Hyperbolic Function Theory

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

We introduce a new notion called the extended hyperbolic metrics, as a hyperbolic metric (i.e. metric of constant curvature − 1) with certain kinds of singularities defined on a Riemann surface, and we give several fundamental properties of such metrics. Extended hyperbolic metrics are closely related to space-like surfaces of constant mean curvature one (i.e. CMC-1 surfaces) in de Sitter 3-space S 1 3. For example, the singular set of a given CMC-1 surface in S 1 3 is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S 1 3 (i.e. weakly complete constant mean curvature 1 surfaces in S 1 3 of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S 1 3-catenoids. Since there is a bijection between the moduli space of S 1 3-catenoids and the moduli space of co-orientable extended hyperbolic metrics with two regular singularities, a classification of such hyperbolic metrics is also given. (Co-orientability of extended hyperbolic metrics is defined in this paper.)

Let Cℓ n be the (universal) Clifford algebra generated by e1,…,e n The Cauchy-Riemann (or Dirac) operator in Cℓ n is defined by $$ D = \sum\nolimits_{{i = 0}}^{n} {{{e}_{i}}} \frac{\partial }{{\partial {{x}_{i}}}}. $$ The functions $$ f{\text{ }}:{\text{ }}\Omega \to C{{\ell }_{{n,}}}{\text{ }}\Omega {\text{ }} \subset {\text{ }}{{\mathbb{R}}^{{n + 1}}} $$ open, satisfying Df = 0, called left monogenic functions, are widely studied in Clifford analysis. The power function xm is not monogenic. The second author noticed that the power function is the solution of the modified Cauchy-Riemann system $$ xnDf + \left( {n - 1} \right)fn = 0 $$ which has strong connections to the hyperbolic metric. We study solutions of the equation $$ xnDf + \left( {n - 1} \right)Q_{{n - 1}}^{'}\left( f \right) = 0, $$ where Qf is the function given by the decomposition $$ f\left( x \right) = Pf\left( x \right) + Qf\left( x \right){{e}_{n}} $$ for $$ Pf\left( x \right),Qf\left( x \right) \notin C{{\ell }_{{n - 1}}}. $$ If $$ f = \sum\nolimits_{{i = 0}}^{n} {{{f}_{i}}} {{e}_{i}} $$ for some real functions f i , then f is the solution of the modified Cauchy-Riemann system stated above.

We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature.

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth ergodic perturbations of certain algebraic systems—including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle—the smooth centralizer is either virtually Zl or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered by Avila, Viana, and the second author: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is equivalent either to Lebesgue or atomic. The second phenomenon, described by the first and third authors, is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation. We employ a variety of techniques, among them a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livsic theory, and nonstationary normal forms.

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC 1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

This theoretical analysis reports on the non-isothermal calendering process of micropolar-Casson fluid and studies the viscoplastic and microrotation effects by utilizing the lubrication approximation (LAT). Exact dimensionless velocity and pressure gradient solutions are achieved. Then a numerical integration technique determined other mechanical quantities. Implementing the finite difference approximations resolved the energy expression. Graphs show how material parameters influence the pressure, pressure gradient, leave-off distance, temperature distribution, force, and power function. Temperature distribution increases with increased coupling number N and decreased Casson parameter [Formula: see text]. Force and power function increase with increased coupling number and decreased Casson parameter. Both Casson and coupling number control the pressure distribution and exiting sheet thickness.

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.

Quantification of soil detachment rates is necessary to establish a basic understanding of soil erosion processes and to develop fundamental‐based erosion models. Many studies have been conducted on the detachment rates of disturbed soils, but very little has been done to quantify the rates of detachment for natural soil conditions. This study was conducted to evaluate the influence of flow discharge, slope gradient, flow velocity, shear stress, stream power, and unit stream power on detachment rates of natural, undisturbed, mixed mesic typical Udorthent soil. Flow rates ranged from 0.25 to 2.0 L s−1 and slope gradient ranged from 8.8 to 46.6%. This study was compared with a previous study that used disturbed soil prepared by static compression. The results indicated that the detachment rates of disturbed soil were 1 to 23 times greater than the ones of natural undisturbed soil. It was necessary to use natural undisturbed soil samples to simulate the detachment process and to evaluate the influence of hydraulic parameter on detachment rate. Along with flow rate increasing, detachment rate increased as a linear function. Detachment rate also increased with slope gradient, but the functional relationship between the two variables depended on flow rate. Stepwise regression analysis indicated that detachment rate could be well predicted by a power function of flow rate and slope gradient (R2 = 0.96). Mean flow velocity was closely correlated to detachment rate (r2 = 0.91). Flow detachment rate was better correlated to a power function of stream power (r2 = 0.95) than to functions of either shear stress or unit stream power.

We report measurements of the stability of magneto-optical traps (MOTs) for neutral atoms in the limit of tight confinement of a single atom. For quadrupole magnetic field gradients at the trap center greater than ∼1 kG/cm, we find that stochastic diffusion of atoms out of the trapping volume becomes the dominant particle loss mechanism, ultimately limiting the MOT size to greater than ∼5 μm. We measured and modeled the diffusive loss rate as a function of laser power, detuning, and field gradient for trapped cesium atoms. In addition, for as few as two atoms, the collisional loss rates become very high for tightly confined traps, allowing the direct observation of isolated two-body atomic collisions in a MOT.

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique, Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This conjecture was proved to be false by Farrell and Jones [6] in every dimension in which exotic spheres exist. They constructed examples of homeomorphisms f : M → N between closed negatively curved manifolds for which f is not homotopic to a diffeomorphism. These counterexamples were later obtained also in dimension six by Ontaneda [21] and later generalized by Farrell, Jones and Ontaneda to all dimensions > 5 [8]. In fact, in [21] and [8] examples are given for which f is not even homotopic to a PL homeomorphism. The fact that f is not homotopic to a PL homeomorphism has several interesting strong consequences that imply certain limitations of well known powerful analytic ethods in geometry [9], [10], [11], [12] (see [13] for a survey). In all the examples mentioned above, one of the manifolds is always a closed hyperbolic manifold. Hence, both manifolds M and N have the homotopy type of a closed hyperbolic manifold (hence the homotopy type of a closed locally symmetric space). We call these examples of hyperbolic homotopy type. In [2], Ardanza also gave counterexamples to the Lawson–Yau conjecture. In his examples, the manifolds M and N are not homotopy equivalent to a closed locally symmetric space; in particular, they are not homotopy

Inclusion functions for complex standard functions have been implemented in the CoStLy C++ class library. Among other functions, CoStLy includes procedures for the exponential function, the logarithm, root and power functions, trigonometric and hyperbolic functions, and their inverse functions.Numerical examples are presented to demonstrate the high level of accuracy in computation that has been achieved with our implementation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The cutting slopes in soft rock of redbed appeared in Yichang-Badong highway often suffer from the instability along weak intercalations, so the creep behaviors of weak intercalated soils are crucially important for the stability of cutting slopes. Because the deformation of weak intercalated soils is significantly affected by water content due to the strong water sensitivity, it is necessary to study the influence of matric suction on the creep behaviors of weak intercalated soils. In order to find out the unsaturated creep characters of weak intercalated soils, a GDS unsaturated triaxial apparatus was used. Then the triaxial creep experiments on weak intercalated soil samples under varying matric suction were conducted to obtain the unsaturated creep curves. The results show that the weak intercalated soils have obvious creep behaviors, and the creep strain is in nonlinear relationship with stress and time. When the matric suction is constant, a larger deviator stress will lead to a larger creep strain; When the deviator stress is constant, a smaller matric suction will lead to a larger creep strain. Based on the Mesri creep model, an improved creep model for weak intercalated soils under varying matric suction was established, in which the relationship of stress-strain was expressed with a hyperbolic function, and the relationship of strain-time was expressed with power functions in stages. Then an unsaturated creep model including stress-matric suction-strain-time for weak intercalated soils was established based on the power function relationship between matric suction and Ed (a parameter of the improved creep model). The comparison of the calculated values of creep model and the experimental values shows that the creep behaviors of weak intercalated soils can be predicted by the unsaturated creep model by and large.

About the determination of the thermal and athermal stress components from stress-relaxation experiments