Y < sub > α </ sub >, then C < sub > α </ sub > is 2 / π.

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6 / π < sup > 2 </ sup > ( see pi ), which is about 61 %.

This probably convinced him that the sum equals π < sup > 2 </ sup > / 6, which he proved in the same year.

To every topological space X with distinguished point x < sub > 0 </ sub >, one can define the fundamental group based at x < sub > 0 </ sub >, denoted π < sub > 1 </ sub >( X, x < sub > 0 </ sub >).

This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π ( X, x < sub > 0 </ sub >) to π ( Y, y < sub > 0 </ sub >).

or simply π ( X, x < sub > 0 </ sub >).

For path-connected spaces, therefore, we can write π < sub > 1 </ sub >( X ) instead of π < sub > 1 </ sub >( X, x < sub > 0 </ sub >) without ambiguity whenever we care about the isomorphism class only.

In this case the probability that the gcd equals d is d < sup >− 2 </ sup >/ ζ ( 2 ), and since ζ ( 2 ) = π < sup > 2 </ sup >/ 6 we have

An example of a projection is the function π < sub > xy </ sub > defined by, which projects an arbitrary point in 3D space to a point on the x – y-plane, where the third coordinate ( z ) is equal to 0.

The inverse limit, A, comes equipped with natural projections π < sub > i </ sub >: A → A < sub > i </ sub > which pick out the ith component of the direct product for each i in I.

The inverse limit of this system is an object X in C together with morphisms π < sub > i </ sub >: X → X < sub > i </ sub > ( called projections ) satisfying π < sub > i </ sub >

The observable ( i. e. past light cone ) universe contains 6 cubic inches (< sup > 4 </ sup >/< sub > 3 </ sub > × π × ( 14 light years in inches )< sup > 3 </ sup >).

X is a Tychonoff space, or T < sub > 3½ </ sub > space, or T < sub > π </ sub > space, or completely T < sub > 3 </ sub > space if it is both completely regular and Hausdorff.

There are 3 fundamental physical units in this equation: time t, mass m, and length l, and 4 dimensional variables, T, M, L, and g. Thus we need only 4 − 3 = 1 dimensionless parameter, denoted π, and the model can be re-expressed as

To build a bijection from T to R: start with the tangent function tan ( x ), which provides a bijection from (− π / 2, π / 2 ) to R. Next observe that the linear function h ( x ) = πx-π / 2 provides a bijection from ( 0, 1 ) to (− π / 2, π / 2 ).

A03 T < sub > 1 </ sub >← π < sub > i </ sub >' B < sub > i '</ sub >

If X is a connected cell complex with homotopy groups π < sub > n </ sub >( X ) = 0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K ( G, 1 ) for G = π < sub > 1 </ sub >( X ).

These " pseudo-Goldstone bosons " are called technipions, π < sub > T </ sub >.

Such huge ETC mass scales imply tiny quark and lepton masses and ETC contributions to M < sub > πT </ sub > of at most a few GeV, in conflict with LEP searches for π < sub > T </ sub > at the Z < sup > 0 </ sup >.

: π < sub > 1 </ sub >( T )= Z < sup > 2 </ sup >,

Frame bundles have a ( canonical or tautological ) solder form which sends a tangent vector X ∈ T < sub > p </ sub > P to the coordinates of dπ < sub > p </ sub >( X ) ∈ T < sub > π ( p )</ sub > M with respect to the frame p.

First note that the fundamental vector fields generating the G action on P provide a bundle isomorphism from the vertical bundle V of P ( with V < sub > p </ sub >= T < sub > p </ sub >( P < sub > π ( p )</ sub >) to.

For any bounded M-measurable function f on X, there is a unique bounded linear operator T < sub > π </ sub >( f ) such that

Later this system of numbering was extended to include T < sub > 0 </ sub >, T < sub > 2½ </ sub >, T < sub > 3½ </ sub > ( or T < sub > π </ sub >), T < sub > 5 </ sub >, and T < sub > 6 </ sub >.

An alternating magnetic field is applied at the atomizer ( graphite furnace ) to split the absorption line into three components, the π component, which remains at the same position as the original absorption line, and two σ components, which are moved to higher and lower wavelengths, respectively ( see Zeeman Effect ).

The π component has to be removed in this case, e. g. using a polarizer, and the σ components do not overlap with the emission profile of the lamp, so that only the background absorption is measured.

The term Enhanced Data Rate ( EDR ) is used to describe π / 4-DPSK and 8DPSK schemes, each giving 2 and 3 Mbit / s respectively.

π Capricorni is a double star with a blue-white hued primary of magnitude 5. 1 and a white-hued secondary of magnitude 8. 3.

This function is undefined at the point π, since tan ( π / 2 ) is undefined ; we will identify this point with our point ∞.

A bounded linear map, π: A → B, between C *- algebras A and B is called a *- homomorphism if

In the case of C *- algebras, any *- homomorphism π between C *- algebras is non-expansive, i. e. bounded with norm ≤ 1.

A bijective *- homomorphism π is called a C *- isomorphism, in which case A and B are said to be isomorphic.

* Radar: The ( monostatic ) radar cross section is defined as 4 π times the radio differential cross section at 180 degrees.

* An equivalence relation ~ on X is the equivalence kernel of its surjective projection π: X → X /~.

The resulting group homomorphism, called the induced homomorphism, is written as π ( f ) or, more commonly,

To calculate π the formula is

This assumes that π is 4 ×( 8 / 9 ) ² ( or 3. 160493 ...), with an error of slightly over 0. 63 percent.

They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3.

It had been redefined during the reign of Queen Anne, in 1706, as 231 cubic inches exactly, which is the result of the earlier definition with π approximated to.

The Gauss – Legendre algorithm is an algorithm to compute the digits of π.

* π is a ring homomorphism which carries involution on A into involution on operators