Boundary Circle Research Articles

The Various Radii Circle Packing Problem in a Triangle

Malfatti’s problem is the problem of fitting three circles into a triangle such that they are tangent to each other and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to include Tn = 1 + 2 + … + n circles inside the triangle with special tangency properties among the circles and triangle sides; this problem is referred to as the extended Malfatti problem or the Tri(Tn) problem. In the extended Malfatti problem, the number of circles in the triangle is a triangle number because the tangency properties between the internal circles and the three sides of the triangle have a special type of structure; that is, the corner circle is tangent to two sides of the triangle and two other circles, the boundary circles are tangent to one side of the triangle and four other circles, and the inner circles are always tangent to six other circles. The circles we find in the extended Malfatti problem have the following property: the smallest and largest radii of the circles differ to a great extent. In the study presented herein, we propose algorithms to solve the problem that the tangency properties between the circles and the sides of the triangle are not fixed, so that the number of circles in the triangle is not necessarily a triangle number. The purpose of this change is to attempt to establish the radii of the circles in the triangle within a small range.

Plastic zone characteristics and rapid stability evaluation indexes of circular tunnels subjected to non-hydrostatic stress: A novel complex variable solution incorporating 3D rock mass strength

The morphological characteristics and distribution range of the plastic zone around tunnels, influenced by the true three-dimensional high geostress environment in deep engineering, constitute fundamental reference data for both tunnel design and the control of surrounding rock stability. Based on the GZZ three-dimensional rock mass strength criteria and the complex variable method, a novel semi-analytical method for solving the plastic zone of a circular tunnel subjected to non-hydrostatic stress is presented. The conformal mapping is employed to intricately transform the complex boundaries of plastic zones characterized by diverse shapes onto the unit circle boundary within the image plane. This framework proves invaluable for the precise analytical process of the complex plastic zone morphology and the plastic influence range within the surrounding rock under complex stress conditions. In contrast to traditional analytical methods, the efficacy and versatility of our proposed approach are substantiated. Notably, it demonstrates unique advantages, particularly in the context of non-hydrostatic stress fields and varied rock mass parameters, with respect to predicting both surrounding rock stability and plastic zone morphology. Through the analysis of numerous cases, the study reveals the systematic patterns of influence that in-situ stress and surrounding rock characteristic parameters have on the shape and maximum depth of the plastic zone. The results indicate that (1) there are notable disparities in the morphology and extent of the plastic zone between the GZZ and GHB criteria; (2) the transition of the plastic zone in surrounding rock from U-shaped to V-shaped is significantly influenced by the geostress-to-strength ratio Kσ; (3) there is a notable linear correlation observed between the maximum plastic zone thickness rmax and both Kσ and the stress concentration factor KSCF, importantly, this correlation is independent of far-field stress and the brittleness characteristics of the surrounding rock. These findings provide a novel discriminative method for the rapid analysis of surrounding rock stability in deep circular tunnels.

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.

Polyharmonic potential theory on the Poincaré disk

We consider the open unit disk D equipped with the hyperbolic metric and the associated hyperbolic Laplacian L. For λ∈C and n∈N, a λ-polyharmonic function of order n is a function f:D→C such that (L−λI)nf=0. If n=1, one gets λ-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for λ-polyharmonic functions. For this purpose, we first determine nth-order λ-Poisson kernels. Subsequently, we introduce the λ-polyspherical functions and determine their asymptotics at the boundary ∂D, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the L2-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the nth-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of λ-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits.

Dimension Evolution of Self-Assembled Organic Microcrystal for Laser and Polarization-Rotation Function.

Multidimensional integrated micro/nanostructures are vitally important for the implementation of versatile photonic functionalities, whereas current material structures still suffer undesired surface defects and contaminations in either multistep micro/nanofabrications or extreme synthetic conditions. Herein, the dimension evolution of organic self-assembled structures 2D microrings and 3D microhelixes for multidimensional photonic devices is realized via a protic/aprotic solvent-directed molecular assembly method based on a multiaxial confined-assisted growth mechanism. The 2D microrings with consummate circle boundaries and molecular-smooth surfaces function as high-quality whispering-gallery-mode microcavities for dual-wavelength energy-influence-dependent switchable lasing. Moreover, the 3D microhelixes with smooth surfaces and natural twistable characteristics act as active photon-transport materials and polarization rotators. These results will broaden the horizon of constructing multidimensional microstructures for integrated photonic circuits.

A symmetry algebra in double-scaled SYK

The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the “chord Hilbert space”. We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the \mathfrak{sl}_2𝔰𝔩2 near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to \mathfrak{sl}_2𝔰𝔩2, but with a non-standard action on the boundary time coordinate. One can make the action of \mathfrak{sl}_2𝔰𝔩2 algebra more standard at the cost of extending the boundary circle to include some “fake” portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.

Generalized complex variable analysis of shallow tunneling through multi-layered ground

The mechanical analysis of shallow tunnels crossing multiple ground layers is a common problem in urban underground engineering. Since the tunnel boundary intersects multiple ground layers, the challenges of mixed boundary value problem for composite boundary and the conformal mapping of irregular banding computational domain are faced in solving the multi-boundary problem. For this, a new solution strategy regarding the generalized complex variable method to calculate the mechanical responses of the multi-layered ground is proposed. The research object is divided into two independent objects including the complete multi-layered ground without tunnel and the complete tunnel boundary. Firstly, the mechanical responses of the complete multi-layered ground are solved through the superposition of multiple single-boundary contributions. Subsequently, the virtual boundary circle and its corresponding analytic functions are introduced in analyzing the tunnel boundary contribution. Finally, the solution for the stress and displacement field of the multi-layered ground is modified by superimposing the contribution of the virtual boundary circle, so that all the given boundary conditions are satisfied. The self-consistency and accuracy of the proposed method are validated by several examples. The characteristics of the displacement propagation of typical multi-layered ground patterns are analyzed based on the analytical solution. The proposed method can be used to analyze the mechanical responses of multi-layered ground with arbitrary intersections between the tunnel and ground boundaries.

Автоматизированное определение дисперсности воздушной фазы в мороженом с применением методов машинного обучения

Ice cream is a popular cold dessert. Its air phase consists of tiny bubbles with an average diameter of 15–60 µm. New ice cream formulations depend on the way the composition and production factors affect the air phase. As a result, ice cream producers need new time-saving and reliable methods to determine dispersion. The research objective was to create a computer program for marking the position of centers and diameter of air bubbles on microscopic images of a bounding circle type.
 The review part included 20 years of Russian and English publications on microscopic research methods in ice cream production indexed in Web of Science and Russian Research Citation Index. Microscopic images of ice cream air phase were obtained using an Olympus CX41RF microscope with a magnification of ×100. The automatic markup program employed the Python programming language, the Keras machine learning library, and the TensorFlow framework. The models were trained using the NVIDIA GTX video accelerator.
 The review showed that the dispersion of ice cream air phase depends on its composition and the freezing parameters whereas bubble formation is usually described in line with the existing foaming theories. A training data set was obtained by manual labeling of microscopic images. The optimal number channels in the convolutional layers of a neural network with LeNet-type architecture was determined, which made it possible to classify images as spheres or non-spheres with an accuracy of ≥ 0.995. The sliding window method helped to determine the limits of the neural network triggering for the sliding window method were determined, which reached 7.5% of the diameter with lateral displacement and 12.5% with scaling. The developed algorithm automatically marked bubbles on microscopic images. The error in determining the average diameter was below 1.8%.
 The new method for automated calculation of the number and diameter of air bubbles in ice cream proved to be user-friendly. It can be found in public domain, and researchers are free to adapt it to solve various computer vision issues.

A Low Switching Frequency Model Predictive Control Method for an Induction Motor Fed by a 3-Level Inverter

Traditional model predictive control (MPC) for the induction motor fed by a three-level inverter needs to explore 27 voltage vectors to obtain the optimal one, which leads to high switching frequency and requires too much computation. To solve this issue, a low switching frequency model predictive control with partition optimization is proposed. First, the reference voltage vector can be gained from the prediction model at the next time, and the space voltage vector plane is divided into 12 sectors for further vector choice. Furthermore, considering inverter constraints, the candidate voltage vectors are determined according to the sector location of the reference voltage vector. In this way, the candidate vectors can be reduced to 3 at most. Then, a boundary circle limit is designed to avoid unnecessary switch changes. If the reference voltage vector is within the boundary limit, the switches do not act, which can reduce the system switching frequency without introducing the extra weight coefficient into the cost function. These selected voltage vectors are substituted into the cost function to determine the optimal one. Finally, the neutral point voltage deviation is controlled by the positive and negative redundant small vectors to realize the multi-objective constraint without weighting coefficients. The simulation results show that the proposed control method can significantly reduce the switching frequency; at the same time, both the dynamic and steady performances can be maintained well, and the cost function has no weight coefficients.

Scaling inequalities for spherical and hyperbolic eigenvalues

Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the length ^{-2} scaling relation in Euclidean space. For a cap of aperture \Theta on the sphere \mathbb{S}^2 , normalizing the k -th eigenvalue by the square of the Euclidean radius of the boundary circle yields that \mu_k(\Theta) \sin^2 \Theta is strictly decreasing, while normalizing by the stereographic radius squared gives that \mu_k(\Theta) 4 \tan^2 \Theta/2 is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that \mu_2(\Theta) 4 \sin^2 \Theta/2 is strictly increasing. Monotonicities of this kind are somewhat surprising, since the Neumann eigenvalues themselves can vary non-monotonically. Cheng and Bandle-type inequalities are deduced by assuming either fixed radius or fixed area and comparing eigenvalues of disks having different curvatures.

The kynurenic acid analog SZR104 induces cytomorphological changes associated with the anti-inflammatory phenotype in cultured microglia

We previously showed the anti-inflammatory effects of kynurenic acid (KYNA) and its brain-penetrable analog N-(2-(dimethylamino)ethyl)-3-(morpholinomethyl)-4-hydroxyquinoline-2-carboxamide (SZR104) both in vivo and in vitro. Here, we identified the cytomorphological effects of KYNA and SZR104 in secondary microglial cultures established from newborn rat forebrains. We quantitatively analyzed selected morphological aspects of microglia in control (unchallenged), lipopolysaccharide (LPS)-treated (challenged), KYNA- or SZR104-treated, and LPS + KYNA or LPS + SZR104-treated cultures. Multicolor immunofluorescence labeling followed by morphometric analysis (area, perimeter, transformation index, lacunarity, density, span ratio, maximum span across the convex hull, hull circularity, hull area, hull perimeter, max/min radii, mean radius, diameter of bounding circle, fractal dimension, roughness, circularity) on binary (digital) silhouettes of the microglia revealed their morphological plasticity under experimental conditions. SZR104 and, to a lesser degree, KYNA inhibited proinflammatory phenotypic changes. For example, SZR104 treatment resulted in hypertrophied microglia characterized by a swollen cell body, enlarged perimeter, increased transformation index/decreased circularity, increased convex hull values (area, perimeter, mean radius, maximum span, diameter of the bounding circle and hull circularity), altered box-counting parameters (such as fractal dimension), and increased roughness/decreased density. Taken together, analysis of cytomorphological features could contribute to the characterization of the anti-inflammatory activity of SZR104 on cultured microglia.

Spectral analysis of the Neumann–Poincaré operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance

We consider the Neumann–Poincaré operator on a planar domain enclosed by two touching circular boundaries. This domain, which is a crescent-shaped domain or touching disks, has a cusp at the touching point of two circles. We analyze the operator via the Fourier transform on the boundary circles of the domain. In particular, we define a Hilbert space on which the operator is bounded, self-adjoint. We then obtain the complete spectral resolution of the Neumann–Poincaré operator. On both the crescent-shaped domain and touching disks, the Neumann–Poincaré operator has only absolutely continuous spectrum on the closed interval [−1/2,1/2]. As an application, we analyze the plasmon resonance on the crescent-shaped domain.

Pupil Detection for Automatic Diagnosis of Eye Diseases Using Optimized Color Mapping

Introduction: Pupil and iris disorders form an important category of eye diseases. Accurate segmentation of the pupil is the first and most important step in the automatic diagnosis of diseases related to the pupil and iris. Most of the existing methods do not have enough accuracy and are sensitive to the effects of noise and specular spot reflection. In addition, the images used in these methods usually have limitations, such as the viewing angle . Method: In the proposed algorithm, a stable method is offered to remove the effects of specular spot reflection in the pupil, and necessary preprocessing is done to detect the exact location of the pupil. An optimized color mapping algorithm is proposed and the mapping is calculated with the help of the LM algorithm to accurately determine the pupil boundary. This method does not impose any restrictions on the eye image and shape, and the angle of the pupil in the image can be in any shape and direction . Results: The proposed method does not assume any specific model as the final pupil boundary (circle or oval) and is robust to noise and specular reflection factors as well. This method has been able to accurately detect the pupil boundary with the accuracy of 98.8% using UBIRIS dataset and 98% using the collected data by authors . Conclusion: The method presented in this paper can be used to increase the accuracy in determining the internal and external border of the iris to diagnose diseases related to the pupil and iris, as well as identity identification based on iris tissue.

A system model for real-time monitoring and geospatial data for the simulation of surveillance of COVID-19 in Makassar, Indonesia

The rapid spread of COVID-19 requires rapid management. Prompt treatment is needed to prevent the spread of this disease, which could be minimized or isolated in one place so that it does not spread to other places. This study was conducted to discover a model of the surveillance system in real time and to analyze the change in its distribution pattern. This study was conducted in the city of Makassar, South Sulawesi, Indonesia, involving 30 volunteers. Two devices were used, the Internet reverse transcription loop-mediated isothermal amplification (iRTLAMP) and IoT button application, to provide spatial data in the form of patient points exposed to COVID-19. Furthermore, three scenarios were applied to see the pattern of data distribution. The data recorded in the cloud database were retrieved with a created application and then analyzed using Kernel Density Estimation (KDE) and Point Pattern Analysis (PPA) to observe the distribution of patterns in real time. The analysis utilizing KDE with the Gaussian kernel function as the kernel revealed significant changes in the probability distribution, which could be seen from color changes in the map. The centrographic analysis revealed that the mean and median points of the three scenarios changed in various ways within approximately 700 m to 1.7 km. Meanwhile, the radius of minimal bounding circle behaved similarly and appeared to change depending on the scenario, from a radius of 5.57 (initial) km to 6.55 km (scenario 1), 5.57 km (scenario 2) and 6.22 km (scenario 3). The standard distance also showed a change from 4.53 km to 4.60 km (scenario 1), 4.70 km (scenario 2) and 5.40 km (scenario 3). Simulations carried out using the developed system showed that the use of internet devices could help monitor people exposed to COVID-19 by changing patterns and distribution points. Therefore, decision makers could take preventive actions earlier so that this disease does not spread quickly.

Double-Center-Based Iris Localization and Segmentation in Cooperative Environment with Visible Illumination.

Iris recognition has been considered as one of the most accurate and reliable biometric technologies, and it is widely used in security applications. Iris segmentation and iris localization, as important preprocessing tasks for iris biometrics, jointly determine the valid iris part of the input eye image; however, iris images that have been captured in user non-cooperative and visible illumination environments often suffer from adverse noise (e.g., light reflection, blurring, and glasses occlusion), which challenges many existing segmentation-based parameter-fitting localization methods. To address this problem, we propose a novel double-center-based end-to-end iris localization and segmentation network. Different from many previous iris localization methods, which use massive post-process methods (e.g., integro-differential operator-based or circular Hough transforms-based) on iris or contour mask to fit the inner and outer circles, our method directly predicts the inner and outer circles of the iris on the feature map. In our method, an anchor-free center-based double-circle iris-localization network and an iris mask segmentation module are designed to directly detect the circle boundary of the pupil and iris, and segment the iris region in an end-to-end framework. To facilitate efficient training, we propose a concentric sampling strategy according to the center distribution of the inner and outer iris circles. Extensive experiments on the four challenging iris data sets show that our method achieves excellent iris-localization performance; in particular, it achieves 84.02% box IoU and 89.15% mask IoU on NICE-II. On the three sub-datasets of MICHE, our method achieves 74.06% average box IoU, surpassing the existing methods by 4.64%.

Billiard with slipping by an arbitrary rational angle

The class of billiards in a disc with slipping along the boundary circle by an angle commensurable with $\pi$ is considered. For such billiards it is shown that an isoenergy surface of the system is homeomorphic to a lens space $L(q,p)$ with parameters satisfying $0 < p <q$. The set of pairs $(q, p)$ such that there exists a billiard in a disc realizing the corresponding lens space $L(q,p)$ is described in terms of solutions of a linear Diophantine equation in two variables. This result also holds for planar billiards with slipping in simply connected domains with smooth boundary, that is, it is not confined to the integrable case. Bibliography: 30 titles.

Poles of cubic differentials and ends of convex $\mathbb{RP}^2$-surfaces

On any oriented surface, the affine sphere construction gives a one-to-one correspondence between convex $\mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist–Hulin, Loftin and Dumas–Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite-volume ends of convex $\mathbb{RP}^2$-surfaces, whereas poles of order $3$ and greater than $3$ correspond to geodesic and piecewise geodesic ends, respectively. Moreover, at each pole of order at least $3$, we construct a natural bordification of the surface such that the $\mathbb{RP}^2$-structure extends to the boundary circle in a metric preserving way.

An Algorithm to Extract the Boundary and Center of EUV Solar Image Based on Sobel Operator and FLICM

An algorithm to extract the disk boundary and center of EUV solar image using the Sobel operator, Fuzzy Local Information C-Means Clustering algorithm (FLICM), and the least square circle fitting method is proposed in this paper. The Sobel operator can determine the solar disk boundary preliminarily, and then the image is processed further using the FLICM algorithm. After the background is removed based on the clustered image and the boundary points can be highlighted, these points are fitted using the least square circle fitting method as the final boundary circle. The solar data used in this paper was from the observation of the Solar Dynamics Observatory Atmospheric Imaging Assembly (SDO/AIA) instrument. The 2523 19.3 nm solar images covering solar minimum, moderate solar activity, and more active suns were calculated using the proposed algorithm to analyze the accuracy statistically. The statistical comparison results demonstrate that the method is accurate and effective. This method can support the processing of solar EUV images and serve the operational system of a space weather forecast.

Steady Normal-Mode Vortices in Circular Cavities

Summary This article establishes elementary families of steady two-dimensional (2D) vortices in circular enclosures filled with inviscid fluid. A normal-mode vortex is a continuous vortex that satisfies a Helmholtz equation for the streamfunction. An individual normal-mode vortex satisfies the steady 2D vorticity equation. A way to satisfy the vorticity equation with two superposed normal-mode vortices is to let one of them have circular streamlines, trivially satisfying the kinematic boundary condition at the circle boundary. The requirement for steady flow is that the wavenumber eigenvalues for the Helmholtz equations are identical. This eigenvalue has to be dictated by a normal-mode vortex that is fully 2D. In a circle sector with a designed angle, it is possible to add two fully 2D normal-mode vortices so that their superposed flow is steady. One such example is demonstrated, where the circle sector has an angle of $63.77^\circ$. While conventional stability analysis does not apply to these families of steady normal-mode vortices, there is a latent algebraic instability in terms of linearly growing offspring vortices evolving from an initial state where a small initial perturbation is added to the steady normal-mode vortex.

A tractroid realization of a 2d black hole vacuum

The two-dimensional black hole vacuum obtained from a spatial slice of the BTZ black hole is mapped explicitly to a tractroid surface minus a bounding circle.