&&&notation 在 计算机软件及计算机应用 分类中
的翻译结果:
查询用时：0.144秒
&在分类学科中查询
&&&&In chapter one, we introduced some basic knowledge, the major achievements in algebraic methods, chief concepts and notation on the theory of cellular automata.
&&&&这部分简单介绍了元胞自动机的基础知识,以及国内外学者利用代数工具对元胞自动机进行研究的一些内容,并给出元胞自动机的基本概念和记号。
&&&&The definition of UML includes two parts: UML Semantic and UML Notation.
&&&&UML的定义包括两个部分：UML语义和UML表示法，其中UML语义定义UML对象建模的抽象文法和语义的元模型；
&&&&Study on general arithmetic conversion of the RMB number representation and Chinese literal notation
&&&&任意人民币数字表示法与中文大写表示法的转换通用研究
&&&&irst, this paper introduces abstract syntax notation one (ASN.1) , ASN.
&&&&本文首先介绍了抽象语法表示法ASN.
&&&&Taking a communications system as an example, it describes and analyzes the specification styles and features of petri net, LOTOS, relational notation, Z and object-Z.
&&&&文中以一个通信系统为例子,论述和分析了Petri网、LOTOS、关系表示法、Z表示法,以及面向对象Z等形式描述技术的形式描述风格及其特征。
&&&&As the smooth transitions from analysis models to design models in software engineering remains a problem to be solved,this paper uses movements of objects between different abstraction levels and dimensions to construct OO analysis models based on hierarchical mappings and refinements,in which an appropriate notation for diagrams and frames is applied.
&&&&针对目前软件工程中存在的分析模型与设计模型之间未能做到平滑过渡的问题，根据对象在不同抽象层和抽象维之间的运动，采用适当的图形和框架表示法，以层次式模型映射构成逐次精化的面向对象的五层分析模型
&&&&Notation for Syntax of Indeterminatly Spatiotemporal Queries
&&&&不确定性时空查询的一种符号表示法
&&&&RECOGNITION AND PLAY SYSTEM OF NUMBERED MUSICAL NOTATION
&&&&音乐简谱的识别与演奏系统
&&&&Applications of the Chinese Character Automatic Phonetic Notation to the Computer Science
&&&&计算机在汉字自动注音中的应用
&&&&This article is about a research of operation system which can support two file systems including FAT16,and FAT32 in Windows98 operation system and three file systems including FAT16, FAT32 and NTFS in Windows2000/XP. This article discusses five questions as follows: (1)the notation
&&&&对Windows 98操作系统可以同时支持FATl6、FAT32两种文件系统和Windows 2000/XP操作系统则支持FATl6、FAT32、NTFS三种文件系统进行了研究,探讨了几个问题1.什么是文件系统;
&&&&Based on studying Business Process Modeling Notation,Operational View aided design method is provided,the conversion relationships of Business Process Diagram and OV5,OV6 are established.
&&&&在深入研究BPM N的基础上,提出了基于BPM N的作战视图辅助设计方法,建立了企业过程图与作战视图产品OV 5、OV 6的转换关系。
&&&&JSON(JavaScript Object Notation) is a lightweight data-interchange format,and it can be easily parsed by browser which supports JavaScript.
&&&&JSON(JavaScript Object Notation)是一种轻量级的数据交换格式,易于被支持JavaScript的浏览器所解析。
查询“notation”译词为用户自定义的双语例句&&&&我想查看译文中含有：的双语例句
为了更好的帮助您理解掌握查询词或其译词在地道英语中的实际用法，我们为您准备了出自英文原文的大量英语例句，供您参考。&&&&&&&&&&&& This paper describes the concrete process of using BASIC to realize theorem proving on a mini-computer PDP-11-03. The only rule of inference used is a refinement algorithm-the Unit Binary Eesolution. A few of technics as follows:1. The subsumption test. This technic can be used to delete some irrelevant and redundant clauses. With the "Fast-Search-Method", some unnecessary teats can be avoided. For instance, let C1, C2, …, Cn be the given old unit clauses and D1, D2, …, Dm be the new ones. If D1, D2, …, Dm.... This paper describes the concrete process of using BASIC to realize theorem proving on a mini-computer PDP-11-03. The only rule of inference used is a refinement algorithm-the Unit Binary Eesolution. A few of technics as follows:1. The subsumption test. This technic can be used to delete some irrelevant and redundant clauses. With the "Fast-Search-Method", some unnecessary teats can be avoided. For instance, let C1, C2, …, Cn be the given old unit clauses and D1, D2, …, Dm be the new ones. If D1, D2, …, Dm. do not subsume each other, m(m-1)/2 times of search can be saved.2. The test of the ordinal number of clauses. To resolve a unit clause, say clause m, against a nonunit clause, say clause n', we first find the last unit clause, say clause m', which is used to obtain clause n'. If m is less than m', we resolve clause m and clause n'; otherwise, they are not resolved. (Using it, we can avoid repeatedly generating the same unit clause.)3. The supporting set. The supporting set can be used to avoid performing any resolution in the basic axioms.4. The programming technic. The nest structure of ALGOL is introduced into BASIC programs so that only one-dimension string array and a few simple variables are needed. Chain structure is used to realize the exchange between the Polish notation and the usual notation of clauses. The dialogue between the user and computer can be used to speed up finding proofs of theorems.本文介绍了在PDP-11-03微型机上,采用BASIC语言实现定理证明的具体过程。所采用的推理方法是近期在国际上引人注目的算法——单位归结法。同时,综合了象“支撑集”,“包括检查”,“判断子句序号”等技巧,并在实现上提出了“快查法”,使程序效率显著提高。程序结构较为紧凑合理,操作过程方便灵活,输出信息简洁明了。此外,还运用了ALGOL嵌套思想,致使内存空间大为节省。 A six papers' series is being published, which is persuing a semantics for communicating processes. As the first paper of this series, a notation for distributed programming is developed, including the constructs: output, input, alternation, parallelism, channel renaming and hiding, and recursion.The rest of this series will be respectively entitled as: "A Deterministic Semanties of Communieating Processes", "Partial Correctness of Protocal","Weakest Environment of Communicating Processes", "A Nondeterministic... A six papers' series is being published, which is persuing a semantics for communicating processes. As the first paper of this series, a notation for distributed programming is developed, including the constructs: output, input, alternation, parallelism, channel renaming and hiding, and recursion.The rest of this series will be respectively entitled as: "A Deterministic Semanties of Communieating Processes", "Partial Correctness of Protocal","Weakest Environment of Communicating Processes", "A Nondeterministic Semantics of Communicating Processes (Ⅰ) " and " (Ⅲ) ".通信的顺序进程(Communicating Sequential Processes)是C.A.R.Hoare教授提出的,简称为CSP,他希望以此作为分布式程序设计的基本机制。 本文以“通信的顺序进程及其研究”作为总称,共分六篇。第一篇以总称为名,介绍CSP的目标及一个CSP式的应用式语言(Applicative Language)。第二篇名为“通信进程的确定性语义学”,该文中给出了这个语言的一种语义,这种语义不考虑CSP中允许的很多非确定现象。语义中同时使用了指称方法(Denotational Semantics)和公理化方法(Axiomatic Semantics)。第三篇为“通信协议的部分正确性”。该文用CSP构造了一个HDLC协议,并用第二篇文章中提供的方法,证明了这个协议的部分正确性。在证明过程中,作者引入了一个类似于顺序程序设计中最弱前提(Weakest Precondition)的最弱环境(Weakest Environment)概念。第四篇中,在一种层次通信结构中,详细地讨论了最弱环境这一概念,文章的名称为“通信进程的最弱环境”。最后两篇讨论CSP的非确定性语义,这种语义考虑了CSP的各种允许...通信的顺序进程(Communicating Sequential Processes)是C.A.R.Hoare教授提出的,简称为CSP,他希望以此作为分布式程序设计的基本机制。 本文以“通信的顺序进程及其研究”作为总称,共分六篇。第一篇以总称为名,介绍CSP的目标及一个CSP式的应用式语言(Applicative Language)。第二篇名为“通信进程的确定性语义学”,该文中给出了这个语言的一种语义,这种语义不考虑CSP中允许的很多非确定现象。语义中同时使用了指称方法(Denotational Semantics)和公理化方法(Axiomatic Semantics)。第三篇为“通信协议的部分正确性”。该文用CSP构造了一个HDLC协议,并用第二篇文章中提供的方法,证明了这个协议的部分正确性。在证明过程中,作者引入了一个类似于顺序程序设计中最弱前提(Weakest Precondition)的最弱环境(Weakest Environment)概念。第四篇中,在一种层次通信结构中,详细地讨论了最弱环境这一概念,文章的名称为“通信进程的最弱环境”。最后两篇讨论CSP的非确定性语义,这种语义考虑了CSP的各种允许的非确定行为。CSP的非确定性语义是用操作语义学(Operational Semantics)和公理化语义学同时给出的。标题为“通信进程的非确定性语义学(上)”及“(下)”。 Using the notation and calculus for describing the behaviour and proving invariant properties of communicating processes presented in [5], [6], a structural approach to the design of communication protocol is developed in this paper, which is illustrated by an HDLC example. 在[5],[6]中,我们陈述了一个用于分布式程序设计的语言原型,这个原型是基于Hoare所提出的通信顺序进程的;并且建议用通道谓词作为分布式程序的功能描述;在发展这个语言的公理语义的同时,也给出了证明程序特性的一种形式途径。本文中,我们将使用上述工具讨论通信协议的结构式设计。本文原拟名为“通信协议的部分正确性”。&nbsp&&&&&相关查询
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&&&↳
android.opengl.Matrix
Class Overview
Matrix math utilities. These methods operate on OpenGL ES format
matrices and vectors stored in float arrays.
Matrices are 4 x 4 column-vector matrices stored in column-major
m[offset +
0] m[offset +
4] m[offset +
8] m[offset + 12]
m[offset +
1] m[offset +
5] m[offset +
9] m[offset + 13]
m[offset +
2] m[offset +
6] m[offset + 10] m[offset + 14]
m[offset +
3] m[offset +
7] m[offset + 11] m[offset + 15]
Vectors are 4 x 1 column vectors stored in order:
v[offset + 0]
v[offset + 1]
v[offset + 2]
v[offset + 3]
Public Constructors
This constructor was deprecated
in API level 19.
All methods are static, do not instantiate this class.
Public Methods
(float[] m, int offset, float left, float right, float bottom, float top, float near, float far)
Defines a projection matrix in terms of six clip planes.
(float[] mInv, int mInvOffset, float[] m, int mOffset)
Inverts a 4 x 4 matrix.
(float x, float y, float z)
Computes the length of a vector.
(float[] result, int resultOffset, float[] lhs, int lhsOffset, float[] rhs, int rhsOffset)
Multiplies two 4x4 matrices together and stores the result in a third 4x4
(float[] resultVec, int resultVecOffset, float[] lhsMat, int lhsMatOffset, float[] rhsVec, int rhsVecOffset)
Multiplies a 4 element vector by a 4x4 matrix and stores the result in a
4-element column vector.
(float[] m, int mOffset, float left, float right, float bottom, float top, float near, float far)
Computes an orthographic projection matrix.
(float[] m, int offset, float fovy, float aspect, float zNear, float zFar)
Defines a projection matrix in terms of a field of view angle, an
aspect ratio, and z clip planes.
(float[] m, int mOffset, float a, float x, float y, float z)
Rotates matrix m in place by angle a (in degrees)
around the axis (x, y, z).
(float[] rm, int rmOffset, float[] m, int mOffset, float a, float x, float y, float z)
Rotates matrix m by angle a (in degrees) around the axis (x, y, z).
(float[] sm, int smOffset, float[] m, int mOffset, float x, float y, float z)
Scales matrix m by x, y, and z, putting the result in sm.
(float[] m, int mOffset, float x, float y, float z)
Scales matrix m in place by sx, sy, and sz.
(float[] sm, int smOffset)
Sets matrix m to the identity matrix.
(float[] rm, int rmOffset, float eyeX, float eyeY, float eyeZ, float centerX, float centerY, float centerZ, float upX, float upY, float upZ)
Defines a viewing transformation in terms of an eye point, a center of
view, and an up vector.
(float[] rm, int rmOffset, float x, float y, float z)
Converts Euler angles to a rotation matrix.
(float[] rm, int rmOffset, float a, float x, float y, float z)
Creates a matrix for rotation by angle a (in degrees)
around the axis (x, y, z).
(float[] m, int mOffset, float x, float y, float z)
Translates matrix m by x, y, and z in place.
(float[] tm, int tmOffset, float[] m, int mOffset, float x, float y, float z)
Translates matrix m by x, y, and z, putting the result in tm.
(float[] mTrans, int mTransOffset, float[] m, int mOffset)
Transposes a 4 x 4 matrix.
Inherited Methods
From class
Creates and returns a copy of this Object.
Compares this instance with the specified object and indicates if they
are equal.
Invoked when the garbage collector has detected that this instance is no longer reachable.
Returns the unique instance of
that represents this
object's class.
Returns an integer hash code for this object.
Causes a thread which is waiting on this object's monitor (by means of
calling one of the wait() methods) to be woken up.
Causes all threads which are waiting on this object's monitor (by means
of calling one of the wait() methods) to be woken up.
Returns a string containing a concise, human-readable description of this
Causes the calling thread to wait until another thread calls the notify() or notifyAll() method of this object.
(long millis, int nanos)
Causes the calling thread to wait until another thread calls the notify() or notifyAll() method of this object or until the
specified timeout expires.
(long millis)
Causes the calling thread to wait until another thread calls the notify() or notifyAll() method of this object or until the
specified timeout expires.
Public Constructors
This constructor was deprecated
in API level 19.
All methods are static, do not instantiate this class.
Public Methods
(float[] m, int offset, float left, float right, float bottom, float top, float near, float far)
(float[] mInv, int mInvOffset, float[] m, int mOffset)
(float x, float y, float z)
multiplyMM
(float[] result, int resultOffset, float[] lhs, int lhsOffset, float[] rhs, int rhsOffset)
multiplyMV
(float[] resultVec, int resultVecOffset, float[] lhsMat, int lhsMatOffset, float[] rhsVec, int rhsVecOffset)
(float[] m, int mOffset, float left, float right, float bottom, float top, float near, float far)
perspectiveM
(float[] m, int offset, float fovy, float aspect, float zNear, float zFar)
(float[] m, int mOffset, float a, float x, float y, float z)
(float[] rm, int rmOffset, float[] m, int mOffset, float a, float x, float y, float z)
(float[] sm, int smOffset, float[] m, int mOffset, float x, float y, float z)
(float[] m, int mOffset, float x, float y, float z)
setIdentityM
(float[] sm, int smOffset)
setLookAtM
(float[] rm, int rmOffset, float eyeX, float eyeY, float eyeZ, float centerX, float centerY, float centerZ, float upX, float upY, float upZ)
setRotateEulerM
(float[] rm, int rmOffset, float x, float y, float z)
setRotateM
(float[] rm, int rmOffset, float a, float x, float y, float z)
translateM
(float[] m, int mOffset, float x, float y, float z)
translateM
(float[] tm, int tmOffset, float[] m, int mOffset, float x, float y, float z)
transposeM
(float[] mTrans, int mTransOffset, float[] m, int mOffset)The Inverse of a Matrix
The Inverse of a Matrix
Section 2.2
The Basics
Linear algebra is perhaps the most powerful and useful math course you can ever take in your entire academic career. Resting at the heart of this course is something called a matrix. A matrix is a rectangular array of numbers organized in rows and columns and encased in brackets. Matrices are very useful due to the fact that they can be easily manipulated. In this page we will explore how to find the inverse of a matrix and its uses.
Much like normal numbers we use the notation A-1 to denote the inverse of matrix A. Some important things to remember about inverse matrices is they are not commutative, and a full generalization is possible only if the matrices you are using a square. (meaning they have the same number of rows and columns, an n x n matrix)
3 x 3 Identity Matrix
An n x n matrix (A) is said to be invertible if there is an n x n matrix (C) such that CA= I and AC= I where I is the n x n identity matrix. An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
Basically A-1 A= I and A A-1 = I where A is an invertible matrix and A-1 is the inverse of A. A matrix is said to be a singular matrix if it is non-invertible. A matrix is a nonsingular matrix if it is an invertible matrix.
A simple formula for finding the inverse of a 2 x 2 matrix is given by Theorem 4:
We call the quantity ad - bc the determinant of the matrix. ( det A = ad - bc ) A 2 x 2 matrix is invertible if and only if (iff) its determinant does not equal 0.
Theorem 5 reveals something else useful about the inverse of matrices. Theorem 5 states that if matrix A is invertible then the equation Ax = b has a unique solution, x. We can find this solution by x = A-1 b. The following example demonstrates this usefulness of this equation. The following proof will help prove theorem 5 by proving 1) that the solution exists and 2) this solution is unique.
1) let x = A-1 b, plug this in to check
A(x) = A A-1 b= Ib = b
2) We will prove this portion of theorem 5 by contradiction.
Remember A(x + y) = Ax = Ay
AB does not equal BA, so assume there are two different solutions x does not equal y with Ax = b and Ay = b, but then Ax - Ay = 0,
A(x-y) = 0
Multiply by A-1 , A-1 0 = A-1 A(x-y) = I(x-y) = x-y
x-y = 0, therefore x = y. However we said in the beginning that x does not equal y, this is a contradiction.
Example of solving Ax=b by using the inverse of A:
More useful properties of inverse matrices are revealed in Theorem 6. This theorem states:
If A is an invertible matrix, then A-1is invertible and (A-1) -1 = A
If A and B are n x n invertible matrices, then so is AB and the inverse of AB is the product of the inverses of A and B in the reverse order. (AB)-1 = B-1 A-1
If A is an invertible matrix, then so is AT , and the inverse of AT is the transpose of A-1 . That is (AT)-1 = (A-1)T
Are you skeptical at all with theorem 6? Well, incase you are, here are the proofs for each part.
To Prove 1: we need to find matrix C so that A-1 C = I and C A-1 = I We already know that these equations will still hold true if we put A in place of C. (see above) Thus A-1 is invertible, and A is its inverse.
To Prove 2: we first must calculate (AB)( B-1 A-1 ) = A(BB-1 ) A-1 = AI A-1 = A A-1 = I
a similar calculation shows that (B-1 A-1 )(AB) = I.
(A-1 )T AT =(A A-1 )T = IT = I. And thus: AT (A-1 )T = IT = I . Thus we have proved that AT is invertible, and its inverse is (A-1 )T
Also, it is useful to remember that the product of n x n invertible matrices is invertible, and the inverse is the product of their inverses in the reverse order.
Elementary Matrices
The usefulness of matrices continues to expand with the introduction of elementary matrices. An elementary matrix is a matrix that is obtained by performing a single elementary row operation to an identity matrix. An elementary row operation is the process of either (1) replacing one row of a matrix with the sum of itself and a multiple of another row (2) Interchanging two rows (3) Multiplying all entries in a row by a nonzero constant. If an elementary row operation is performed on an
matrix A, the resulting matrix can be written as E A, where the
matrix E is created by performing the same row operation on Im . The following example demonstrates this concept:
It should also be noted that each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.
Finally theorem 7 gives us a way to visualize an inverse matrix and helps us develop a method of finding inverse matrices. Theorem 7 says that an
matrix, called A is invertible iff (if and only if) A is row equivalent to In , and any sequence of elementary row operations that reduces A to
In also transforms In into A-1.
An Algorithm for Finding A-1
Say if we placed a matrix
A and its identity matrix I next to each other and formed an augmented matrix. Row operations done to this matrix would produce the same results
on both A and I. The following is an algorithm for finding A-1 or the inverse of matrix A. First row reduce the augmented matrix [ A
I ] . If I and A are row equivalent then the matrix [ A
I ] is row equivalent to [ I
A-1 ]. If not the A does not have an inverse.
Another View of Matrix Inversion
Finally, this section gives us another way to view inverse matrices. This new way to view matrices also introduces a new trick to us. We see that the "super augmented" matrix [ A
I ] which is matrix A and its identity matrix row reduces to the matrix [ I
Now how any why does this work!? Well, in general the matrix [ A
B ] row reduces to [ I A-1 B ]
B ] row reduces to [ I
x ] then x = A-1 b, Ax = b
If [ A b1 b2 ] row reduces to [ I x1 x2 ] then Ax = b1 , Ax1 = b1 , Ax2 = b2
Where bn are the columns of matrix B.
Section 2.3
The Invertible Matrix Theorem
Let A be a square n x n matrix.
Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
(a) A is an invertible matrix.
(b) A is row equivalent to the n x n identity matrix.
(c) A has n pivot positions.
(d) The equation Ax=0 has only the trivial solution.
(e) The columns of A form a linearly independent set.
(f) The linear transformation x-->Ax is one-to-one.
(g) The equation Ax=b has at least one solution for each b in Rn.
(h) The columns of A span Rn.
(i) The linear transformation x-->Ax maps Rn onto Rn.
(j) There is an n x n matrix C such that CA=I.
(k) There is an n x n matrix D such that AD=I.
(l) AT is an invertible matrix.
Notation note: If the truth of statement (a) always implies that statement (j) is true, we say that (a) implies (j) and write (a) => (j).
If (a) is true, meaning A is an invertible matrix, then A-1 works for C in statement (j) of the theorem, so (a)=>(j).
Next, (j)=>(d) and (d)=>(c) as previously shown.
Furthermore, if A is a square, n x n, matrix and has n pivots must lie on the main diagonal line, each being one down and one right from the one before it.
In this case the reduced echelon form of A is In.
Which means (c)=>(b).
Also, (b)=>(a) by the previous theorem.
This completes the explanation for the circle diagram.
(a)=>(k) because A-1 works for D.
Also, (k)=>(g) and (g)=>(a) as previously shown.
This shows that (k) and (g) are linked to the circle.
Furthermore, (g), (h), and (i) are equivalent for any matrix.
Therefore, (h) and (i) are linked through (g) to the circle.
Since (d) is linked to the circle so are (e) and (f), because (d), (e), and (f) are equivalent for any matrix A.
In conclusion, (a)=>(l) by the theorem previously stated and (l)=>(a)
by the same theorem with A and AT interchanged.
This explains the diagram.
A linear transformation T: Rn • Rn is said to be invertible if there exists a function S:
Rn • Rn such that:
S(T(x)) = x
for all x in Rn
T(S(x)) = x
for all x in Rn
This theorem shows that if S exists, it is unique and must be a linear transformation.
S is the inverse of T and is written as T-1.
Invertible Linear Transformations Theorem
Let T: Rn • Rn be a linear transformation and let A be the standard matrix for T.
Thus T is invertible if and only if A is and invertible matrix.
In that case, the linear transformation S is given by S(x) = A-1x is the unique function satisfying (1) and (2).
If we suppose that T is invertible, then (2) shows that T is onto Rn, because if b is in Rn and
x=S(b), then T(x)= T(S(b))=b, so each b is in the range of T.
Therefore, A is invertible, by the Invertible Matrix Theorem, statement (i).
Now suppose that A is invertible.
Let S(x)=A-1x.
is a linear transformation and S satisfies (1) and (2).
S(T(x))=S(Ax)=A-1(Ax)=x
T is invertible.