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In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (a v η x v z ′ v) ′ + q v F x g v = 0, where v ≥ v 0 and the corresponding function z = x + p x ∘ h . Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler's differential equation introduces the well-known sharp oscillation criterion. [ABSTRACT FROM AUTHOR]

We derive asymptotic estimates for the projection of the vorticity onto principal directions of material stretching in 3D flows. In flows with pointwise bounded vorticity, these estimates predict vorticity alignment with Lyapunov vectors along trajectories with positive Lyapunov exponents. Specifically, we find that in inviscid flows with conservative body forces, the vorticity exactly aligns with the intersection of the planes orthogonal to the dominant forward and backward Lyapunov vectors along trajectories with positive Lyapunov exponent. Furthermore, we derive asymptotic estimates for the vorticity alignment with the intermediate eigenvector of the rate-of-strain tensor for viscous flows under general forcing. We illustrate these results on explicit solutions of Euler's equation and on direct numerical simulations of homogeneous isotropic turbulence. [ABSTRACT FROM AUTHOR]

Context. High-precision ephemerides are not only useful in supporting space missions, but also in investigating the physical nature of celestial bodies. This paper reports an update to the orbit and rotation model of the Martian moon Phobos. In contrast to earlier numerical models, this paper details a dynamical model that fully considers the rotation of Phobos. Here, Phobos' rotation is first described by Euler's rotational equations and integrated simultaneously with the orbital motion equations. We discuss this dynamical model, along with the differences with respect to the model now in use. Aims. This work is aimed at updating the physical model embedded in the ephemerides of Martian moons, considering improvements offered by exploiting high-precision observations expected from future missions (e.g., Japanese Martian Moons exploration, MMX), which fully supports future studies of the Martian moons. Methods. The rotational motion of Phobos can be expressed by Euler's rotational equations and integrated in parallel with the equations of the orbital motion of Phobos around Mars. In order to investigate the differences between the two models, we first reproduced and simulated the dynamical model that is now used in the ephemerides, but based on our own parameters. We then fit the model to the newest Phobos ephemeris published by Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Based on our derived variational equations, the influence of the gravity field, the Love number, k2, and the rotation behavior were studied by fitting the full model to the simulated simple model. Our revised dynamic model for Phobos was constructed as a general method that can be extended with appropriate corrections (mainly rotation) to systems other than Phobos, such as the Saturn and Jupiter systems. Results. We present the variational equation for Phobos' rotation employing the symbolic Maple computation software. The adjustment test simulations confirm the latitude libration of Phobos, suggesting gravity field coefficients obtained using a shape model and homogeneous density hypothesis should be re-examined in the future in the context of dynamics. Furthermore, the simulations with different k2 values indicate that it is difficult to determine k2 efficiently using the current data. [ABSTRACT FROM AUTHOR]

In this research article, we study the (2+1)-dimensional Korteweg-de Vries equation (KdV), which is derived from Euler's equations related to ideal fluid systems. Our primary goal is to obtain explicit solutions utilizing novel methodologies, particularly the new auxiliary equation and the simplest equation methods. We concentrate on the search for solitary wave solutions inside the (2+1)-dimensional KdV, which is relevant in domains such as fluid dynamics, plasma physics and nonlinear waves. Our objective is to increase knowledge of this equation and give insight into the behavior of solitary waves by employing a novel mathematical technique. The bifurcation analysis is also presented for the model problem. This will be accomplished by displaying our findings in 2D and 3D graphics. [ABSTRACT FROM AUTHOR]

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler's equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler's angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body's parameters have on the motion's behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body's motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance. [ABSTRACT FROM AUTHOR]

Mineral exploration projects can make considerable use of a variety of geophysical techniques and datasets, including magnetic and gravity data. The interpretation of large quantities of data can be very time consuming, so semi-automatic interpretation techniques are often used to provide initial estimates of the parameters (primarily the location and depth) of the sources of anomalies. Euler deconvolution is a commonly used interpretation method for potential fields which has a number of advantages over many other techniques, such as working in the presence of remanent magnetisation, and not being restricted to a particular model such as a contact. A second-order version of Euler's equation is introduced here, which is much less affected by trends in the data than the standard method and additionally produces depth parabolas, which simplify the interpretation of results. The method was applied to aeromagnetic data from a mineral exploration project in Southern Africa and provided plausible results. [ABSTRACT FROM AUTHOR]

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body's middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions' extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications. [ABSTRACT FROM AUTHOR]

We consider a liquid layer of a finite depth described by Euler's equations. The ice cover is geometrically modeled by a nonlinear elastic Kirchhoff–Love plate. We determine the trajectories of liquid particles under an ice cover in the field of a nonlinear surface traveling wave rapidly decaying at infinity, namely, a solitary wave packet (a monochromatic wave under the envelope, with the wave velocity equal to the envelope velocity) of a small but finite amplitude. Our analysis is based on the use of explicit asymptotic expressions for solutions describing the wave structures at the water–ice interface of a solitary wave packet type, as well as asymptotic solutions for the velocity field generated by these waves in the depth of the liquid. [ABSTRACT FROM AUTHOR]

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler's Diophantine equation A 3 + B 3 = C 3 + D 3 . In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation A 4 + B 4 + C 4 + D 4 + E 4 = F 4 in the spirit of Ramanujan. [ABSTRACT FROM AUTHOR]

Purpose: Composite sandwich structures are becoming an essential part of engineering appliances due to their excellent mechanical and structural properties, high strength-to-weight as well as stiffness-to-weight ratios. And so, they are widely used in aerospace and marine applications, such as airplane wings, flight control surfaces, turbine blades, ship interiors, and small boats and yachts. Despite their superior properties, the sandwich plates consisting of an aluminum core and carbon fiber-reinforced plastic face sheets exhibit low-frequency resonant modes when subjected to external excitation. Therefore, in this paper, a nonlinear energy sink (NES) is attached to the thermally induced sandwich plate to attenuate the resonant peaks passively in a broad frequency range. Methods: The equation of motion of the sandwich plate-NES system is obtained by utilizing Lagrange Euler's equation followed by Galerkin discretization. The analytical solution of the system is obtained by the complexification averaging method and arc-length continuation method. The nonlinear and stability analyses are also conducted for the present system. Results: The results show that an optimum NES obtained by the sensitivity analysis of the response curve attenuates the resonant peaks effectively and is applicable in a wide frequency range at various temperatures. Conclusion: Optimum parameters and the location of the NES help to eliminate the complex nonlinear dynamical phenomena at the first resonance frequency in various thermal environments. [ABSTRACT FROM AUTHOR]

Background: Cardiac dysfunction due to cardiotoxicity from anthracycline chemotherapy is a leading cause of morbidity and mortality in childhood cancer survivors (CCS), and the cumulative incidence of cardiac events has continued to increase. This study identifies an adequate indicator of cardiac dysfunction during long-term follow-up. Procedure: In total, 116 patients (median age: 15.5 [range: 4.7–40.2] years) with childhood cancer who were treated with anthracycline were divided into three age groups for analysis (C1: 4–12 years of age, C2: 13–18 years of age, C3: 19–40 years of age), and 116 control patients of similar ages were divided into three corresponding groups (N1, N2, and N3). Layer-specific strains were assessed for longitudinal strain (LS) and circumferential strain (CS). The total and segmental intraventricular pressure gradients (IVPG) were also calculated based on Doppler imaging of the mitral inflow using Euler's equation. Results: Conventional echocardiographic parameters were not significantly different between the patients and controls. All layers of the LS and inner and middle layers of the basal and papillary CS in all ages and all IVPGs in C2 and C3 decreased compared to those of corresponding age groups. Interestingly, basal CS and basal IVPG in CCS showed moderate correlation and both tended to rapidly decrease with aging. Furthermore, basal IVPG and anthracycline dose showed significant correlations. Conclusions: Basal CS and total and basal IVPGs may be particularly useful indicators of cardiotoxicity in long-term follow-up. [ABSTRACT FROM AUTHOR]

The article presents a summarised history of the equations governing fluid motion, known as the Navier–Stokes equations. It starts with the work of Castelli, who established the continuity equation in 1628. The determination of fluid flow resistance was a topic that involved the brightest minds of the 17th and 18th centuries. Navier's contribution consisted of the incorporation of molecular attraction effects into Euler's equation, giving rise to an additional term associated with resistance. However, his analysis was not the only one. This continued until 1850, when Stokes firmly established the boundary conditions that must be applied to the differential equations of motion, specifically stating the non-slip condition of the fluid in contact with a solid surface. With this article, the author wants to commemorate the bicentennial of the publication of "Sur les Lois du Mouvement des Fluides" by Navier in the Mémoires de l'Académie Royale des Sciences de l'Institut de France. [ABSTRACT FROM AUTHOR]

Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle. In contrast, Gauss' principle of least constraint is a true minimum principle. In this paper, we apply Gauss' principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). It turns a fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving four classical problems in fluid mechanics using the PMPG without resorting to Navier–Stokes' equation. In some cases, the PMPG minimization approach is not any more efficient than solving Navier–Stokes'. However, in other cases, it is more insightful and efficient. In fact, the inviscid version of the PMPG allowed solving the long-standing problem of the aerohydrodynamic lift over smooth cylindrical shapes where Euler's equation fails to provide a unique answer. The PMPG transcends Navier–Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g., electromagnetic). [ABSTRACT FROM AUTHOR]

Copyright of Journal of University of Anbar for Pure Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider the Nernst--Planck equations describing the nonlinear time evolution of multiple ionic concentrations in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to either the Euler or Darcy's equations, both forced nonlinearly by the electric forces generated by the presence of charged ions. We address the global well-posedness and Gevrey regularity of the resulting electrodiffusion models in the periodic setting. [ABSTRACT FROM AUTHOR]

The artillery firing process will instantly produce high-temperature and high-pressure gunpowder gas, this process will produce shock waves. The gunpowder gas has a limited effect on the projectile during the firing and ballistic after-effects period, however, it has a very obvious effect on the stability of the gun body, and the reduction of the stability of the gun body directly affects the firing accuracy and the safety of the firing personnel. Based on the method of Computational Fluid Dynamics (CFD), numerical simulation is carried out, and the structure and flow parameters of the muzzle flow field are obtained by using three-dimensional Euler's control equation, gas equation of state, and k-epsilon model, as well as dynamic mesh technology. By comparing the flow parameters of the brake before and after optimization, and analyzing the results obtained from the 8-round firing experiments, the efficiency of the optimized brake is increased by 8.2%, and the deviation between the experimental data and the simulation results is only 10.5%, which not only verifies the accuracy of the numerical simulation calculations but also verifies the optimized brake's good retracting effect. The results of the study can provide a reference for the optimization and design of the double-chamber brake. [ABSTRACT FROM AUTHOR]

For image segmentation, level set models are frequently employed. It offer best solution to overcome the main limitations of deformable parametric models. However, the challenge when applying those models in medical images stills deal with removing blurs in image edges which directly affects the edge indicator function, leads to not adaptively segmenting images and causes a wrong analysis of pathologies wich prevents to conclude a correct diagnosis. To overcome such issues, an effective process is suggested by simultaneously modelling and solving systems’ two-dimensional partial differential equations (PDE). The first PDE equation allows restoration using Euler’s equation similar to an anisotropic smoothing based on a regularized Perona and Malik filter that eliminates noise while preserving edge information in accordance with detected contours in the second equation that segments the image based on the first equation solutions. This approach allows developing a new algorithm which overcome the studied model drawbacks. Results of the proposed method give clear segments that can be applied to any application. Experiments on many medical images in particular blurry images with high information losses, demonstrate that the developed approach produces superior segmentation results in terms of quantity and quality compared to other models already presented in previeous works. [ABSTRACT FROM AUTHOR]

The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel's Equation of the second kind and, under certain conditions, to equations specified in Gambier's family. The symmetry analysis for the force-free term leads to Euler's Equation and to a system of reduced second-order odes for which singularity analysis is performed to determine their integrability. [ABSTRACT FROM AUTHOR]

Variational active contour seeks to segment or extract desired object boundaries for further analysis. The model can be divided into global segmentation and selective segmentation. Selective segmentation, which focuses on segmenting a particular object, is preferable to the global model. Recently, a number of selective segmentation models have been developed to precisely extract an object on grayscale images. Nevertheless, if the input image is vector-valued (colour), these models merely convert it to a grayscale image, resulting in data loss owing to the reduction in image dimension. Furthermore, they may have poor segmentation performance due to the intensity inhomogeneous images. Therefore, a new model on variational selective active contour for segmenting vector-valued images has been proposed that incorporates the concepts of local image fitting and distancebased fitting terms into a variational minimization energy functional. Moreover, a Gaussian function was used as a regularizer to replace the computationally expensive Total Variation term. Then, the proposed model's Euler Lagrange equation has been provided to solve the model. When segmenting an object in inhomogeneous intensity images, the result of the proposed model was about 30% more accurate based on the Jaccard value and about 3 times faster than other existing methods. [ABSTRACT FROM AUTHOR]

Inviscid, incompressible liquid is released from rest by a sudden dam break, accelerating under gravity over a uniformly sloping impermeable plane bed. The liquid flows downhill or up a beach. A linearised model is derived from Euler's equations for the early stage of motion, of duration $2\sqrt {H/g}$ , where H is the depth scale and $g$ is the acceleration due to gravity. Initial pressure and acceleration fields are calculated in closed form, first for an isosceles right-angled triangle on a slope of $45^{\circ }$. Second, the triangle belongs to a class of finite-domain solutions with a curved front face. Third, an unbounded domain is treated, with a curved face resembling a steep-fronted breaking water wave flowing up a beach. The fluid goes uphill due to a nearshore pressure gradient. In all cases the free-surface-bed contact point is the most accelerated particle, exceeding the acceleration due to gravity. Physical consequences are discussed, and the pressure approximation of shallow water theory is found poor during this early stage, near the steep free surface exposed by a dam break. [ABSTRACT FROM AUTHOR]

In this study, the motion of Lagrange's gyro about its fixed point in the presence of a perturbed torque, a gyroscopic torque, and a varied restoring one is searched. We assume sufficiently small angular velocity components in the direction of the principal axes that differ from the dynamical symmetry one and a restoring torque that is considered to be greater than the perturbing one. In this manner, we replace the familiar small parameter that was used in previous works with a large one. In such cases, the gyro equations for motion (EOM) are formulated in the form of a two-degrees-of-freedom (DOF) autonomous system. We average the obtained system to get periodic solutions and motion's geometric interpretation of the problem using the large parameter. The regular precession and the pure rotation of the motion are obtained. A numerical study is evaluated for asserted the used techniques and showed the influence of the changing parameters of motion on the gyro behavior. The trajectories of the motions and their stabilities are discussed and analyzed. The novelty of this work lies in how to adapt the method of large parameter (MLP) to solve the rigid body problem, especially since it has been assumed initially that its angular velocity or its initial energy are very small. MSC (2000): 70E20, 70E17, 70E15, 70E05 [ABSTRACT FROM AUTHOR]

Whereas rational approximations for have been explored with the best known approximation of 245850922/78256779 which has an accuracy of 7.8179366199075 × 10 SP -17 sp , this is far from the case for Euler's number I e i [[1]-[4], [8], [11]]. Graph B Sir, b The quest for transcendental numbers, and Euler's constant I e i , has a long history ever since antiquity, as both numbers have intrigued mathematicians and theoretical scientists for several centuries [[1]-[9]]. References 1 Wilson R. Euler's Pioneering Equation: The most beautiful theorem in mathematics (illustrated ed.). [Extracted from the article]

In this work, general formula of set solutions for system of Euler's equation in dimension m are derived by using Elzaki transformation. Also, supported examples are presented utilizing formula derived. [ABSTRACT FROM AUTHOR]

Euler's equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler's equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler's equation for rigid-body rotation. [ABSTRACT FROM AUTHOR]

• A hybrid analytical method is built to study vibrations of the sandwich plate with z-direction reinforcements and cavities. • Both the sandwiches with hard and soft cores are considered in the analytical method that utilizes the layerwise model. • For the soft core sandwich, the layerwise model shows a much better accuracy than the existing equivalent single-layer model. • The accuracy of present method decreases at the high frequency since the homogenization of core neglects local vibrations. This paper presents a multi-functional sandwich plate with z-direction reinforcements and cavities that has the potential to meet the pressure resistance and the acoustic stealth requirement of the underwater equipment in deep sea. In order to predict its free and forced vibrations in both vacuum and water, a hybrid analytical method is proposed by combining a multi-level homogenization model and a layerwise model. The core with z-direction reinforcements and cavities is homogenized and effective elastic constants are calculated by the Mori-Tanaka formula. Vibration equations of the sandwich plate are derived utilizing the layerwise model based on the first-order shear deformation theory. The water effect is taken into account by the Euler's equation. Vibration results are solved under simply supported boundaries. Validation studies and parameter analyses are then conducted. It is found that different from the existing theoretical approach based on the equivalent single-layer model that is only applicable for the sandwich with a hard core material, the present hybrid analytical method is well suitable for vibration predictions of the sandwich plate with z-direction reinforcements and cavities, no matter the hard or soft core. Besides, due to the homogenization of the three-phase core and the neglecting of local vibrations in the layerwise model, the present analytical method only predicts accurate free vibrations when the half wave numbers are less than numbers of cavities in x and y directions, and also just shows the high precision in force vibration calculations as the maximum analysis frequency is less than the natural frequency of the mode corresponding to numbers of cavities. [ABSTRACT FROM AUTHOR]

The article provides a bibliography listing various references cited in the issue of journal.

In Mediterranean semi-arid regions, shallow aquifers are generally over-exploited. However, several deeper groundwater boreholes exhibit serious problems of low hydraulic yield and dry wells. The existence of reliable information about the regional geological structure can enhance the success of deep well programmes. Geophysics has proved very useful as a fast and efficient tool to understand the structuring of deep hydrogeology aquifers. Indeed, the identification and characterization of the deep structures of the Bir Drassen–Sidi Toumi region (Cap-Bon, northeastern Tunisia) contribute to a better quantification of the groundwater potential, allowing more sustainable management. This province can be considered as the SW extension of the NE–SW Jebel Abderrahman Atlasic anticline, one of the most prominent geological structures in northeastern Tunisia. To identify the Miocene and Oligocene reservoirs, a geophysical study was carried out using advanced gravity analysis, supported by boreholes, and seismic and geoelectrical calibration. Different gravity filters were applied to reveal the sub-basin architecture and faulting network in the region. By calculating the maxima of the total horizontal gravity derivative and the distribution solutions for Euler's equations, a set of deep faults of different directions was divulged. Time-domain electromagnetic mapping illustrated the influence of these fault zones on the regional arrangement of sub-basin aquifers and their recharge processes. Finally, a 3D residual gravity inversion was carried out to better image the tectonically controlled depocentres associated with prolific thick fluvial–deltaic reservoirs. These 'deep aquifer valley' can be suggested as strategic targets for deep hydrogeological exploration planning. [ABSTRACT FROM AUTHOR]

This research presents a practical study of active disturbance rejection control law (ADRC) for the control of robotic manipulators in the presence of uncertainties. The control objective of the proposed control law is to track the trajectory accurately and rejects the disturbance caused by robotic manipulators. First, Euler Lagrange's equations are used to model the dynamic behavior of a rotary flexible joint manipulator system (RFJMS). Secondly, the ADRC is designed for the rejection of lump disturbances (modeling uncertainties, nonlinearities, and external disturbances) of the manipulator; to estimate the lump disturbances a fifth-order extended state observer is constructed. Furthermore, a state error feedback control law with disturbance compensation is designed, which needs only a few control parameters to be adjusted. Furthermore, we demonstrate the robustness and effectiveness of the proposed control law by comparing the experimental results with those of LQR and PID. Experimental results show that high precision position control as well as vibration suppression can be achieved with the proposed controller, which is superior to LQR and PID controllers. In spite of uncertainties and nonlinearity of the flexible joint, QUANSER's RFJMS exhibits excellent tracking behavior and disturbance rejection. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the general three-dimensional equilibrium solutions for steady-state equations of inviscid fluids. Incompressible, homogeneous, inhomogeneous and compressible flows motion in a gravitational potential are considered. General three-dimensional formulas for the gas pressure and the gravitational potential are obtained. For incompressible flows, the vector and scalar potentials of the velocity field are used to derive general formulas for general three-dimensional solutions. To verify our results, some examples are presented. For compressible flows, a class of three-dimensional solutions is constructed. [ABSTRACT FROM AUTHOR]

The analysis of the derivation of the Riemann Analytic Continuation Formula from Euler's Quadratic Equation is presented in this paper. The connections between the roots of Euler's quadratic equation and the Analytic Continuation Formula of the Riemann Zeta equation are also considered. The method of partial summation is applied twice on the resulting series, thus leading to the Riemann Analytic Continuation Formula. A polynomial approach is anticipated to prove the Riemann hypothesis; thus, a general equation for the zeros of the Analytic Continuation Formula of the Riemann Zeta equation based on a polynomial function is also obtained. An expression in Terms of Prime numbers and their products is considered and obtained. A quadratic function, G(tn), that is required for Euler's quadratic equation (EQE) to give the Analytic Continuation Formula of the Riemann Zeta equation (ACF) is presented. This function thus allows a new way of defining the Analytic Continuation Formula of the Riemann Zeta equation (ACF) via this equivalent equation. By and large, the Riemann Zeta function is shown to be a type of L function whose solutions are connected to some algebraic functions. These algebraic functions are shown and presented to be connected to some polynomials. These Polynomials are also shown to be some of the algebraic functions' solutions. Conclusively, &(z) is redefined as the product of a new function which is called H(tn; z) and this new function is shown to be dependent on the polynomial function, G(tn). [ABSTRACT FROM AUTHOR]

This paper proposes two analytical methods to predict free and forced vibrations of symmetric sandwich plates in vacuum and water. The application scopes of the two methods are investigated. Displacement fields of the plates are described by the first-order shear deformation theory. The two methods are, respectively, built via the equivalent single-layer (ESL) theory and layerwise (LW) theory. The equations of motion are derived through Hamilton's principle. The Helmholtz and Euler's equations are utilized to represent the acoustic pressure in water domain and the continuities on acoustic-structure coupling faces. Through assuming the submerged sandwich as a thin plate, the formulas of transverse vibration in the two analytical models are reconstructed. The Navier method is employed to solve the new equations of the plates with simply supported boundaries at four sides. The obtained analytical nature frequencies and forced vibration responses are compared with the existing ones in the literature, and numerical results from the finite element method (FEM) and coupling finite element and boundary element method (FEM/BEM). Accuracies of the ESL and LW methods are studied for the hard or soft core case. The application scope of the ESL method is further analyzed by taking the fundamental frequency of the plates in vacuum as the evaluation object. Through five-level discussions which consider effects of the Young's modulus ratio of core to panel, the thickness ratio of core to panel, the length–thickness ratio and the length–width ratio of the sandwich plate, a general quantitative critical dimensionless parameter is designed to determine the application scope of the ESL method and distinguish the hard or soft core. [ABSTRACT FROM AUTHOR]

A fast and reliable turbine design method is necessary in aeroengine development practice. As Q3D time-marching method iteration process can be time consuming and unstable, a novel S2 stream surface approximation approach coupled with flow tangential condition is proposed to reduce the computation requirement. The surface approximation uses a quadratic function of axial coordinate x to formulate distribution of circumferential coordinate θ. The flow tangential condition inherently represents inviscid blade force effect, and the formulation of inviscid blade force is not needed. A corresponding personal Q3D turbine computer code was developed, which was suitable for both design and analysis applications. The Q3D design method solves Euler's equation through third-order Godunov's scheme with TVD property in finite volume method. Semi-implicit Crank-Nicolson's temporal scheme is implemented. Profile, secondary, and tip clearance energy loss models are added to predict viscous losses. Blade row exit swirl is required as the convergence target for Q3D design. A two-stage turbine and a single-stage high-pressure turbine were designed and analyzed by the Q3D method. Viscous 3D CFD was utilized to check design performances. The results showed that the Q3D method could finish a design case within 1 min. At design point, Q3D mass flow rate error was no more than 1.25%, expansion ratio error was no less than -0.022, and isentropic efficiency error was no more than 0.37 percentage points. The Q3D design method is fast and accurate. The stream surface approximation approach is suitable for Q3D design and analysis. [ABSTRACT FROM AUTHOR]

The authors analyze the Fréedericksz transition phenomenon of reorientation of molecules in the plane liquid crystal layer in inhomogeneous electric field generated by parallel plate capacitor using the Oseen–Frank mathematical model. The capacitor is a set of short metal plates periodically located on both sides of the extended liquid crystal layer, connected in series in two circuits. When electric charges of different signs appear on opposite plates of the capacitor, the electric field in the periodicity cell turns out to be significantly inhomogeneous both in length and in thickness of the layer. Governing equations of the model represent Euler's variational equations in the problem on minimization of the potential energy of liquid crystal. Numerical solution of the equations uses the authorial variational difference scheme. The Laplace equation for electric potential in the exterior of the liquid crystal layer is solved using the method of lines. The electric field and the patterns of orientation angles of molecules inside the layer are determined by iterating with numerical solution of Poisson's equation using the fast Fourier transform at each iteration step. The algorithm is implemented by means of CUDA technology for computing systems with graphics accelerators. The scale of the mesh to ensure the computation accuracy is selected from calculation of Tsvetkov's order parameter in a rectangular cell pattern. Verification of the algorithm and program is implemented by way of comparing the computational results and the exact solution of the problem on the homogenous electric field with the constant initial orientation angle of liquid crystal molecules. The computations accomplished yield the patterns of the electric potentials and orientation angles of molecules depending on the position of the capacitor plates at uniform and s‐shaped initial distribution of the orientation angles through the thickness of the liquid crystal layer. [ABSTRACT FROM AUTHOR]

An analytical model is developed in this article to study the effects of liquid sloshing on dynamics of articulated vehicle carrying liquid in turning maneuver. To investigate the interaction between the fluid and the vehicle, a four-axle tractor-semitrailer tanker with 32 degrees of freedom has been used. The method of separation of variables is used to solve Laplace equation together with the wall and free surface boundary conditions for fluid sloshing. Based on the results, the effects of fluid density, fill volume of the tanker, vehicle speed, and steer angle on lateral stability of the vehicle have been studied. Simulation results showed that decrease in the fill volume and fluid density leads to decrease in the normalized dynamic normal load factor of the rear axle of the semitrailer. According to the results, the frequency of the normalized roll angle of the semitrailer is independent of vehicle velocity and steering angle. [ABSTRACT FROM AUTHOR]

The depth from extreme points (DEXP) method can be used for estimating source depths and providing a rough image as a starting model for inversion. However, the application of the DEXP method is limited by the lack of prior information regarding the structural index. Herein, we describe an automatic DEXP method derived from Euler's Homogeneity equation, and we call it the Euler–DEXP method. We prove that its scaling field is independent of structural indices, and the scaling exponent is a constant for any potential field or its derivative. Therefore, we can simultaneously estimate source depths with different geometries in one DEXP image. The implementation of the Euler–DEXP method is fully automatic. The structural index can be subsequently determined by utilizing the estimated depth. This method has been tested using synthetic cases with single and multiple sources. All estimated solutions are in accordance with theoretical source parameters. We demonstrate the practicability of the Euler–DEXP method with the gravity field data of the Hastings Salt Dome. The results ultimately represent a better understanding of the geometry and depth of the salt dome. [ABSTRACT FROM AUTHOR]

In this work, I provide a new rephrasing of Fermat's Last Theorem, based on an earlier work by Euler on the ternary quadratic forms. Effectively, Fermat's Last Theorem can be derived from an appropriate use of the concordant forms of Euler and from an equivalent ternary quadratic homogeneous Diophantine equation able to accommodate a solution of Fermat's extraordinary equation. Following a similar and almost identical approach to that of A. Wiles, I tried to translate the link between Euler's double equations (concordant/discordant forms) and Fermat's Last Theorem into a possible reformulation of the Fermat Theorem. More precisely, through the aid of a Diophantine equation of second degree, homogeneous and ternary, solved not directly, but as a consequence of the resolution of the double Euler equations that originated it, I was able to obtain the following result: the intersection of the infinite solutions of Euler's double equations gives rise to an empty set and this only by exploiting a well-known Legendre Theorem, which concerns the properties of all the Diophantine equations of the second degree, homogeneous and ternary. The impossibility of solving the second degree Diophantine equation thus obtained is possible using well-known techniques at the end of 18th century (see Euler, Lagrange and Legendre) and perhaps present in Fermat's brilliant mind. [ABSTRACT FROM AUTHOR]

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We review well-balanced methods for the faithful approximation of solutions of systems of hyperbolic balance laws that are of interest to computational astrophysics. Well-balanced methods are specialized numerical techniques that guarantee the accurate resolution of non-trivial steady-state solutions, that balance laws prominently feature, and perturbations thereof. We discuss versatile frameworks and techniques for generic systems of balance laws for finite volume and finite difference methods. The principal emphasis of the presentation is on the algorithms and their implementation. Subsequently, we specialize in hydrodynamics' Euler equations to exemplify the techniques and give an overview of the available well-balanced methods in the literature, including the classic hydrostatic equilibrium and steady adiabatic flows. The performance of the schemes is evaluated on a selection of test problems. [ABSTRACT FROM AUTHOR]

This paper presents an optimized stochastic multiplier to implement Izhikevich spiking neuron model. A novel stochastic number generator is used as the main block for stochastic computing to design a stochastic multiplier. The designed multiplier was used to describe and implement Izhikevich neuron model equation which is capable to produce variant responses of spiking neurons. To the best of the author's knowledge, this work is the pioneer using a designed multiplier for computing all of the multiplication expressions in spiking neuron equation. The effect of increasing number of bit streams on the accuracy of stochastic computing is studied. Furthermore, some numerical intervals that are more prone to the error are found and illustrated upon stochastic computing usage, and a method is proposed to improve the error prone numerical intervals. In addition, the response accuracy is optimized by changing time step parameter in solving ordinary differential Izhikevich neuron model equation using Euler's method. Hence, a 13‐bit stochastic multiplier is designed. It is found that variations in parameters a and b have a profound effect on the accuracy of recovery variable u and, consequently, on the responses. Out of 20 responses of Izhikevich neuron model, 11 responses are generated using the proposed stochastic computing method and compared with piecewise linear method and some previous architectures of Izhikevich neuron model implementations. [ABSTRACT FROM AUTHOR]

The Dzhanibekov effect is the phenomenon by which triaxial objects like a spinning wing bolt may continuously flip their rotational axis when initially spinning around the intermediate axis of inertia. This effect is closely related to the Tennis Racket theorem that establishes that the intermediate axis of inertia is unstable. Over time, however, dissipation ensures that a torque free spinning body will eventually rotate around its major axis, in a process called precession relaxation, which counteracts the Dzhanibekov effect. Euler's equations for a rigid body effectively describe the Dzhanibekov effect, but cannot account for the precession relaxation effect. A dissipative generalization of Euler's equations displays two dissipative mechanisms: orientational diffusion and viscoelasticity. Here we show through numerical simulations of the dissipative Euler's equations that orientational diffusion, rather than viscoelasticity, primarily drives precession relaxation and effectively suppresses the Dzhanibekov effect. [Display omitted] • Numerical solution of the dissipative Euler's equations. • The Dzhanibekov effect is strongly affected by dissipation through precession relaxation. • Precession relaxation is governed by orientational diffusion. • Viscoelasticity is not responsible for precession relaxation. [ABSTRACT FROM AUTHOR]

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well. • We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. • The method uses the non-divergence formulation of the MHD system. • The method preserves positivity of density, minimum principle of the specific entropy, energy-stability, and involution constraints. • Euler equations and MHD source system are treated separately allowing for significant freedom in the choice of solvers. • CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. [ABSTRACT FROM AUTHOR]

With this notice, the Mathematics Editorial Office states its awareness of concerns regarding the findings of the published manuscript [[1]]. Concerns have been raised by readers about the validity and accuracy of the results. In order to maintain the highest scientific standards, the Editorial Office has decided to issue this expression of concern while an investigation is conducted.The author of this publication has been contacted for further clarification. The investigation is being coordinated by the Editorial Board and Editor-in-Chief. An update will be provided following the conclusion of the investigation. [Extracted from the article]

Quasisimple wave solutions of Euler's system of equations for ideal gas are investigated under the assumption of spherical and cylindrical symmetries. These solutions are proved to be stabilized into sound wave solutions and cavitation. It is proved that if initial conditions from outside the invariant region approach to transitional solution, then reciprocal of the self-similar parameter goes to infinity. However, when initial conditions stabilize into sound waves or cavitation, then reciprocal of self-similar parameter approaches finite value. Further, it is proved that initial conditions can be parametrized so that some of the initial conditions stabilize into sound wave solutions. The rest of the initial conditions are proved to be stabilized into cavitation. This extends the work of G. I. Taylor to the case of cavitation. It is proved that quasisimple wave solutions exist for the balance laws comprised of Euler's system of equations in the case of cylindrically and spherically symmetric cases. The description applies to the motion of cylindrical and spherical piston in real life. In particular, self-similar description of appearance of vacuum in the motion of cylindrical and spherical piston is given. [ABSTRACT FROM AUTHOR]