Euler Form Research Articles

Teachers’ knowledge of different forms of complex numbers through quantitative reasoning

ABSTRACT This study investigated teachers’ conceptualizations of different forms of complex numbers in a professional development study (PD) conducted with classroom teaching experiments focusing on quantitative reasoning. We report from five secondary school mathematics teachers’ post-interview data and the post-written sessions upon completion of the PD. Results showed that all the participants could relate the formal definition of complex numbers with the roots of quadratic equations both algebraically and geometrically. Participants could also explain the Cartesian and polar form relationship using vectors with connections to the roots of quadratic equations. They further explained the Euler form by pointing out that the polar form of any complex number on a circle determines a function of Θ from R to C. Albeit making connections, results also pointed to some teachers’ difficulties in certain points. Altogether, results suggest that quantitative reasoning might lay a foundation for connecting different forms of complex numbers. We discuss implications for curriculum, teaching, and teacher education.

On homological properties of the category of [formula omitted]-representations over a linear quiver of type [formula omitted

Let Q be a quiver of type An with linear orientation and rep(Q,F1) the category of representations of Q over the virtual field F1. It is proved that rep(Q,F1) has global dimension 2 whenever n≥3 and it is hereditary if n≤2. As a consequence, the Euler form 〈L,M〉=∑i=0∞dim⁡Exti(L,M) is well-defined. However, it does not descend to the Grothendieck group of rep(Q,F1). This yields negative answers to questions raised by Szczesny (2012) [4].

Closed Loop Vector Formulation in Eulers Complex Numbers for Multi Loop Planar Mechanisms With N bars A Novel Modeling Approach and Algorithm

This paper presents a novel iterative algorithm incorporated in a user-friendly GUI for modeling the kinematics of multiple looped N-bar closed-loop mechanisms. Past research works have used custom coding or expensive commercial software to analyze the mechanisms of specific applications. The proposed algorithm focuses on kinematics and offers a quick, easy-to-use, cost-effective solution to analyze a wide range of generic mechanisms, reducing the need for custom coding and lowering computational costs. The algorithm employs algebraic equations, such as solving complex closed-loop vector equations using the Euler form of complex numbers, to simulate and derive the unknowns necessary to characterise any generic closed-loop mechanism. The Python code implemented in the algorithm adapts to various scenarios by utilising available information on the position, velocity, and acceleration variables of the mechanisms. The simulation tool can display real-time color contour plots (RGB color scale) for linear and angular velocities and accelerations, simulate mechanisms with multiple loops and switch configurations, and find inverse mechanisms. The approach for solving multiple loop problems and the algorithm utilized to solve the configurations, methods, equations used and GUI features implementation are all described in this study. The case study considered for a four-bar mechanism indicates a strong agreement between the results obtained from the proposed kinematics-based simulator and ANSYS software. These results demonstrate the simulator’s effectiveness in providing low-cost and user-friendly simulation results for various generic mechanisms involving multiple interconnected loops.

Global exponential stability of discrete-time almost automorphic Caputo–Fabrizio BAM fuzzy neural networks via exponential Euler technique

Exponential Euler discrete schemes have been widely employed in the studies of Caputo fractional order differential equations, but almost no literature concerns the Caputo–Fabrizio case. In current work, exponential discrete form has been set up to study Caputo–Fabrizio fuzzy BAM neural networks(CF-FBAMNNs). The research findings tell someone that (1) the exponential discrete form characterizes the continuous CF-FBAMNNs superior to the classical Euler discrete technique; (2) this type discrete technique pertains to the implicit Euler form, which can be calculated by PECE algorithm. Furthermore, the existence of a unique bounded asymptotically almost automorphic sequence solution and global exponential stability of the proposed discrete-time models are investigated. More importantly, the current works make up for the lacks in the existing literatures and build a set of new theories and methods in studying discrete-time Caputo–Fabrizio models in the fields of science and engineering.

Колебательные режимы атмосферы Юпитера и Венеры

Introduction. The article studies the oscillatory modes of the atmosphere of two planets (Jupiter, Venus), taking into account the influence of axial rotation, which affects the occurrence of beating and resonance, and comparing their parameters with the parameters of the Earth’s atmosphere. Materials and research methods. We consider fluctuations in the atmosphere, which was initially in a state of statics, caused by the initial temperature rise near the earth's surface and the daily rotation of the earth. To describe them, a system is used that includes the equation of motion in the Euler form and expressions that take into account the distribution of temperature and air pressure with height. The case when the rotation of the Earth around its axis leads to significant periodic temperature changes is considered in detail. Research results and their discussion. The values of the main parameters of the atmosphere of Jupiter and Venus are found in order to study the oscillatory regimes of the atmospheres on these planets and build an appropriate mathematical model. At the beginning, we find the standard parameters of velocity amplitudes, temperatures, Brunt-Väisälä frequencies in order to calculate the quantities included in the system of equations describing the fluctuations of air in the atmosphere and build the corresponding graphs of these equations for the purpose of their further analysis. Solutions for oscillatory processes are obtained when daily changes in air temperature are taken into account. Conclusions. It is shown that in the standard atmosphere the axial rotation of the planet does not affect the oscillatory processes in the atmosphere, while the terrain is taken into account. However, there is a case when the difference between the above frequencies is extremely small or zero (this is possible at). If the frequency of Brent – Väisälä coincides with the frequency of the daily rotation of the Earth or their difference is insignificant, then phenomena such as resonance or beats occur, which are considered in detail.

The Faddeev-LeVerrier algorithm and the Pfaffian

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(nβ+1) where n is the size of the matrix and O(nβ) is the cost of multiplying n×n-matrices, β∈[2,2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

Model of Radiation-Induced Thermomechanical Effects in Heterogeneous Finely Dispersed Materials

A complex model for supercomputing the parameters of radiation-induced thermomechanical fields in heterogeneous media of a complex dispersed structure is developed. A technique for calculating the parameters of the photon-electron cascade generated in the object by the interaction of radiation with matter is constructed. A geometric model of the medium with a direct resolution of its microstructure is worked out. The model of the detecting system for the statistical evaluation of the energy deposit of radiation is part of the geometric description of the medium. The continuum mechanics equations taken in the Euler form of the conservation laws are the base for calculating thermomechanical processes. The results of trial simulations in the form of the calculated thermomechanical fields are presented.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

Modelling the Motion of a Single Solid Bead in a Newtonian Fluid by Two-Phase CFD Methods

Sedimentation is important in many processes for example in cement, chemical and food industries. Investigation of the sedimentation process of solid bodies showed a phenomenon as settling velocity is not the constant value as it is expected from the analytical deduction; rather it has fluctuating behaviour. Seeking an explanation, detailed two phase hydrodynamic models are developed. In the present work, two-dimensional incompressible flow equation in conservative Euler form is solved numerically on PC by an implicit finite difference method, and the fluid-solid interaction is described by the Immersed Boundary Method. For additional model validation, Particle Tracing module of COMSOL is applied as well. The self-developed simulator based on an a priori (white box) model, and the 3D model by the commercial CFD software is used for modelling the two-phase system and the particle motion in the fluid column. Settling velocity is calculated and compared to the velocity of a particle recorded by a high-speed camera measurement system in the laboratory scale sedimentation equipment. The results are promising, and after improvements, the simulator can be a useful tool in supporting energetically optimal industrial design and operation.

Simulation of Gas Solid Flow in Quasi Two Dimensional Fluidized Bed by Immersed Boundary Method

Gas solid two phase flow occurs in multiple fields of industry. Therefore, a study researching the phenomena of this type of systems is highly adequate. In the present work subsonic, compressible flow equation in conservative Euler form is solved numerically on PC by a two step finite difference method, which is validated by the analytical solution of Sod’s shock tube problem. Fluid solid interaction is described by Immersed Boundary Method, where body force is calculated by direct forcing algorithm. Two dimensional simulation of a single solid circle shaped particle motion in the gas flow is calculated and compared to the trajectory of a particle recorded by a high speed camera measurement system in a laboratory scale quasi two dimensional fluidized bed. Image processing algorithms are implemented to track the motion of a chosen single. Comparison of experimental measurement and simulation resulted in the same magnitude of particle velocities. The possible reasons of differences and opportunities of improving the results are discussed in the study. The results are promising, and the presented preliminary simulator can be a useful tool in supporting industrial design and operation after further developments.

Pitching effect on transonic wing stall of a blended flying wing with low aspect ratio

Numerical simulation of the pitching effect on transonic wing stall of a blended flying wing with low aspect ratio was performed using improved delayed detached eddy simulation (IDDES). To capture the discontinuity caused by shock wave, a second-order upwind scheme with Roe’s flux-difference splitting is introduced into the inviscid flux. The artificial dissipation is also turned off in the region where the upwind scheme is applied. To reveal the pitching effect, the implicit approximate-factorization method with sub-iterations and second-order temporal accuracy is employed to avoid the time integration of the unsteady Navier–Stokes equations solved by finite volume method at Arbitrary Lagrange–Euler (ALE) form. The leading edge vortex (LEV) development and LEV circulation of pitch-up wings at a free-stream Mach number M = 0.9 and a Reynolds number Re = [Formula: see text] is studied. The Q-criterion is used to capture the LEV structure from shear layer. The result shows that a shock wave/vortex interaction is responsible for the vortex breakdown which eventually causes the wing stall. The balance of the vortex strength and axial flow, and the shock strength, is examined to provide an explanation of the sensitivity of the breakdown location. Pitching motion has great influence on shock wave and shock wave/vortex interactions, which can significantly affect the vortex breakdown behavior and wing stall onset of low aspect ratio blended flying wing.

Construction of non-commutative surfaces with exceptional collections of length 4

Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\geq 2$ there are no commutative surfaces. In this paper we construct sheaves of maximal orders on surfaces having these Euler forms, giving a geometric construction for their numerical blowups.

Asymptotic behavior of impulsive neutral delay differential equations with positive and negative coefficients of Euler form

In this paper, we are concerned with asymptotic properties of solutions for a class of neutral delay differential equations with forced term, positive and negative coefficients of Euler form, and constant impulsive jumps of the form \t\t\t{[x(t)−C(t)g(x(τ(t)))]′+P(t)tf(x(αt))−Q(t)tf(x(βt))=h(t),t≥t0>0,t≠tk,x(tk+)−x(tk)=αk,k∈Z+.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} [x(t)-C(t)g(x(\\tau(t)))]'+ \\frac{P(t)}{t}f(x(\\alpha t))-\\frac {Q(t)}{t}f(x(\\beta t))=h(t),\\quad t\\geq t_{0}>0, t\\neq t_{k},\\\\ x(t_{k}^{+})-x(t_{k})=\\alpha_{k},\\quad k\\in{\\mathbb{Z}_{+}}. \\end{cases} $$\\end{document} By constructing auxiliary functions and applying the technique of considering asymptotic properties of nonoscillatory and oscillatory solutions we establish some sufficient conditions to guarantee that every solution of the system tends to zero as tto+infty.

ON NEW CLASSES OF ANALYTIC FUNCTIONS IMPOSED VIA THE FRACTIONAL ENTROPY INTEGRAL OPERATOR

In this paper, we aim to introduce some geometric properties of analytic functions by utilizing the concept of fractional entropy in a complex domain. We extend the fractional entropy, type Tsallis entropy in the complex z-plane, by using some analytic functions. Established by this diffusion,we state specic new classes of analytic functions (type Schwarz function). Other geometric properties are validated in the sequel. Our development is completed by the Euler form Lemma and Jack Lemma.

An Euler–Lagrange approach for studying blood flow in an aneurysmal geometry

To numerically study blood flow in an aneurysm, the development of an approach that tracks the moving tissue and accounts for its interaction with the fluid is required. This study presents a mathematical approach that expands fluid mechanics principles, taking into consideration the domain’s motion. The initial fluid equations, derived in Euler form, are expanded to a mixed Euler–Lagrange formulation to study blood flow in the aneurysm during the cardiac cycle. Transport equations are transformed into a moving body-fitted reference frame using generalized curvilinear coordinates. The equations of motion consist of a coupled and nonlinear system of partial differential equations (PDEs). The PDEs are discretized using the finite volume method. Owing to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that velocity is substantially influenced by the pulsating wall. Intensification of polymorphic flow patterns is observed. Increments of Reynolds and Womersley numbers are evident as pulsatility increases. The pressure field reveals areas of a lateral pressure gradient at the aneurysm. As pulsatility increases, the diastolic flow vortex shifts towards the aortic wall, distal to the aneurysmal neck. Wall shear stress is amplified at the shoulders of the moving wall compared with that of the rigid one.

On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

Abstract We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve Y with bi-degree (2,2) in a product of projective lines ℙ1 × ℙ1. We calculate two differenent monodromy representations of period integrals for the affine variety X(2,2) obtained by the dual polyhedron mirror variety construction from Y. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on Y with respect to the Euler form.

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

We prove that the Euler form of a metric connection on a real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$.

Numerically finite hereditary categories with Serre duality

Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.

A stochastic Gauss–Bonnet–Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle \(E\) over compact manifold \(M\) canonically defines a metric on \(E\) together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle \(E\) and the manifold \(M\) are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.

A Gabriel-type theorem for cluster tilting

We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization of BGP reflection functors.