the other cases, the Supreme Court condensed all the cases to one case. The court ruled on April 21, 2009 that the 2% [[election threshold]] unconstitutional, and stipulated that for every five legislative districts created, one seat for sectoral representatives should be created; this thereby increased the sectoral seats in the [[14th Congress of the Philippines|14th Congress]] from 22 to 55; the Supreme Court, however, upheld the 3-seat cap.<ref>{{cite news|url=http://opinion.inquirer.net/inquireropinion/columns/view/20090425-201449/Party-list-imponderables Party-list imponderables |first=Artemio |last=Panganiban |authorlink=Artemio Panganiban |title=Party-list imponderables |work=[[Philippine Daily Inquirer]] |work=2009-04-25 |accessdate=2011-04-14}}</ref>

To determine the number of seats for sectoral representatives, the formula for the quotient is: <center> <math>S = \left(\frac{D}{0.8}\right) \times 0.2</math> </center>
::where:
:::* ''S'' is the number of seats allocated for sectoral representation,
:::* ''D'' is the total number of district representatives, and
:::* ''D / 0.8'' is the total number of members of the House.

To get the first guaranteed seat, a sectoral party or organization should at least get 2% of the total votes cast for partly list elections. The formula for the quotient is: <center> <math>g = \frac{V}{P}</math></center>
::where:
:::* ''g'' is the percentage of votes garnered by the sectoral organization,
:::* ''V'' is the total number of votes cast in the party list representation, and
:::* ''P'' is the total number votes of the sectoral organization.
::Therefore: <center> <math>R_1 = 1~ \mbox{if}~g\ >= 0.02 </math> </center>

If the total number of guaranteed seats awarded is less than the total number of seats reserved for sectoral representatives (''S''), the unassigned seats will awarded in the second round of seat allocation. To get the number of additional seats, this formula will be followed. <center> <math>R_2 =(S-T_1) \times g </math></center>
::where:
:::* ''<math>{R_2}</math>'' is the total number of additional seats awarded to the sectoral organization,
:::* ''S'' is the number of seats allocated for party-list representatives,
:::* ''<math>{T_1}</math>'' is the total number awarded seats <math>({R_1})</math> in the first round of seat allocation, and
:::* ''g'' is the percentage of votes garnered by the sectoral organization.
::''Note:'' <math>{R_2}</math> should appear as whole integer.

If the total number of seats awarded after two rounds is still less than the total number of seats reserved for sectoral representatives (''S''), the remaining seats will be assigned to sectoral organizations next in rank (one seat each organization) whose <math>{R_2}</math> result is 0 until all available seats are completely distributed. <center> <math>T_3 =(S-T_1-T_2)\ </math></center>
::where:
:::* ''<math>{T_3}</math>'' is the total number of sectoral organizations next in rank (in Round 2) to be given with one seat,
:::* ''S'' is the number of seats allocated for party-list representatives,
:::* ''<math>{T_1}</math>'' is the total number awarded seats in the first round of seat allocation, and
:::* ''<math>{T_2}</math>'' is the total number awarded seats in the second round of seat allocation.

This is essentially a [[Hare quota]], with the following exceptions:
*The 2% [[election threshold]] automatically awards parties one seat; this means that the total seats that will be disputed is the difference of the number of party-list seats and the number of parties that surpassed the threshold.
*The fractional remainder is disregarded. The seats that could've been distributed from the fractional remainders are given to parties that quotas less than 1 after the threshold.
*The party cannot win more than three seats. With the large amount of parties contesting, this means the share of the votes the parties get are small—in 2010, the party with the most votes ([[Ako Bicol Political Party]]) won 5.20% of the vote—the only way a party's [[wasted vote|votes can be wasted]] is if its quota after the threshold is 4 or more. This can be affected if several parties surpassed the threshold (thus lessening the amount of seats to be distributed), or if a party wins via a landslide. In 2010, AKB's quota after threshold was 2.33, or, disregarding decimals, 2. This entitled them to 2 additional seats aside from the automatic 1 seat they've won by surpassing the threshold.

Senator [[Joker Arroyo]] criticized the ruling of the Supreme Court, saying that the court "overreached itself and engaged in judicial legislation." Arroyo later compared with parties with between "155,000 to 197,000 votes... a measly 1 percent to 1.24 percent of the votes" to a city which needs a population of 250,000 or more to obtain its own legislative district.<ref>{{cite news|url=http://www.mb.com.ph/node/204097 |title=Supreme Court and judicial legislation |first=Joker |last=Arroyo |authorlink=Joker Arroyo |work=[[Manila Bulletin]] |date=2009-04-28 |accessdate=2011-04-14}}</ref>

===Summary===
{|class=wikitable
|-
!Method !! First seat !! Second seat !! Third seat
|-
|R.A. 7941 || 2% of vote || 4% of vote || 6% of vote
|-
|rowspan=2|''VFP vs. COMELEC'' ||rowspan=2|2% of the vote ||Party with most votes: 4% of the vote ||Party with most votes: 6% of the vote
|-
|colspan=2|Other parties: Total votes divided by votes of the party with most votes; quotient will be multiplied by the number of seats the party with the most votes have. Product, disregarding decimals, is the number of seats.
|-
|rowspan=2|''BANAT vs. COMELEC'' || 2% of the vote ||colspan=2 rowspan=2|[[Hare quota]], without decimals, from the seats that are not yet allocated.
|-
|If quota has not been met, parties with less than 1% of the preferences will get one seat until quota is met.
|}

===Example===
In [[Philippine House of Representatives party-list election, 2010|2010]], there are 57 party-list seats being contested, with 29,311,294 valid votes cast, and 12 parties having at least 2% of the vote.

[[Ako Bicol Political Party]] topped the vote, receiving 1,524,006 votes or 5.20% of the vote.
*First round:
:<math>R_1 = 1~ \mbox{since}~0.0519 \ >= 0.02 </math>
*Second round:
:<math>R_2 =(57-12) \times 0.0519 </math>
:<math>R_2 =45 \times 0.0519 </math>
:<math>R_2 =2.3397</math>
:Disregarding decimals, <math>R_2 =2</math>
*Both rounds:
:<math>S =1+2=3</math>
*Hence, AKB won three seats in the House of Representatives.

[[Akbayan Citizens' Action Party]] received 1,061,947 votes or 3.62% of the vote.
*First round:
:<math>R_1 = 1~ \mbox{since}~0.0362\ >= 0.02 </math>
*Second round:
:<math>R_2 =(57-12) \times 0.0362 </math>
:<math>R_2 =45 \times 0.0362 </math>
:<math>R_2 =1.6303</math>
:Disregarding decimals, <math>R_2 =1</math>
*Both rounds:
:<math>S =1+1=2</math>
*Hence, Akbayan won two seats in the House of Representatives.

[[Alagad]] received 227,281 or 0.78% of the vote.
*First round:
:<math>R_0 = 0~ \mbox{since}~0.0078 \ >= 0.02 </math>
*Second round: At this point, 35 seats have already been awarded.
:<math>R_2 =(57-35) \times 0.0078</math>
:<math>R_2 =45 \times 0.0078</math>
:<math>R_2 =0.3489</math>
:Disregarding decimals, <math>R_2 =0</math>
*Both rounds:
:<math>S =0+0=0</math>
*However, not all seats have been distributed. Therefore: Alagad won one seat in the House of Representatives.

==Nominations==
While, the party-list system has been used by some sectors which had not been able to participate in government to have a voice in Congress, allegations from left-leaning party-lists say that several parties had been used as fronts by then [[Gloria Macapagal-Arroyo]] administration to further their interests. Parties such as 1-UTAK, that represents transport groups, PACYAW, that claims to advocate sports development and others have government officials as nominees.<ref>{{cite news|url=http://www.gmanews.tv/story/187170/arroyo-to-use-party-list-seats-to-win-as-house-speaker |title=‘Arroyo to use party-list seats to win as House Speaker’ |first=Andreo |last=Calonzo |work=GMANews.tv |date=2010-03-27 |accessdate=2010-04-08}}</ref> The number one nominee of [[Ang Galing Pinoy]], a group claiming to represent security guards and [[Motorized tricycle (Philippines)|tricycle]] drivers, is outgoing [[Pampanga]] [[Legislative districts of Pampanga#2nd District|2nd district]] representative [[Mikey Arroyo]], the president's son; Arroyo won a seat in the [[Philippine House of Representatives party-list election, 2010|2010 election]].<ref>{{cite news|url=http://www.gmanews.tv/story/214076/sc-its-final-mikey-can-represent-security-guards-tricycle-drivers-in-congress |title=SC: It’s final, Mikey can represent tricycle drivers in Congress |work=GMANews.tv |date=2011-02-27 |accessdate=2011-04-14}}</ref>

On the other hand, a disqualification case had been brought up against the left-leaning parties in the [[Bagong Alyansang Makabayan]] (New Patriotic Alliance) ([[Bayan Muna]] (Nation First), [[Kabataan Party-list]] (Youth Party-list), [[GABRIELA Women's Party]], and [[Anakpawis]]), as they are just pursuing "ideological objectives" like overthrowing the ruling system through "bloody means."<ref>{{cite news|url=http://www.gmanews.tv/story/182530/disqualification-of-leftist-party-list-groups-eyed |title=Disqualification of leftist party-list groups eyed |work=GMANews.tv |date=2010-01-16 |accessdate=2011-04-14}}</ref>

===''Ang Bagong Bayani-OFW Labor Party vs. COMELEC''===
On 2002, the Supreme Court ruled in ''Ang Bagong Bayani-OFW Labor Party vs. COMELEC'' that nominees "must be Filipino citizens belonging to marginalized and unrepresented sectors, organizations and parties, as the constitution intended to give genuine power to the people, not only by giving more law to those who have less in life, but more so by enabling them to become veritable lawmakers themselves."<ref>{{cite news|url=http://opinion.inquirer.net/inquireropinion/columns/view/20070506-64338/Another_slap_on_the_Comelec |first=Artemio |last=Panganiban |authorlink=Artemio Panganiban |title=Another slap on the Comelec |work=[[Philippine Daily Inquirer]] |work=2007-05-06 |accessdate=2011-04-14}}</ref>

===''BANAT vs. COMELEC''===
In the same ''BANAT vs. COMELEC'' case stated above, while the Supreme Court, in an 8–7 vote, continued to prohibit major political parties from participating in the party-list election, Associated Justice [[Antonio Carpio]] noted that "it is not necessary that the party-list organization's nominee 'wallow in poverty, destitution and infirmity' as there is no financial status required by the law."<ref>{{cite news|url=http://newsinfo.inquirer.net/inquirerheadlines/nation/view/20090422-200725/SC-increases-House-party-list-reps-to-55 |first=Vincent|last=Cabreza |title=SC increases House party-list reps to 55 |work=[[Philippine Daily Inquirer]] |work=2007-05-06 |accessdate=2011-04-14}}</ref> This effectively allowed anyone to be nominated by a party participating in the party-list election.

==Results==
===2001===
{{See|Philippine House of Representatives elections, 2001}}

===2004===
{{See|Philippine House of Representatives elections, 2004}}

===2007===
{{See|Philippine House of Representatives elections, 2007}}

===2010===
{{See|Philippine House of Representatives party-list election, 2010}}

===2013===
{{See|Philippine House of Representatives party-list election, 2013}}

==References==
{{reflist}}

==Further reading==
*[http://sc.judiciary.gov.ph/jurisprudence/2000/oct2000/136781.htm ''Veterans Federation Party, et. al. vs. COMELEC'']: rules the 2%-4%-6% allocation of seats as unconstitutional, replacing it with a formula where the number of seats won depends on the lead of the party that finished first.
*[http://www.lawphil.net/judjuris/juri2001/jun2001/gr_147589_2001.html ''Ang Bagong Bayani-OFW Labor Party vs. COMELEC''], on which parties can participate in the party-list election.
*[http://sc.judiciary.gov.ph/jurisprudence/2009/april2009/179271.htm ''Barangay Association for National Advancement and Transparency vs. COMELEC''], declares the 2% threshold unconstitutional, proscribes the use of the [[Hare quota]] in determining the number of seats won, while still allowing the 3-seat cap.
*[http://sc.judiciary.gov.ph/jurisprudence/2013/april2013/203766.pdf ''Atong Paglaum vs. COMELEC''], opens the participation in the party-list election of major parties, as long as they are sectoral wings of it, separate and distinct from the mother party, and is linked to the latter via a coalition agreement.

==See also==
Methods of determining winners in [[party-list proportional representation]]:
*[[Highest averages method]]
** [[D'Hondt method]]
** [[Sainte-Laguë method]]
*[[Largest remainder method]]
** [[Hare quota]]
** [[Droop quota]]
** [[Imperiali quota]]

{{DEFAULTSORT:Sectoral Representation In The House Of Representatives Of The Philippines}}
[[Category:Party-list PR]]
[[Category:House of Representatives of the Philippines]]