Dual space of exponential vector space

Complex numbers

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

and Khintchine [2] has shown that the measure of all real S-numbers with 7=1 is zero. On the other hand, Mahler [3] proved that almost all (real or complex) numbers are S-numbers. His proof shows that this statement is true with 7=4, and he conjectured that for almost all real numbers one can takey = 1 +e, and that for almost all complex numbers one can take y=1/2 +e, for arbitrary e >0. Let yr be the infimum of all numbers y' such that almost all real numbers are S-numbers with y 1/; i.e., he proved the conjecture for m = 2. (It is well known for m = 1.) By combining a theorem due to Fel'dman [6] with Mahler's original argument, it is shown here that y, 5 2, yc < 3/2. Lemma 5 of Fel'dman's paper is as follows: Let f(z) = ao+ + +amzm, where ao, * * *, am are rational integers with Iai( <a, let t1y .. * *,(m be its zeros, which are supposed distinct, and let r be an arbitrary complex number. If 5 =mini (1 v-(4), then

CHAPTER VII - COMPLEX NUMBERS

Presently, factorials of real negative numbers and imaginary numbers, except for zero and negative integers are interpolated using the Euler’s gamma function. In the present paper, the concept of factorials has been generalised as applicable to real and imaginary numbers, and multifactorials. New functions based on Euler’s factorial function have been proposed for the factorials of real negative and imaginary numbers. As per the present concept, the factorials of real negative numbers, are complex numbers. The factorials of real negative integers have their imaginary part equal to zero, thus are real numbers. Similarly, the factorials of imaginary numbers are complex numbers. The moduli of the complex factorials of real negative numbers, and imaginary numbers are equal to their respective real positive number factorials. Fractional factorials and multifactorials have been defined in a new perspective. The proposed concept has also been extended to Euler’s gamma function for real negative numbers and imaginary numbers, and beta function.

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.

It is well-known that if a, b are irrational numbers, then a b need not be an irrational number. Let M be a set of real numbers. In this note it is proved that if M is any of (i) the set of all irrational real numbers, (ii) the set of all transcendental real numbers, (iii) the set of all non-computable real numbers, (iv) the set of all real normal numbers, (v) the set of all real numbers of irrationality exponent equal to 2, (vi) the set of all real Mahler S-numbers, (vii) or indeed any subset of R of full Lebesgue measure, then, for each positive real number s ≠ 1 , there exist a , b ∈ M such that s = a b . The analogous result for complex numbers is also proved. These results are proved using measure theory.

Conventionally the solution of any quadratic equation in one unknown quantity (say x) is represented in real or complex numbers. Instead of finding the solution of any quadratic equation in one unknown in complex numbers, the author introduced Bhattacharyya's Theorem – 1 &amp; 2 to find its solution in real numbers only. Bhattacharyya's Theorem – 1 states that the square root of any negatively directed number is a negatively directed number and Bhattacharyya's Theorem -2 states that the square of any negatively directed number is a negatively directed number. Both theorems are based on the newly invented concept of the Theory of Dynamics of Numbers by the author. To find the root of any quadratic equation it must satisfy the two criteria: 1) The value of x must satisfy the equation (as conventional method). (2) The inherent nature of x must satisfy the quadratic equation (new concept). The inherent nature of x may be a positively directed number, a negatively directed number, or a neutral number which can be determined depending on the constant term, c&lt;0, c&gt;0, or c= 0 respectively of the quadratic equation. The author states that the quadratic expression which is factorizable into two linear functions may be defined as a pseudo-quadratic equation but all factorizable quadratic equations are not pseudo-quadratic equations. Using the unique concept of the Theory of Dynamics of Numbers the solution of the quadratic equation, ax2+bx+c=0, in one unknown quantity (say x) can be determined in real numbers only even if the discriminant, b2 – 4ac &lt; 0, without using the concept of the complex numbers.

In numerical computations mainly real and complex numbers, intervals as well as matrices and vectors with such components occur. It is well known that the arithmetic operations with real numbers, complex numbers etc. can be carried over to real floating-point numbers, complex floating-point numbers etc. using roundings. This proceeding results in agreeable arithmetic-, order- and compatibility-properties for an abundance of numerical data types and the accompanying arithmetic operations. Most programming languages however only provide real floating-point numbers; all the other data types and operations have to be simulated, e. g. in the form of arrays and procedure calls, which often causes loss of accuracy and arithmetic properties. Furthermore the complicate notation makes programs difficult to read. Therefore in this article an extension of PASCAL is presented which serves as an example for the way these numerical data types can be embedded into the syntax of a programming language.

It is well known that the advantage of complex number over real number is that it can express vector, while real number can only express scalar. two real numbers can describe two-dimensional coordinates, and a complex number can describe two-dimensional coordinates; three real numbers can describe three-dimensional coordinates, and four real numbers can describe four-dimensional coordinates, but a quaternion can describe two-dimensional, three-dimensional or four-dimensional coordinates, so a complex or quaternion containing variables can reduce an independent variable when replacing the equation. In this paper, the complex number containing variables is used to describe a circle, a two-dimensional circle, and a quaternion containing variables is used to describe a three-dimensional the surface of the ball and a three-dimensional sphere, using quaternion with variables to describe four-dimensional the surface of the ball and four-dimensional sphere . It is also found that the complex number containing a variable can describe the plane golden spiral and the quaternion containing a variable can describe three-dimensional golden spiral. It is also found that if ab=aC can not deduce b=c when the exponent is a complex number.

In Chapters 7 and 8, general theorems were proved about the structure of a single linear transformation. For certain types of linear transformations, and matrices corresponding to them, over the fields of real or complex numbers, sharper theorems about the eigenvalues and eigen-vectors can be proved. This chapter contains some of these results, for orthogonal and symmetric transformations on vector spaces over the real numbers, with applications to quadratic forms, and for unitary, self-adjoint, and normal transformations on vector spaces over the complex numbers. Further results, about the exponential of a matrix, and the Perron-Frobenius theorem on the eigenvalues of positive real matrices, with applications to systems of differential equations and Markov chains, are proved using analytic methods in Sections 34 and 35.

A complex number z is a number of the form a + bi where a and b are real numbers and i 2 = − 1. Cardona, who was a sixteenth century Italian mathematician, introduced complex numbers, and he used them to solve cubic equations. The set of complex numbers is denoted by ℂ, and each complex number has two parts namely the real part Re(z) = a, and the imaginary part Im(z) = b. The set of complex numbers is a superset of the set of real numbers, and this is clear since every real number is a complex number with an imaginary part of zero. A complex number with a real part of zero (i.e. a = 0) is termed an imaginary number. Complex numbers have many applications in physics, engineering and applied mathematics.

Damping in acoustics means sound absorption. In this paper, a theoretical analysis of the one-dimensional wave propagation in the damping medium is discussed. First, it is shown that density in complex number is derived by adding the viscous damping force and the inertial force of the fluid. This results in that the propagation velocity and the wave number are also complex numbers, and the imaginary part of the complex wave number is always negative. Therefore, distance decay occurs in the damping medium. In case of undamped medium, natural modes are formed while traveling wave and the reflected wave are superimposed in one-dimensional acoustic tube. Both the eigenvalues and natural modes are real numbers. If the damping is assumed to be Rayleigh damping, complex eigenvalues and real natural modes occur. However, if the excitation vibration pattern is not valid, natural mode is never excited. So, excitation vibration pattern is considered to excite the natural modes. Numerically calculations are achieved in different two ways of measuring propagation speed. One is time decay of damped natural mode, and the other is distance decay of forced vibration.

The model reference adaptive system (MRAS) is generally described in the field of real numbers. Therefore, the MRAS cannot be directly applicable to the system described by complex numbers such as three-phase ac motors.In order to apply the MRAS to the discrete-time linear time-invariant system whose coefficients and variables are complex numbers, the authors expand the theory of the MRAS into the field of complex numbers. The hyperstability theory in the field of complex numbers is used to give the asymptotic stability of the suggested algorithm.The effectiveness of the suggested algorithm is proved by simulation results for the parameter identification. The results show that the convergence time of the parameters using the proposed method can be reduced to one tenth compared with that using the MRAS in the field of real numbers.

For many years, vector space models have been used in information retrieval and computational linguistics to represent terms, queries, and documents, using vector addition as a simple operator to model semantic composition. Though surprisingly successful, many aspects of meaning including word order, typed relationships, and nested structures are not captured by this modelling process.