formulasearchengine - User contributions [en]

Simple public-key infrastructure

2014-08-13T20:08:58Z

199.106.103.53: Correct the url for the last external link. It was missing a final / and without it one gets HTTP 404 error.

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Meet-in-the-middle attack

2013-09-13T19:38:27Z

199.106.103.57: /* MITM (1D-MITM) */ - Reworded to use gender-neutral language

In [[mathematics]], a '''Vitali set''' is an elementary example of a set of [[real number]]s that is not [[Lebesgue measure|Lebesgue measurable]], found by {{harvs|txt|authorlink=Giuseppe Vitali|first=Giuseppe |last=Vitali|year=1905}}. The '''Vitali theorem''' is the [[existence theorem]] that there are such sets. There are [[uncountably many]] Vitali sets, and their existence is proven on the assumption of the [[axiom of choice]].

== Measurable sets ==
Certain sets have a definite 'length' or 'mass'. For instance, the [[interval (mathematics)|interval]] [0, 1] is deemed to have length 1; more generally, an interval [''a'', ''b''], ''a'' ≤ ''b'', is deemed to have length ''b''−''a''. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.

There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of [[rational number]]s, given that the mass of the interval [0, 1] is 1. The rationals are [[Dense_set|dense]] in the reals, so any non negative value may appear reasonable.

However the closest generalization to mass is [[sigma additivity]], which gives rise to the [[Lebesgue measure]]. It assigns a measure of ''b'' − ''a'' to the interval [''a'', ''b''], but will assign a measure of 0 to the set of rational numbers because it is [[countable]]. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using [[Carathéodory's extension theorem]]) does not make it obvious whether there exist non-measurable sets. The answer to that question involves the [[axiom of choice]].

== Construction and proof ==
A Vitali set is a subset <math>V</math> of the [[interval (mathematics)|interval]] <nowiki>[0, 1]</nowiki> of [[real number]]s which for each real number ''r'' contains exactly one number <math>v \in V</math> such that ''v''−''r'' is a [[rational number]]. Vitali sets exist because the rational numbers '''Q''' form a [[subgroup]] of the real numbers '''R''' under [[addition]], and this allows the construction of the additive [[quotient group]] '''R'''/'''Q''' of these two groups which is the group formed by the [[coset]]s of the rational numbers as a subgroup of the real numbers under addition. This group '''R'''/'''Q''' consists of [[disjoint sets|disjoint]] "shifted copies" of the rational numbers in the sense that each element of this quotient group is a set of the form {{nowrap|'''Q''' + ''r''}} for some ''r'' in '''R'''. The [[uncountable set|uncountably many]] elements of '''R'''/'''Q''' [[partition of a set|partition]] '''R''', and each element is [[dense set|dense]] in '''R'''. Each element of '''R'''/'''Q''' intersects [0, 1], and the [[axiom of choice]] guarantees the existence of a subset of [0, 1] containing exactly one representative out of each element of '''R'''/'''Q'''. A set formed this way is called a Vitali set.

Every Vitali set ''V'' is uncountable, and ''v''−''u'' is irrational for any <math>u,v \in V, u \neq v</math>.

A Vitali set is non-measurable. To show this, we assume that ''V'' is measurable and we derive a contradiction. Let ''q''1, ''q''2, ... be an enumeration of the rational numbers in [−1, 1] (recall that the rational numbers are [[countable]]). From the construction of ''V'', note that the translated sets <math>V_k=V+q_k=\{v+q_k : v \in V\}</math>, ''k'' = 1, 2, ... are pairwise disjoint, and further note that <math>[0,1]\subseteq\biguplus_k V_k\subseteq[-1,2]</math>. (To see the first inclusion, consider any real number ''r'' in [0, 1] and let ''v'' be the representative in ''V'' for the equivalence class [''r'']; then ''r''−''v'' = ''q'' for some rational number ''q'' in [-1, 1].)

Apply the Lebesgue measure to these inclusions using [[sigma additivity]]:
:<math>1 \leq \sum_{k=1}^\infty \lambda(V_k) \leq 3.</math>

Because the Lebesgue measure is translation invariant, <math>\lambda(V_k) = \lambda(V)</math> and therefore
:<math>1 \leq \sum_{k=1}^\infty \lambda(V) \leq 3.</math>

But this is impossible. Summing infinitely many copies of the constant λ(''V'') yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in [1, 3]. So ''V'' cannot have been measurable after all, i.e., the Lebesgue measure λ must not define any value for λ(''V'').

==See also==
*[[Non-measurable set]]
*[[Banach–Tarski paradox]]

==References==
* {{cite book|last=Herrlich|first=Horst|title=Axiom of Choice|page=120|publisher=Springer|year=2006}}
* {{cite journal|last=Vitali|first=Giuseppe|authorlink=Giuseppe Vitali|year=1905|title= Sul problema della misura dei gruppi di punti di una retta|journal=Bologna, Tip. Gamberini e Parmeggiani}}

[[Category:Sets of real numbers]]
[[Category:Measure theory]]
[[Category:Articles containing proofs]]

Biorthogonal wavelet

2013-06-10T22:07:10Z

199.106.103.54: side by side for comparison

{{Unreferenced|date=December 2009}}
A '''modular elliptic curve''' is an [[elliptic curve]] ''E'' that admits a parametrisation ''X''0(''N'') → ''E'' by a [[modular curve]]. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The [[modularity theorem]], also known as the [[Taniyama–Shimura conjecture]], asserts that every elliptic curve defined over the rational numbers is modular.

==Modularity theorem==
The [[theorem]] states that any [[elliptic curve]] over '''Q''' can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]]

:<math>X_0(N)\ </math>

for some integer ''N''; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level ''N''. If ''N'' is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the ''conductor''), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level ''N'', a normalized [[newform]] with integer ''q''-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].

The modularity theorem implies a closely related analytic statement: to an elliptic curve ''E'' over '''Q''' we may attach a corresponding [[L-series of an elliptic curve|L-series]]. The ''L''-series is a [[Dirichlet series]], commonly written

:<math>L(s, E) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>

The [[generating function]] of the coefficients <math>a_n</math> is then

:<math>f(q, E) = \sum_{n=1}^\infty a_n q^n.</math>

If we make the substitution

:<math>q = e^{2 \pi i \tau}\ </math>

we see that we have written the [[Fourier series|Fourier expansion]] of a function <math>f(\tau, E)</math> of the complex variable ''τ'', so the coefficients of the ''q''-series are also thought of as the Fourier coefficients of <math>f</math>. The function obtained in this way is, remarkably, a [[modular form|cusp form]] of weight two and level ''N'' and is also an eigenform (an eigenvector of all [[Hecke operator]]s); this is the '''Hasse–Weil conjecture''', which follows from the modularity theorem.

Some modular forms of weight two, in turn, correspond to [[holomorphic differential]]s for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible [[Abelian varieties]], corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is [[Elliptic curve#Isogeny|isogenous]] to the original curve (but not, in general, isomorphic to it).

{{DEFAULTSORT:Modular Elliptic Curve}}

==References==
*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=1995 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=141 | issue=3 | pages=443–551}}
*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) | publisher=Birkhäuser | location=Basel, Boston, Berlin | mr=1403925 | year=1995 | chapter=Modular forms, elliptic curves, and Fermat's last theorem | pages=243–245}}

[[Category:Elliptic curves]]