Ordinary Complex Numbers Research Articles

Ideal structure of rings of analytic functions with non-Archimedean metrics

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

Generalized complex numbers over near-fields

The construction of the complex numbers over the reals has been generalized in many ways leading, amongs others, to the 2-dimensional elliptical complex numbers (= ordinary complex numbers), the parabolic complex numbers and the hyperbolic complex numbers. One can extend these to higher dimensions and also to using an arbitrary ring as the base ring. Here we propose a construction of generalized complex numbers over a near-field and investigate some of the structural properties of this near-ring.

Multi-Fuzzy Complex Numbers and Multi-Fuzzy Complex Sets

This paper presents a method to construct more general fuzzy complex numbers and sets from ordinary fuzzy complex numbers by introduced the ordered sequences of membership functions. It is concluded that multi-fuzzy complex set is an extension of Buckley fuzzy complex set. Also, two types of multi-fuzzy complex numbers based on the forms and are investigated.

Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.

Hypercomplex numbers and the description of spin states

A family of hypercomplex numbers is introduced in which multiplication is commutative and members can have up to eight components. In particular, the eight basis elements {E} contain those for ordinary complex numbers, E**=E, as well as new elements where E**=−E; the operation * being the generalization of complex conjugation. This family lends itself to the description of quantum mechanical spin states in that it offers a simple treatment of time reversal, representations with the same conjugation properties as underlying operators, and explicit continuous-angle spherical harmonic functions Zsm(θ,φ) analogous to the Ylm(θ,φ) for orbital angular momentum. The new elements are especially well suited for half-integral spin states, whereas conventional complex numbers remain useful for integral spin states.

Algebraic classification of four-dimensional locally Euclidean spaces

The algebraic classification of locally Euclidean four-dimensional spaces (having positive-definite metric) is considered in terms of dual complex variables, which, like ordinary complex numbers, are a type of hypercomplex numbers. It is shown that only four space subtypes exist: 1, 1a, D, 0 and there are no wave types. Transition between space subtypes is described by a CUSP catastrophe analogous to a second-order phase transition in a solid. Introducing an imaginary coordinate, we obtain the usual four-dimensional space—time, and our classification reduces to the well-known Petrov—Penrose algebraic classification of gravitational fields.

Quaternions as a tool for the analysis of molecular systems

Quaternions are generalized complex numbers and represent rotations in space as ordinary complex numbers represent rotations in a plane. In the context of molecular dynamics (MD) simulations they have been ‘rediscovered’ for the integration of the rotational equation of motion of rigid molecules since they allow one to write down these equations in a singularity-free form. In this paper applications to the analysis of molecular systems are described.

P-adic analyticity and virasoro algebras for conformal theories in more than two dimensions

The field of p-adic complex numbers has a much richer structure than the field of ordinary complex numbers. This is used in order to extend the powerful tools of two-dimensional conformal field theories to higher dimensions. It is thus proposed that critical systems in more than two dimensions be first studied over the p-adics and then, if possible, recovered by the adelic construction. It is further argued that this higher-dimensional p-adic analyticity may be the key to membrane theories. A natural ansatz for three-brane tree-scattering amplitudes, where p-adic analyticity is instrumental, is given as an explicit example.

Generation of new solutions of the stationary axisymmetric Einstein equations by a double complex function method

A new approach to the solution of certain differential equations, the double complex function method, is developed, combining ordinary complex numbers and hyperbolic complex numbers. This method is applied to the theory of stationary axisymmetric Einstein equations in general relativity. A family of exact double solutions, double transformation groups, and n-soliton double solutions are obtained.

Proposed Test for Complex versus Quaternion Quantum Theory

If scattering amplitudes are ordinary complex numbers (not quaternions) then there is a universal algebraic relationship between the six coherent cross sections of any three scatterers (taken singly and pairwise). A violation of this relationship would indicate either that scattering amplitudes are quaternions, or that the superposition principle fails. Some experimental tests are proposed, involving neutron diffraction by crystals made of three different isotopes, neutron interferometry, and ${K}_{S}$-meson regeneration.

Hypernumbers—II. further concepts and computational applications

In a previous paper [1] on hypernumbers it was suggested that there were significant ramifications to be explored and exemplified. In particular, the following important areas were only mentioned in passing, but are now more fully examined: the hypernumber use of idempotents, zero divisors, and nilpotents; the concept of bimatrices; and a survey, necessarily brief, of hypernumbers beyond ε and i, that is, beyond the proper square and fourth roots of unity respectively. The term hypernumber was introduced into mathematics by the present author in 1966 [2] to denote domains of number including or beyond the arithmetic of ordinary numbers, complex numbers, quaternions, matrices, or octaves (Cayley-Graves numbers); and the term has since been used by Kline [3] and by Spencer and Moon [4], but unnecessarily restricted to forms of i and ε, and too often only to matrix forms.

A General Theory of Limits

I. As to Notions.-In this paper we investigate a simple general limit of which, as will appear in ? 5, the various classical limits of analysis are actually instances.* The general limit in question is an obvious generalization of the following two limits: 1. An infinite sequence {an} of real or (ordinary) complex numbers an (n = 1, 2, *..) converges to a number a as a limit, in notation: Lnoo an = a,-as clearly defined about a century ago,-in case for every positive number e there exists a positive integer ne of such a nature that for every integer n i ne it is true that in absolute value ana is. at most e. Here the numerical sequence {an } may be considered as a numerically valued function a (an In) of the positive integer n (or on the range [n] of positive integers n), viz., a(n) an for every n. 2. Relative to a general (i.e., any particular) class e[q] of general elements q and the class ( = [s] of all finite classes s of elements q, a numerically valued function a (a (s) Is) on the range S converges to a number a as limit, in notation: L,a(s) = a, in case for every positive number e there exists a class Se of such a nature that for every class s including Se it is true that in absolute value a(s) a is at most e. The limit (2) belongs to General Analysis, i.e., to that doctrine of analysis in which a general class, here {D, plays a fundamental role. The limit (2), introducedt in 1915 by the senior author, plays a central role in his second theory$ of Linear Integral Equations in General Analysis. This general theory has as notable instances: (a) Hilbert's theory of limited quadratic forms in a denumerable infinitude of variables; here the Hilbert space [a] of infinite sequences a (a (n) I n) of real numbers with convergent Ina (n) a (n) plays a central role; (b) an analogous theory in which the corresponding role

A symbolic theory of formal modular covariants

1. Prologue. Thus far, very little has been published on the general theory of formal modular invariants or covariants. Workers have, on the whole, obtained results for special, more or less isolated, cases; and although some beautiful and important general theorems have been proved, they are more or less unrelated. This is, of course, only natural in any division of knowledge in its formative state. Nevertheless, no worker in the field could fail to be conscious of a certain uniformity common to the special cases that have been studied in detail; though (alas!) this uniformity usually appeared to be broken ruthlessly in the next case studied. This breaking of an apparent law signified, however, merely that we did not know these special cases with a sufficient thoroughness of illuminating detail, or were trying unwittingly to make the laws conform to certain standards, unconsciously preconceived. This latter handicap was laid on us naturally enough by our thorough knowledge of algebraic invariants and the fact that this newer kind of covariants is, in many ways, strikingly like the older, classic covariants, though so tantalisingly different. Their similarity and their difference show themselves in the very beginning of the study: in the definitions, in the simplest examples. Perhaps the differences that first come to mind are those which are inherent in the fields of definition, which, in the case of classic covariants, is the field of reals or ordinary complex numbers and, in the case of modular covariants, is a Galois field, GF [pn] , of order pn . These differences are too obvious to mention in detail, but one who has studied the beautiful proofs given by the old masters of invariant theory has been forced to the conclusion that most of the proofs seemed to use the properties of a field of characteristic zero, not in some accidental manner, but rather in veriest necessity. Growing from the surface differences between the two fields are two very important distinguishing characteristics of the two kinds of covariants. It * Part II was presented to the Society, September 7, 1920; Part III, December 28, 1921; Parts IV and V, December 27, 1922. The reading of the literature in connection with this paper was much facilitated by the purchase of books with a grant made by the American Association for the Advancement of Science and this help is herewith gratefully acknowledged. 286

On the theory of associative division algebras

1. Relation to the literature. There is a famous theorem to the effect that the only linear associative algebras over the field of all real numbers in which division is uniquely possible are the field of real numbers, the field of ordinary complex numbers, and real quaternions. The first published proof of this was that given in 1878 by Frobenius in his fundamental memoirt on bilinear forms. Since this proof, there have been numerous otbers,t the most recent being one by Professor Dickson.? In the last mentioned proof, the theorem is a special case of a more general theorem of the same nature for a certain class of algebras, which Dickson calls Type A. He defines an algebra of this type as a linear associative algebra A, the coordinates of whose numbers range over any given algebraicll field F, and for which the following properties hold: (a) There exists in A a number i satisfying an equation 4 (x) = 0 of degree n with coefficients in F and irreducible in F. (b) Any number of A which is commutative with i is in F (i). (c) There exists in A a number j, not in F (i), such that ji = Oj + , where 0 and a are in F (i). All three of these conditions are satisfied by real quaternions, and the first two by any linear associative division algebra D over F, where we take i so that the degree of the irreducible equation in F satisfied by i is the maximum. Dickson showed? that every algebra of Type A over F has a subalgebra

On the Rational, Integral Invariants of Nilpotent Algebras

?1. Relation to literature. In the study of linear algebras, their invariantive classification is one of the most important problems. Peirce, Study, Scheffers, Hawkes, and others* have classified, over the field C of ordinary complex numbers, linear associative algebras in a small number of units; and associative algebras having less than four units, with a modulus, are easily characterized invariantively by invariants obtainable from the characteristic determinants,t but such invariants are not sufficient for the characterization of quaternary associative algebras with a modulus.: In view of Wedderburn's? generalization of a theorem of Cartan, 11 we can go a long way toward characterizing the general linear algebra if we can characterize three special kinds of algebras; namely, matric algebras, division algebras, and nilpotent algebras. Over the field C, there is only a finite number of classes of the first two kinds of algebras of a given order; whereas, over the field C, there is an infinite number of classes of nilpotent algebras of any order greater than two; and hence, we centre our attention on nilpotent algebras. In this paper, we consider rational, integral invariants of such algebras. Now all invariants obtainable from the characteristic determinants are zero for nilpotent algebras; and furthermore, there are no rational, integral invariants (other than constants) for n-ary nilpotent algebras under the total n-ary linear group. We accordingly consider invariants under the group which leaves unaltered the canonical form. For such rational, integral invariants this paper proves theorems analogous to theorems about invariants of algebraic forms, and in particular proves their finiteness. The fundamental invariants are found for the simpler cases.

Finite groups represented by special matrices

The direct object of this paper is to prove that every possible finite group which contains an abelian subgroup of half its own order can be represented by square matrices all of whose elements are equal to zero with the exception of those which appear in one of the diagonals, and these are all ordinary complex numbers. The fact that every finite group which can be represented by such matrices must contain an abelian subgroup of half its own order is at once evident. The proof of the stated theorem may be based upon several interesting theorems relating to a possible choice of the independent generators of an abelian group. As these auxiliary theorems seem to be quite fundamental, we proceed to develop them in a somewhat more general form than would be necessary for the particular application in view.

Linear associative algebras and abelian equations

1. In the recent development of the theory of linear associative algebras the coordinates of whose numbers range over the field of ordinary complex numbers, an important role is played by the so-called simple algebras, the only ones being the matric algebras with r2 units. For the more general case of algebras the coordinates of whose numbers range over any field (Korper) K of finite or infinite order, a corresponding importance is to be attached to simple algebras; an algebra is simple t if and only if it is the direct product of a matric algebra and a primitive algebra (one in which every element has an inverse). The problem of primitive algebras is the chief outstanding problem in the theory of algebras over a field K. Aside from real quaternions the only known primitive linear associative algebras are fields. We proceed to exhibit a primitive algebra L with r2 units over K. Let K (X)-O be a uniserial abelian equation of degree r with respect to the field K, i. e., let the equation have as its coefficients numbers in K, be irreducible in K, and have as its roots

Two-dimensional chains and the associated collineations in a complex plane

The recogllition of the abstract identity of geonletry and analysis, which results from the llotion of coordinates on the one hand and from the classical work of VON STAUDT t on the algebra of throws on the other, and which recent work oll the foundations of geometry has ftllly established, has brought with it a broader conception of the colltellt of geometry. It has meant llot only the introduction of imaginary elements alld the resulting conception of a complex space (of any number of dimensions), but it has also led to the consideration of geometries with respect to any number-system (finite or infinite), i. e., of spaces the elemellts of which may be determined by sets of numbers (coordinates) belonging to a givell number-system. t An important result of the recognition of the identity referred to is the etnphasis it places on the possibility of using geometric or synthetic methods in the solution of analytic problems. It seems likely that the fact that such methods have received comparatively little attention hitherto has resulted in a loss of power. The present paper, it is hoped, will telld to substantiate this assertion. We are llere concerned with certaill fundamental problems in the projective geotnetry on a comptex plane, i. e., a plane the points of which are determined by sets of llomogeneous coordinates (xl, X2, X3), where the xi are any ordinary complex numbers. not all zero. Though the illvestigation is essentially geometric alld the results are susceptible of immediate application to problems of importance in geoInetry, these results are of even greater interest in the theory of functions of two complex variables. To make this clear we shall glance briefly at the corresponding problems on a complex line, tlle results of which are well-known ill the theory of functions

The Geometry of Chains on a Complex Line

Introduction. This paper forms an elementary chapter in the projective geometry on a complex line, i. e., of a line whose points are isomorphic with the system of ordinary complex numbers and infinity. It furnishes a synthetic treatment of certain well-known topics in the theory of functions of a complex variable; it has close contact with recent work on the foundations of projective geometry; finally, it forms the basis for certain generalizations, which offer a powerful synthetic approach to various analytic problems. To this last aspect of the paper I shall return in detail on a future occasion. t The first two aspects I discuss briefly in this introduction. But first, in view of the fact that the notion of a chain of points on a line does not appear to be generally familiar, it seems desirable to describe it briefly. This is most readily done in analytic language. Any point of a projective line may be conveniently represented by a single coordinate x. If the points of a line are hereby brought into one-to-one correspondence with the ordinary real numbers and infinity, we are wont to speak of a real line; if on the other hand the points of a line are hereby made isomorphic with the system of ordinary complex numbers and infinity we speak of a complex line. A real representation of the points of a complex line may then be obtained by the usual method for representing complex numbers by the real points of an Argand plane. The points of the line with real coordinate x form a subset (E0 of points on the line in the representation mentioned this subset q0 is represented by the points of the real axis. By means of a projective transformation on the line the set Go is transformedinto another or the same set (E. Any such set of points ( on the line which is projective with the set G3 of real points on the line is called a chain on the line. IIn the Argand representation the chains on the line evidently correspond to the circles (and straight lines) of the plane.

On hypercomplex number systems

1. The usual theory relates to systems of numbers =,a,e, in which the co6rdinates ai range independently over all real nulmbers or else over all ordinary complex numbers; for example, the real quaternion system, or the complex quaternion system. As an obvious generalization,t the co6rdinates may range independently over all the marks of any field F; for example, the rational quaternion system. As a further generalization, the sets of co6rdinates al, *, a in the various numbers of a system may include only a part of the sets bl, -, b , each b ranging independently over F; for example, the integral quaternion system. The various coordinates a1, , a. need not have the same range; for example, the numbers (a + 2b V/2) el + (c + 4d 12 ) e2 (a, b, c, d arbitrary integers)