Euler's Formula Research Articles

Bounding the Number of Edges of Matchstick Graphs

A matchstick graph is a crossing-free unit-distance graph in the plane. Harborth conjectured in 1981 that the maximum number of edges of a matchstick graph with $n$ vertices is $\lfloor 3n-\sqrt{12n-3}\rfloor$. Using the Euler formula and the isoperimetric inequality, it can be shown that a matchstick graph with $n$ vertices has no more than $3n-\sqrt{2\pi\sqrt{3}\cdot n}+O(1)$ edges. We improve this upper bound to $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\sqrt{3}})$. The main tool in the proof is a new upper bound for the number of edges that takes into account the number of nontriangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.

Identification of dynamic parameters of closed-chain industrial robots considering motor couplings

Closed-chain robots have great advantages in machining complex components, especially under high-speed and heavy-load conditions. Accurate dynamic model is fundamental importance to obtain good performance. However, most existing works focus on the excitation trajectories and identification methods, but few works of literature deal with the accuracy and simplification of non-singular dynamic modelling. In this paper, we propose a general scheme to identify the non-singular dynamic model considering motor couplings, which also simplifies the elimination of redundant parameters. At first, a symbolic model is proposed for motor couplings by analysing the mapping relationships of joint rotations, and the coefficients of the model can be identified by the presented numerical method of least squares curve fitting (LSCF). Then, a unified approach is proposed to determine non-singular dynamic model by performing adjacent link mapping (ALM) with Newton–Euler formulas, which is more convenient and applicable to both serial and tree structures. Finally, a refined model of closed-chain robot is given for identification, by considering motor couplings and applying the simplified elimination of redundant parameters. Compared to the existing schemes, the proposed motor-couplings based dynamic model is more accurate, and the ALM has better applicability and efficiency. Experimental studies are conducted to validate the proposed methodology on a closed-chain robot.

Euler Formula Derivation

This paper discusses the derivation of Euler's formula. To obtain this model, the writer derives Euler's formula from ex+iy by first finding the norm and argument of ex+iy. In this derivation we substitute the norm and argument of ex+iy on complex numbers in polar coordinates, until we get the derivation of Euler's formula.

The Euler characteristic as a basis for teaching topology concepts to crystallographers.

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V - E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss-Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

Local certification of graphs on surfaces

A proof labelling scheme for a graph class C is an assignment of certificates to the vertices of any graph in the class C, such that upon reading its certificate and the certificates of its neighbors, every vertex from a graph G∈C accepts the instance, while if G∉C, for every possible assignment of certificates, at least one vertex rejects the instance. It was proved recently that for any fixed surface Σ, the class of graphs embeddable in Σ has a proof labelling scheme in which each vertex of an n-vertex graph receives a certificate of at most O(log⁡n) bits. The proof is quite long and intricate and heavily relies on an earlier result for planar graphs. Here we give a very short proof for any surface. The main idea is to encode a rotation system locally, together with a spanning tree supporting the local computation of the genus via Euler's formula.

The failure behavior of buckling pin valve: Theoretical, numerical, and experimental study

At present, buckling pin in the bypass of piping as pressure relief valve has been gradually utilized in the low-concentration coal-bed methane (CBM), which bends to release pressure when the main valve fails leading to pipeline blockage. However, current researches mainly focused on the buckling behavior of hydraulic cylinder rod or rod string, and less consideration was given to the operational reliability of buckling pin valves. This paper deduced the calculation formula of the critical failure load based on Euler formula in the buckling pin under buckling load. Besides, three finite element models (FEM) based on Johnson−Cook constitutive model were compared to predict failure strength of buckling pin which were verified by experiment. In addition, the defect sensitivity analysis of the buckling pin under different initial geometric defects rate was carried out. The results showed that a) the experimental value of the critical failure load in the buckling pin was 206.04 N and the bending position was in the middle of the buckling pin; b) the analysis result adopting explicit dynamic method was in best agreement with the experimental results within deviation of 0.24%; and c) the initial geometric defect of buckling pin should be controlled within 1%. This study provides an important reference to predict the critical failure load of the buckling pin valve and achieve safe transportation of low-concentration CBM.

Euler's Formula Determines the Minimum Number of Splits in Maps of Election Districts

Euler's Formula Determines the Minimum Number of Splits in Maps of Election Districts

Dynamic response analysis of isolated curved beam bridge under multi-dimensional non-stationary random excitation considering the torsion component

The coupling effect of bending and torsion caused by plane irregularity of the isolated curved beam bridge makes its seismic response more complex than that of the isolated straight beam bridge. In order to study the influence of multi-dimensional non-stationary random excitation considering torsion component on seismic response of tqiongw@mls.sinanet.com he isolated curved beam bridge. The classical Bouc-Wen model was used to simulate the hysteretic characteristics of the curved beam bridge, and the nonlinear dynamic equation considering the eccentricity of superstructure was established. The pseudo excitation was transformed into harmonic external load by Euler formula, and combining with the precise integration method, the precise integration forms of the special solution under different modulation functions were derived. By using these integral forms to solve the response at each time, the time-varying variance of displacement of the isolated curved beam bridge under multi-dimensional non-stationary random excitation was obtained, and the law of time-varying variance of displacement, including the upper and lower structures of the isolated curved beam bridge, was analyzed with different deck width, curvature radius and yield ratio of bearing. The results show that the multi-dimensional non-stationary random excitation with torsion component has significant influence on the seismic response of the isolated curved beam bridge. Under the rare non-stationary random excitation, the dynamic response of the isolated curved beam bridge presents strong non-stationarity, and the time-lag phenomenon is apparent. For the isolated curved beam bridge with large deck width and small yield ratio of bearing, the influence of non-stationary random excitation on the structure is particularly obvious. The influence of non-stationary random excitation considering torsion component on dynamic response of the isolated curved beam bridge with large deck width, small curvature radius and large yield ratio of bearing is more significant.

Overall stability design of T-shaped irregular concrete-filled steel tubular columns under axial compression

This paper presents investigations into the overall stability design of irregularly-shaped concrete-filled steel tubular columns with T-shape (T-ICFSTCs). The T-ICFSTC consists of I-steel, U-steel, and infilled concrete, forming a multi-cell steel tubular column. They are applied in the T-shaped sectional corner of the intersection between two walls in the residential buildings structures. The overall instability of the slender T-ICFSTCs may still be a significant concern in their strength design when they are placed in a higher spacing between two adjacent floors. This paper mainly focuses on establishing a design method for predicting the overall stability capacity of T-ICFSTCs subjected to axial load. Firstly, experimental investigations of an end-hinged T-ICFSTC and an end-fixed T-ICFSTC under axial load were carried out for checking its overall stability bearing capacity. Subsequently, the refined finite element (FE) model of the T-ICFSTCs is established and validated by the experimental results. In addition, the elastic buckling behavior of T-ICFSTC is studied by employing finite element (FE) eigenvalue buckling analysis, and the corresponding normalized slenderness ratio is obtained in the form of the Euler formula. Finally, based on the FE model validated experimentally, the parametric analyses of T-ICFSTCs are performed to investigate their overall stability capacities through changing the effects of steel ratio, sectional size, and material strength combination between concrete and steel. With these FE parametric analytic results, the overall stability coefficient φ, as well as φ-λ curves of T-ICFSTCs, are then obtained. These curves form the overall stability design formula of T-ICFSTCs loaded axially and apply in a common range of the percentage of load-carrying capacity shared by concrete. This design formula provides fundamentals to further develop a comprehensive design method of the T-ICFSTCs subjected to combined axial loads and moments.

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.

High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L2-gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.

3D cell neighbour dynamics in growing pseudostratified epithelia.

During morphogenesis, epithelial sheets remodel into complex geometries. How cells dynamically organise their contact with neighbouring cells in these tightly packed tissues is poorly understood. We have used light-sheet microscopy of growing mouse embryonic lung explants, three-dimensional cell segmentation, and physical theory to unravel the principles behind 3D cell organisation in growing pseudostratified epithelia. We find that cells have highly irregular 3D shapes and exhibit numerous neighbour intercalations along the apical-basal axis as well as over time. Despite the fluidic nature, the cell packing configurations follow fundamental relationships previously described for apical epithelial layers, that is, Euler's polyhedron formula, Lewis' law, and Aboav-Weaire's law, at all times and across the entire tissue thickness. This arrangement minimises the lateral cell-cell surface energy for a given cross-sectional area variability, generated primarily by the distribution and movement of nuclei. We conclude that the complex 3D cell organisation in growing epithelia emerges from simple physical principles.

Dynamic parameter identification method for robotic arms with static friction modelling

This paper presents an identification method for robotic manipulators. It demonstrates how a dynamic model can be constructed with the help of the modified Newton–Euler formula. To model the friction of the joints, static friction modelling is used, in which the friction behaviour depends only on the actual velocity of the given joint. With these techniques, the model can be converted into a linear-in-parameters form, which can make the identification process easier. Two estimators are introduced to solve the identification problem, the least-squares and the weighted least-squares estimators, and the determination of the independently identifiable parameter vector to make the regression matrix maximal column rank is presented. The Frobenius norm is used as the condition of the regression matrix to optimise the excitation trajectories, and the form of the trajectories has been selected from the finite Fourier series. The method is tested in a simulated environment to achieve a three-degrees-of-freedom manipulator.

Analysis of Euler Characteristic

Usually, people first learn about Euler characteristic in three-dimensional space, related to the famous Euler formula. This invariant is explained in a top-down manner. In particular, there is an invariant first defined on general smooth manifold. Then, looking at different properties of this invariant leads to the discover of coincidence with Euler formula in lower dimension case. This paper extends the definition of Euler characteristic to more general geometries by glueing, and seeks for some possible applications. Result shows that Euler Charisteristic is an useful invariant that ties many subject in topology together. By studying Euler Characteristic in various cases, one may found interesting relationship between objects that are not seemingly related.

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

Phase-field crystal study of topological evolution during grain annihilation process in polycrystalline materials

The topological evolution and annihilation mechanism of the three-, four- and five-sided grains are studied by using the phase field crystal (PFC) model. Both the large-angle and small-angle grain boundaries are discussed during the grain annihilation process. The results show that for the large-angle grain boundaries, grain annihilation is achieved through grain boundaries migration and absorption by triple junction. For the small-angle grain boundaries, the grain annihilation is achieved by dislocation slip, reaction, and absorption by triple junction. However, when the number of sides of a grain increases to seven, the grains show obvious growth instead of annihilation. Two kinds of growth mechanisms for the seven-sided grain are obtained by simulating the growth process. The effects of the number of grain sides on the evolution of grain annihilation are explored. These results are consistent with the Von Neumann-Mullins model and Euler formula.

On Infinite Series and with Their Some Applications to Euler's Summation Formula

Infinite series is still used in engineering, mathematical and physical sciences.In the modern area a great progress is made in the sciences of calculus in addition to what was accompanied by advances in infinite series and their some applications .The aims of this paper is to develop and introduces the infinite series and their some applications to Euler's summation , also we show and explain how to apply the infinite series in Euler's summation .We followed the induction mathematical method and found that : The relationship through the Euler's summation focused on the physical link questioner to make the study as an application for Euler's summation so it can be the beginning of advanced study in concept of the infinite series and their some applications to Euler's summation.

On the Colored and the Set-Theoretical Yang–Baxter Equations

This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied.

Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations

In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form to where is the mesh parameter. Finally, some numerical results are given to support the theoretical results.

Испытание гибкого стержня на сжатие

The article considers experimental and theoretical research of longitudinal stability of a flexible steel bar design under axial compression. The bar is a flat thin-walled element, pivotally fixed at the ends. The experimental study was carried out on a Zwick/Roell Z100 installation using special equipment that simulates geometric boundary conditions. During the loading process, a diagram of the deformation of a real bar with initial shape imperfections was automatically constructed. The experimental critical force was determined from the deformation diagram. This force was compared with the critical forces obtained from calculations using two schemes: the Euler formula and the dynamic analysis methodology. In the second scheme, in contrast to the first one, the initial imperfection, established by the measurements of the tested structure, was taken into account. The design calculation errors for both schemes were determined.