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The standard configuration of a sonic black hole (SBH) is a discrete set of rings with back cavities at a duct termination, whose inner radii decrease according to a power-law. Although several practical realizations of this type of slow-sound waveguide have been investigated, the continuous problem in which the number of ring/cavity sets goes to infinity has not been analyzed in depth. Understanding its performance is interesting in itself and could provide clues for new designs. For this purpose, we transform the generalized Webster equation for the acoustic pressure inside the ideal SBH into a spatially varying wavenumber Helmholtz equation for a new locally scaled pressure. We eliminate the singularity of the original problem at the SBH termination by considering a rigid residual surface that will be shown to play a role analogous to that of the acoustic black hole (ABH) residual thickness in beams and plates. The variational formulation of the Helmholtz equation is presented and solved by expanding the unknown scaled pressure in terms of a basis of Gaussian functions. The same framework is used to set up an eigenvalue problem that provides the SBH modes and thereby a modal decomposition for the pressure within the SBH. Once the theoretical foundations are established, the variation of the speed of sound and wavenumber in the SBH is presented. The dependence of the shape and distribution of the modes on the residual surface and damping helps to clarify the occurrence and disappearance of oscillations in the input admittance and reflection coefficient of the SBH at different frequencies. For the latter, a detailed parametric study is carried out. Alternative profiles to the power law are also investigated, some showing reflection coefficient values similar to those of the classical SBH. • Sonic black holes at the end of ducts provide minimal wave reflection. • A theoretical approach is proposed to solve the SBH governing equation. • The modal distribution in the SBH is strongly affected by the residual radius. • The modal distribution explains the oscillations in the reflection coefficient. • Alternative SBH profiles to the power law are studied. [ABSTRACT FROM AUTHOR]

In this paper, we study a Riemann-Liouville-type fractional Riemann-Stieltjes integral boundary value problem under some conditions regarding the spectral radius of the relevant linear operator. The existence of nontrivial solutions is obtained using topological degree, and our results improve and generalize some results in the literature. [ABSTRACT FROM AUTHOR]

We present the observational mass–radius (M–R) relation for a sample of 47 magnetized white dwarfs (WDs) with the magnetic field strength (B) ranging from 1 to 773 MG, identified from the SDSS data release 7 (DR7). We derive their effective temperature, surface gravity (log g), luminosity, radius, and mass. While atmospheric parameters are derived using a Virtual Observatory Spectral Energy Distribution Analyzer (VOSA), the mass is derived using their location in the HR diagram in comparison with the evolutionary tracks of different masses. We implement this mass measurement instead of a more traditional method of deriving masses from log g , which is unreliable as is based on SED and generates errors from other physical parameters involved. The main disadvantage of this method is that we need to assume a core composition of WDs. As it is complicated to identify the exact composition of these WDs from low-resolution spectra, we use tracks for the masses 0.2 to 0.4 M⊙ assuming a He-core, 0.5 to 1.0 M⊙ assuming CO core, and above M⊙ assuming O–Ne–Mg core. We compare the observed M–R relation with those predicted by the finite temperature model by considering different B , which are well in agreement considering their relatively low-surface fields, ≲109 G. Currently, there is no direct observational detection of magnetized WDs with B > 109 G. We propose that our model can be further extrapolated to higher B , which may indicate the existence of super-Chandrasekhar mass (M > 1.4 M⊙) WDs at higher B. [ABSTRACT FROM AUTHOR]

Establish a new classification system of distal radius fracture based on computed tomographic (CT), and evaluate its reliability and reproducibility preliminarily, and provide a new theoretical reference for clinicians to use the clinical classification system. The imaging data and clinical data of 204 inpatients with distal radius fracture during 6 years from January 1, 2014 to January 1, 2019 in orthopaedic department were analyzed retrospectively and classified based on CT. Three observers evaluated the image data of 48 randomly selected cases based on CT at different time nodes of T1 and T2. Cohen's kappa was used to calculate the consistency. At the last follow-up, patients' Disabilities of the Arm, Shoulder and Hand (DASH), Patient Rated Wrist Evaluation (PRWE), and VAS scores were collected. Among 204 cases, there were 12 cases of type 1, including 6 cases of type 1-D, 4 cases of type 1-V and 2 cases of type 1-R. There were 6 cases of type 2, including 2 cases of type 2-DV, 2 cases of type 2-DR and 2 cases of type 2-VR. There were 186 cases of type 3, including 32 cases of type 3–0, 127 cases of type 3–1 and 27 cases of type 3–2. There was no significant difference in DASH, PRWE and VAS scores among all types (P > 0.05). The results of interobserver reproducibility were kappa = 0.985, ICC = 0.984 in the first evaluation, kappa = 0.986, ICC = 0.986 in the second evaluation. The results of intraobserver reproducibility were O1 = 0.991, O2 = 0.991, O3 = 0.989 respectively. The new classification system of distal radius fracture based on CT has theoretical and practical significance for incision selection, fracture reduction and internal fixation. 123 classification system is clear, comprehensive, easy to understand and remember. Moreover, it has higher interobserver reliability and intraobserver reproducibility than other systems reported at present. [ABSTRACT FROM AUTHOR]

Let 0 < ρ ≤ 2 and w ρ (X) be the operator radius of a bounded linear Hilbert space operator X. In this paper we present characterizations of operators satisfying w ρ (X) ≤ 1. We also give an expression for the ρ -radii based on the numerical radius of a certain 2 × 2 block matrix. This enables us to investigate the properties of the operator radii w ρ (⋅). In particular, we obtain lower and upper bounds for the operator radii. In addition, we define a norm on Hilbert space operators, which generalizes the operator radii and prove some basic properties of this norm. Our results extend or improve some theorems due to Ando [4] , Bhatia–Jain [9] and Kittaneh [19]. [ABSTRACT FROM AUTHOR]

Ignition electrodes have an immense impact on the accurate measurement of the flame propagation spherical radius. In this study, a flame-radius calculation method is designed. The method is able to eliminate effects due to the ignition electrodes. The adaptability and optimization effects of the proposed method are analyzed. The results show that the ratio of the angle is affected by the ignition electrodes under the Han II method. There are three obvious divisions include a high-value area, a sharp-variation area, and a mild-variation area. The ratio of the angle affected by the ignition electrodes is only applicable to the mild-variation region when the flame presents respective convex and concave distributions. For these distributions, the increment rate of the mean radius is 0.4–0.85% and 0.42–3.19%. The reduced rate of the standard deviation of the radius extraction value is 11.91–22.1% and 5.13–17.99%, and the reduced rate of the radius extraction value range is 20.32–39.51% and 0.32–8.09%. [ABSTRACT FROM AUTHOR]

We propose an effective version of the lift by derivation, an invariant that allows us to provide the classification of $ B(5,6,8) = RM(6, 8)/RM(4, 8) $. The main consequence is to establish that the covering radius of the Reed-Muller $ RM(4,8) $ is equal to 26. [ABSTRACT FROM AUTHOR]

In this work, we give an improved new argument to prove that the solution of the Cauchy problem for the nonlinear spatially homogeneous Landau equation with hard potentials with $ L^2(\mathbb{R}^3) $ initial datum enjoys a analytic Gelfand-Shilov regularizing effect in the class $ S^1_1(\mathbb{R}^3) $, the evolution of analytic radius is similar to the heat equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the q-numerical radius $ \omega _{q}(.) $ ω q (.) of bounded linear operators on Hilbert spaces. More precisely, first, we show that $ \omega _{q}(.) $ ω q (.) defines a norm which is equivalent to the operator norm. Next, the following compatible generalization of Kittaneh's inequality $$\begin{align*} \frac{1}{4}\left(\frac{q}{2-q^{2}}\right)^{2}\parallel T^{*}T +TT^{*}\parallel & \leq \omega_{q}^{2}(T) \leq \frac{(q+2\sqrt{1-q^2})^2}{2}\\ & \quad\times \Vert T^{*}T+TT^{*}\Vert. \end{align*}$$ 1 4 (q 2 − q 2 ) 2 ∥ T ∗ T + T T ∗ ∥ ≤ ω q 2 (T) ≤ (q + 2 1 − q 2 ) 2 2 × ‖ T ∗ T + T T ∗ ‖. is obtained. Finally, some generalizations of q-numerical radius inequalities for composition of operators are established. [ABSTRACT FROM AUTHOR]

We present several upper bounds for the numerical radii of 2 × 2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then for every r ≥ 1 and α ∈ [0, 1]. This substantially improves on the existing inequality. Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively. [ABSTRACT FROM AUTHOR]

The interface tends to behave as a circular shape under the effect of surface tension in the two-dimensional two-phase fluid flow. In this study, based on the coupled volume-of-fluid and level set (VOSET) method, an accurate circle-based interface reconstruction (CIR) algorithm for structured meshes is proposed, which features a straightforward implementation procedure. A portion of the standard circle is employed to approximate the curve interface, without the need for complex classification, coordinate transformation, and equation transformation. The radius of this circle is computed simply by the curvature and signed distance at the mixed cell center, and the center of this circle is determined using a straightforward dichotomy method under the mass conservation constraints. Results indicate that the coupled VOSET and CIR (VOSET-CIR) method maintains superior computational accuracy in the signed distance, interface curvature, and dynamic interface reconstruction compared to the coupled VOSET and piecewise linear interface calculation (VOSET-PLIC) method, with accuracy improvements ranging from 34% to 1490% across different test cases. Furthermore, the VOSET-CIR method outperforms the efficient least squares volume-of-fluid interface reconstruction (ELVIRA) and PLIC algorithms in reconstructing random circles and surpasses the quadratic spline based interface reconstruction (QUASI) algorithm in reconstructing random squares. In terms of computational efficiency, except for the initial PLIC algorithm, the time costs of the VOSET-CIR method are substantially lower than those of the ELVIRA and QUASI algorithms, as well as the VOSET-PLIC method. [ABSTRACT FROM AUTHOR]

The convergence-confinement method (CCM) is a standard design tool to study the ground-structure interaction. Constitutive model selection is a critical issue in the correct application of the CCM to represent the real behavior of rock mass and plastic zone. In this paper, the post-failure behavior of rock mass is formulated and incorporated by a numerical approach and the results are compared with the experimental observations. Elastic perfectly plastic (EPP) and strain softening (SS) models, are used and compared for a circular tunnel to be applied in the CCM method. The results show that elastic parts of the ground reaction curve and the longitudinal deformation profiles for both models are similar. But when the rock failure occurs and tunnel face exceeds 0.5D, differences in the curves are significant. Based on the results, the maximum displacement in different amount of K (in-situ stress ratios) for the SS model is more than 3 times of the EPP model. Plastic radius in the SS model is about 2 times the radius in the EPP model. In addition to precisely identifying the plastic zone and its distribution, the modified numerical approach in this paper, can determine the critical support pressure within residual and softening regions. [ABSTRACT FROM AUTHOR]

In this work, we give several new upper and lower bounds for the p-numerical radii of operators as well as for n × n operator matrices. An application to 2-nilpotent operators is provided, and a p-numerical radius power inequality is also given. [ABSTRACT FROM AUTHOR]

The normalized distance Laplacian of a graph G is defined as D L (G) = T (G) − 1 / 2 (T (G) − D (G)) T (G) − 1 / 2 where D (G) is the matrix with pairwise distances between vertices and T (G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius. [ABSTRACT FROM AUTHOR]

Numerical simulation of fatigue crack growth (FCG) is presented, based on the application of the extended finite element method (XFEM). Couple of case studies are presented, as a review of recent experience of the Belgrade school of XFEM simulations, established at the Faculty of Mechanical Engineering. Case studies comprised Charpy specimen with initial crack at the notch tip, oil rig pipe with an axial surface crack, pressure vessels with a crack at the welded joint with connecting pipe, turbine shaft with a crack at the transition radius, optimization of a wing spar in respect to FCG, the effect of mesh size on fatigue life estimation of a stringer panel, fatigue life estimation of wing‐to‐fuselage connecting lug, geometry effect on fatigue life of orthopedic plates and fatigue life of hip implant. Based on these case studies, XFEM is discussed in respect to its accuracy and efficiency. It is concluded that XFEM is a versatile tool for simulation of FCG, providing an excellent option for precise and reliable fatigue life of a cracked component. [ABSTRACT FROM AUTHOR]

This study presents the effect of the flow velocity, which varies exponentially with time, on solute transport in a finite porous medium with radial geometry. The two-dimensional advection-dispersion equation in the cylindrical coordinate system is solved by using finite difference method. The concentration profiles are established for a duration of four hours of infiltration. It appears that the concentrations are lower in the case of the decreasing exponential velocity. The analysis of the effect of the injection radius showed that the retention of pollutants decreases when the injection radius increases. These findings can be used to design the size of a filter system for drinking water provision, and the environment protection. [ABSTRACT FROM AUTHOR]

In this paper, a semi-analytical technique based on Taylor's series method namely DTM has been used to solve the differential equation which governs the motion of three types of annular FGM plates. The differential equation has been obtained using Hamilton principle and classical plate theory. The mechanical properties of the plate (Young's modulus and density) are considered to be graded in thickness direction and vary following the power-law. The behaviour of volume fraction index and radii ratio has been investigated onto first three modes of frequency parameter for all three plates. Moreover, the novelty of this paper is the application of the versatile technique DTM to study the effect of radii ratio and volume fraction index on three different types of annular FGM plates. A comparison has been made between the obtained numerical results and the results are available in the literature. A good agreement of the results verifies the accuracy of the present technique. Three-dimensional mode shapes for all three plates are also presented. [ABSTRACT FROM AUTHOR]

In Geometric function theory, the Ma-Minda class of starlike functions has a unique place as it unifies various subclasses of starlike functions. There has been an vivid interplay between special functions and their geometric properties, like starlikeness. In this article, we establish certain special function's radius of Ma-Minda starlikness. As an application, we obtain conditions on parameters for these special functions to be in the Ma-Minda class. Further, we focus on certain convolution properties for the Ma-Minda class that are not done so far, and study their applications in radius problem. Finally, we prove a variational problem of Goluzin, namely, the region of variability for the Ma-Minda class. Our results simplify and generalize the already-known ones. [ABSTRACT FROM AUTHOR]

The performance of tactile and optical surface sensors for nano and micro coordinate measuring machines is currently limited by the lack of precisely characterised micro spheres, since established strategies have mainly been developed for spheres in the range of millimetres or above. We have, therefore, recently focused our research efforts towards a novel strategy for the characterisation of spheres in the sub-millimetre range. It is based on a set of atomic force microscope (AFM) surface scans in conjunction with a stitching algorithm. To obtain an uncertainty statement, the uncertainty about the shape of the reference surface needs to be propagated via the shape of the AFM tip to the actual measurement object. However, the sampling process of an AFM is non-linear and the processing of AFM scans requires complex algorithms. We have, therefore, recently begun to model the characterisation of micro spheres through simulations. In this contribution, this model is extended by the influence of the tip and reference surface. The influence of the tip's shape and reference surface is investigated through virtual and real experiments. The shape of the tip is varied by using tips with mean radii of 200 nm and 2 μm while sampling the same ruby sphere with a mean radius of 150 μm. In general, the simulation results imply that an uncertainty of less then 10 nm is achievable. However, an experimental validation of the model is still pending. The experimental investigations were limited by the lack of a suitable cleaning strategy for micro parts, which demonstrates the need for further investigations in this area. Although the characterisation of a full sphere has already been demonstrated, the investigations in this contribution are limited to equator measurements. [ABSTRACT FROM AUTHOR]

Any 3D AFM image is a convolution of the geometry of the AFM tip and the profile of the scanned sample, especially when the dimensions of the scanned sample are comparable to those of the AFM tip shape. The precise profile of the scanned sample can be extracted from the 3D AFM image if the geometry of the AFM tip is known. Therefore, in order to separate the geometry of the AFM probe tip from the 3D AFM image of a diffraction grating with a rectangular profile and to correct for the topographic convolutions induced by the AFM probe tip, a method is used to quantitatively evaluate the geometry of the AFM probe tip, including the tip radius and the included angle. A model for reconstructing the measured AFM image is proposed to correct topography convolutions caused by the AFM tip shape when scanning a diffraction grating with rectangular profiles. A series of experiments were performed to verify the effectiveness of the proposed AFM tip geometry evaluation method, and comparison experiments were conducted to demonstrate the feasibility and reliability of the proposed reconstruction model. [ABSTRACT FROM AUTHOR]

Traffic noise mapping frequently employs Kriging, Inverse Distance Weighted (IDW), and Triangular Irregular Networks (TIN) spatial interpolations. This study uses the Henk de Kluijver noise model to evaluate the performance of spatial interpolations. Effective traffic noise mapping requires that noise observation points (Nops) be designed as 2 m grids. The upper and lower slopes function as noise barriers to reduce sound levels. Therefore, assessment of accuracy is essential for visualising noise levels in undulating and level terrain. In addition, this work gives an accurate comparison of traffic noise interpolation in undulating areas. The elements of spatial interpolations, such as the weighting factor, variogram, radius, and number of points influence the interpolation accuracy. The Kriging with a Gaussian variogram, where the radius is 5 m and the number of points is 12 demonstrates the highest level of precision. However, there is no direct relationship between accuracy validation and cross-validation. In cross-validation, however, the accuracy of the Gaussian variogram with a 7 m radius and 18 points is more accurate. In addition, this study demonstrates that Kriging is superior for extrapolating noise levels in undulating regions. Accurate visualising traffic noise levels requires a prior understanding of spatial interpolations. [ABSTRACT FROM AUTHOR]

For a system of nonlinear integral equations on the semiaxis, we study a boundary value problem whose matrix kernel has unit spectral radius. This boundary value problem has applications in various areas of physics and biology. In particular, such problems arise in the dynamical theory of -adic strings for the scalar field of tachyons, in the mathematical theory of spread of epidemic diseases, in the kinetic theory of gases, and in the theory of radiative transfer. The questions of the existence, absence, and uniqueness of a nontrivial solution of this boundary value problem are discussed. In particular, it is proved that a boundary value problem with a zero boundary conditions at infinity has only a trivial solution in the class of nonnegative and bounded functions. It is also proved that if at least one of the values at infinity is positive, then this problem has a convex nontrivial nonnegative bounded and continuous solution. At the end of this paper, examples of the matrix kernel and nonlinearity are provided that satisfy all the conditions of the proved theorems. [ABSTRACT FROM AUTHOR]

Simple Summary: Traumatic brain injuries, a primary global health concern, can lead to severe disabilities. Acute subdural haematoma is one type of traumatic brain injury resulting from the rupture of bridging veins in the brain. Existing numerical models often oversimplify these structures. This study's main goals were to understand how the cerebral vein's cross-sectional shape affects its response, more akin to real-life testing, and to introduce a hyperelastic model for these veins that captures their non-linear behaviour. The results reveal that the vein's shape significantly influences its response to stretching, highlighting the need for more realistic geometries in modelling. Additionally, hyperelastic models, like the Marlow and polynomial models, were successfully implemented. This research provides valuable insights into the mechanical properties of bridging veins and their response to tensile loading, potentially improving our understanding of traumatic brain injuries. Traumatic brain injury (TBI) is a serious global health issue, leading to serious disabilities. One type of TBI is acute subdural haematoma (ASDH), which occurs when a bridging vein ruptures. Many numerical models of these structures, mainly based on the finite element method, have been developed. However, most rely on linear elasticity (without validation) and others on simplifications at the geometrical level. An example of the latter is the assumption of a regular cylinder with a constant radius, or the geometry of the vein acquired from medical images. Unfortunately, these do not replicate the real conditions of a mechanical tensile test. In this work, the main goal is to evaluate the influence of the vein's geometry in its mechanical behaviour under tensile loading, simulating the real conditions of experimental tests. The second goal is to implement a hyperelastic model of the bridging veins where it would be possible to observe its non-linear elastic behaviour. The results of the developed finite element models were compared to experimental data available in the literature and other models. It was possible to conclude that the geometry of the vein structure influences the tensile stress–strain curve, which means that flattened specimens should be modelled when validating constitutive models for bridging veins. Additionally, the implementation of hyperelastic material models has been verified, highlighting the potential application of the Marlow and reduced polynomial (of fourth and sixth orders) constitutive models. [ABSTRACT FROM AUTHOR]

The median location problem concerns finding locations of one or several new facilities that minimize the overall weighted distances from the existing to the new facilities. We address the problem of locating one new facility with a radius r on networks. Furthermore, the radius r is flexible and the objective function is the conic combination of the traditional 1-median function and the value r. We call this problem an extensive 1-median problem with radius on networks. To solve the problem, we first induce the so-called finite dominating set, that contains all points on the underlying network and radius values which are candidate for the optimal solution of the problem. This helps to develop a combinatorial algorithm that solves the problem on a general network G = (V,E) in O(|E||V|³) time. We also consider the underlying problem with improved algorithm on trees. Based the convexity of the objective function with variable radius, we develop a linear time algorithm to find an extensive 1-median with radius on the underlying tree. [ABSTRACT FROM AUTHOR]

In this paper, we introduced s-index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum H-eigenvalue of s-index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of s-index weakly positive tensors. The final numerical example shows the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This article presents a new DIRECT-type SCABALL (scattering balls) algorithm with a new partition method for derivation-free optimization problems. It does not focus on dividing the region of interest into specific geometric shapes, but rather scatters several balls to cover it. In SCABALL, several potential optimal regions are selected at each iteration, and they are covered by smaller balls sequentially. In this way, the SCABALL ensures the everywhere dense convergence. The center points and radii of the scattered balls significantly influence the efficiency of SCABALL; therefore, the minimax designs are used in the initial and sequential stages to obtain better coverage. The SCABALL parameters, including the number of balls and their radii, were analyzed by numerical investigation. We provided the empirical choices for those parameters and found that the balls' radii can be contracted to balance efficiency and global convergence. Numerical experiments show that the SCABALL algorithm is locally biased and robust. [ABSTRACT FROM AUTHOR]

The effect of cutting edge radius (rß) on the chip formation mechanism during the finish turning of Ti6Al4V is studied. An increase in rß decreases the cutting to thrust force ratio and produces lower chip thickness due to the increase in plowing zone depth. Larger radius tools produce thinner loosely curled chips whereas tightly curled thicker chips are formed for the unprepared edge tool (36 µm). As rß increases, the chip flow angle (ß) increases and free-flowing chips with reduced up-curl but increased side and lateral curl and consequently larger resultant chip curl radius are produced. The chip serration height to overall chip thickness ratio increases whereas chip core thickness reduces owing to the increased plowing action as rß increases. Chip segmentation analysis reveals increased serration height, thermal instability, and shear localisation in the shear zone due to the increased ploughing effect beyond ~60 µm edge radius. [ABSTRACT FROM AUTHOR]

The univalent function ϕ SG (z) : = 2 / (1 + e − z) maps the open unit disk | z | < 1 onto a sigmoid domain | log (w / (2 − w)) | < 1. The sharp radii constants have been computed for the classes involving ratios f / g and g / (z p) of analytic functions f , g and p defined in | z | < 1 such that p is subordinate to ϕ SG , the other two ratios are subordinate to 1 + z , e z or z + 1 + z 2 and the quantity z f ′ / f lies in a certain region of the right half plane. [ABSTRACT FROM AUTHOR]

The motion of planetesimals initially located in the feeding zone of the planet Proxima Centauri c, at distances of 500 AU from the star to the star's Hill sphere radius of 1200 AU was considered. In the analyzed non-gaseous model, the primary ejection of planetesimals from most of the feeding zone of an almost formed planet c to distances greater than 500 AU from the star occurred during the first 10 million years. Only for planetesimals originally located at the edges of the planet's feeding zone, the fraction of planetesimals that first reached 500 AU over the time greater than 10 million years was more than half. Some planetesimals could reach the outer part of the star's Hill sphere over hundreds of millions of years. Approximately 90% of the planetesimals that first reached 500 AU from Proxima Centauri first reached 1200 AU from the star in less than 1 million years, given the current mass of the planet c. No more than 2% of planetesimals with aphelion orbital distances between 500 and 1200 AU followed such orbits for more than 10 million years (but less than a few tens of millions of years). With a planet mass equal to half the mass of the planet c, approximately 70–80% of planetesimals increased their maximum distances from the star from 500 to 1200 AU in less than 1 million years. For planetesimals that first reached 500 AU from the star under the current mass of the planet c, the fraction of planetesimals with orbital eccentricities greater than 1 was 0.05 and 0.1 for the initial eccentricities of their orbits eo = 0.02 and eo = 0.15, respectively. Among the planetesimals that first reached 1200 AU from the star, this fraction was approximately 0.3 for both eo values. The minimum eccentricity values for planetesimals that have reached 500 and 1200 AU from the star were 0.992 and 0.995, respectively. In the considered model, the disk of planetesimals in the outer part of the star's Hill sphere was rather flat. Inclinations i of the orbits for more than 80% of the planetesimals that first reached 500 or 1200 AU from the star did not exceed 10°. With the current mass of the planet c, the percentage of such planetesimals with i > 20° did not exceed 1% in all calculation variants. The results may be of interest for understanding the motion of bodies in other exoplanetary systems, especially those with a single dominant planet. They can be used to provide the initial data for models of the evolution of the disk of bodies in the outer part of Proxima Centauri's Hill sphere, which take into account gravitational interactions and collisions between bodies, as well as the influence of other stars. The strongly inclined orbits of bodies in the outer part of Proxima Centauri's Hill sphere can primarily result from bodies that entered the Hill sphere from outside. The radius of Proxima Centauri's Hill sphere is an order of magnitude smaller than the radius of the outer boundary of the Hills cloud in the Solar System and two orders of magnitude smaller than the radius of the Sun's Hill sphere. Therefore, it is difficult to expect the existence of a similarly massive cloud around this star as the Oort cloud around the Sun. [ABSTRACT FROM AUTHOR]

The multi-objective optimisation of high-speed rail profile with small radius curve is studied in the paper. A multi-objective mathematical model for rail profile optimisation of high-speed railway is established. The CN60 rail profile is parameterised into a series of generalised functions of design variables. In order to guarantee the smoothness of the rail profile and meet the maximum grinding depth of rail in China, the constraints are employed. The wheel-rail vertical clearance and equivalent conicity of wheelset are taken as objective functions, and a rail wear prediction programme is compiled. Contact line method is employed to complete the detection algorithm of wheel-rail contact points. Finally, NSGA-II Algorithm is adopted to solve the Pareto-optimal front of the optimisation model. A set of solutions are retrieved from the Pareto optimal front solution as the optimised profile. The optimised rail profile and the original rail profile are matched with the LMA wheel profile (a certain worn type of wheel profiles for EMU in China) respectively. It is testified that the rail profile could effectively reduce the rail wear and improve curving performance. The new method proposed in this paper can provide some reference for the optimisation design of high-speed rail profile with small radius curve. [ABSTRACT FROM AUTHOR]

For a given polynomial ... of degree n with complex coefficients, the Cauchy radius r0 is a unique positive root of the equation... It refers to a radius of the circular region z ≤ r0 in which all the zeros of p(z) lie. The basic aim has been to determine the smallest radius, thereby, minimizing the area of the circular region. In this present paper, we have obtained a result which gives an improvement of the Cauchy radius. Also, we produce an annular region whose center is different from the origin in which the zeros of p(z) lie. Moreover, in many cases, our results give better approximations for estimating the region of polynomial zeros than that obtained from many other well-known results. [ABSTRACT FROM AUTHOR]

A single elliptical or ellipsoidal inclusion with an arbitrary uniform eigenstrain is known to achieve a constant stress field when embedded in an elastic medium provided the edge of the medium is sufficiently far from the inclusion (i.e. the interaction between the inclusion and the edge of the medium is negligible). In this paper, we aim to answer the question as to whether there exists an inclusion of certain configuration (with a uniform eigenstrain) that remains to possess a constant stress when embedded in a bounded medium whose edge interacts significantly with it. Specifically, we consider the anti-plane shear case of an inclusion with a uniform eigenstrain in a circular medium with a traction-free edge. We derive a sufficient and necessary condition ensuring the uniformity of the stress within the inclusion, which further leads to a nonlinear system of equations with respect to an infinite group of parameters characterizing the shape of the inclusion. We obtain convergent solutions for the truncated version of the nonlinear system using numerical techniques, and illustrate the corresponding shape of the inclusion in a few numerical examples. Our results for the case corresponding to small inclusion size and small edge-inclusion distance (relative to the radius of the medium) are well-consistent with the existing results for an inclusion with uniform stress in a semi-infinite medium with a traction-free surface, while those for centrally placed inclusions achieving uniform stress capture the classical case of centric circular inclusion accurately. The results presented in this paper provide a strong evidence for the existence of inclusions possessing uniform stress in an elastic bounded domain subjected to common external boundary conditions under anti-plane shear deformation. [ABSTRACT FROM AUTHOR]

To explore the curvature dependence of solid–fluid interfacial thermodynamics, we calculate, using Grand Canonical Monte Carlo simulation, the surface free energy for a 2d hard-disk fluid confined in a circular hard container of radius R as a function of the bulk packing fraction η and wall curvature C ̄ = − 1 / R. (The curvature is negative because the surface is concave.) Combining this with our previous data [Martin et al., J. Phys. Chem. B 124, 7938–7947 (2020)] for the positive curvature case (a hard-disk fluid at a circular wall, C ̄ = + 1 / R), we obtain a complete picture of surface thermodynamics in this system over the full range of positive and negative wall curvatures. Our results show that γ is linear in C ̄ with a slope that is the same for both positive and negative wall curvatures, with deviations seen only at high negative curvatures (strong confinement) and high density. This observation indicates that the surface thermodynamics of this system is consistent with the predictions of so-called morphometric thermodynamics at both positive and negative curvatures. In addition, we show that classical density functional theory and a generalized scaled particle theory can be constructed that give excellent agreement with the simulation data over most of the range of curvatures and densities. For extremely high curvatures, where only one or two disks can occupy the container at maximum packing, it is possible to calculate γ exactly. In this limit, the simulations and density functional theory calculations are in remarkable agreement with the exact results. [ABSTRACT FROM AUTHOR]

In desert regions like the Middle East, dust has a profound impact on the environment, climate, air quality, and solar devices. The size of dust particles determines the extent of these effects. Dust deposition (DD) measurements show that coarse dust particles with geometric radius r > 10 μm comprise most of the deposited mass. Still, these particles are not represented in the current models that are tuned to fit the observed aerosol optical depth (AOD). As a result, the existing models and reanalysis products underestimate DD and dust emission (DE) almost 3 times. This is the first study to constrain the dust simulations by both AOD and DD measurements to quantify the effect of coarse and fine dust using the WRF‐Chem model. We found that, on average, coarse dust contributes less than 10% to dust shortwave radiative forcing (RF) at the surface but comprises more than 70% of DE. Annual mean net RF over the Arabian Peninsula and regional seas locally reaches −25 W m−2. Airborne fine dust particles with radii r < 3 μm are mainly responsible for the significant dimming (5%–10%) of solar radiation, cooling the surface, and hampering solar energy production. However, dust mass deposition is primarily linked to coarse particles, decreasing the efficiency of photovoltaic panels by 2%–5% per day. Therefore, incorporating coarse dust in model simulations and data assimilation would improve the overall description of the dust mass balance and its impact on environmental systems and solar devices. Plain Language Summary: Most dust emission schemes used in aerosol chemical transport models are constrained primarily by observed aerosol optical depth (AOD) and do not generate enough coarse dust particles. This study represents the first attempt to constrain the dust emission scheme using AOD and dust deposition (DD) measurements conducted at King Abdullah University of Science and Technology (KAUST). By incorporating both AOD and DD observations, we aimed to enhance the accuracy of the dust emission scheme and provide a more comprehensive understanding of dust mass balance and dynamics in the region. We found that accounting for generating coarse and giant dust particles led to a tripling of dust emissions compared with the previous estimates. We assessed the size distribution of dust particles and their impact on shortwave and longwave radiation. We quantified how particles of different sizes contributed to the regional radiative forcing and examined the implications of DD on the performance and efficiency of solar panels for optimizing solar energy systems in dust‐prone regions. Key Points: Models and reanalysis products underestimate coarse dust emission and dust deposition by 2–3 timesFine dust affects radiation, but coarse dust dominates mass deposition ratesAtmospheric dust dims solar radiation, and coarse dust causes the soiling of solar panels [ABSTRACT FROM AUTHOR]

In light of the Alexander transformation, the class of spirallike functions is significant. The characteristics of special functions also appear very frequently in Geometric function theory. In this paper, we find the radii of γ -spirallike and convex γ -spirallike of order α of certain special functions. [ABSTRACT FROM AUTHOR]

A fractional matching of a graph G is a function f : E (G) → [ 0 , 1 ] such that for any v ∈ V (G) , ∑ e ∈ E G (v) f (e) ≤ 1 , where E G (v) = { e ∈ E (G) : e is incident with v in G }. The fractional matching number of G is μ f (G) = max { ∑ e ∈ E (G) f (e) : f is a fractional matching of G }. Let k ∈ (0 , n) is an integer. In this paper, we prove a tight lower bound of the spectral radius to guarantee μ f (G) > n − k 2 in a graph with minimum degree δ , which implies the result on the fractional perfect matching due to Fan et al. (2022) [6]. For a set { A , B , C , ... } of graphs, an { A , B , C , ... } -factor of a graph G is defined to be a spanning subgraph of G each component of which is isomorphic to one of { A , B , C , ... }. We present a tight sufficient condition in terms of the spectral radius for the existence of a { K 2 , { C k } } -factor in a graph with minimum degree δ , where k ≥ 3 is an integer. Moreover, we also provide a tight spectral radius condition for the existence of a { K 1 , 1 , K 1 , 2 , ... , K 1 , k } -factor with k ≥ 2 in a graph with minimum degree δ , which generalizes the result of Miao et al. (2023) [10]. [ABSTRACT FROM AUTHOR]

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi, Phys. Rev. Fluids , vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers. [ABSTRACT FROM AUTHOR]

We review recent developments concerning the soft factorization of scattering amplitudes that arise in the large radius limit of four dimensional Anti-de Sitter (AdS4) spacetimes. This includes the presence of AdS radius dependent corrections of known flat spacetime soft factors and their implication on the relationship between soft theorems and Ward identities of the boundary conformai field theory. [ABSTRACT FROM AUTHOR]

We explore computationally tractable deformations of the SYK model. The deformed theories are described by the sum of two SYK Hamiltonians with differing numbers, q and q ~ , of interacting fermions. In the large N limit, employing analytic and numerical tools, we compute finite temperature correlation functions and thermodynamic quantities. We identify a novel analytically solvable model in the large q limit. We find that, under certain circumstances, the thermal RG flow in the strongly coupled infrared phase exhibits two regions of linear-in-temperature entropy, which we interpret in terms of Schwarzian actions. Using conformal perturbation theory we compute the leading relevant correction away from the intermediate near-conformal fixed point. Holographic spacetimes in two spacetime dimensions that reproduce the thermodynamics of the microphysical theory are discussed. These are flow geometries that interpolate between two Euclidean near-AdS2 spacetimes with different radii. The Schwarzian soft mode corresponding to the AdS2 region in the deep interior resides entirely within the geometric regime. [ABSTRACT FROM AUTHOR]

A solution of the Laplace equation in cylindrical coordinates is presented for a bounded cylinder with the known height and radius which is subject to inhomogeneous mixed boundary conditions of the third and second kinds on the surface. On the other surface, unmixed boundary conditions of the first or second kind are given. Through separation of variables, the Hankel integral transform, and the dual series equations, the solution of the mixed problem is reduced to solving the Fredholm integral equation of the second kind. [ABSTRACT FROM AUTHOR]

Most works on Support Vector Regression (SVR) focus on kernel or loss functions, with the corresponding support vectors obtained using a fixed-radius ε -tube, affording good predictive performance on datasets. However, the fixed radius limitation prevents the adaptive selection of support vectors according to the data distribution characteristics, compromising the performance of the SVR-based methods. Therefore, this study proposes an "Alterable ε i -Support Vector Regression" (A ε i -SVR) model by applying a novel ε , named "Alterable ε i ," to the SVR model. Based on the data point sparsity at each location, the model solves the different ε i at the corresponding position, and thus zoom-in or zoom-out the ε -tube by changing its radius. Such a variable ε -tube strategy diminishes noise and outliers in the dataset, enhancing the prediction performance of the A ε i -SVR model. Therefore, we suggest a novel non-deterministic algorithm to iteratively solve the complex problem of optimizing ε i associated with every location. Extensive experimental results demonstrate that our approach can improve the accuracy and stability on simulated and real data compared with the baseline methods. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , the generalized distance matrix D α (G) is defined as D α (G) = α Tr (G) + (1 − α) D (G) , 0 ≤ α ≤ 1 , where Tr (G) and D (G) are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of G. In this paper, we will study the spectral radius of the generalized distance matrix D α (G) of a connected graph G. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature. [ABSTRACT FROM AUTHOR]

We give some new bounds for the numerical radii of the off-diagonal parts of 2 × 2 operator matrices. A refinement of the triangle inequality for the operator norm is also given. [ABSTRACT FROM AUTHOR]

In this paper, a model based on constraint convex optimization is proposed for the radiation pattern synthesis in cylindrical and spherical microstrip conformal array antennas. The optimization algorithm is modeled to decrease the Euclidean distance between the obtained radiation pattern of the deformed array and the desired radiation pattern of the planar array by calculating the suitable amplitude and phase values for the individual array patches. Four prototypes of 2 × 2 , 3 × 3 , 4 × 4 , and 5 × 5 microstrip patch array antenna are used as the radiating elements which are deformed from planar arrangement and placed on the faces of prescribed spherical and cylindrical shapes with various radii of curvatures. The results reveal that the performance of optimization algorithm performs better in 5 × 5 array, followed by 4 × 4 array, 3 × 3 array, and 2 × 2 array. However, 5 × 5 array provides more degrees of freedom for tuning the radiation pattern in desired form. The 5 × 5 array computes suitable amplitude and phase values for the radiating elements of the array, which restore the radiation pattern in terms of mainlobe reconstruction, null locations, sidelobe levels, up to the deformation of 20   cm in both spherical and cylindrical configurations of conformal array. The proposed optimization technique when applied to conformal antenna array offers fast computational speed and good convergence accuracy for radiation pattern synthesis, which can be useful for various engineering applications. [ABSTRACT FROM AUTHOR]

We propose a method for controlling the formation of microarray structures on the surface by using an electric field. In this study, a phase field model is employed to describe the formation of microstructures under the influence of the electric field. Initially, we utilize Cahn–Hilliard equation, Navier–Stokes equation, and Gaussian theory to construct the phase field model. Subsequently, we obtain the theoretical solution for the partial differential equations and apply numerical methods such as semi-implicit Fourier spectral and semi-implicit difference method. Finally, we establish the numerical model to observe the morphological changes on the polymer surface by changing the electric field strengths, the distribution and the radius of the electrode plate. Simulated observations show that the electric field induce the formation of microarray structures on the polymer surface successfully, and the microarray structures match the shape of the electric field. Additionally, higher electric field strength promotes the formation of morphological changes obviously. The distribution of microarray structures is controlled by changing the electric field distribution. These studies provide a theoretical basis for the fabrication of microarray structures. [ABSTRACT FROM AUTHOR]

Recent observations demonstrate that misalignments and other out-of-plane structures are common in protoplanetary discs. Many of these have been linked to a central host binary with an orbit that is inclined with respect to the disc. We present simulations of misaligned circumbinary discs with a range of parameters to gain a better understanding of the link between those parameters and the disc morphology in the wave-like regime of warp propagation that is appropriate to protoplanetary discs. The simulations confirm that disc tearing is possible in protoplanetary discs as long as the mass ratio, μ, and disc–binary inclination angle, i , are not too small. For the simulations presented here, this corresponds to μ > 0.1 and i ≳ 40°. For highly eccentric binaries, tearing can occur for discs with smaller misalignment. Existing theoretical predictions provide an estimate of the radial extent of the disc in which we can expect breaking to occur. However, there does not seem to be a simple relationship between the disc properties and the radius within the circumbinary disc at which the breaks appear, and furthermore the radius at which the disc breaks can change as a function of time in each case. We discuss the implications of our results for interpreting observations and suggest some considerations for modelling misaligned discs in the future. [ABSTRACT FROM AUTHOR]

We develop a self-consistent and accurate halo model by partitioning matter according to the depletion radii of haloes. Unlike conventional models that define haloes with the virial radius while relying on a separate exclusion radius or ad hoc fixes to account for halo exclusion, our model distributes mass across all scales self-consistently and accounts for both the virialized and non-virialized matter distribution around each halo. Using a cosmological simulation, we show that our halo definition leads to very simple and intuitive model components, with the one-halo term given by the Einasto profile with no truncation needed, and the halo–halo correlation function following a universal power-law form down to the halo boundary. The universal halo–halo correlation also allows us to easily model the distribution of unresolved haloes as well as diffuse matter. Convolving the halo profile with the halo–halo correlation function, we obtain a complete description of the halo–matter correlation across all scales, which self-consistently accounts for halo exclusion at the transition scale. Mass conservation is explicitly maintained in our model, and the scale dependence of the classical halo bias is easily reproduced. Our model can successfully reconstruct the halo–matter correlation function within an accuracy of 9 per cent for halo virial masses in the range of 1011.5 h −1 M⊙ < M vir < 1015.35 h −1 M⊙ at z  = 0, and covers the radial range of 0.01 h −1 Mpc < r < 20 h −1 Mpc. We also show that our model profile can accurately predict the characteristic depletion radius at the minimum bias and the splash-back radius at the steepest density slope locations. [ABSTRACT FROM AUTHOR]

Extending certain scalar and norm inequalities, we present new inequalities for the numerical radius, which generalize and refine some known results. Applications of the obtained inequalities include a new original proof of the matrix arithmetic-geometric mean inequality and certain extensions of some well-established results from the literature for products of matrices. [ABSTRACT FROM AUTHOR]