Conformal Mapping Research Articles

Bowen’s Formula for a Dynamical Solenoid

More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections (fn:X→X)n∈N defined on a compact metric space (X,d). Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection f:X→X of a compact

Explicit construction of Penrose diagrams for black hole to white hole transition with spacelike thin shells.

In this article, we explicitly construct the coordinates associated with the Penrose diagram in spacetimes connected via a spacelike thin shell in the following two examples: the generalized black-to-white hole bounce with mass difference and the Schwarzschild-to-de Sitter transition. We point out the issue of the first junction condition in the Penrose diagram constructed by cutting and pasting analytically known metrics with spherical symmetry by a static spacelike thin shell. With the goal of a global conformal coordinate chart associated with the corresponding Penrose diagram without discontinuity at the thin shell, we give a procedure consisting of three conformal transformations that serve different purposes. The first two of them are used to generate a continuous coordinate patch covering the entire thin shell, and therefore, the Penrose diagram can be constructed properly by patches with overlapping. The third transformation removes any coordinate singularity reintroduced by the first two transformations at the event horizons.

Notes on conformal and non-conformal transformation metrics of generalized quasi-Einstein manifolds

Notes on conformal and non-conformal transformation metrics of generalized quasi-Einstein manifolds

Probing the Dark Universe with Gravitational Waves

Gravitational waves (GWs) are expected to interact with dark energy and dark matter, affecting their propagation on cosmological scales. To model this interaction, we derive a gauge-invariant effective equation and action valid for all GW polarizations. This is achieved by encoding the effects of GW interactions at different orders of perturbation into a polarization-, frequency-, and time-dependent effective speed. The invariance of perturbations under time-dependent conformal transformations and the gauge invariance of GWs allow us to derive the unitary gauge effective action in any conformally related frame, thereby clarifying the relationship between the Einstein and Jordan frames. Tests of the polarization and frequency dependencies in the propagation time and luminosity distance of different GW polarizations allow us to probe the dark Universe, which acts as an effective medium, modeled by the GW effective speed.

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

Myosin supramolecular self-assembly: The crucial precursor that manipulates the covalent aggregation, emulsification and rheological properties of myosin

The transformation of molecular conformation and self-assembly properties of myosin during the heating process at different ionic strengths (0.2 M, 0.4 M and 0.6 M NaCl) and its effect on rheological behavior and emulsification properties were investigated. Under incubation temperatures between 40 °C and 50 °C, myosin underwent a supramolecular self-assembly stage dominated by noncovalent forces (hydrogen bonding, ionic bonding and hydrophobic interactions). Higher ionic strength facilitated molecular rearrangement through enhanced swelling of myosin heads and head-to-head assemblies, which contributed to enhanced ordering and homogeneity of myosin covalent aggregates (above 60 °C) and manifested itself macroscopically as enhanced gel viscoelasticity and emulsion stability. In contrast, at lower ionic strength, the tail-to-tail assemblies of myosin led to the preferential formation of covalent cross-links in the tails, which resulted in the inability of molecular rearrangement and the formation of disordered aggregates and finally led to the deterioration of the gel and the destabilization of the emulsion. In conclusion, the supramolecular self-assembly behavior of myosin, as an intermediate process in myosin’s sol–gel transition, is crucial for the orderliness of myosin assemblies, gel network strengthening, and emulsion stability. The obtained insight provides a reference for the precise implementation of quality improvement strategies for meat products.

An Improved Conformal Mapping Method for Electromagnetic Vibration Analysis in PMSMs With Rotor Eccentricity

An Improved Conformal Mapping Method for Electromagnetic Vibration Analysis in PMSMs With Rotor Eccentricity

Analytical solutions for steady groundwater flow through aquifer folds and faults

Deformations of the earth’s crust create tortuous paths for groundwater flow, altering pressure distributions and flow lines. A solution for steady groundwater flow through a deformed aquifer is derived by applying the singular point method using a rectangular reference plane. The singular point method is used to develop conformal mappings with complex geometries using basic principles of groundwater mechanics including superposition, the method of images, and stagnation point analysis. The derived solution contains three parameters that can be chosen to simulate flow through a variety of deformed aquifers, including flow through a normal fault with a 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {90}^\\circ $$\\end{document} dip, flow through a fold, and flow through a relay ramp.

Global loss of promoter-enhancer connectivity and rebalancing of gene expression during early colorectal cancer carcinogenesis.

Although three-dimensional (3D) genome architecture is crucial for gene regulation, its role in disease remains elusive. We traced the evolution and malignant transformation of colorectal cancer (CRC) by generating high-resolution chromatin conformation maps of 33 colon samples spanning different stages of early neoplastic growth in persons with familial adenomatous polyposis (FAP). Our analysis revealed a substantial progressive loss of genome-wide cis-regulatory connectivity at early malignancy stages, correlating with nonlinear gene regulation effects. Genes with high promoter-enhancer (P-E) connectivity in unaffected mucosa were not linked to elevated baseline expression but tended to be upregulated in advanced stages. Inhibiting highly connected promoters preferentially represses gene expression in CRC cells compared to normal colonic epithelial cells. Our results suggest a two-phase model whereby neoplastic transformation reduces P-E connectivity from a redundant state to a rate-limiting one for transcriptional levels, highlighting the intricate interplay between 3D genome architecture and gene regulation during early CRC progression.

Conformal Yang-Mills field in (A)dS space

Ordinary-derivative (second-derivative) Lagrangian formulation of classical conformal Yang-Mills field in the (A)dS space of six, eight, and ten dimensions is developed. For such conformal field, we develop two gauge invariant Lagrangian formulations which we refer to as generic formulation and decoupled formulation. In both formulations, the usual Yang-Mills field is accompanied by additional vector and scalar fields where the scalar fields are realized as Stueckelberg fields. In the generic formulation, the usual Yang-Mills field is realized as a primary field, while the additional vector fields are realized as auxiliary fields. In the decoupled formulation, the usual Yang-Mills field is realized as massless field, while the additional vector fields together with the Stueckelberg are realized as massive fields. Some massless/massive fields appear with the wrong sign of kinetic terms, hence demonstrating explicitly that the considered models are not unitary. The use of embedding space method allows us to treat the isometry symmetries of (A)dS space manifestly and obtain conformal transformations of fields in a relatively straightforward way. By accompanying each vector field by the respective gauge parameter, we introduce an extended gauge algebra. Levy-Maltsev decomposition of such algebra is noted. Use of the extended gauge algebra setup allows us to present concise form for the Lagrangian and gauge transformations of the conformal Yang-Mills field. Higher-derivative representation of the Lagrangian is also obtained.

Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension

AbstractFor large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property under conformal transformations. In terms of the co‐polyharmonic Gaussian field , we define the Liouville Quantum Gravity measure, a random measure on , heuristically given as and rigorously obtained as almost sure weak limit of the right‐hand side with replaced by suitable regular approximations . In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's ‐curvature: we give a rigorous meaning to the Polyakov–Liouville measure and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2‐dimensional Riemannian manifolds, all compact non‐negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even‐dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.

Theoretical development and validation of asphalt concrete density measurement using non-contact interdigital coplanar capacitive sensor

Asphalt concrete (AC) density serves as a critical criterion for quality control and quality assurance in asphalt pavement construction. Current non-destructive testing (NDT) approaches have shown potential for AC density measurement, but they present disadvantages such as inefficiency and unstable accuracy. This paper developed a novel NDT approach for AC density measurement based on non-contact interdigital coplanar capacitive sensor (ICCS). The feasibility of this approach was evaluated through both numerical and experimental validations. Firstly, the theoretical model of non-contact ICCS for AC density measurement was established based on conformal mapping technique, partial capacitance technique, and dielectric mixing model. Secondly, numerical simulations were performed to validate the developed theoretical model under two scenarios. Thirdly, laboratory experiments were conducted to measure capacitance data and predict density for three common types of ACs. Finally, the error of non-contact ICCS was discussed and compared with that of other NDT instruments. The results indicated that the developed theoretical model could establish the relationship between AC density and capacitance value, which provided the basis for AC density measurement using non-contact ICCS. It was also observed that the capacitance data obtained by the proposed model closely matched those computed by numerical simulations. The maximum errors between theoretical and numerical data were 4.10% and 1.08% for two cases respectively, which proved the feasibility of the theoretical model. Furthermore, experimental results indicated that the capacitance value of AC specimens increased by about 1 pF as the rolling time increases due to a decrease in the volume of air in AC, which reaffirms the validity of AC density measurement using non-contact ICCS. Additionally, the non-contact ICCS demonstrated excellent accuracy with a maximum error of 6.69%, which is comparable to the current mainstream NDT approaches. And it has the advantages of lower cost, simpler data processing and higher efficiency. The findings of this paper offer a new and reliable method and tool for non-destructive and large-scale AC density measurement.

Salinity-responsive hyperaccumulation of flavonoids in Spirodela polyrrhiza, resultant maneuvering in the structure and antimicrobial as well as azo dye decontamination profile of biofabricated zinc oxide nanoentities.

Duckweeds (Spirodela polyrrhiza) are free-floating macrophytes that grow profusely in nutrient-rich waters. Under ideal conditions, they exhibit a rapid growth rate and can absorb a substantial amount of nutrients, macromolecules, and pollutants from bodies of water. Zinc oxide nanoparticles (ZnO NPs) synthesized from plant extracts, particularly under stress conditions, have opened new research avenues in the field of nanotechnology. Under salinity stress, the accumulation of flavonoids in duckweeds can affect the structure of ZnO NPs, helping researchers ascertain their antimicrobial role. In our study, we exposed mid-log phase duckweed monocultures to 75 mM NaCl in a full-strength Murashige and Skoog medium for 7 days, followed by a 15-day recovery period. We observed significant overexpression of superoxide and hydrogen peroxide as reactive oxygen species. As a result, chlorophyll and certain metabolites were produced in lesser amounts, while flavonoid and phenol content increased by 12% and 8%, respectively. This overproduction persisted up to 10 days into the recovery treatment period but dropped by 8% and 5%, respectively, by the 15th day. The flavonoid coating transformed the NPs into rosette clusters, which exhibited reduced antimicrobial activity against Aeromonas hydrophila, a Gram-negative, fish-pathogenic bacterium. Herein, we discuss potential mechanisms for the conformational transformation of ZnO NPs into finer dimensions in response to NaCl-induced oxidative stress in duckweed. In this study, the azo dye degradation capacity of salinity-treated plants increased as the flavonoid profile became enriched. Zinc oxide nanoparticles, both prior to and after salinity treatment, were found to be efficient in scavenging azo dye and mitigating its toxicity, as evidenced by improved germination, growth, and overall plant morphometry.

Sustainable di-functional stimuli-responsive imprinted nanozymes for reversible colorimetric sensing of tetracycline

Regenerable nanozymes with high catalytic specificity and sustainability may serve as substitutes for naturally occurring enzymes, but their application is hindered by inadequate and nonselective catalytic activity. In this study, we constructed a photo-responsive molecularly imprinted layer on the Fe3O4 nanozyme surface (P-MIPs) using an ATRP method and obtained high-specificity reversible colorimetric sensing of tetracycline (TC). Due to the photo-responsive properties of P-MIPs, the imprinted nanozyme undergoes reversible release and uptake of TC under alternating irradiation at 365/440 nm based on the photo-regulated conformational transformation of the 4-[(4-methacryloyloxy)phenylazo] benzenesulfonic acid (MAPASA) at the polymer receptor sites, which may be confirmed by calculating the electron densities of the trans-/cis-MAPASA forms. Simultaneously, these processes can be visualized through the change in the color of the substrates. Highly specific P-MIPs-based nanozyme also exhibit adequate stability and sustainable catalytic activity after multiple cycles. This method can be used to achieve nanozymes with high selectivity for the enrichment, separation, and detection of hazardous substances and contribute to material sustainability.

Ramachandran-like Conformational Space for DNA.

DNA's ability to exist in a wide variety of structural forms, subforms, and secondary motifs is fundamental to numerous biological processes and has driven the development of biotechnological applications. Major determinants of DNA flexibility are the multiple torsional degrees of freedom around the phosphodiester backbone. This high complexity can be rationalized by using two pseudotorsional angles linking atoms P and C4', from which Ramachandran-like plots can be built. In this contribution, we explore the distribution of η (eta: C4'i-1-Pi-C4'i-Pi+1) and θ (theta: Pi-C4'i-Pi+1-C4'i+1) angles in known experimental structures retrieved from the Protein Data Bank (PDB), subdividing the conformational space into different datasets. After the removal of the canonical/helical conformations typical of the B-form, we find the existence of a conformational map with clearly permitted and forbidden regions. Some of these regions are populated with specific DNA forms, like Z- or A-DNA, or by specific secondary motifs, like G-quadruplexes and junctions. We evaluated the sequence dependency and energy relationship among the high-density regions identified in the η-θ space. Furthermore, we analyzed the effect produced by proteins and cations when bound to DNA, finding that specific proteins produce some nonhelical conformations, while other regions appear to be stabilized by divalent cations.

Compressive behaviors and deformation mechanism of carbon fiber reinforced epoxy composites: A microscopic and molecular dynamics perspective

AbstractInvestigating the macro‐mechanical properties of polymer composites at the nanoscale is a challenging topic. Here we report the compressive performances of carbon fiber‐reinforced epoxy composites using molecular dynamics (MD) simulations. The evolution of microstructure and microscopic energy, including free volume, radius of gyration, dihedral angle, potential energy, and interaction energy, has been investigated to characterize compressive behaviors at the nanoscale. Molecular chains gradually bend, and the movement space of the molecular chains decreases during compression loading. The radius of gyration and dihedral angle are deformed into a state of irreversible plasticity to accommodate the microstructural transformation. The interaction energy increases and subsequently declines as the carbon fiber/epoxy interface gets closer, reflecting the transition of the internal structure. The inhomogeneity and interfacial concentration distribution of stress, and local molecular stiffness under various strains have been obtained to reveal the nanoscopic mechanism of epoxy failure and interface delamination during compression. This study reveals the compression behaviors and deformation mechanism of carbon fiber‐reinforced epoxy composites at the molecular level, providing directions and possibilities for molecular design and structural optimization of composites.Highlights The nanoscopic compression deformation of CFRP composites is studied. Microstructural evolution and conformation transformation are revealed. The evolution and transition of microscopic energy are analyzed. Micro‐mechanism of epoxy deformation and interface delamination is elucidated.

Acid-triggered rattan ball-like β-glucan carrier embedding doxorubicin to synergistically alleviate precancerous lesions of gastric cancer via p53 and PI3K pathways

The early intervention of precancerous lesions of gastric cancer (PLGC) is crucial for improving the survival of patients with gastric cancer. Traditional pharmaceuticals for the treatment of PLGC are limited by side effects, thus developing innovative drug carrier that are more efficient but without the undesirable side effects is required. Here, we proposed an acid-triggered mushroom-derived β-glucan carrier embedding doxorubicin (DOX) to circumvent drug cytotoxicity and synergistically alleviate PLGC based on the controlled conformational transformation. The triple helix β-glucan extracted from Dictyophora rubrovolvata (DRP) loaded doxorubicin driven by pH and DMSO regulation, forming two rattan ball-like nanoparticles (DRP-DOX(pH) and DRP-DOX(DMSO)) via its collapse and recombination of triple-helix conformation. The findings revealed that DRP-DOXs achieved acid-triggerable and sustained drug delivery with an average particle size of 500 nm and 550 nm. In vitro evaluation of GES-1 cells showed DRP-DOXs reduced reactive oxygen species (ROS) production and altered mitochondrial membrane potential. Compared to DRP-DOX(DMSO) and DRP, DRP-DOX(pH) could more effectively downregulate cellular oxidative stress and inflammation to eventually alleviate PLGC, by regulating the p53 and PI3K pathways to mitigate gastric mucosa damage. Consequently, the nature-derived β-glucan delivery nanovesicle holds great promising applications in reducing drug toxicity and suppressing the development of PLGC.

Failure Prediction of an Airfoil-Type Crack in Thermoelectric Coupling Materials

Airfoil-type cracks with curved surfaces and sharp tips are frequently encountered in modern engineering applications and the aerospace industry. However, such cracks significantly threaten structures due to stress concentration and further cause damage. This study investigates the theoretical analysis and failure prediction of airfoil-type cracks within thermoelectric coupling materials under the interaction of remote mechanical loads, heat flux and electric current density. By employing complex variable theory, conformal mapping methods, and the analytic continuation theorem, the full-field stress and stress intensity factors of an airfoil-type crack were determined under three different loading conditions. The results indicate that when the airfoil-shaped crack tip is subjected to horizontal compression, vertical tensile mechanical load, and both heat flux and current density parallel to the crack, the crack tip would reach a dangerous state. Additionally, the full-field stress distribution reveals stress concentration around the crack tip, which explains variations in SIFs under different external forces and shape factors. Four critical situations suppressing crack propagation under mixed loading conditions are proposed. Furthermore, using an acrylic plate and bismuth telluride alloy as examples, the critical values of external loading for the mixed-mode fracture are determined based on the strain energy density criterion. These critical values can be used to predict failure initiation, thus reducing the risk of structural failure due to airfoil cracks within thermoelectric coupling materials.

Self-assembly of amphiphilic homopolymers grafted onto spherical nanoparticles: complete embedded minimal surfaces and a machine learning algorithm for their recognition.

By means of computer modelling, the self-assembly of amphiphilic A-graft-B macromolecules, grafted onto a spherical nanoparticle, is studied. In a solvent, that is poor for side pendants, the macromolecules self-assemble into thin membrane-like ABBA bilayers deviated from spherical nanoparticles. The bilayers form morphological structures that depend on the grafting density and macromolecular polymerization degree and can be referred to as the classical family of complete embedded minimal surfaces. The plane disk, catenoid, helicoid, Costa and Enneper surfaces as well as "double" helicoid and "complex minimal surface" were identified, and the fields of their stability were defined. The surfaces can be grouped according to the sequences of conformal transformations that transform them into each other. These surfaces arise in different experiments situationally. Results are summarized in a pie diagram constructed using a machine learning algorithm based on matching grafting points with a specially created planar graphic image.

Mechanism model for effect of eccentricity on capacitive control rod position indicators

The continuous identification technology for position signals based on the capacitive method has a good application potential in the field of the control rods position measurement in nuclear reactors. The main source for error of the Capacitive Control Rod Position Indicators (CCRPIs) comes from the effect of eccentricity. The conformal mapping technique was used in this paper, and the geometry of the CCRPIs could be transformed. A model for revealing the mechanism of the effect of eccentricity was established, and it was tested and verified by the method of numerical calculation. The characteristics of the distribution of the sensitive field was clarified by the model. The results indicate that the maximum error of the model is smaller than 13 %, and the eccentric effect of the indicator can be described by the model.