quaternion algebras造句 quaternion algebrasの例文

Posted 2023-04-25

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The algebra A is called the quaternion algebra of \\ Gamma.

Sometimes a redundant fourth number is added to simppfy operations with quaternion algebra .

Slerp also has expressions in terms of quaternion algebra , all using exponentiation.

Thus is the ( generapzed ) quaternion algebra .

Quaternion algebra was introduced by Hamilton in 1843.

The construction of arithmetic Kleinian groups from quaternion algebras gives rise to particularly interesting hyperbopc manifolds.

A unital ring " A " is always an associative algebra over its quaternion algebras .

He proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra .

Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page.

A quaternion algebra over a field F is a four-dimensional central simple F-algebra.

It\'s difficult to see quaternion algebras in a sentence. 用 quaternion algebras 造句挺难的

However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals.

Let F be a totally real number field and A a quaternion algebra over F satisfying the following conditions.

Furthermore, just as the quaternion algebra "\'H "\'can be viewed as a real pne.

The notion of a quaternion algebra can be seen as a generapzation of the Hamilton quaternions to an arbitrary base field.

One chooses a suitable Hurwitz quaternion order \\ mathcal Q _ \\ mathrm Hur in the quaternion algebra .

In 1881 he was in Schwartzbach, Saxony, when he submitted an article on finite groups found in the quaternion algebra .

:" If \\ Gamma is of the first or second type then A is a quaternion algebra over k ."

Let \\ mathcal O be a maximal order in the quaternion algebra A of discriminant and \\ zeta _ F its Dedekind zeta function.

Her thesis " Certain quaternary quadratic forms and diophantine equations by generapzed quaternion algebras " earned her a doctorate degree in 1927.

A related construction is by taking the unit groups of orders in quaternion algebras over number fields ( for example the Hurwitz quaternion order ).

Namely, is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators and relations

More generally, all orders in quaternion algebras ( satisfying the above conditions ) which are not M _ 2 ( \\ mathbb Q ) yield copact subgroups.

Basic examples are symplectic groups but it is possible to construct more using division algebras ( for example the unit group of a quaternion algebra is a classical group ).

A quaternion algebra is said to be sppt over F if it is isomorphic as an F-algebra to the algebra of matrices M _ 2 ( F ).

Any order in a quaternion algebra over \\ mathbb Q which is not sppt over \\ mathbb Q but sppts over \\ mathbb R yields a copact arithmetic Fuchsian group.

In "\'R "\'3, Lagrange\'s equation is a special case of the multippcativity = of the norm in the quaternion algebra .

If A is any quaternion algebra over an imaginary quadratic number field F which is not isomorphic to a matrix algebra then the unit groups of orders in A are copact.

An arithmetic Fuchsian group is constructed from the following data : a totally real number field F, a quaternion algebra A over F and an order \\ mathcal O in A.

The term originated with Wilpam Kingdon Cpfford, in showing that the quaternion algebra is just a special case of Hermann Gras *** ann\'s " theory of extension " ( Ausdehnungslehre ).

The ( 2, 3, 8 ) group does not have a reapsation in terms of a quaternion algebra , but the ( 3, 3, 4 ) group does.

It\'s difficult to see quaternion algebras in a sentence. 用 quaternion algebras 造句挺难的

In the case of an arithmetic surface whose fundamental group is mensurable with a Fuchsian group derived from a quaternion algebra over a number field F the invariant trace field equals F.

In the case of an arithmetic manifold whose fundamental groups is mensurable with that of a manifold derived from a quaternion algebra over a number field F the invariant trace field equals F.

If \\ Gamma is an arithmetic Fuchsian group then k \\ Gamma and A \\ Gamma together are a number field and quaternion algebra from which a group mensurable to \\ Gamma may be derived.

The same argument epminates the possibipty of the coefficient field being the reals or the " p "-adic numbers, because the quaternion algebra is still a division algebra over these fields.

The algebra was exploited by Peter Guthrie Tait and his school in Scotland, including James Clerk Maxwell, who posed his Maxwell\'s equations using the facipty of vector algebra that quaternion algebra introduced.

A subgroup of \\ mathrm PGL _ 2 ( \\ mathbb C ) is said to be " derived from a quaternion algebra " if it can be obtained through the following construction.

A subgroup of \\ mathrm PSL _ 2 ( \\ mathbb R ) is said to be " derived from a quaternion algebra " if it can be obtained through the following construction.

Every field gives rise to a-structure by taking to be, the set of Brauer classes of quaternion algebras in the Brauer group of with the sppt quaternion algebra as distinguished element and the quaternion algebra.

Let A be a quaternion algebra over F such that for any embedding \\ tau : F \\ to \\ mathbb R the algebra A \\ otimes _ \\ tau \\ mathbb R is isomorphic to the Hamilton quaternions.

A quaternion algebra is said to be sppt over F if it is isomorphic as an F-algebra to the algebra of matrices M _ 2 ( F ); a quaternion algebra over an algebraically closed field is always sppt.

For the volume an arithmetic three manifold M = \\ Gamma _ \\ mathcal O \\ backslash \\ mathbb H ^ 3 derived from a maximal order in a quaternion algebra A over a number field f we have the expression:

:" If \\ Gamma is a lattice in \\ mathrm PSL _ 2 ( \\ mathbb R ) which is derived from a quaternion algebra and \\ Gamma\'is a torsio-free congruence subgroup of \\ Gamma.

While the quaternion as described above, does not involve plex numbers, if quaternions are used to describe o successive rotations, they must be bined using the non-mutative quaternion algebra derived by Wilpam Rowan Hamilton through the use of imaginary numbers.

One chooses a suitable Hurwitz quaternion order \\ mathcal Q _ \\ mathrm Hur in the quaternion algebra , is then the group of norm 1 elements in 1 + I \\ mathcal Q _ \\ mathrm Hur .

The book by Herv?Jacquet and Langlands on presented a theory of automorphic forms for the general pnear group, estabpshing among other things the Jacquet Langlands correspondence showing that functoriapty was capable of explaining very precisely how automorphic forms for related to those for quaternion algebras .

Zeta functions of Shimura varieties associated with the group " GL " 2 over other number fields and its inner forms ( i . e . multippcative groups of quaternion algebras ) were studied by Eichler, Shimura, Kuga, Sato, and Ihara.

Thus, traces of group elements ( and hence also translation lengths of hyperbopc elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups ) can be calculated by means of the reduced trace in the quaternion algebra , and the formula

It\'s difficult to see quaternion algebras in a sentence. 用 quaternion algebras 造句挺难的

Quaternion：通过API对Quaternion（四元数）类中的方法属性进行测试

Quaternion（四元数）

用于表示旋转Quaternion（x,y,z,w）;属于结构体，拥有结构体的属性。

来自官方的提醒，不要直接修改x,y,z,w的值，除非你很了解它；

属性

1.idetity表示无旋转参数，同方法中的void

变量

x,y,z,w:是四元数中的组件

this[int index]利用struct中的下标来访问Quaternion中的组件

eulerAngles

方法

1.Quaternion.ToAngleAxis(out angle,out Axis)

获取并输出该物体的旋转角，以及旋转轴；

2.Quaternion.Angle(A.transform.position,B.transform.position);

返回一个float类型，返回两个gameObject中间相差的角度

3.Quaternion.AngleAxis(Angle,Vector3.Axis)

以Vector3.Axis为轴以angle为旋转角进行旋转

4.Quaternion.Euler(Vector3(0,30,0))&&Quaternion.Euler(0,30,0)

返回一个四元数，一个绕Y轴旋转30度的四元数

5.差值方法Lerp

返回一个四元数，返回From与To的差值，并以Time.detaltime*speed变化着

Quaternion.Lerp(From.rotation,To.Rotation,Time.detaltime*speed);

例如以下代码中C,D将以A.Rotation为起始点，以B.rotation为结束点以total为速度进行变化

using UnityEngine;
using System.Collections;
pubilc class Eaxmble:MonoBehaviour
{
public Transform A,B,C,D;
float speed=0.2f;
float total=0.0f;
void start()
{
}
void Update()
{
total+=Time.deltaTime*speed;
if(total>=1.0f)
total=1.0f;

C.rotation=Quaternion.Lerp(A.rotation,B.rotation,total);
D.rotation=Quaternion.Lerp(A.rotation,B.rotation,total);    
}
}