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A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

The problem of the interaction of a shock wave with an increased-density gas bubble in the neighborhood of the wall is investigated on the basis of numerical simulation of Euler’s equations in the two-dimensional plane formulation. The process of shock wave refraction and focusing, namely, reflection of transverse shocks from the plane of symmetry of flow and the wall, is described. It is found that qualitatively different flow regimes, in which the wave is focused in the plane of symmetry before or after the beginning of wave reflection from the wall, can be implemented depending on the constitutive parameters of the problem. It is shown that the presence of a heavy bubble in the neighborhood of the wall strengthens multiply the pulse shock-wave loading on the wall. The maximum pressure reached on the wall is found as a function of the impinging wave Mach number, the bubble gas density, and the initial distance between the bubble and the wall. In some cases such a dependence is essentially nonmonotonic with respect to the bubble gas density and the distance between bubble and wall.

Using the models of elastic and dielectric continuum, the system of differential equations is obtained, the exact analytical solutions of which describe the elastic displacement of the medium for nitride-based semiconductor nanostructure and the piezoelectric effect, which is caused by shear acoustic phonons. The theory of the shear acoustic phonons spectrum caused by the piezoelectric potential was developed. It is shown that shear acoustic phonons do not interact with electrons due to the deformation potential, but such interaction can occur due to the piezoelectric potential. Using the method of temperature Green’s functions and Dyson equation, expressions that describe the temperature dependences of the electronic level shifts and their decay rates are obtained. Calculations of the spectra of electrons, acoustic phonons, and characteristics that determine their interaction at different temperatures were carried out using the example of physical and geometric parameters of typical AlN/GaN nanostructure, which can function as an element of a separate cascade of a quantum cascade laser or detector.

This article presents a simple and effective decoupling approach based on the common mode (CM) and differential mode (DM) analysis to solve decoupling problem between two symmetrical antenna elements. Especially, the concept of the self-resonance frequencies of CM and DM is utilized in antenna decoupling analysis. It is found that when the self-resonance frequencies of CM and DM are the same, strong mutual coupling between two symmetrically and closely placed two-port antennas can be completely eliminated. The difference between self-resonance frequency and resonance frequency is discussed; and the method of obtaining the self-resonance frequencies of CM and DM is given. Since the proposed antenna decoupling judgment condition is straightforward and simple, it can effectively deal with any kind of closely placed antenna system with a plane of symmetry; and obtain good decoupling effect simply and easily. The effectiveness, feasibility and advantage of the proposed decoupling condition for the CM and DM analysis are demonstrated through several symmetrical two-port antenna examples. The decoupling approach proposed in this article can undoubtedly make the common mode and differential mode analysis for solving antenna decoupling problems more effective and practical.

All the previous studies on the cavity flow are confined to either the study of its 2D or its 3D configuration in isolation. In this study, we endeavour to gain some physical insight into the corner vortices from the perspective of the flow topology in the 2D vis a vis 3D driven cavity by employing some recent developments in the field of topological fluid dynamics. The computed flow is post-processed to identify critical points in the flow field leading to the prediction of separation, reattachment and vortical structures in the flow. The limit cycles in the plane of symmetry of the 3D flow representing the vortices are found to be stable ones. The Poincare–Bendixson formula is used to validate the computed flow, i.e., the possible number of critical points in the 2D cavity identified by us from the computation. The topology of the corner vortices in actual 3D flow and its 2D idealization has also been compared in detail.

The dispersion of surface jets in crossflows such as rivers or channels can cause critical environmental problems in the form of chemical or thermal pollution of these water bodies. The turbulent flow structures occurring in such crossflows play an important role in the mixing of surface jets with the surrounding water bodies. In this study, experimental measurements of the time history of the 3-D velocity field were conducted to better understand the flow structure of surface jets in crossflow conditions. Stereoscopic Particle Image Velocimetry was used to measure the instantaneous spatial and temporal velocity distribution downstream of the jet’s discharge point. In addition to the mean velocity distribution, turbulent flow characteristics such as the turbulent kinetic energy ( $$k$$ ), turbulent kinetic energy dissipation rate ( $$\epsilon$$ ), and turbulent eddy viscosity ( $$\nu_{t}$$ ) were calculated. The formation and evolution of a vortex in the surface jet’s flow structure was detected over the measurement zone. The vortex in the surface jets in crossflow resembled to half of the vortices in a counter-rotating vortex pair (CVP) of submerged jets in crossflows. It can be inferred that the water surface performed like a plane of symmetry.

Current developments in colloidal science include the assembly of anisotropic colloids with broad geometric diversity. As the complexity of particle assemblies increases, the need for ubiquitous algorithms that quantitatively analyze images of the assemblies to deliver key information such as quantification of crystal structures becomes more urgent. This contribution describes algorithms capable of image analysis for classifying colloidal structures based on abstracted interparticle relationship information and quantitatively analyzing the abundance of each structure in mixed pattern assemblies. The algorithm parameters can be adjusted, allowing for the algorithms to be adapted for different image analyses. Three different ellipsoidal particle assembly images are presented to demonstrate the effectiveness of the algorithms: a one-dimensional (1D) particle chain assembly and two two-dimensional (2D) polymorphic crystals each consisting of assemblies of two distinct plane symmetry groups. Angle relationships between neighbouring particles are calculated and neighbour counts of each particle are determined. Combining these two parameters as rules for classification criteria allows for the labeling and quantification of each particle into a defined symmetry class within an assembly. The algorithms provide a labelled image comprising classification results and particle counts of each defined class. For multiple images or individual frames from a video, the script can be looped to achieve automatic processing. The yielded classification data allow for more in-depth image analysis of mixed pattern particle assemblies. We envision that these algorithms will have utility in quantitative analysis of images comprising ellipsoidal colloidal materials, nanoparticles, or biological matter.

One of the most challenging issues in the automotive industry is to reliably predict the fracture limits across a broad range of the deformation modes subjected to various forming processes. Sheet metals in general exhibit anisotropy in both their deformation and fracture behaviors, and accordingly the stress and strain states at fracture strongly depend on both the loading condition and direction. A primary concern here is that there exists no global tendency of the anisotropic fracture limits over the wide range of the loading conditions, which gives rise to a difficulty in modeling the anisotropic fracture criterion. In this paper, a solution to the challenge of modeling anisotropic fracture behavior is proposed based on a decoupled formulation of anisotropic fracture limits with an employment of a Lagrangian interpolation function and an L 2 -norm of the principal stress vector. The anisotropic fracture criterion developed based on the proposed modeling approach was successfully applied to accurately represent the fracture forming severity of the DP980 with a thickness of 1.2 mm over a wide range of the loading condition including in-plane shear, uniaxial tension, plane strain tension, and equi-biaxial tension. Comparison of the experimental results with the predictions of the proposed criterion revealed that the proposed criterion has a remarkable capability in describing the anisotropic strain- and stress-based fracture limits. In addition, it was also theoretically discussed how the proposed modeling approach can be extended to predict fracture even for fully anisotropic materials without any plane of symmetry on the fracture initiation, and for dealing with various physical factors such as the strain rate, temperature, etc. in describing a phenomenological fracture criterion.

Publisher Summary The maximum loads on the components of an aircraft's structure generally occur when the aircraft is undergoing some form of acceleration or deceleration such as in landings, take-offs, and maneuvers within the flight and gust envelopes. Therefore, before a structural component can be designed, the inertia loads corresponding to these accelerations and decelerations must be calculated. This chapter considers the calculation of aircraft loads corresponding to the flight conditions specified by flight envelopes. The next section determines aircraft loads corresponding to a given maneuver load factor. Clearly, it is necessary to relate this load factor to given types of maneuver. Two cases arise: the first involves a steady pull-out from a dive and the second, a correctly banked turn. Although the latter is not a symmetric maneuver in the strict sense of the word, it gives rise to normal accelerations in the plane of symmetry and is, therefore, included. The chapter considers aircraft loads resulting from prescribed maneuvers in the longitudinal plane of symmetry. Other types of in-flight load are caused by air turbulence. The movements of the air in turbulence are generally known as gusts and produce changes in wing incidence, thereby subjecting the aircraft to sudden or gradual increases or decreases in lift from which normal accelerations result. These may be critical for large, high-speed aircraft and may possibly cause higher loads than control initiated maneuvers.

A pseudo-five-component reaction involving double decarboxylative 1,3-dipolar cycloaddition of azomethine ylides with olefinic oxindoles for the diastereoselective synthesis of bispiro[oxindole-pyrrolidine]s is developed. The major diastereomers are unique butterfly shaped compounds with a plane of symmetry. Recyclable zeolite HY acid catalyst is used to promote the reaction, and only CO2 and H2O are generated as byproducts.

We proposed a method to automatically eliminate the obscuration in the design of off-axis mirror systems without plane of symmetry. Off-axis settings and local surface expression are designed to simplify the system representation. Obscuration error function is established based on the positional relation between the proposed mirror boxes and ray boxes, and through function optimization, the obscuration is eliminated. The method is demonstrated by two examples, indicating its ability to eliminate obscuration for totally off-axis mirror system and the searching for such systems from co-axial starting points.

Στην παρούσα εργασία παρουσιάζεται μία ολοκληρωμένη μεθοδολογία για τον προσδιορισμό της δυναμικής απόκρισης τρισδιάστατων εύκαμπτων οδοστρωμάτων σε κινούμενα με σταθερή ταχύτητα οχήματα. Η ανάλυση πραγματοποιείται με τη μέθοδο των πεπερασμένων στοιχείων στο πεδίο του χρόνου μέσα στο πλαίσιο του εμπορικού προγράμματος ANSYS.Ο πεπερασμένος χώρος της κατασκευής του οδοστρώματος καθορίζεται από τέσσερα επίπεδα με ιξωδοελαστικούς απορροφητήρες ή απλές κυλίσεις, την οριζόντια επιφάνεια του οδοστρώματος και το επίπεδο συμμετρίας που διέρχεται από τον άξονα της οδού. Η διακριτοποίηση του χώρου αυτού γίνεται με ορθογώνια παραλληλεπίπεδα στοιχεία. Τα κινούμενα φορτία (τροχοί) του οχήματος προσομοιώνονται με τον καθορισμό τιμών φορτίου συναρτήσει του χρόνου σε όλους τους κόμβους της επιφάνειας κατά μήκος του άξονα κίνησης του οχήματος, οι οποίες ενεργοποιούνται στο χρόνο που χρειάζεται κάθε φορτίο να διανύσει την απόσταση από την αρχή της κίνησης μέχρι τον κάθε κόμβο του άξονα. Μελετώνται τόσο συγκεντρωμένα όσο και διανεμημένα κινούμενα φορτία. Αρχικά γίνεται η υπόθεση ότι το οδόστρωμα αποτελείται από ομογενές γραμμικά ελαστικό υλικό για το οποίο υπάρχουν αναλυτικές λύσεις για συγκεντρωμένο και διανεμημένο κινούμενο φορτίο. Έτσι είναι δυνατόν μέσω μελετών σύγκρισης και σύγκλισης να προσδιορισθεί εκείνος ο χώρος, εκείνη η διακριτοποίησή του, εκείνο το είδος συνοριακών συνθηκών και εκείνο το χρονικό βήμα ολοκλήρωσης των εξισώσεων κίνησης, ώστε να επιτευχθεί απόκριση αποδεκτού λάθους με λογικό υπολογιστικό φορτίο. Αυτές οι μελέτες σύγκρισης και σύγκλισης επαναλαμβάνονται και στην περίπτωση οδοστρωμάτων που αποτελούνται από τρεις ελαστικές στρώσεις και διατυπώνονται τελικές υποδείξεις για την βέλτιστη εφαρμογή της μεθόδου.Ακολούθως προσδιορίζεται η απόκριση οδοστρώματος αποτελούμενου από τρεις ή περισσότερες στρώσεις με ανελαστική συμπεριφορά υλικού σε κινούμενο διανεμημένο φορτίο με διάφορες ταχύτητες. Εξετάζονται οι περιπτώσεις η επιφανειακή στρώση να είναι ελαστική ή ιξωδοελαστική ή ιξωδοπλαστική και οι υπόλοιπες στρώσεις ελαστικές ή ελαστοπλαστικές τύπου Drucker-Prager με επιφάνεια διαρροής κωνικής μορφής. Η απόκριση περιλαμβάνει κυρίως την κατακόρυφη μετατόπιση στην επιφάνεια της ασφαλτικής στρώσης και τάσεις και παραμορφώσεις στην κάτω επιφάνεια της ασφαλτικής στρώσης και στην άνω επιφάνεια της στρώσης του υπεδάφους, που είναι απαραίτητες για σχεδιαστικούς σκοπούς. Γίνονται συγκρίσεις με άλλες αριθμητικές λύσεις και με πειραματικά αποτελέσματα πεδίου και αποδεικνύεται ότι η ανωτέρω μεθοδολογία είναι ικανή να προσεγγίσει με ικανοποιητική ακρίβεια την πραγματικότητα. Εκτελούνται επίσης διάφορες παραμετρικές μελέτες και βρίσκεται ότι οι μέγιστες τιμές της ανελαστικής απόκρισης μειώνονται με αύξηση της ταχύτητας σε συμφωνία με το πείραμα και σε αντιδιαστολή με την περίπτωση που όλες οι στρώσεις είναι ελαστικές για την οποία συμβαίνει ακριβώς το αντίθετο. Επί πλέον παρατηρείται ότι οι μέγιστες τιμές της ανελαστικής απόκρισης είναι πάντα μεγαλύτερες από τις αντίστοιχες τιμές της περίπτωσης για την οποία όλες οι στρώσεις είναι ελαστικές. Τέλος, η αναπτυχθείσα μεθοδολογία εφαρμόζεται στη δημιουργία εμπειρικών σχέσεων κόπωσης ασφαλτικού υλικού οδοστρωμάτων με τη βοήθεια υπαρχουσών μετρήσεων σε εύκαμπτα οδοστρώματα.

The radiation character of nonlinear Thomson scattering is investigated in the interaction of Laguerre-Gaussian circularly polarized laser pulses with a single electron in the angular plane. With theoretical analysis and numerical calculation, it is shown that the angular radiation distributions have annular structures with great fourfold or plane symmetry in pulses characterized by comparatively lower laser intensity (a0 < 6), prolonged pulse duration (τ > 50fs)or wide beam waist (b0 > 5μm). In other circumstances, a vortex radiation pattern is found for the first time on the basis of the electron dynamics. Further, by increasing the initial phase of laser pulse, the overall angular radiation has an interesting counter-clockwise rotating trend with a cycle of Δξ0 = 2π. These results would help the understanding of nonlinear Thomson scattering and push forward the research of twisted X/γ-ray generation in optical laboratory.

For a perfectly plane symmetric object, we can find two views—mirrored at the plane of symmetry—that will yield the exact same image of that object. In consequence, having one image of a plane symmetric object and a calibrated camera, we automatically have a second, virtual image of that object if the 3-D location of the symmetry plane is known. We propose a method for estimating the symmetry plane from a set of projection images as the solution of a consistency maximization based on epipolar consistency. With the known symmetry plane, we can exploit symmetry to estimate in-plane motion by introducing the X-trajectory that can be acquired with a conventional short-scan trajectory by simply tilting the acquisition plane relative to the plane of symmetry. We inspect the symmetry plane estimation on a real scan of an anthropomorphic human head phantom and show the robustness using a synthetic dataset. Further, we demonstrate the advantage of the proposed method for estimating in-plane motion using the acquired projection data. Symmetry breakers in the human body are widely used for the detection of tumors or strokes. We provide a fast estimation of the symmetry plane, robust to outliers, by computing it directly from a set of projections. Further, by coupling the symmetry prior with epipolar consistency, we overcome inherent limitations in the estimation of in-plane motion.

The flow in the three-dimensional laminar boundary layer in the vicinity of the plane of symmetry of a semi-infinite planar wing having a bend of the leading edge is investigated in the case of strong interaction with the external hypersonic flow. The flow functions are asymptotically expanded in power series in the angular coordinate in the vicinity of the plane of symmetry. The corresponding boundary value problem for the principal terms of the expansion is formulated and solved. The possibility of the existence of several solutions in the vicinity of the plane of symmetry is shown. The sweep angle effect on the distinctive features of the flow is considered.

This paper develops a novel and robust semi-analytical model to investigate the bending response of an SMA composite beam during a loading-unloading cycle based on an improvement of Brinsons model for an Euler–Bernoulli beam. The improvement of Brinsons model is taken into account to consider the asymmetric behavior of SMAs in tension and compression. In the proposed semi-analytical solution first a linear distribution of strain along the cross section is assumed and then employing Newton–Raphson method, an iterative numerical procedure is implemented to satisfy the classical equilibrium equations. The proposed method in this paper is applicable to any Euler–Bernoulli beam with any material and layup as well as any cross sectional geometry (with a plane of symmetry). To validate the results of the proposed model a homogeneous SMA beam is studied and the results are compared to the results of a 2D finite element (FE) solution. The results of the bending response of the beam are reported for both the symmetric and asymmetric behavior and also for both FE and semi-analytical solution. The results for both approaches are generally in good agreement with each other, and a maximum error of 3.6% is observed between FE and semi-analytical solution. Further, a Bi-layer composite beam consisting of two different SMAs is studied. Accordingly, Load-deflection, moment-curvature and also variation of the location of the neutral axis with respect to the center line of the cross-section is investigated. In one of the case studies there is 15% error in the prediction of the symmetric model with respect to the asymmetric model which is a more actual prediction, proves the importance of employing the asymmetric model. Moreover, for the same case the maximum deviation of the neutral axis with respect to the centroid is 17% for the asymmetric model and 24% for the symmetric model. Furthermore, hysteresis inner loops for the Bi-layer composite beam is investigated. The proposed formulation is appropriate for design or optimization of smart structures consisting of multilayer SMA components under bending where a large number of simulations are necessary.

The rotational fingerprint of the thiophene-water complex was investigated for the first time using Fourier transform microwave spectroscopy (7-20 GHz) aided by quantum mechanical calculations. Transitions for a single species were observed, and the rotational constants for the parent and 18O isotopomers are consistent with a geometry that is highly averaged over a barrierless large-amplitude motion of water that interconverts two equivalent forms corresponding to the global minimum (B2PLYP-D3(BJ)/def2-TZVP). In this effective geometry, the water lies above the thiophene ring close to its σv plane of symmetry. The observed transitions are split by a second water-centered tunneling motion that exchanges its two protons by internal rotation about its C2 axis with a calculated barrier of ∼2.7 kJ mol-1 (B2PLYP-D3(BJ)/def2-TZVP). Based on quantum theory of atoms in molecules, noncovalent interaction, and symmetry-adapted perturbation theory analyses, the observed geometry enables two intermolecular interactions (O-H···π and O-H···S) whose electrostatic and dispersive contributions favor formation of the thiophene-water complex.

Introduction: The locomotion behavior of an organism involves the integration of aspects like body symmetry, sensory and locomotor systems. Furthermore, various ecological factors seem to be related to locomotion characteristics, such as foraging strategy, migration trends, response to predators and competitors, and environmental stress. Objective: To analyze locomotion and the influence of body symmetry in the crawling and righting movements of the sea star Asterina stellifera. Methods: We carried out laboratory experiments in aquariums in the presence/absence of water current and on a horizontal and vertical surface. Results: The speed is similar to speed in other species of similar size. Both the speed and linearity of displacement were independent of individual body size. A water current leads to faster crawling and straight paths, but there is no rheotaxis: streams do not affect locomotion. Speed and linearity of displacement were independent of individual body size. The displacement pattern described here may be an adaptation of organisms that present dense populations in communities with high prey abundance, as is the case of A. stellifera. Conclusions: Like other asteroids, this species did not show an Anterior/Posterior plane of symmetry during locomotion, or righting movement: it does not tend to bilaterality. Resumen Introducción: El comportamiento de locomoción de un organismo implica la integración de aspectos como la simetría corporal, los sistemas sensorial y locomotor. Además, varios factores ecológicos parecen estar relacionados con las características de la locomoción, como la estrategia de alimentación, las tendencias migratorias, la respuesta a los depredadores y competidores y el estrés ambiental. Objetivo: Analizar el patrón general de locomoción y la influencia de la simetría corporal en la locomoción y enderezamiento de la estrella de mar Asterina stellifera. Métodos: Realizamos experimentos de laboratorio en acuarios en presencia / ausencia de corriente de agua y en superficie horizontal y vertical. Resultados: La velocidad es similar a la velocidad en otras especies de tamaño similar. Tanto la velocidad como la linealidad del desplazamiento fueron independientes del tamaño corporal individual. Una corriente de agua conduce a una velocidad de desplazamiento mayor y a trayectorias más rectas, pero no hay reotaxis: una corriente de agua no afecta el patrón de locomoción. La velocidad y la linealidad del desplazamiento fueron independientes del tamaño corporal individual. El patrón de desplazamiento aquí descrito puede ser una adaptación de organismos que presentan densas poblaciones en comunidades con alta abundancia de presas, como es el caso de A. stellifera. Conclusiones: Al igual que otros asteroides, esta especie no mostró un plano de simetría Anterior / Posterior durante la locomoción o el movimiento de enderezamiento: no tiende a la bilateralidad.

Chapter 4 describes the complex plane which provides a graphical representation for complex numbers. The chapter also contains historical information about the complex plane’s invention, and complements similar historical events associated with quaternions. Polar representation of a complex number is described and how it provides a useful mechanism to visualize rotations in the plane. The chapter summarises key formulae and contains some useful worked examples.

The theories of pure bending of beams are restricted to beams having a plane of symmetry through their longitudinal axis and loads applied act in that plane. Bending takes place about an axis (called neutral axis), which is perpendicular to the plane of symmetry. It is assumed that beams are made of linearly elastic materials, which implies that neutral axis of the beam’s cross-section passes through its centroid.

Practically all commercial LaB6 cathodes make use of (100) crystalline plane symmetry, stability, and low workfucntion. However, in real gun, this plane may suffer dramatic transformations, including surface damage, partial re-crystallization, etc. We studied self-healing of LaB6 cathodes suffered (100) plane damage.

Lecture Notes in Computer Science, 12370, ISSN:0302-9743, ISSN:1611-3349, Computer Vision – ECCV 2020 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part V, ISBN:978-3-030-58594-5, ISBN:978-3-030-58595-2

During buckling of column, it is assumed that the column would buckle as the cross section bends in the plane of symmetry. But in some problems of buckling failures of column, it would be either due to twisting or due to combined effect of bending and twisting. Such a combined effect of bending and twisting in a structure is known as torsional buckling. In the present work, a thin-walled bar of cross section (b × t) with the length ‘l’ is studied by applying uniform axial compression. The differential equation for the deflection curve and the differential equation for torsional buckling are presented. The expressions for total moment, torque and torque per unit length are derived and finally the expressions for the critical stresses and critical load for torsional buckling failure are derived. A numerical example is solved. The critical stress and critical load are calculated.

Numerical simulations were performed to investigate the re-initiation mechanism of a diffracted detonation wave near the critical channel width for a weakly unstable gas. Two scenarios were examined: diffraction of a planar detonation wave and of a cellular detonation wave inside the inlet channel. The results revealed that the critical channel width predicted using a cellular detonation wave is smaller than that predicted using a planar detonation wave. The re-initiation mechanisms are described in detail by tracing massless particles along both the plane of symmetry and the re-initiation path. For planar detonation diffractions, a compression wave is formed in the far field behind the diffracted shock. Re-initiation is closely related to the amplification of this compression wave and its coalescence with the diffracted shock. Depending on the inlet channel width, the strength of the reflected rarefaction wave is responsible for weakening the strength of the compression wave and its coalescence with the diffracted shock, consequently hindering the reaction of particles behind the diffracted shock wave. In cellular cases, the continuous collisions of transverse waves, which generate local explosion sites, sustain detonation wave propagation.

The midsagittal plane (MSP) is the foundation for 3-dimensional (3D) cephalometric analyses. This article aimed to provide a protocol to construct a reliable and accurate MSP for 3D cephalometric craniofacial analysis.Cone-beam computed tomography data of 16 adult patients without obvious bilateral asymmetry were collected. The model of the anterior cranial base and sphenoid bone was constructed, and the candidate MSP was determined on the basis of the symmetry of this model. Intraclass correlation coefficients were used to assess intra- and interexaminer reliability of the candidate MSP. To investigate the accuracy of this candidate MSP, we constructed a true plane of symmetry of craniomaxillofacial structure and a control plane on the basis of 3 cranial midline points. We then compared these with the candidate MSP.This candidate MSP resembled the true plane of symmetry with all the mean absolute errors 1 mm, and all the absolute errors for the candidate MSP were significantly smaller than the control plane (P ≤0.002). The relative intra- and interexaminer reliability for this candidate MSP was almost perfect (intraclass correlation coefficients 0.9).The candidate MSP constructed using this method was thought to be reliable and accurate for 3D cephalometric analysis in patients without obvious cranial asymmetry.

In this paper the motion of the plane symmetric Bricard 6R mechanism is studied. A simple method based on the dimensions of intersecting varieties in the Study quadric is used to show that the mechanism is mobile. The degree of the motion of the third link (the one adjacent to the plane of symmetry) relative to the plane of symmetry is found. The degree and genus of the motion of the third link relative to the first link is also found. This curve in the Study quadric is given as the intersection of the variety generated by the RR dyad formed by second and third joints with the variety of displacements that keep the fourth joint axis in the special linear line complex whose axis is the axis of the first joint. Finally, the motion of the symmetry plane when the second link is fixed is considered. The symmetry planes comprise the common tangent planes to a pair of circularly symmetric hyperboloids.

We formulate necessary and sufficient conditions for a unit vector $\pmb{\nu }$ to generate a plane or axial symmetry of a constitutive tensor. For the elasticity tensor, these conditions consist of two polynomial equations of degree lower than four in the components of $\pmb{\nu }$ . Compared to Cowin–Mehrabadi conditions, this is an improvement, since these equations involve only the normal vector $\pmb{\nu }$ to the plane symmetry (and no vector perpendicular to $\pmb{\nu }$ ). Similar reduced algebraic conditions are obtained for linear piezo-electricity and for totally symmetric tensors up to order 6.

In this paper we investigate the vortex structure and dynamics formed in the near field of a turbulent axisymmetric jet subjected to transverse acoustic forcing. Full three-dimensional phase-averaged velocity measurements were obtained to elucidate the coherent structures formed when the jet is positioned at the pressure node of a plane standing wave oriented transversely to the streamwise flow direction, which creates a plane symmetry about the nodal line dissecting the jet exit. Due to the change in phase that occurs across the nodal line, it was found that axisymmetry is broken and the jet undergoes a periodic transverse flapping motion consistent with a sinuous mode. This was accompanied by a periodic train of interconnected vortex structures, resembling inverted hairpin (or horseshoe) vortices, formed as the shear layers rolled up in anti-phase either side of the jet, and propagated a few diameters downstream before breaking up. An inviscid vortex model employing inverted hairpin line vortices is shown to capture both the dynamics of the vortex structures and the fluctuating velocity fields. Overall, the jet response and resulting vortex dynamics observed represent a significant departure from the axisymmetric flow structures observed with conventional longitudinal forcing and more closely resemble the phenomenon of bifurcating jets. c The Author(s) 2019 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, the roughness characteristics of the face milled surface were investigated made with a parallelogram insert (κr = 90°) on C45 steel. Changes in surface roughness were analyzed for the specific surface created by the milling movement conditions (rotational main movement and linear feed motion). The topography was created by a tool moving in the plane of symmetry of the workpiece, which produced double cutting-edge grooves. We found that the roughness varies in different parts of the resulting pattern, and we analyzed its nature and the magnitude of the variance based on the average over the entire surface. Also, in parallel measurements, the Rz parameter was at its maximum in the symmetry plane, and in the perpendicular direction roughness values are the lowest in the middle and increase with distance. У цій статті представлено дослідження шорсткості обробленої торцевим фрезеруванням поверхні сталевої заготовки фрезою з твердосплавною пластиною. Для вимірювання змін в значеннях шорсткості були взяті 5 площин вимірювання на однаковій відстані одна від одної, і 5 точок вимірювання були розміщені на однаковій відстані уздовж кожної площини. Спочатку вимір робився в площинах, паралельних напрямку подачі, одна з яких - це площина симетрії заготовки, де вісь інструменту рухається вздовж напрямку подачі. Згодом зміни вимірювали в напрямку, перпендикулярному подачі, де середні точки центральних вимірювань були вирівняні по площині симетрії. Середнє арифметичне значення шорсткості виміряних точок було взято в якості базового характеристичного значення шорсткості. Значення точок вимірювання також були проаналізовані щодо цієї бази. Можна констатувати, що базове (середнє) значення шорсткості збільшується при вимірах, паралельних напрямку подачі, в той час як базове значення зменшується при перпендикулярних вимірах. Крім того, діаграми середньої шорсткості (Ra) і висоти шорсткості (Rz) в кожному напрямку вимірювання показують дуже схожі характеристики. Експерименти підтвердили, що значення параметрів шорсткості змінюються на поверхні фрезерованій поверхні в залежності від напрямку і розташування вимірювання. Експерименти підтвердили, що площини, паралельні подачі, показують різницю в залежності від їх положення і відстані від площини симетрії, яка в основному визначається положенням ріжучої кромки на стороні входу або виходу. На підставі досліджень середнє значення шорсткості, виміряне в точках вимірювання, було рівномірно розподілено по поверхні. Крім того, значення шорсткості, виміряні в площині симетрії, можуть бути рекомендовані для оцінки шорсткості фрезерованої поверхні.

In this paper, a symmetry detection algorithm for three-dimensional point cloud model based on weighted principal component analysis (PCA) is proposed. The proposed algorithm works as follows: first, using the point element’s area as the initial weight, a weighted PCA is performed and a plane is selected as the initial symmetry plane; and then an iterative method is used to adjust the approximate symmetry plane step by step to make it tend to perfect symmetry plane (dominant symmetry plane). In each iteration, we first update the weight of each point based on a distance metric and then use the new weights to perform a weighted PCA to determine a new symmetry plane. If the current plane of symmetry is close enough to the plane of symmetry in the previous iteration or if the number of iterations exceeds a given threshold, the iteration terminates. After the iteration is terminated, the plane of symmetry in the last iteration is taken as the dominant symmetry plane of the model. As shown in experimental results, the proposed algorithm can find the dominant symmetry plane for symmetric models and it also works well for nonperfectly symmetric models.

Reconstruction model of aerodrome pavement with the use of new design of precast concrete prestressed slab is proposed. Numerical study of stress–strain state of strip pavement was performed. The aim of the article is studying stress–strain state of the proposed design of precast concrete aerodrome pavement when interacting with soft soil conditions. The design of the jump joint, which maintains jointless connection of pavement, is offered. To study stress–strain state of aerodrome pavement constructed from precast concrete prestressed slabs, numerical technique is used in the article. In furtherance of numerical technique, it was defined the equivalent cross-section unit of the three-layer thin slab construction. The scheme with one plane of symmetry for discrete model fragment of flight strip prestressed cover slabs was constructed. Based on the results of numerical calculations, nodal displacements plot, driving bending moments and nodal reactions of soil base are made. It is established that conditions of aerodrome pavement structure strength and deformability are satisfied. Practical importance of the work is that, with sufficient feasibility study, a significant reduction in material capacity of coating structure can be achieved with the necessary durability and reliability, even for heavy-duty oversized aircraft.

A mathematical classification of two-dimensional ornaments into 17 plane symmetry groups is presented, which were created by five medieval cultural groups of Middle East. The data are considered representative for the cultural groups. By applying a correlation algorithm on the individual use of symmetry classes by each cultural group, the strength of the interaction between the pairs of groups are quantitatively determined. The analysis shows that the strongest similarity in the creation of periodic ornaments is between Rum Seljuks and Arab Muslims. It is also found that the Rum Seljuks, followed by Armenians, are the most interactive cultures. This report is the first attempt to quantify cultural communication by mathematical means., Journal of Mathematical Sciences and Modelling, 3 (2), ISSN:2636-8692

This article introduces a method for morphing a parametric, rectangular and structured 3D template mesh such that it embodies: (1) a given (half) hull surface properly aligned with a pre-defined air-water interface zone, (2) a high-quality boundary layer region near the hull, and (3) a smooth transition of the mesh to the template mesh structure away from the hull. The performance of the proposed method is successfully validated on three widely studied benchmark cases, namely the parabolic Wigley hull, the Series 60 hull and the DTMB 5415 combatant hull. The proposed automated mesh generation method is useful in shape optimization for computational fluid dynamics, among others. Highlights A new method for creating structured meshes for marine CFD applications is presented. The proposed method is parametric and thus fully programmable. The method includes an automatic resolution of CAD geometry errors. The resulting meshes are strictly orthogonal near the inter-phase zone and the hull. The main application of the proposed method is in CFD-based shape optimization.

Local fi ber direction is decisive for both strength and stiffness in timber. In-plane fi ber direction on surfaces of timber can be determined using the so-called tracheid effect which is frequently used in both research and industry applications. However, a similar established method does not exist for measuring the out-of-plane angle, also known as diving angle. The purposes of this article were to evaluate if the tracheid effect can also be used to determine, with reasonable accuracy, the out-of-plane angle in Norway spruce and to verify an existing mathematical model used to calculate the fiber direction in the vicinity of knots. A newly developed laboratory laser scanner was applied for assessment of fiber directions in a single Norway spruce specimen containing a knot. It was assumed that the specimen had a plane of symmetry through the center of the knot, and by splitting the specimen through this plane into two parts, it was possible to make measurements on orthogonal planes. The results showed that the out-of-plane angle could not be determined with very high accuracy and the difficulties related to this objective were analyzed. Regarding the mathematical model of fiber direction in the vicinity of a knot, fiber directions calculated on the basis of this model agreed well with experimentally obtained fiber directions, but successful application of the model requires that the geometry of the knot is known in detail.

Three-dimensional problems of synthesis and analysis of polyfocal constrained lenses are considered. It is shown that such lenses can have up to five focal points on each side. The problem of synthesis of a pentafocal lens in the general case is reduced to three transcendental equations and, of a quadrifocal lens, to two transcendental equations. In the case of a pentafocal lens with three focuses in the symmetry plane of the lens, forming plane wave fronts at the output, the synthesis problem is reduced to one transcendental equation; in all other cases of plane front formation, an analytical solution of the problem is obtained. An analytical solution is also obtained for the problem of synthesis of a quadrifocal lens with three symmetry planes. An analytical solution is also found for the problem of synthesis of a quadrifocal lens with two symmetry planes forming in the plane of symmetry four plane fronts with simultaneous fulfillment of the aplanatic conditions (Abbe sines) in the orthogonal plane. Examples of solving the synthesis problems with the optimization of parameters in order to minimize the mean square aberration of the eikonal are given. In the two-dimensional case, the dependence of the mean square aberration of the eikonal on focal distances is studied and it is shown that there is an optimal ratio of focal distances providing its minimum.

With respect to the Aharonov-Bohm effect, we consider only solenoids of zero radius and their action on its plane of symmetry. We find all self-adjoint extensions in case of finite length solenoids and compare them with the case of infinitely long one. It is then shown a convergence in the uniform resolvent sense for each angular momentum sector of such self-adjoint extensions as the solenoid length goes to infinity.

We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial data, is stable under perturbations of the initial data that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression. Specifically, the vorticity remains uniformly bounded, while the vorticity divided by the density exhibits even more regular behavior: the ratio remains uniformly Lipschitz relative to the standard Cartesian coordinates. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations (derived in a companion article) exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds., National Science Foundation (U.S.) (Grant DMS-1162211), National Science Foundation (U.S.). Career Grant (DMS-1454419)