Exponentials and Logarithms Properties in an Extended Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

Exponentials and Logarithms Properties in an Extended Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we proove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

Exponentials and Logarithms Properties in an Extended Complex Number Field

In this study we demonstrate the complex logarithm and exponential multivalued results and identity failures are not induced by the exponentiation and logarithm operations, but are solely induced by the definition of complex numbers and exponentiation as in C. We propose a new definition of the complex number set, in which the issues related to the identity failures and the multivalued results resolve. Furthermore the exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. There is a cost as some algebraic properties of the addition and substraction will be diminished, though remaining valid to a certain extent. Finally we attempt a geometric and algebraic formalization of the new complex numbers set. It will appear clearly the new complex numbers system is a natural and harmonious complement to the C field.

Exponentials and Logarithms Properties in an Extended Complex Number Field

In this study we demonstrate the complex logarithm and exponential multivalued results and identity failures are not induced by the exponentiation and logarithm operations, but are solely induced by the definition of complex numbers and exponentiation as in C. We propose a new definition of the complex number set, in which the issues related to the identity failures and the multivalued results resolve. Furthermore the exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. There is a cost as some algebraic properties of the addition and substraction will be diminished, though remaining valid to a certain extent. Finally we attempt a geometric and algebraic formalization of the new complex numbers set. It will appear clearly the new complex numbers system is a natural and harmonious complement to the C field.

Preliminaries

In addition to giving a very brief reminder of set notation and basic set operations, this chapter provides a brief refresher on basic mathematical concepts. The natural, rational and real number systems are taken for granted. However, it does develop at length the Cauchy criterion and its equivalence to the completeness of the real line, and the Bolzano-Weierstrass theorem, as well as the complex number field, including its completeness. Embryonic manifestations of completeness and compactness can be seen in this chapter. Examples include the nested interval theorem and the uniform continuity of continuous functions on compact intervals, and the proof of the Heine-Borel theorem in chapter 4 is squarely based on the Bolzano-Weierstrass property of bounded sets.

FRACTAL SETS IN THE FIELD OF p-ADIC ANALOGUE OF THE COMPLEX NUMBERS

The [Formula: see text]-adic number field [Formula: see text] and the [Formula: see text]-adic analogue of the complex number field [Formula: see text] have a rich algebraic and geometric structure that in some ways rivals that of the corresponding objects for the real or complex fields. In this paper, we attempt to find and understand geometric structures of general sets in a [Formula: see text]-adic setting. Several kinds of fractal measures and dimensions of sets in [Formula: see text] are studied. Some typical fractal sets are constructed. It is worthwhile to note that there exist some essential differences between [Formula: see text]-adic case and classical case.

Local Superderivations on Basic Classical Lie Superalgebras

In this paper, we prove that every local superderivation on basic classical Lie superalgebras except for A(1, 1) over the complex number field ℂ is a superderivation. Furthermore, we give an example of a class of nilpotent Lie superalgebras with local superderivations which are not superderivations.

ON NUMBER FIELDS WITHOUT A UNIT PRIMITIVE ELEMENT

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

MORE CONSTRUCTIONS OF APPROXIMATELY MUTUALLY UNBIASED BASES

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.