Assessment of Complex System Dynamics via Harmonic Mapping in a Multifractal Paradigm

In the present paper, nonlinear behaviors of complex system dynamics from a multifractal perspective of motion are analyzed. In the framework of scale relativity theory, by analyzing the dynamics of complex system entities based on continuous but non-differentiable curves (multifractal curves), both the Schrödinger and Madelung scenarios on the holographic implementations of dynamics are functional and complementary. In the Madelung scenario, the holographic implementation of dynamics (i.e., free of any external or internal constraints) has some important consequences explicated by means of various operational procedures. The selected procedures involve synchronous modes through SL (2R) transformation group based on a hidden symmetry, coherence domains through Riemann manifold embedded with a Poincaré metric based on a parallel transport of direction (in a Levi Civita sense). Other procedures used here relate to the stationary-non-stationary dynamics transition through harmonic mapping from the usual space to the hyperbolic one manifested as cellular and channel type self-structuring. Finally, the Madelung scenario on the holographic implementations of dynamics are discussed with respect to laser-produced plasma dynamics.

A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

Right Ventricular Shape Feature Quantification for Evaluation of Pulmonary Hypertension: Feasibility and Preliminary Associations with Clinical Outcome Submitted for Publication

This study aimed to demonstrate feasibility of statistical shape analysis techniques to identify distinguishing features of right ventricle (RV) shape as related to hemodynamic variables and outcome data in pulmonary hypertension (PH).Cardiovascular magnetic resonance images were acquired from 50 patients (33 PH, 17 Non-PH). Contemporaneous right heart catheterization data was collected for all individuals. Outcome was defined by all-cause mortality and hospitalization for heart failure. RV endocardial borders were manually segmented, and 3D surfaces reconstructed at end-diastole and end-systole. Registration and harmonic mapping were then used to create a quantitative correspondence between all RV surfaces. Proper orthogonal decomposition was performed to generate modes describing RV shape features. The first 15 modes captured over 98 % of the total modal energy. Two shape modes, 8 (free wall expansion) and 13 (septal flattening), stood out as relating to PH state (Mode 13: r=0.424, p=0.002. Mode 8: r=0.429, p=0.002). Mode 13 was significantly correlated with outcome (r = 0.438, p = 0.001), more so than any hemodynamic variable. Shape analysis techniques can derive unique RV shape descriptors corresponding to specific, anatomically meaningful features. The modes quantify shape features that had been previously only qualitatively related to PH progression. Modes describing relevant RV features are shown to correlate with clinical measures of RV status, as well as outcomes. These new shape descriptors lay the groundwork for a non-invasive strategy for identification of failing RVs, beyond what is currently available to clinicians.

“Holographic Implementations” in the Complex Fluid Dynamics through a Fractal Paradigm

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

Planar harmonic mappings in a family of functions convex in one direction

Let D := {z ? C : |z| < 1} be the open unit disk, and h and 1 be two analytic functions in D. Suppose that f = h + ?g is a harmonic mapping in D with the usual normalization h(0) = 0 = g(0) and h'(0) = 1. In this paper, we consider harmonic mappings f by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.

Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

A learning-based harmonic mapping: Framework, assessment, and case study of human-to-robot hand pose mapping

Harmonic mapping provides a natural way of mapping two manifolds by minimizing distortion induced by the mapping. However, most applications are limited to mapping between 2D and/or 3D spaces owing to the high computational cost. We propose a novel approach, the harmonic autoencoder (HAE), by approximating a harmonic mapping in a data-driven way. The HAE learns a mapping from an input domain to a target domain that minimizes distortion and requires only a small number of input–target reference pairs. The HAE can be applied to high-dimensional applications, such as human-to-robot hand pose mapping. Our method can map from the input to the target domain while minimizing distortion over the input samples, covering the target domain, and satisfying the reference pairs. This is achieved by extending an existing neural network method called the contractive autoencoder. Starting from a contractive autoencoder, the HAE takes into account a distance function between point clouds within the input and target domains, in addition to a penalty for estimation error on reference points. For efficiently selecting a set of input–target reference pairs during the training process, we introduce an adaptive optimization criterion. We demonstrate that pairs selected in this way yield a higher-performance mapping than pairs selected randomly, and the mapping is comparable to that from pairs selected heuristically by the experimenter. Our experimental results with synthetic data and human-to-robot hand pose data demonstrate that our method can learn an effective mapping between the input and target domains.

The Harmonic Mapping Whose Hopf Differential Is a Constant

Suppose that h(z) is a harmonic mapping from the unit disk D to itself with respect to the hyperbolic metric. If the Hopf differential of h(z) is a constant c>0, the Beltrami coefficient μ(z) of h(z) is radially symmetric and takes the maximum at z=0. Furthermore, the mapping γ:c→μ(0) is increasing and gives a homeomorphism from (0,+∞) to (0,1).

Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.