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R. Lyons, R. Pemantle and Y. Peres (Ann. Probab. 24 (4), 1996, 1993–2006) conjectured that for a Cayley graph G with a growth rate gr(G) > 1, the speed of a biased random walk exists and is positive for the biased parameter λ ∈ (1, gr(G)). And Gabor Pete (Probability and geometry ´ on groups, Chaper 9, 2024) sheds light on the intricate relationship between the spectral radius of the graph and the speed of the biased random walk. Here, we focus on an example of a Cayley graph, a free product of complete graphs. In this paper, we establish the continuity of the spectral radius of biased random walks with respect to the bias parameter in this class of Cayley graphs. Our method relies on the Kesten-Cheeger-Dodziuk-Mohar theorem and the analysis of generating functions. [ABSTRACT FROM AUTHOR]

A dominating set S in a graph G is called a radius dominating set if for each vertex V ∈ v -- S there is at least one vertex in u ∈ S S such that d(u, v) = rad(G). The cardinality of the minimum radius dominating set is called the radius domination number of G, denoted by γrd(G). In this article, we study the properties of this parameter. The radius domination number of some families of standard graphs are obtained and the bounds are estimated. [ABSTRACT FROM AUTHOR]

In the presence of advection at a constant flow rate in a rectilinear geometry, the properties of planar A + B → C reaction fronts feature the same temporal scalings as in the pure reaction–diffusion case. In a radial injection geometry where A is injected into B radially at a constant flow rate Q, temporal scalings are conserved, but the related coefficients depend on the injection flow rate Q and on the ratio γ of initial concentrations of the reactants. We show here that this dependence of the front properties on the radial velocity allows us to tune the amount of product obtained in the course of time by varying the flow rate. We compare theoretically the efficiency of the rectilinear and radial geometries by computing the amount of product C generated in the course of time or per volume of reactant injected. We show that a curve γc(Q) can be defined in the parameter space (γ, Q) below which, for similar experimental conditions, the total amount of C is larger in the radial case. In addition, another curve γ*(Q) < γc(Q) can be defined such that for γ < γ*, the total amount of C produced is larger in the radial geometry, even if the production of C per unit area of the contact interface between the two reactants is larger in the rectilinear case. This comes from the fact that the length of the contact zone increases with the radius in the radial case, which allows us to produce in fine more product C for a same injected volume of reactant or in reactors of a same volume than in the rectilinear case. These results pave the way to the geometrical optimization of the properties of chemical fronts. [ABSTRACT FROM AUTHOR]

Poor coordination between geometric elements of horizontal curves can lead to unsafe speed. This study evaluated the effect of horizontal curve geometry on vehicle speed distribution for a four-lane divided highway and developed prediction models for mean speed and standard deviation. Spot speed data of passenger cars (PC), heavy commercial vehicles and light commercial vehicles were collected at 24 horizontal curves. Statistical analysis of the speed data of randomly selected 10 sites revealed PC as the critical vehicle category. Hence, the effects of horizontal curve geometry on PC speed were analyzed and radius was identified as the most influencing parameter. Radius beyond 400 m had similar influence on speed distributions. Appropriate models were developed using regression analysis. Degree of curvature was found as the most significant predictor. Benefits of the proposed models over traditional 85th percentile speed in designing and evaluating horizontal curves for consistency and safety were discussed. [ABSTRACT FROM AUTHOR]

Gear hobbing is a well-established manufacturing process for cylindrical spur gears. The cutting edge of a hobbing tool is, among others, characterized by the cutting edge radius and the form-factor K. The magnitude of these parameters is ideally chosen based on the machining conditions given by the workpiece and cutting material and the cutting parameters as well as the gear and tool geometry. However, the influence of the cutting edge geometry on tool life and wear behavior is hardly known, which complicates an optimized tool design. Furthermore, the preparation process regarding the coating thickness distribution on the wear behavior is equally relevant. Therefore, the objective was to identify the influence of the cutting edge radius, the form-factor K, and the preparation process on the wear behavior of gear hobbing tools made of powder metallurgical high-speed steel (PM-HSS). Fly-cutting trials were performed as an analogy process for gear hobbing in order to study the wear behavior and identify the respective tool lives. The trials indicated that the form-factor K influences the wear behavior, while a variation of the cutting edge radius did not have a significant effect. A homogenous coating thickness could extend the tool life significantly. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ > 0 , proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple (ϵ , ϕ , g , f). Our aim is to study how the solution depends on (ϵ , ϕ , g , f) , especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple (ϵ , ϕ , g , f) multiplied by the factor 1 / ϵ n − 2 . In case of dimension n = 2 , we have to add log ϵ times the integral of f / 2 π. [ABSTRACT FROM AUTHOR]

This paper presents a novel geometrical design of inserts to realize high-efficiency smooth-surface high-chatter-stability turning. Inserts with a nose radius are used in turning to realize a smooth surface since the theoretical surface roughness R t h = f 2 / (8 r) where f is the feed rate and r is the radius of the nose part. Since the size of the nose radius is limited because of the chatter stability, i.e. a large nose radius tends to cause chatter, the feed rate has to be kept small so that the acquired roughness is acceptable. Hence, the efficiency is limited. In this study, a novel turning insert geometry is proposed to realize high chatter stability along with a high efficiency and a smooth surface. The proposed insert geometry is verified by analyses and experiments. • Insert geometry to realize high efficiency, smooth surface, and high chatter stability. • Large nose radius and small effective regenerative cutting width are combined. • Higher gain margin and same surface roughness is realized compared to conventional inserts. [ABSTRACT FROM AUTHOR]

The renormalized quantum stress–energy tensor 〈 T μ ν 〉 ren for a massive spinor field around global monopoles is constructed within the framework of Schwinger–DeWitt approximation, valid whenever the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results obtained show that the quantum massive spinor field in the global monopole spacetime violates all the pointwise energy conditions. [ABSTRACT FROM AUTHOR]

Swirling gas flow is an important topic of research that helps in the design of rockets, atomizers,and gas turbine combustors. In the present work, swirl flow inside annular geometries with constant and varying cross-sectional areas are examined using computational fluid dynamics (CFD). Effects of changing the flow and geometric parameters on the swirl behaviour were studied. Reynolds number was found to increase the swirl number in straight and diverging cross-sectional geometries while no significant effect of Reynolds number was observed in converging cross-sectional geometry. Radius ratio, defined as the ratio of the inner to the outer radius of the annular geometry, was found to have a significant effect on the swirl number. Decreasing the radius ratio, in straight and diverging annular geometry decreases the swirl number, however, an opposite trend was observed in the case of converging annular geometry due to significant increase in axial velocity as a result of reduced cross-sectional area. Increasing the swirler vane angle increased the swirl number. At higher vane angles of 60° and 70°, a recirculation zone is developed near the exit of the swirler. Using small cone angles was found to lower the swirl decay rate in converging and diverging nozzles. [ABSTRACT FROM AUTHOR]

Abstract Let λ (D) be the A α spectral radius of digraph D , and let G n r be the set of digraphs with order n and dichromatic number r. In this paper, we characterize the digraph which has the maximal A α spectral radius in G n r. Moreover, we also determine the unique digraph which attains the maximum (resp. minimum) A α spectral radius among all strongly connected bicyclic digraphs. [ABSTRACT FROM AUTHOR]

Introduction: There has been a curiosity about the spheroid geometry for PET scanners developments since several years ago, therefore in this study, we are aiming to evaluate the performance of this geometry and compare its performance with cylindrical geometry using Monte Carlo simulation. Methods: We simulated a spheroid geometry with a radius of 199 mm, patient bore with of radius of 175 mm, which is compatible with brain size. In second design, cylindrical geometry was simulated with transaxial FOV and ring radius of 175 mm as well. Photon detection efficiency (PDE), NEMA line source sensitivity, spatial resolution and Derenzo phantom image quality were analyzed. Results: We obtained PDE about 21.7% versus 23.8% in 250-750 keV and 19.5% versus 21.3% in 410-613 keV for point source in center of FOV for spheroid and cylindrical PET respectively. The results of NEMA sensitivity measurements indicate 3.29 kcps/MBq versus 3.64 kcps/MBq for spheroid and cylindrical designs. The spatial resolution (FWHM) calculations using MLEM reconstruction algorithm show around 1.6 mm for transvers and axial resolution for point source placed in center of FOV for both scanners. Also we found for spheroid and cylindrical designs 4.8 and 2.7 mm versus 4 and 3.6 mm as transvers and axial mean resolution for off-center point sources. Conclusion: Performance evaluation study indicates that the spheroid geometry delivers better axial resolution whereas cylindrical design can still provide higher sensitivity and transvers spatial resolution than the spheroid geometry PET with same scanner bore size. [ABSTRACT FROM AUTHOR]

Let ra; rb; rc be the radii of the tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In this article, we prove some relations for the numbers ra; rb; rc. [ABSTRACT FROM AUTHOR]

In this study we investigate the sharp radius of starlikeness of subclasses of Ma and Minda class for the ratio of analytic functions which are related to limaçon functions. This survey is connected also to the first-order differential subordinations. In this context, we get the condition on β for which certain differential subordinations associated with limaçon functions imply Ma and Minda starlike functions. Simple corollaries are provided for certain examples of our results. Finally, we present several geometries related to our study. [ABSTRACT FROM AUTHOR]

Mathematical model for designing the surface geometry of the Roots pump rotor based on the Cassini oval principle was derived. The polar coordinate system was used, and the radius vector, the direction of which was set by the φ angle, characterizes the location of the point on the surface of the rotor. The distance of this point from the axis of rotor rotation was set by the calculated value of the ρR polar radius vector. The γ angle of rotors rotation characterizes their mutual orientation in the plane of rotation. Peculiarities of the choice of the a and b parameters that satisfy the shape of the rotor surface geometry are considered. An example of rotor geometry is given for rotor radius R = 50 mm, rotor rounding radius r = 20 mm, parameters a = 33.166 and b = 28. Rotor geometry depends on normalized parameters of a and b, which are constant for a given shape of the surface and constructive dimensions. A mathematical model of the usable cross-sectional area of the pump has been developed. The usable cross-sectional area of the pump was simulated by the geometry of the rotors. The area of the rotor was determined by the geometry of the surface, which was described by an elliptic integral of the 2nd kind. The usable cross-sectional area for the given parameters is modelled. The results of simulation in the form of graphical dependences are given. Parameters a and b must meet the condition of √2/2 < b/a < 1. Under such conditions, the geometry of the rotor surface will be a Cassini oval. The rotation of the two rotors against each other will be by rolling one surface over another. [ABSTRACT FROM AUTHOR]

Exact and approximate mathematical models for the effects of sample transparency on the powder diffraction intensity data are examined. Application of the formula based on the first-order approximation about the deviation angle is justified for realistic measurement and computing systems. The effects of sample transparency are expressed by double convolution formulas applying two different scale transforms, including three parameters, goniometer radius R, penetration depth μ−1, and thickness of the sample t. The deconvolutional treatment automatically recovers the lost intensity and corrects the peak shift and asymmetric deformation of peak profile caused by the sample transparency. [ABSTRACT FROM AUTHOR]

One of the challenging problems from the geometry of (normed or) Minkowki spaces is the question of whether the unit ball must be an ellipsoid if it is a solution of the corresponding isoperimetric problem. The inner and outer radii of the unit ball with respect to the corresponding isoperimetrix (represented in terms of cross-section measures) will be used to establish a result on this problem for a specific measure. Some new minmax and maxmin inequalities for centered convex bodies will also be established. [ABSTRACT FROM AUTHOR]

Measuring ear-canal absorbance and compensating for effects of the ear-canal acoustics on otoacoustic-emission measurements using an ear probe rely on accurately determining its acoustic source parameters. Using pressure measurements made in several rigid waveguides and models of their input impedances, a conventional calibration method estimates the ear-probe Thévenin-equivalent source parameters via a least-squares fit to an over-determined system of equations. Such a calibration procedure involves critical considerations on the geometry and number of utilized calibration waveguides. This paper studies the effects of calibration-waveguide geometry on achieving accurate ear-probe calibrations and measurements by systematically varying the lengths, length ratios, radii, and number of waveguides. For calibration-waveguide lengths in the range of 10–60 mm, accurate calibrations were generally obtained with absorbance measurement errors of approximately 0.02. Longer waveguides resulted in calibration errors, mainly due to coincident resonance frequencies among waveguides in the presence of mismatches between their assumed and actual geometries. The accuracy of calibrations was independent of the calibration-waveguide radius, except for an increased sensitivity of wider waveguides to noise. Finally, it is demonstrated how reformulating the over-determined system of equations to return the least-squares reflectance source parameters substantially reduces calibration and measurement errors. [ABSTRACT FROM AUTHOR]

A trajectory-front for a surface magnetic field is formed by terminuses of trajectory-segments of given arc-radius which are emanating from a given point. In order to show how trajectories are spreaded we give estimates of their arc-lengths of trajectory-fronts. [ABSTRACT FROM AUTHOR]

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We consider the notion of cosmological symmetry, i.e. spatial homogeneity and isotropy, in the field of teleparallel gravity and geometry, and provide a complete classification of all homogeneous and isotropic teleparallel geometries. We explicitly construct these geometries by independently employing three different methods, and prove that all of them lead to the same class of geometries. Further, we derive their properties, such as the torsion tensor and its irreducible decomposition, as well as the transformation behavior under change of the time coordinate, and derive the most general cosmological field equations for a number of teleparallel gravity theories. In addition to homogeneity and isotropy, we extend the notion of cosmological symmetry to also include spatial reflections, and find that this further restricts the possible teleparallel geometries. This work answers an important question in teleparallel cosmology, in which so far only particular examples of cosmologically symmetric solutions had been known, but it was unknown whether further solutions can be constructed. [ABSTRACT FROM AUTHOR]

Interstellar Boundary Explorer (IBEX) observations of the "ribbon" of enhanced energetic neutral atom (ENA) fluxes show that it is a persistent feature that approximately forms a circle in the sky, likely formed from secondary ENAs whose source lies outside the heliopause. The IBEX ribbon's geometry (radius and center) depends on ENA energy and is believed to be influenced by the draping of the ISMF and the latitudinal structure of the SW. In this study, we demonstrate that the ribbon's geometry also depends on the pitch-angle scattering rate of ions outside the heliopause, which we simulate under strong and weak-scattering limits. The ribbon radius in the weak-scattering model is ∼4° larger than IBEX observations at most energies, and the strong-scattering model produces radii statistically consistent with IBEX at 1.1–2.7 keV. The simulated ribbon center is shifted between ∼2° and 5° along the B–V plane away from the IBEX center for the weak and strong limits, respectively, suggesting that the pristine ISMF far from the heliosphere is shifted ∼2°–5° away from our simulated ISMF toward the VLISM inflow direction. However, the magnitude needs to be decreased from ∼3 to 2 μG for the weak-scattering model to be consistent with the IBEX ribbon radius, which seems unlikely. We also find that the presence of interstellar He does not significantly affect the ribbon in the strong-scattering limit but yields weaker agreement with data in the weak limit. Our results slightly favor the strong-scattering limit for the ribbon's origin. [ABSTRACT FROM AUTHOR]

In this paper, we shall give an n-dimensional generalization of a geometric inequality posed by Jan Ciach [1]. [ABSTRACT FROM AUTHOR]

Considering the movement track of the cutting head of the longitudinal roadheader directly affected the shape characteristics of the formed surfaces of the roadway, this paper analyzed the formation mechanism of the outer contours of the roadway obtained by the cutting head cutting the roadway, determined the coordinates of the cutting head of the roadheader in the spatial coordinate system of the roadway, and onstructed the mathematical model of the cutting head cutting the outer contours of the roadway. Through numerical calculation, the influences of feed rate, cutting angle, cutting head shape, cutting head radius and cutting head cone angle on the cutting surface topography were analyzed. The results show that the surfaces of the tunnel are the most uneven when they are cut by the ball crown + cyiinder type cutting head, and the tunnel surfaces are the most flat when they are cut by the ball crown +cone +cyiinder type cutting head. With the decrease of cutting angle, the decrease of cutting feed, the increase of ball crown radius of cutting head and the increase of cone angle of cutting head, the flatness of roadway obtained by cutting ss higher. The larger the cutting angle Is, the more necessary the roadway flatness controls accurately. [ABSTRACT FROM AUTHOR]

This paper focuses on analyzing the effect of geometrical parameters on structural performance of the ventilated brake disc. Multi-objective optimization through response surface methodology was deployed for improving the structural performance of ventilated brake discs. Simulation runs were designed based on central composite design technique. The second order regression models correlating the geometry parameters with maximum deformation and equivalent stress were developed. ANOVA was performed to test the significance of disc geometry parameters. The deformation and equivalent stress were influenced by flange outer peripheral radius. While the spigot radius had a significant effect on the deformation but not on equivalent stress. Also, the mounting surface radius influenced the equivalent stress developed on the ventilated brake disc rotor. The multi-objective optimization of geometrical characteristics for minimum deformation (4.2332 µm) and minimum equivalent stress (4.00989 MPa) yielded significant reduction in total deformation and equivalent stress i.e., 10.28% and 9.12% respectively at optimal levels of geometrical parameters. [ABSTRACT FROM AUTHOR]

We study the Gibbs measure of mixed spherical p‐spin glass models at low temperature, in (part of) the 1‐RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius Nρ*, where ρ*2 is the rightmost point in the support of the overlap distribution. We also show that the relevant critical points are pairwise orthogonal for two different low temperatures. This allows us to explain why temperature chaos occurs for those models, in contrast to the pure spherical models. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The main objective of this study is to characterize the effects of the pre-hole geometry on the countersinking process. The final shape of the countersunk hole and the punch load were investigated. A finite element model was developed using ABAQUS/Standard. A configuration with maintained blank-holder was used for this study. The adopted model was performed with an elastoplastic behavior and isotropic hardening rule for the workpiece. In this study, an approach based on the definitive screening design (DSD) was performed to perform a parametric study and a statistical analysis to investigate the effects of the geometrical parameters of the pre-hole and their interactions on the countersinking process. The evolution plot of the parameters of the final countersunk hole shape, during the process, was an additional tool to better understand the obtained results. It was concluded that the inner radius of the pre-hole had the most important impact on this final shape. In addition, it was found that the punch load was more affected by the fillet radius of the pre-hole than the other parameters. Concerning the thickness of the workpiece, it had the lowest influence on the process. The experiments were carried out and the obtained results were in good agreement with the numerical ones. [ABSTRACT FROM AUTHOR]

Las pistas cubiertas que se usan en la temporada invernal de atletismo tienen una peculiaridad, y es que su parte curva está peraltada. Por ello, este artículo fija su atención en la forma que tienen dichas pistas, compuestas por rectas, arcos de circunferencia y pequeños tramos de transición que sirven para suavizar la curvatura. Es ahí donde se utiliza la clotoide, una curva que permite hacer transiciones graduales entre una recta y una circunferencia. [ABSTRACT FROM AUTHOR]

It is well-known that the reciprocal of the radius of the inscribed circle of a triangle is equal to the sum of the reciprocals of the radii of the three escribed circles. I generalize this theorem for an n-dimensional simplex. In three or higher dimensions, there are many types of spheres tangent to a simplex. I characterize the existence and uniqueness of such spheres and derive formulas connected with their radii. [ABSTRACT FROM AUTHOR]

The apex field enhancement factor (AFEF) γ a of a cylindrical emitter depends sensitively on its end-cap geometry. The hemispherical end-cap is well studied due to its simplicity, but, in general, a cylindrical emitter may terminate in a variety of end-cap shapes. It is well known that the AFEF depends on the ratio h / R a , where h is the total height of the emitter and R a is the apex radius of curvature. The authors show here that there can be a large variation in γ a as the end-cap geometry is altered while keeping h / R a fixed. They carry out a systematic numerical study and determine an approximate formula for γ a in terms of measurable end-cap geometry parameters such as its height H , the radius of the cylinder R , and the apex radius of curvature R a. They show that the formula is robust and can predict the net field emission current with errors generally less than 40 %. [ABSTRACT FROM AUTHOR]

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds X that can be expressed as a semidirect product of R² with R endowed with a left invariant metric. For any such compact minimal surface M, we provide an a priori radius estimate which depends only on the maximum distance of points of the boundary ∂M to a vertical geodesic of X. We also give a generalization of the classical Radó theorem in R³ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in X, and we study the geometry, existence, and uniqueness of this type of Plateau problem. [ABSTRACT FROM AUTHOR]

We investigate a family of regression problems in a semisupervised setting. The task is to assign real-valued labels to a set of n sample points provided a small training subset of N labeled points. A goal of semisupervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius ε(n), to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider discrete p-Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases while the number of training points remains fixed. A delicate interplay between the regularizing nature of the functionals and the nonlocality inherent to the graph constructions is uncovered. Rigorous, almost optimal, ranges on the scaling of ε(n) for asymptotic consistency are obtained and in these admissible ranges it is shown that mini-mizers of the discrete functionals converge uniformly to the desired continuum limit. Furthermore, we discover that, for the standard model, there is a restrictive upper bound on how quickly ε(n) must converge to zero as n → ∞. A new model is introduced, which is as simple as the original model, but overcomes this restriction. [ABSTRACT FROM AUTHOR]

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its centers and a radius function r(t) and canal surface is parameterized through Frenet frame of the spine curve C(t). If the radius function r(t) = r is a constant, then the canal surface is called a tube or tubular surface. In this work, we investigate tubular surface with Bishop frame in place of Frenet frame and afterwards give some characterizations about special curves lying on this surface [ABSTRACT FROM AUTHOR]

The bony geometry of the distal femoral condyles may have a significant influence on knee joint kinematics. The aim of this study was to analyze the relationship between the size of the medial and lateral femoral condyles in different planes. Seventy-four three-dimensional (3D) CT reconstructions of 37 patients with ACL intact and contralateral ACL reconstructed knees were used and the data were imported into a graphical software program. The radii of the medial and lateral femoral condyles were analyzed in the sagittal, coronal, and axial planes by digitally reconstructed circular arcs along the bony condylar profiles marked with multiple digital surface points. Intra- and interobserver testing was performed. In the intact knees the average sagittal radius of the distal medial and lateral femoral condyles was similar. There was a significant difference between the radii of the distal medial femoral condyles compared to lateral femoral condyles in the coronal plane (22.4 vs. 27.8 mm, P < 0.001) as well as between the radii of the medial femoral condyles in the axial plane in 90° knee flexion compared to the lateral femoral condyles (21.3 vs. 18.3 mm, P < 0.001). The average radius of the medial femoral condyles was significantly smaller in extension compared to 90° of flexion (21.2 vs. 22.4 mm, P = 0.05) and the average radius of the lateral femoral condyles was significantly larger in extension compared to 90° of flexion (27.8 vs. 18.3 mm, P < 0.001). The 37 ACL reconstructed knees demonstrated similar radii in all three planes when compared to the intact knees without any significant difference. The described method of assessing the architecture of the distal femoral condyles is non-invasive, reproducible, and provides reliable geometric parameters necessary for the 3D reconstruction of the femoral geometry in vivo. The radii of the FC were similar in the sagittal planes but demonstrate a significant asymmetry in the axial and coronal planes. The average radius of the lateral femoral condyles was significantly larger in extension whereas the radius of the medial femoral condyles was significantly larger in flexion. We did not find any significant difference in the shape of the femoral condyles in ACL intact and contralateral ACL reconstructed knees indicating that the geometry of the femoral condyles might not influence the injury mechanism of ACL rupture. The asymmetry between the femoral condyles may be considered when designing new anatomical femoral components in knee arthroplasty. [ABSTRACT FROM AUTHOR]

Optimization approach to define geometry (parameters) of the radial matching section in an RFQ accelerator is suggested. This approach gives wider opportunities for the design of the radial matching section because it does not have certain prescribed laws of variation of focusing strength along the section. [ABSTRACT FROM AUTHOR]

We determine upper bounds for the least radius of a ball in which a given set is a Pompeiu set (the set considered is a half right circular cone). The obtained estimates significantly improve known results. [ABSTRACT FROM AUTHOR]

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points. [ABSTRACT FROM AUTHOR]

Formulas for an acoustic channel are obtained for an angle probe and a reflector in the form of dihedral angle, which is formed by the end and cylindrical surfaces of a specimen—a round cylinder. It has been shown that, when the wave path in a prism is much shorter than the cylinder’s radius, the decrease that occurs in the amplitude of the echo-signal from the dihedral angle (without additional repeated reflections) owing to a wave-front divergence is the same as that observed during the reflection from a plane surface. The calculated and experimental data are compared. [ABSTRACT FROM AUTHOR]

The article offers information on the radius of a circle. It presents the quadratic formula of a circle which gives one the option on which to reject since the other one is positive. It notes that not all integral values will yield a positive integral value for the radius. Also, an integral solution is called primitive if all the integral values are equal to one. Several charts and diagrams depicting a radius of a circle are presented.

Abstract: In this paper, we study the performance of Saitou and Nei''s neighbor-joining method for phylogenetic reconstruction. We show that the edge radius of the method is . This improves an result by Atteson [The performance of neighbor-joining methods of phylogenetic reconstruction, Algorithmica 25 (1999) 251–278] and Xu et al. [A lower bound on the edge radius of Saitou and Nei''s method for phylogenetic reconstruction, Inform. Process. Lett. 94(5) (2005) 225–230]. Previously, only an upper bound and a lower bound were known. [Copyright &y& Elsevier]

We introduce the concept of (k, l)-radius of a graph and prove that for any fixed pair k, l the (k, l)-radius is equal to 2(2k) - (2l) for almost all graphs. Since for k = 2 and l = 0 the (k, l)-radius is equal to the diameter, our result is a generalization of the known fact that almost all graphs have diameter two. [ABSTRACT FROM AUTHOR]

Abstract: Let T (n, d) be the set of trees on n vertices with diameter d. In this paper, the first spectral radii of trees in the set T (n, d) (3⩽ d ⩽ n −4) are characterized. [Copyright &y& Elsevier]

Focuses on early Greek trigonometry. Evidence for Greek use of the scale of radius; Models used by astronomer Hipparchus to analyze a trio of eclipses; Summary of the geometrical solution of the lunar eclipse trio problem.

Abstract: In this paper we consider the following problem: Of all trees of order n with k pendant vertices (n, k fixed), which achieves the maximal spectral radius? We show that the maximal spectral radius is obtained uniquely at Tn,k, where Tn,k is a tree obtained from a star K1,k and k paths of almost equal lengths by joining each pendant vertex of K1,k to an end vertex of one path. We also discuss the spectral radius of Tn,k and get some results. [Copyright &y& Elsevier]