conformal mapping Research Articles (Page 1)

Effects of conformational transformation on foam stability of ovalbumin-naringenin complexes: Molecular interactions and differential mechanisms of air-water interface assembly

Fast electrostatic evaluation via conformal mapping and 3D image approximation

The primary objective of the article is to investigate the symmetry and pseudosymmetry properties of the Reissner-Nordström-de Sitter (briefly, RNdS) spacetime. The secondary aim of the paper is to explore the notion of Ricci solitons in RNdS spacetimes. The study is important due to the conceding of almost Ricci soliton and almost Ricci Yamabe soliton of the RNdS spacetime. The analysis shows that this spacetime satisfies multiple types of symmetric and pseudosymmetric conditions. It is interesting to note that RNdS spacetime revealed pseudosymmetry, Weyl conformal pseudosymmetry, Weyl projective pseudosymmetry, conharmonic pseudosymmetry, and concircular pseudosymmetry. Also the RNdS spacetime is pseudosymmetric due to conformal, projective, conharmonic curvature tensors. Furthermore, in the RNdS spacetime, the second order commutator tensor R · R is linearly dependent on Q(Ric, R) and Q(g, C), and the tensor R · C does not commute with the tensor C · R. The investigation also shows that RNdS spacetime is a Roter type. It is demonstrated that the RNdS spacetime is 2-quasi Einstein and an Ein(2) space with recurrent conformal 2-forms. Riemann compatibility preserved under geodesic mapping and it has a great geometric insight in determining the nature of Pontryagian forms. We derive the general form of the compatible tensors of the RNdS spacetime. The energy-momentum tensor of the RNdS spacetime is shown to be pseudosymmetric, and also the energy momentum tensor is pseudosymmetric due to conformal, conharmonic, concircular and projective curvature tensor. We note that geodesic maps do not necessarily preserve Weyl compatibility. However such kind of compatibility remains invariant under conformal maps. The presence of a Weyl-compatible vector ensures that the Weyl tensor assumes an algebraically special form, and this condition is both necessary and sufficient for the vanishing of its magnetic component. The energy momentum tensor is compatible with projective, conharmonic, concircular, Riemann and conformal curvature. It is shown that the energy momentum tensor of the RNdS spacetime is Weyl compatible and hence the Weyl tensor of a such a spacetime is algebraically special and hence its magnetic component vanishes. Consequently the Weyl tensor in RNdS spacetime is purely electric. We study a generalized notion of curvature inheritance and find that, with respect to the non-Killing vector fields ∂/∂r and ∂/∂θ, the RNdS spacetime does not satisfy such inheritance condition. However, the RNdS spacetime is shown to admit an almost Ricci soliton and an almost η-Ricci Yamabe soliton with respect to the non-Killing vector field ∂/∂r, but with respect to the non-Killing vector field ∂/∂θ the spacetime does not admit such notions. Finally, we present a comparison between the RNdS and Vaidya-Bonner-de Sitter (VBdS) spacetimes.

Abstract A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights is established. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. In particular, a Littlewood-Paley estimate recently proved by Peláez and the second author (Peláez and Rättyä Adv. Math. , 391 , 70, 2021) is improved in part. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map f , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under f of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

This paper presents a solution to the plane doubly periodic problem of loading an infinite, isotropic, elastic plane with a hexagonal packing of elliptical inclusions. The plane is subjected to one of three types of remote loading: uniaxial tension along one of the inclusion’s axes or pure shear. The regular hexagonal unit cell contains a single elliptical inclusion with its axes perpendicular to the cell boundaries. The size of the inclusion is significantly larger than the plate thickness. The solution is constructed by reducing the problem to finding complex potentials from the boundary conditions, which are derived from the equality of normal forces and displacements in the matrix and the inclusion. This is achieved using conformal mapping techniques and the method of Muskhelishvili. The influence of non-central inclusions is accounted for via a small parameter method. As a result, a system of linear algebraic equations is derived for solving the considered doubly periodic problem, and solutions for several specific cases are obtained. The results were verified against a numerical solution obtained by the finite element method in the Abaqus software package. The solution to this problem serves as a model for the loading of a fibrous composite, which makes it highly relevant. A relatively small number of works in mechanics are devoted to fibrous composites, with most publications focusing on the analysis of experimental studies or numerical solutions. Therefore, this analytical solution possesses significant scientific value.

We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$ . We distinguish between the cases $p \gt 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$ . We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$ .

Beyond elemental composition and molecular structure, ordered multiscale architectures (self-assembled structure, crystal structure, phase structure, etc.) play a comparably important role in determining material functions, especially the optoelectronic properties. Light, a unique external stimulus with superior spatiotemporal resolution and non-invasive nature, provides unparalleled capabilities for dynamically constructing and manipulating such ordered structures. Since previous reviews on light-responsive materials have generally overlooked how light can govern multiscale structural ordering, this review comprehensively surveys the rapidly advancing field of photo-responsive chemical systems designed to achieve and control multiscale structural ordering. We begin by elucidating the fundamental photochemical and photophysical mechanisms at the molecular level, which serve as the primary triggers for molecular structural, configurational, or conformational transformations. Based on these events that initiate molecular perturbation, the core of the review critically examines several key light-mediated strategies for constructing and manipulating ordered hierarchical architectures, notably photo-triggered polymerization-induced self-assembly, liquid crystal-assisted phase structural amplification, photo-induced self-assembly mediated by host-guest interaction, and the emerging paradigm of photoexcitation-induced assembly (PEIA). We discuss how these strategies successfully translate molecular or nanoscale events into well-defined, macroscopic ordered states in real time and in situ. Finally, we provide a forward-looking perspective on the existing challenges and promising future research directions.

We investigate the scattering behavior of global energy solutions to the inhomogeneous defocusing generalized Hartree equation $ i\partial_t \varphi+\Delta \varphi = |x|^{-{\varrho}}|\varphi|^{\beta-2}\bigl(J_\gamma *|\cdot|^{-{\varrho}}|\varphi|^{\beta}\bigr)\varphi, \qquad {\varrho}>0,\ \beta>2. $ This work complements our earlier study (Acta Appl. Math., 177, no. 6 (2022)) on the radial case by removing the spherical symmetry assumption under the additional condition $ x\varphi_0\in L^2 $. The novelty is to establish the scattering in weighted Sobolev spaces within the mass-subcritical range$ \beta_0<\beta\leq \frac{\gamma+2-2{\varrho}}{d}+1, $where $ \beta_0 $ denotes the Strauss exponent. The analysis relies on decay properties of global solutions, nonlinear estimates in Strichartz norms, and conformal transformation techniques. This result provides a refined description of the asymptotic behavior of energy solutions, showing that the nonlinear interaction becomes negligible at large times and that the dynamics approach those of the associated linear Schrödinger equation. The complementary case $ {\varrho} < 0 $ will be addressed in future work.

The bending problem of an infinitely large thin anisotropic plate with an elliptical or linear elastic inclusion inserted into a hole without initial tension and under perfect mechanical contact with the plate matrix is solved. The plate at infinity is subjected to constant bending moments. The solution is obtained by employing the formalism of generalized complex potentials, expansions of functions into Laurent series and Faber polynomials, as well as conformal mapping techniques to transform the exterior of the unit circle into the exterior of an ellipse. An exact analytical solution for the case of an elliptical inclusion, providing expressions for bending moments and transverse forces both in the plate matrix and in the inclusion, is presented. For the case when the elliptical inclusion reduces to a line, formulas for calculating the moment intensity factors (MIF) at its ends are derived. This approach accurately captures the singular behavior of bending moments and identifies conditions under which MIF values are significant. Numerical studies were conducted for plates made of isotropic material (CAST–V) and anisotropic material (skew-wound glass-fiber-reinforced plastic) under various values of the inclusion’s relative stiffness and axis ratio. It was found that decreasing the inclusion’s stiffness leads to an increase in bending moments in certain contact zones, with higher moment concentrations in anisotropic plates compared to isotropic ones. For linear inclusions, significant MIF values arise only for substantially stiff or soft inclusions; when the stiffnesses of the plate and inclusion differ by less than a few times, MIF values are negligible, and it is inappropriate to discuss bending moment singularities. Isotropic plates are treated as a special case of anisotropic ones, enabling the extension of these results to a broad class of engineering problems involving composites and structures with embedded elements.

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Space-division multiplexing (SDM) has emerged as a promising technology for both classical and quantum communications. Twisted light, particularly in the form of orthogonal optical vortex beams, offers significant potential for SDM by enhancing the capacity and performance of optical communication systems. A critical component in such systems is a twisted light (de)multiplexer, which enables scalable and efficient (de)multiplexing of twisted light to meet the demands of high-capacity optical communications. In this work, we propose and experimentally demonstrate quasi-wavelet conformal mapping microstructures for efficient twisted light demultiplexing. These microstructures, specifically designed and fabricated for this experiment, successfully demultiplex 11 optical vortex beams with a low crosstalk below −20.9 dB at the target wavelength. Moreover, the microstructures exhibit a broad operational bandwidth (1525–1570 nm), maintaining crosstalk levels under −13 dB. To further validate their performance, the microstructures are also tested in the demultiplexing of various configurations of multiplexed two-channel optical vortex beams. The presented quasi-wavelet conformal mapping microstructures with compact size and exceptional demultiplexing performance hold great promise for applications in high-capacity optical communication systems using twisted light.

To reveal the biochemical mechanism of pyruvic acid under the bioelectric currents, the effect of the external electric field on the intermolecular double-proton cooperativity migration was investigated for the dimer, accompanied by a comparison of the conformation transformation. The results indicate that the influences of the external electric fields on the transition state of the proton migration, i.e., the changes in the molecular structures, atoms in molecules, surface electrostatic potentials, and potential barriers, are closely related to the directions of the fields. The spatial order parameters of the conformation transformation of pyruvic acid can be quickly converged through the umbrella sampling and parameter averaging, thereby obtaining the potential energy surface of the potential of mean force. Under the strong external electric field, the barrier heights of the double-proton cooperativity migration are close to the free energy change of the conformation transformation, indicating the competition between the proton cooperativity transfer and conformation transformation. This study is of importance for revealing furthermore the essence of the competition between the proton migration and conformation transformation of the biomolecules stimulated by the bioelectric currents. The effects of external electric fields on the double-proton cooperativity migration reaction pathway, potential barrier, and rate constant of the pyruvic acid dimer were studied using the M06-2X/6-311++G**, M06-2X/aug-cc-pVTZ, and CCSD(T)/6-311++G(2d,p) methods by using Gaussian 09 packages. A chaotic finite temperature string model with enhanced sampling for molecular conformation transformation was constructed, and the free energy change of conformation transformation under external electric field was studied by the finite temperature string (FTS) method with the CHARMM22 force field from NAMD software packages and PLUMED program.

We investigate a standard minimally-coupled scalar-field inflationary scenario, which is based on a new potential, with suitably generalized plateau features, that leads to extra small tensor-to-scalar ratios. In particular, we consider a specific three-parameter potential, which has a flatter plateau and a steeper well compared to the Starobinsky potential in the Einstein frame. We study the inflationary realization and we show that it guarantees a prolonged period of slow-roll inflation and a successful exit. Additionally, the steeper minimum leads to significantly suppressed tensor perturbations, and thus to an extra-small tensor-to-scalar ratio r, and we show that we are able to obtain r values less than 10-5. Moreover, we calculate the reheating temperature showing that in order to be in agreement with observations one of the potential parameters should remain within specific bounds. Finally, performing an inverse conformal transformation to the Jordan frame we show that the considered potential corresponds to higher-order corrections to Starobinsky potential in the Einstein frame, and these corrections are the reason for the improved behavior of the tensor-to-scalar ratio.

Aiming at the problem that obstacle avoidance flexibility and formation integrity are difficult to coexist in multi-robot formation motion, a path-deformation mapping mechanism is proposed, which deeply integrates artificial potential field and affine transformation, and drives formation adaptive adjustment in real time through path information. By using the non-uniform scaling characteristics of the affine transformation, the limitation of traditional conformal transformation is broken through, and the unity of flexibility and integrity is realized. The effectiveness of the algorithm is verified by experiments, which provide a practical solution for cooperative obstacle avoidance of multi-robot systems in complex environments. In order to verify the performance of the algorithm, a numerical simulation is carried out, and an experimental platform composed of seven omnidirectional mobile robots is built for physical verification. The simulation and experimental results show that the formation can complete the obstacle avoidance task in the complex static obstacle environment, and the average formation tracking error is maintained below 0.05 m. Compared with the traditional local obstacle avoidance or formation switching method, this algorithm significantly improves the fluency of the obstacle avoidance process and the integrity of the formation while ensuring a success rate of 100% obstacle avoidance.

ABSTRACT Excavating tunnels in anisotropic rock induces greater stress concentrations at the excavation boundary compared to isotropic rock. However, existing analytical solutions for the shallow tunnel are founded on the simplifying assumption of isotropic rock masses. To clarify the deformation mechanism and mechanical behavior of the shallow tunnel, we proposed an analytical method for solving stress and displacement of the arbitrary‐shaped shallow tunnel excavated in orthotropic rock mass, incorporating the effects of body forces and anisotropy of rock mass. Proposing the specific forms of the analytical functions for the shallow tunnel in anisotropic rock, which can reflect body forces and the anisotropy of the rock mass. A linear equation system, derived from stress boundary conditions at the surface and tunnel excavation, is solved by the boundary collocation method. In the solution process, conformal transformation techniques are employed to construct three polar coordinate systems, which aid in resolving boundary conditions. Subsequently, we analyzed the effects of tunnel depth and anisotropic parameters on the stress and displacement of a horseshoe‐shaped tunnel and verified the correctness of the results through ANSYS software.

Compressed Air Energy Storage (CAES) in abandoned coal mines offers a cost-effective and sustainable solution for large-scale energy storage. This study presents a comprehensive analytical framework to evaluate the thermo-mechanical behavior and long-term structural stability of three-layer underground chambers with circular and horseshoe-shaped cross-sections. Based on the theory of complex elasticity and conformal mapping, analytical solutions for stress and displacement are derived under steady-state thermal conditions. The framework incorporates both mechanical loads-such as in-situ stress and internal gas pressure-and temperature-induced thermal stresses, which are often overlooked in traditional models. Using the Cao Zhuang Coal Mine in Shandong Province as a case study, the analytical results are validated against finite element simulations performed in COMSOL 6.2. The validation demonstrates strong agreement in both temperature and stress distributions across different burial depths. Comparative results reveal that while circular chambers maintain relatively uniform stress profiles, horseshoe-shaped chambers are prone to localized stress concentrations, especially near the lower arch and corner regions. This makes them more vulnerable to structural failure under thermal cycling conditions. The findings underscore the importance of incorporating thermal effects in underground energy storage system design. The developed methodology offers a computationally efficient alternative to fully numerical simulations, enabling rapid scenario assessment and structural optimization. This work provides theoretical and practical insights into the safe reuse of abandoned mine tunnels for energy storage, contributing to the long-term feasibility of compressed air energy storage systems and the advancement of clean energy infrastructure.

Abstract Based on the principles of continuum mechanics and the Gurtin‐Murdoch (GM) surface/interface theoretical model, the mechanical behavior characteristics of a nanoscale lip‐shaped orifice with two asymmetric nano‐cracks in one‐dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) is studied, taking into account surface effects. By employing conformal mapping techniques, the electro‐elastic field distribution is obtained. Under partially electrically permeable boundary conditions, the precise analytical solutions are derived for the intensity factors of electro‐elastic fields and the energy release rate (ERR) at the crack tips. The obtained solutions can be reduced to existing results to verify the effectiveness of the method. Meanwhile, the effects of surface effects on stress intensity factors (SIFs), electric displacement factor (EDIF), and ERR are analyzed in detail through numerical examples, and the physical mechanism is also explained. The results indicate that the electro‐elastic field intensity factors exhibit significant size dependence. The surface effect usually acts as a “shield” and attenuates the intensity of the crack tip field. Appropriately increasing the electrical conductivity can effectively inhibit crack propagation. The findings of this study provide invaluable theoretical guidance for the design of smart devices in micro‐electromechanical systems and nano‐electromechanical systems.

Abstract This study investigates the conditions under which various curves maintain their characteristics when mapped between two regular surfaces in four-dimensional Euclidean space E 4 {{\mathbb{E}}}^{4} . We particularly focus on rectifying, osculating, and normal curves, examining their behavior under isometric, conformal, and homothetic conformal maps. By analyzing their behavior, this study reveals conditions under which these curves remain invariant when mapped between two surfaces. The findings contribute to a deeper understanding of geometric transformations in four-dimensional spaces.

The lemniscatic functions and their inverses play a vital role in various fields of applied science. In classical mechanics, they appear in the exact solutions of the nonlinear pendulum equation and describe the trajectories of particles in central force fields. In electromagnetism, they are involved in the modeling of electric and magnetic field distributions with elliptic symmetry. Moreover, in engineering mathematics and optical system design, lemniscatic functions arise in problems involving wave propagation, signal transmission, and conformal mappings. Their mathematical properties make them valuable tools in both theoretical analysis and computational modeling. Lemniscatic functions, such as Gauss’ arc lemniscate sine and the hyperbolic arc lemniscate sine, arise from the incompletely symmetric elliptic integral of the first kind. In 2007, Neuman introduced additional lemniscatic functions including the arc lemniscate tangent and its hyperbolic counterpart. These functions have attracted considerable attention due to their deep connections with elliptic integrals and their applications in inequality theory. In this paper, we build upon recent work by Wei, He, and Wang (2020), who established Shafer–Fink type inequalities for these lemniscatic functions. Employing methods from real analysis, we present new and sharper refinements of these inequalities. Specifically, we establish double inequalities for Gauss’ arc lemniscate sine, the hyperbolic arc lemniscate sine, arc lemniscate tangent and its hyperbolic counterpart bounded by rational expressions involving radical terms. These results not only improve the existing Shafer–Fink type inequalities but also provide further insight into the monotonicity and convexity properties of the involved lemniscatic functions. Our approach offers a unified framework for deriving tight bounds for all four Neuman lemniscatic functions, contributing to the broader understanding of special function inequalities. These results represent significant refinements over previous known estimates and open avenues for further generalizations.

It is known that the limit of a sequence of (quasi)conformal mappings is either a constant or a (quasi)conformal mapping. In this paper, we prove that in the case of Heisenberg-type Carnot groups, a similar property is valid for mappings that are quasiconformal in the mean, i.e., for homeomorphisms with finite distortion and a distortion function integrable to an appropriate degree. This result is applied to solving model problems of nonlinear elasticity theory on Carnot groups.