Multiplicative Group Of Complex Numbers Research Articles

Real mixed Hodge structures

We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real manifold, and C^* is the multiplicative group of complex numbers considered as a real group.

The model theory of the field of reals with a subgroup of the unit circle

We describe definable sets in the field of reals augmented by a predicate for a finite rank multiplicative group of complex numbers contained in the unit circle . This structure interprets the quotient-space / which, for infinite cyclic, is related to the quantum torus. Every definable set is proved to be a Boolean combination of existentially definable sets. We give a complete set of axioms for the theory of such a structure

The equivalence problem for higher-codimensional CR structures

The equivalence problem for CR structures can be viewed as a special case of the equivalence problem for G-structure. This paper uses Cartan’s methods (in modernized form) to show that a CR manifold of codimension 3 or greater with suitably generic Levi form admits a canonical connection on a reduced structure bundle whose group is isomorphic to the multiplicative group of complex numbers. As corollaries, it follows that the CR manifold admits a canonical ane connection, and consequently that the automorphisms of the CR manifold constitute a Lie group. The most dicult technical step is to construct a smooth moduli space for generic vector-valued hermitian forms, which is tied to the CR manifold via the Levi map. The techniques used to construct this space are drawn from the classical invariant theory of complex projective hypersurfaces.

ON ITO-NISIO TYPE THEOREMS FOR DS-GROUPS

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be veriÞed by means of characteristic function- als. Analogous results are obtained also for almost everywhere con- vergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed. 1. Preliminaries. Throughout the paper N denotes the set of natu- ral numbers; Q;R, and C are, respectively, the Þelds of rational, real, and complex numbers with the ordinary (Euclidean) metric, and T denotes the multiplicative group of complex numbers of modulus 1 with the metric in- duced from C. For a topological abelian group X we denote by X 0 the topological dual group which consists of all continuous (unitary) characters h : X !T; the group operation in X 0 is the natural pointwise multiplication. No topology in X 0 is speciÞed. A topological abelian group X is called dually separated or DS-group if X 0 separates the points of X; in other words, X is a DS-group if for any dierent x1;x2 2 X there is a character h 2 X 0 such that h(x1) 6 h(x2). Hausdorlocally compact abelian (LCA-) groups and any additive subgroup of any Hausdorlocally convex space are examples of DS-groups. Below we shall see another type of examples too. Let X be a completely regular Hausdortopological space. Denote by Mt(X) the set of all Radon probability measures n deÞned on the Borel o-algebra of X. In Mt(X) we consider only the weak topology (for all the notions unexplained here the reader is referred to (1) and (2)). For Þxed x 2 X we denote by ex the Dirac measure concentrated at x. The Prokhorov theorem says that a subset M o Mt(X) is relatively compact if

On Ito–Nisio Type Theorems for DS-Groups

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be veriÞed by means of characteristic function- als. Analogous results are obtained also for almost everywhere con- vergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed. 1. Preliminaries. Throughout the paper N denotes the set of natu- ral numbers; Q;R, and C are, respectively, the Þelds of rational, real, and complex numbers with the ordinary (Euclidean) metric, and T denotes the multiplicative group of complex numbers of modulus 1 with the metric in- duced from C. For a topological abelian group X we denote by X 0 the topological dual group which consists of all continuous (unitary) characters h : X !T; the group operation in X 0 is the natural pointwise multiplication. No topology in X 0 is speciÞed. A topological abelian group X is called dually separated or DS-group if X 0 separates the points of X; in other words, X is a DS-group if for any dierent x1;x2 2 X there is a character h 2 X 0 such that h(x1) 6 h(x2). Hausdorlocally compact abelian (LCA-) groups and any additive subgroup of any Hausdorlocally convex space are examples of DS-groups. Below we shall see another type of examples too. Let X be a completely regular Hausdortopological space. Denote by Mt(X) the set of all Radon probability measures n deÞned on the Borel o-algebra of X. In Mt(X) we consider only the weak topology (for all the notions unexplained here the reader is referred to (1) and (2)). For Þxed x 2 X we denote by ex the Dirac measure concentrated at x. The Prokhorov theorem says that a subset M o Mt(X) is relatively compact if

Closures of orbits of C and C^* actions

Let G be a 1-dimensional complex linear algebraic group. In [7] Horrocks has shown that when G is the additive group C of complex numbers acting regularly on a normal (complex) projective variety, say X, or an algebraic variety X which is algebraically locally factorial, then the closure C of each G-orbit of is nonsingular. Moreover, Mabuchi [9] has shown that, if C touches a 1codimensional component in X of the set X of fixed points, then C is nonsingular and it intersect X transversally for any complex manifold X. On the other hand, when G=C*, the multiplicative group of complex numbers, Horrocks [7] showed that on a variety X as above the closure of each orbit is locally irreducible. If x is any point of X, this is equivalent to saying that either xeX, or 0(X)/oo(x), where 0(x) = limt^0t(x) and oo(x) = lim f_> 0 0/(x) for teC*. Furthermore, this fact was generalized as follows (cf. [3], [4]). Let X be a complex manifold on which G = C* acts biholomorphically and meromorphically. Then a sequence of points xί9 9xs9 s>l, of X is said to generate a quasi-cycle if xte X—X for each /, and co(xt) and 0(^ί+1) are contained in one and the same connected component of X for ^defining the action extends to a meromorphic map Px X^> X with respect to the natural inclusion G<—»P, where P denotes the complex projective line. The purpose of this note is then to generalize the above results in the following two theorems:

Simultaneous weighted sums of elements of finitely generated multiplicative groups

Let {G j} jεJ be a finite set of finitely generated subgroups of the multiplicative group of complex numbers C x. Write H=∩ jεJ G j . Let n be a positive integer and a ij a complex number for i = 1, ..., n and j ε J. Then there exists a set W with the following properties. The cardinality of W depends only on {G j} jεJ and n. If, for each jεJ, α has a representation α = Σ i n = 1a ijg ij in elements g ij of G j , then α has a representation a= Σ k=1 n w kh k with w kεW, h k εH for k = 1,..., n. The theorem in this note gives information on such representations.

Soluble gauge theory of a noncompact group

It is shown that the gauge theory of the multiplicative group of complex numbers is the theory of two free vector mesons only one of which is massless.Received 6 April 1979DOI:https://doi.org/10.1103/PhysRevD.20.2636©1979 American Physical Society

Lattice and complex extensions of point groups. I

The formulae enabling in a simple way to determine the explicit form of factors of non-equivalent extensions of the point groups by lattice groups and multiplicative group of complex numbers are derived. It is shown that the number of all non-equivalent extensions of the groups C 2 h , D 2 h , D 4 h by lattice groups Γ m , Γ o , Γ q is equal 88, an and the explicit form of the factors of these extensions is determined. Also the explicit form of the factors of all non-equivalent extensions of the groups C nh , D n , D nh by the multiplicative group of complex numbers is derived.

The punctured plane is isomorphic to the unit circle

This note shows that the multiplicative group of non-zero complex numbers and the multiplicative group of complex numbers of absolute value 1 are isomorphic as groups.

Locally compact topologies for groups

The investigations leading to this paper were suggested by the papers of Hewitt [3] and Ross [5]. In [3] Hewitt was interested in proving that if an abelian group is locally compact in two topologies, one strictly stronger than the other, there is a character continuous in one topology and discontinuous in the other (actually a special case of a theorem of Kaplansky-see Theorem 1.1 of [2]). Actually Hewitt proved a stronger result. His arguments were based on the fact that both the additive group of reals, and the multiplicative group of complex numbers of absolute value 1 have the property that every stronger locally compact group topology is discrete. A natural question to ask is what other groups have this property. The answer is very simple (2.1 of this paper). We consider in the second section of this paper the obvious generalization. Namely which groups have the property that there are only finitely many stronger locally compact group topologies. The investigations in the first section of this paper were suggested by the paper of Ross [5]. Ross was considering the same question as Hewitt, and was led to consider the relationship between two locally compact group topologies on a group G such that G has the same closed subgroups in the two topologies. We investigate this further in the first section of this paper, and are able to say that many of the properties of G as a topological group can be recovered once we know the closed subgroups.

Linear functions on almost periodic functions

Our aim is, first, to present two realizations of the space 1* of all bounded complex linear functionals on 21, and, second, to use these realizations for the study of positive definite functions which are not necessarily continuous. In this fashion, we obtain a generalization of Bochner's representation theorem for continuous positive definite functions as well as various facts concerning positive definite functions and their structure. 0.2 Throughout the present paper, the symbol R denotes the real numbers, considered either as an additive group or as a field; K the field of complex numbers; T the multiplicative group of complex numbers of absolute value 1; Tm the complete Cartesian product of m groups each identical with T, m being any cardinal number greater than 1. The characteristic function of a subset B of a set X is denoted by XB. If G is any locally compact Abelian group, we denote the group of all continuous characters of G by the symbol G]. G] is given the usual compact-open topology. If X is any topological space, we denote the set of all complex-valued continuous functions on X which are bounded in absolute value by the symbol C(X). The space of all trigonometric polynomials 0.1.1 is denoted by 93; the space of all almost periodic continuous functions on R by 2f. For a normed complex linear space V, we denote the space of all bounded complex linear functionals on V by the symbol V*.