Post-hoc analysis

2013-11-04T17:51:32Z

194.57.165.4: /* List of post hoc tests */

:''There also is [[Brauer's theorem on induced characters]].''

In [[mathematics]], '''Brauer's theorem''', named for [[Richard Brauer]], is a result on the representability of 0 by forms over certain [[field (mathematics)|fields]] in sufficiently many variables.<ref>R. Brauer, ''A note on systems of homogeneous algebraic equations'', Bulletin of the American Mathematical Society, '''51''', pages 749-755 (1945)</ref>

==Statement of Brauer's theorem==
Let ''K'' be a field such that for every integer ''r'' > 0 there exists an integer ψ(''r'') such that for ''n'' ≥ ψ(r) every equation

:<math>(*)\qquad a_1x_1^r+\cdots+a_nx_n^r=0,\quad a_i\in K,\quad i=1,\ldots,n</math>

has a non-trivial (i.e. not all ''x''''i'' are equal to 0) solution in ''K''.
Then, given homogeneous polynomials ''f''1,...,''f''''k'' of degrees ''r''1,...,''r''''k'' respectively with coefficients in ''K'', for every set of positive integers ''r''1,...,''r''''k'' and every non-negative integer ''l'', there exists a number ω(''r''1,...,''r''''k'',''l'') such that for ''n'' ≥ ω(''r''1,...,''r''''k'',''l'') there exists an ''l''-dimensional [[affine subspace]] ''M'' of ''Kn'' (regarded as a vector space over ''K'') satisfying

:<math>f_1(x_1,\ldots,x_n)=\cdots=f_k(x_1,\ldots,x_n)=0,\quad\forall(x_1,\ldots,x_n)\in M.</math>

==An application to the field of p-adic numbers==
Letting ''K'' be the field of [[p-adic number]]s in the theorem, the equation (*) is satisfied, since <math>\mathbb{Q}_p^*/\left(\mathbb{Q}_p^*\right)^b</math>, ''b'' a natural number, is finite. Choosing ''k'' = 1, one obtains the following corollary:

:A homogeneous equation ''f''(''x''1,...,''x''''n'') = 0 of degree ''r'' in the field of p-adic numbers has a non-trivial solution if ''n'' is sufficiently large.

One can show that if ''n'' is sufficiently large according to the above corollary, then ''n'' is greater than ''r''2. Indeed, [[Emil Artin]] conjectured<ref>''Collected papers of Emil Artin'', page x, Addison–Wesley, Reading, Mass., 1965</ref> that every homogeneous polynomial of degree ''r'' over '''Q'''''p'' in more than ''r''2 variables represents 0. This is obviously true for ''r'' = 1, and it is well known that the conjecture is true for ''r'' = 2 (see, for example, J.-P. Serre, ''A Course in Arithmetic'', Chapter IV, Theorem 6). See [[quasi-algebraic closure]] for further context.

In 1950 Demyanov<ref>{{cite journal| last=Demyanov | first=V. B. | year=1950 |title=На кубических форм дискретных линейных нормированных полей |trans_title=On cubic forms over discrete normed fields | journal=[[Doklady Akademii Nauk SSSR]] | volume=74 | pages=889–891}}</ref> verified the conjecture for ''r'' = 3 and ''p'' ≠ 3, and in 1952 [[D. J. Lewis]]<ref>D. J. Lewis, ''Cubic homogeneous polynomials over p-adic number fields'', Annals of Mathematics, '''56''', pages 473–478, (1952)</ref> independently proved the case ''r'' = 3 for all primes ''p''. But in 1966 [[Guy Terjanian]] constructed a homogeneous polynomial of degree 4 over '''Q'''2 in 18 variables that has no non-trivial zero.<ref>Guy Terjanian, ''Un contre-exemple à une conjecture d'Artin'', C. R. Acad. Sci. Paris Sér. A–B, '''262''', A612, (1966)</ref> On the other hand, the [[Ax–Kochen theorem]] shows that for any fixed degree Artin's conjecture is true for all but finitely many '''Q'''''p''.

== References ==
{{reflist}}
* {{cite book | zbl=1125.11018 | last=Davenport | first=Harold | authorlink=Harold Davenport | title=Analytic methods for Diophantine equations and Diophantine inequalities | others=Edited and prepared by T. D. Browning. With a preface by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman | edition=2nd | series=Cambridge Mathematical Library | publisher=[[Cambridge University Press]] | year=2005 | isbn=0-521-60583-0 }}

[[Category:Diophantine equations]]
[[Category:Theorems in number theory]]

Belyi's theorem

2013-10-16T08:26:54Z

194.57.88.29:

== Description ==
The '''Physical Coding Sublayer''' (PCS) further helps to define physical layer specifications (speed and Duplex modes, etc.) for networking protocols like [[Fast Ethernet]], [[Gigabit Ethernet]] and [[10 Gigabit Ethernet]].

The Ethernet PCS is part of the Ethernet PHY layer. The hierarchy is as follows:

* [[Data Link Layer]] (Layer 2)
** LLC ([[Logical Link Control]] Sublayer)
** MAC ([[Media Access Control]] Sublayer)
*** RS (Reconciliation Sublayer) - This sublayer processes PHY Local/Remote Fault messages and handles DDR conversion
* [[physical layer | PHY Layer]] (Layer 1)
** PCS (Physical Coding Sublayer) - This sublayer performs [[autonegotiation]] and coding such as [[8b/10b encoding]]
** PMA (Physical Medium Attachment Sublayer) - This sublayer performs PMA framing, octet synchronization/detection, and <math>x^7+x^6+1</math> scrambling/descrambling
** PMD ([[Physical Medium Dependent]] Sublayer) - This sublayer consists of a transceiver for the physical medium

== Physical Coding Sublayer (PCS) specifications ==

===10 Gigabit Ethernet===

*10GBASE-R (LAN) is the serial encoded PCS that allows for Ethernet framing at a rate of approximately 10.3 Gbit/s (MAC = 10,000 Gbit/s, overhead = 64 B/66 B effective rate = 10,000 * 66/64 = 10,312.5 - see also [[64b/66b encoding]]). This rate does not match the rate 9.953 Gbit/s used in [[SONET]] and SDH and is not supported over a WAN based on SONET or SDH.
*10GBASE-X (LAN) uses similar coding methods as 10GBASE-R but is only used in the definition of 10GBASE-LX4. This is mainly because LX4 operates on both single and multimode fibers, giving it a unique set of specifications as defined in its [[Physical Media Dependent|PMD]].
*10GBASE-W (WAN) defines WAN encoding for 10GbE, it encodes the frames so that they are compatible with SONET STS-192c data rates and SDH VC-4-64 transmission standards allowing for 10 Gbit/s transmission across a WAN. It does this by wrapping the 64/66b payload into a SONET frame, making the effective rate 9.95 Gbit/s.

=== Lattice Semiconductor multi-protocol ===

"PCS logic can be configured to support numerous industry-standard, high-speed serial data transfer protocols."{{cite journal|title=LatticeECP3 SERDES/PCS Usage Guide|journal=Lattice semiconductor Corporation|pages=p 8–1|url=http://www.latticesemi.com/documents/tn1176.pdf}}
Link is dead

== References ==

{{cite journal|last=Barbieri|first=Alessandro|title=10 GbE and Its X Factors|journal=Packet: Cisco Systems Users Magazine|volume=17|issue=3|pages=25–28|url=http://www.cisco.com/asiapac/campaigns/metroethernet/files/10ge_po_pack_mag_arti.pdf|accessdate=2007-12-31}}

== External links ==
* [http://grouper.ieee.org/groups/802/3/ae/public/mar00/figueira_1_0300.pdf IEEE 802.3 Meeting]
* [http://www.xilinx.com/systemio/1gbsx_phy/basics.htm Ethernet 1000BASE-X PCS/PMA Technology Basics]

[[Category:Networking standards]]
{{compu-network-stub}}

formulasearchengine - User contributions [en]

Post-hoc analysis

Belyi's theorem