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We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

Background Quantitative areas is of great measurement of wound significance in clinical trials, wound pathological analysis, and daily patient care. 2D methods cannot solve the problems caused by human body curvatures and different camera shooting angles. Our objective is to simply collect wound areas, accurately measure wound areas and overcome the shortcomings of 2D methods. Results We propose a method with 3D transformation to measure wound area on a human body surface, which combines structure from motion (SFM), least squares conformal mapping (LSCM), and image segmentation. The method captures 2D images of wound, which is surrounded by adhesive tape scale next to it, by smartphone and implements 3D reconstruction from the images based on SFM. Then it uses LSCM to unwrap the UV map of the 3D model. In the end, it utilizes image segmentation by interactive method for wound extraction and measurement. Our system yields state-of-the-art results on a dataset of 118 wounds on 54 patients, and performs with an accuracy of 0.97. The Pearson correlation, standardized regression coefficient and adjusted R square of our method are 0.999, 0.895 and 0.998 respectively. Conclusions A smartphone is used to capture wound images, which lowers costs, lessens dependence on hardware, and avoids the risk of infection. The quantitative calculation of the 3D wound area is realized, solving the challenges that 2D methods cannot and achieving a good accuracy.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Conformal mapping functions have significant applications in mechanics and other fields, and their computation methods have drawn considerable attention. We propose an iterative algorithm to compute the conformal mapping from the unit disk to physical domains with regular boundaries, defined by having only prime ends of the first kind. The mapping function is expanded into a Laurent series and use its truncated partial sum as an approximation. The Schwarz–Christoffel mapping formula provides the initial estimates for the series coefficients, which are then iteratively optimized. This algorithm efficiently handles complex domain shapes, such as winding orifices and slits, with high computational speed. Moreover, it offers valuable insights for designing algorithms to solve other types of conformal mapping problems and has practical significance in applications involving conformal mappings.

For spatially varying multiscale configurations (SVMSCs) that increasingly gain engineering perspective, a design scheme enabling the direct tuning of their microstructural layouts is still desired. Challenges lie behind the fact that an arbitrary setting on the (field) distributions about the microstructural geometric features is likely to violate the underlying conditions of total differential, and the article is aimed to resolve the issue systematically. This is done by representing (two-dimensional) SVMSCs from the viewpoint of quasi-conformal mapping, where the microstructural profile can be summarised by a (complete) family consisting of four geometric feature fields. By modifying the so-called Beltrami equations that automatically satisfy the total differential condition, the relationships among these four geometric feature fields can be explicitly derived, and their degrees of freedom are reduced to two. Different choices for which two field quantities as the free design variables result in various geometric-feature-based modes, which almost cover all scenarios of SVMSC design by means of directly tuning their microstructural profiles. Furthermore, we can now directly manipulate the stretch ratio, one of the four field variables, to avoid ill microstructural distortion, which has been a challenging issue for SVMSCs representation. Fueled by an asymptotic homogenisation scheme (Zhu et al. in J Mech Phys Solids 124:612–633, 2019, https://doi.org/10.1016/j.jmps.2018.11.008 ), the present method is then employed for geometric-feature-based compliance design of SVMSC.

In this paper, we propose the iterative numerical methods to calculate the conformal preimage domains for the specified logarithmic spiral slit regions and develop the applications of conformal mappings in the simulations of the flow around bodies. Firstly, we postulate that the boundaries of the preimage domains mapped onto logarithmic spiral slits are ellipses. The lengths of the long axes of ellipses and the coordinates of the centers are calculated using our iterative methods. Secondly, each type of the presented iterative method calculates numerical conformal mappings via solving the boundary integral equation with the generalized Neumann kernel. Finally, numerical examples show the convergence and availability of our iterative methods and display the simulations of the flow around the bodies as an application.

In this study, based on the plane elastic complex variable theory, employing image technique and conformal mapping technique, an analytical solution for antiplane scattering of SH waves by a lined tunnel in an exponentially graded half-space is derived, and the dynamic stress concentration factor (DSCF) around the tunnel is investigated. The medium is a bimaterial consisting of a semi-infinite homogeneous space and an exponentially inhomogeneous half-space with a lined circular tunnel. The governing equation is normalized into a Helmholtz equation with constant coefficients in complex coordinates based on the plane complex variable theory. The conformal mapping technique is used to convert the physical plane with two half-spaces including a lined tunnel into an image region consisting of three concentric circles. By applying the boundary conditions and the principle of orthogonality of trigonometric functions, a series of infinite algebraic equation are constructed and the unknown coefficients for the scattered wave functions are calculated. The numerical calculations are performed by considering the several parameters of medium and various conditions, and then, the influences of medium parameters on the dynamic response of the tunnel are analyzed based on the calculation results.

In freeform optical metrology, wavefront fitting over non-circular apertures is hindered by the loss of Zernike polynomial orthogonality and severe sampling grid distortion inherent in standard conformal mappings. To address the resulting numerical instability and fitting bias, we propose a unified framework curve-shortening flow (CSF)-guided progressive quasi-conformal mapping (CSF-QCM), which integrates geometric boundary evolution with topology-aware parameterization. CSF-QCM first smooths complex boundaries via curve-shortening flow, then solves a sparse Laplacian system for harmonic interior coordinates, thereby establishing a stable diffeomorphism between physical and canonical domains. For doubly connected apertures, it preserves topology by computing the conformal modulus via Dirichlet energy minimization and simultaneously mapping both boundaries. Benchmarked against state-of-the-art methods (e.g., Fornberg, Schwarz–Christoffel, and Ricci flow) on representative irregular apertures, CSF-QCM suppresses area distortion and restores discrete orthogonality of the Zernike basis, reducing the Gram matrix condition number from >900 to <8. This enables high-precision reconstruction with RMS residuals as low as 3×10−3λ and up to 92% lower fitting errors than baselines. The framework provides a unified, computationally efficient, and numerically stable solution for wavefront reconstruction in complex off-axis and freeform optical systems.

The purpose of this paper is to present a new Hybrid Analytical Model (HAM) based on Winding Function Theory (WFT) for electromagnetic analysis of the performance of one typical Unbalanced Two-Phase Induction Motor (UTPIM). Different indexes of electromagnetic modeling, such as winding distribution, slotting effect, and magnetic saturation, can be accurately considered by using the proposed HAM. To obtain this new hybrid technique, WFT is reformulated to consider magnetic saturation in addition to the influence of slotting and winding distribution. The Conformal Mappings (CMs) are used to calculate the slotted air-gap length accurately. The Magnetic Equivalent Circuit (MFC) model is used to consider the Magneto-Motive Force (MMF) drop in stator and rotor cores due to the excitation of one phase-winding. The results obtained from CMs and MEC are then utilized in reformulated WFT to calculate the inductances of the respective phase-winding. Transient analysis is then done to calculate the indexes of performance, such as air-gap magnetic field, phase currents, electromagnetic torque, and rotor speed, by using the lookup table of inductances while considering different capacitors in the auxiliary phase. In each step, the accuracy of analytical results is confirmed by comparing with corresponding results obtained from the Finite Element Method (FEM).

In the fiberglass industry, Pt–Rh bushings made of platinum and rhodium have very good characteristics, such as high temperature resistance, corrosion resistance, oxidation resistance, and creep resistance. In this paper, a semi-infinite lath structure model is constructed, and the expression of the surface temperature distribution of a Pt–Rh alloy plate with a circular through hole is obtained based on the non-Fourier heat conduction equation, complex function method and conformal mapping method. At the same time, the influence of the position of the circular through hole in the Pt–Rh bushing and the parameters of the incident light source (Non-diffusion incident wave number and relative thermal diffusion length) on the surface temperature distribution of the Pt–Rh bushing is studied by using this formula. It is found that: 1. heat concentration and fracture are occur easily at the through hole; 2. when the through hole is in the asymmetric center, the greater the asymmetry, the smaller the maximum temperature amplitude; 3. when the buried depth of the through hole increases, the maximum temperature amplitude decreases; 4. when the incident wave number and the relative thermal diffusion length of the incident light source are larger, the maximum temperature amplitude is smaller. The numerical results are almost consistent with those of ANSYS thermal simulation. The expression of the surface temperature distribution of the semi-infinite lath structure proposed in this paper can effectively reduce the loss of precious metal materials and the time of thermal simulation in the experimental process, as well as provide important significance for structural design, quality inspection, process optimization, and service life improvement of Pt–Rh bushings.