Hyperbolic Complex Research Articles

Extended hyperbolicity

The concept of Brody hyperbolicity is interpreted in terms of homotopy theoretic structures. We extend the definition of Brody hyperbolicity to simplicial sheaves of sets over the site of complex spaces with the strong topology. Imitating one possible definition of homotopy groups for a topological space, we defined the holotopy groups for a simplicial sheaf and showed that their vanishing in “positive” degrees is a necessary condition for a sheaf to be Brody hyperbolic. A partial converse to this theorem is proved at the end of the paper. We deduce that if X is a complex space with a nonzero holotopy group in positive degree, then X cannot be weakly equivalent (in a particular sense) to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results along with a topological realization functor, constructed in the previous section, to prove that complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces.

Considerations on the hyperbolic complex Klein–Gordon equation

This article summarizes and consolidates investigations on hyperbolic complex numbers with respect to the Klein–Gordon equation for fermions and bosons. The hyperbolic complex numbers are applied in the sense that complex extensions of groups and algebras are performed not with the complex unit, but with the product of complex and hyperbolic unit. The modified complexification is the key ingredient for the theory. The Klein–Gordon equation is represented in this framework in the form of the first invariant of the Poincaré group, the mass operator, in order to emphasize its geometric origin. The possibility of new interactions arising from hyperbolic complex gauge transformations is discussed.

On the boundary of a special class of hyperbolic two-dimensional simplicial complexes

A class of hyperbolic two-dimensional complexes X is defined. It is shown that the limit set of the action of π1(X) on the universal covering \({\widetilde{X}}\) of X is equal to the visual boundary \({\partial \widetilde{X}}\) of and that \({\partial \widetilde{X}}\) is path connected and locally path connected. A kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.

Links between quantum physics and thought.

Quantum mechanics (QM) provides a variety of ideas that can assist in developing Artificial Intelligence for healthcare, and opens the possibility of developing a unified system of Best Practice for inference that will embrace both QM and classical inference. Of particular interest is inference in the hyperbolic-complex plane, the counterpart of the normal i-complex plane of basic QM. There are two reasons. First, QM appears to rotate from i-complex Hilbert space to hyperbolic-complex descriptions when observations are made on wave functions as particles, yielding classical results, and classical laws of probability manipulation (e.g. the law of composition of probabilities) then hold, whereas in the i-complex plane they do not. Second, i-complex Hilbert space is not the whole story in physics. Hyperbolic complex planes arise in extension from the Dirac-Clifford calculus to particle physics, in relativistic correction thereby, and in regard to spinors and twisters. Generalization of these forms resemble grammatical constructions and promote the idea that probability-weighted algebraic elements can be used to hold dimensions of syntactic and semantic meaning. It is also starting to look as though when a solution is reached by an inference system in the hyperbolic-complex, the hyperbolic-imaginary values disappear, while conversely hyperbolic-imaginary values are associated with the un-queried state of a system and goal seeking behavior.

The Tricomi and Frankl problems for generalized Chaplygin equations in multiply connected domains

The present paper deals with Tricomi and Frankl problems for generalized Chaplygin equations in multiply connected domains. We first give the representation of solutions of the Tricomi problem for the equations, and then prove the uniqueness and existence of solutions for the problem by a new method, i.e. the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used. Finally we discuss the Frankl problem for generalized Chaplygin equations in multiply connected domains.

Inverse scattering method and soliton solution family for the Einstein–Maxwell theory with multiple Abelian gauge fields

A Hauser–Ernst-type extended hyperbolic complex linear system given in our previous paper [Gao Y J 2004 Chin. Phys. 13 602] is slightly modified and used to develop a new inverse scattering method for the stationary axisymmetric Einstein–Maxwell theory with multiple Abelian gauge fields. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the inverse scattering method be fine and effective in practical application. As an example, a concrete family of soliton solutions for the considered theory is obtained.

Sous-groupes discrets de PU(2,1) engendrés par n réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane \({{\mathbb{H}}^2_{\mathbb{C}}}\) and let Gn be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in \({{{\mathbb{H}}^2_{\mathbb{C}}}}\) . Under certain conditions, we prove that Gn is discrete. We construct representations ρ of the fundamental group Γg of the compact surface Σg of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.

Oblique derivative problems for second order nonlinear mixed equations with degenerate line

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampèreequations (CMA and HCMA) provide a lift of non-invariant solutions of three- and twodimensionalreduced equations, i.e., a lift of invariant solutions of the original CMA andHCMA equations, to non-invariant solutions of the latter four-dimensional equations. Thelift is applied to non-invariant solutions of the two-dimensional Helmholtz equation to yieldnon-invariant solutions of CMA, and to non-invariant solutions of three-dimensional waveequation and three-dimensional hyperbolic Boyer-Finley equation to yield non-invariant solutionsof HCMA. By using these solutions as metric potentials, it may be possible to constructfour-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that havenon-zero curvature tensors and no Killing vectors.

Double Superpotential and Generation of New Solutions of Stationary Axisymmetric Gravitational Fields

In this paper COX’s superpotential is extended to double complex form firstly, then how to get hyperbolic complex superpotential from the known ordinary complex superpotential is discussed, finally, new solutions of stationary axisymmetric gravitational fields are shown with a few specific cases.

Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics

We demonstrate that partner symmetries provide a lift of noninvariant solutions of the three-dimensional Boyer–Finley equation to noninvariant solutions of the four-dimensional hyperbolic complex Monge–Ampère equation. The lift is applied to noninvariant solutions of the Boyer–Finley equation, obtained earlier by the method of group foliation, to yield noninvariant solutions of the hyperbolic complex Monge–Ampère equation. Using these solutions we construct new Ricci-flat ultra-hyperbolic metrics with non-zero curvature tensor that have no Killing vectors.

Reconstructing Equation of State for Dark Energy in Double ComplexSymmetric Gravitational Theory

We propose to study the accelerating expansion of the universe inthe double complex symmetric gravitational theory (DCSGT). Theuniverse we live in is taken as the real part of the wholespacetime ℳ4C(J), which is double complex. Byintroducing the spatially flat FRW metric, not only the doubleFriedmann equations but also the two constraint conditions pJ = 0and J2 = 1 are obtained. Furthermore, using parametric DL(z)ansatz, we reconstruct the ω′(z) and V(ϕ) for darkenergy from real observational data. We find that in the two casesof J = i, pJ = 0, and J = ε, pJ≠0, thecorresponding equations of state ω′(z) remain close to-1 at present (z = 0) and change from below -1 to above -1. The resultsillustrate that the whole spacetime, i.e. the double complexspacetime ℳ4C(J), may be either ordinary complex(J = i, pJ = 0) or hyperbolic complex (J = ε,pJ≠0). And the fate of the universe would be Big Rip in the future.

Expansion complexes for finite subdivision rules. II

This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0 0 ) has an invariant partial conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.

Some Physics Questions in Hyperbolic Complex Space

In hyperbolic complex space, the Clifford algebra is isomorphic to that of a corresponding Minkowski geometry. We define the hyperbolic imaginary unit j (j2 = 1, j ≠ ± 1, j* = − j) to generate a class of Clifford algebras. We can introduce a class of non-Euclidean spaces and discuss the general form of 4-dimensional Lorentz transformation, and related special relativistic physics.

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein–Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex (EHC) function method is used to study further the stationary axisymmetric Einstein–Maxwell theory with p Abelian gauge fields (EM-p theory, for short). Two EHC structural Riemann–Hilbert (RH) transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-p theory. This symmetry group is verified to have the structure of semidirect product of Kac–Moody group SU(+1,1) and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author's paper by a different scheme. This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.

Ritt’s theorem and the Heins map in hyperbolic complex manifolds

Let X be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt’s theorem: every holomorphic self-map f:X→X such that f(X) is relatively compact in X has a unique fixed point τ(f) ∈ X, which is attracting. Furthermore, we shall prove that τ(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.

The Tricomi problem for second order linear equations of mixed type with parabolic degeneracy

In Bers, 1958, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York: Wiley); Bitsadze, 1988, Some Classes of Partial Differential Equations (New York: Gordon and Breach); Rassias, 1990, Lecture Notes on Mixed Type Partial Differential Equations (Singapore: World Scientific); Salakhitdinov and Islomov, 1987, The Tricomi problem for the general linear equation of mixed type with a nonsmooth line of degeneracy. Soviet Math. Dokl., 34, 133–136; Smirnov, 1978, Equations of Mixed type (Providence, RI: American Mathematical Society), the authors posed and discussed the Tricomi problem of second order equations of mixed type with parabolic degeneracy, which possesses important application to gas dynamics. The present article deals with the Tricomi problem for general second order equations of mixed type with parabolic degeneracy. Firstly the formulation of the problem for the equations is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension. In this article, we use the complex method, namely the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used (see Wen, 2002, Linear and Quasilinear Equations of Hyperbolic and Mixed Types (London: Taylor and Francis)).

Flows and joins of metric spaces

We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H^n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X-bar= X union bdry X. They are continuous, Isom(X)-invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g in Isom(X). For any hyperbolic complex X, the symmetric join *X-bar of X-bar and the (generalized) metric d* on it are defined. The geodesic flow space F(X) arises as a part of *X-bar. (F(X),d*) is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex X and has sharp properties. We also give a construction of the asymmetric join X*Y of two metric spaces. These concepts are canonical, i.e. functorial in X, and involve no `quasi'-language. Applications and relation to the Borel conjecture and others are discussed.

Hyperbolic phase transformation group and its applicaton to four-dimensional relativistic spacetime

The hyperbolic complex space RH defined by the Clifford algebra and the hyperbolic phase transformation group U4(H) acting on RH are endowed with definite physical meaning in this paper. The hyperbolic complex space RH is isomorphic to the 4-dimensional(4D) Minkowski spacetime, and the hyperbolic phase transformation group U4(H) in RH is just Lorentz transformation group on 4D relativistic spacetime. Furthermore, the general expressions of Lorentz transformation and the velocity transformation on 4D Minkowski spacetime are naturally derived using the composite transformations of the group U4(H). Hence, the well-known special Lorentz transformation in the special relativity(SR) is contained as a special case in our discussions.

Complexn-dimensional manifolds with a realn 2-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n2-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dimℝ Aut (M) ≽ (dimℂM)2.