Indefinite Equations Research Articles

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Accurate error compensation method for multi-axis parallel machine via singularized jacobi geometric parameter correction and coupling error evaluation

The Jacobian model is a prevalent tool for error compensation in multi-axis parallel mechanisms. However, discrepancies between the model's nominal and actual geometrical parameters, combined with equivalent replacements and high-order rounding in the modeling process, lead to equation solving challenges and modeling errors. These inaccuracies result in residual errors in the Jacobian model compensation. To address these problems, this paper proposes an optimal Jacobian correction approach. This is based on a geometrical parameter singularized Jacobian correction model, and a module for the evaluation of coupling errors for multi-axis parallel mechanisms was incorporated. Instead of relying on iterative processes, a singularized geometrical error solution method (SESM) was developed. Through this method, precise derivation of the Jacobian correction parameters is ensured, effectively addressing the indefinite equation challenge and partial posture non-solution problem. Moreover, modeling errors resulting from equivalent infinitesimal replacements and the overlooking of high-order minor values are compensated for by the SESM. It was observed that varying singularized geometrical parameters in the Jacobian model can produce different coupling effects and compensation outcomes. Therefore, a sensitivity-based error predictive evaluation method (EPEM) was introduced. By this method, the optimal correction parameter of the Jacobian model across the entire workspace is identified, ensuring precise pose error compensation. The proposed method was validated using a three-axis parallel mechanism. Through these tests, its superior efficacy was revealed. In comparison to the traditional uncorrected Jacobian compensation, reductions in position and orientation errors by 64.93% and 55.29%, respectively, were achieved. This method provides a new approach for error modeling, equation solving, and parameter correction for multi-axis mechanism error compensation and precision equipment development.

The Solution of the Indefinite Equation by the Method of Euclidean Algorithm

The Solution of the Indefinite Equation by the Method of Euclidean Algorithm

Erratum to “Solutions of Indefinite Equations”, [Advances in Pure Mathematics Vol. 10, No. 9, (2020) 540-544

The original online version of this article (Zengyong Liang (2020) Solutions of Indefinite Equations, Volume 10(9), 540-544, doi: https://doi.org/10.4236/apm.2020.109033) unfortunately contains some mistakes. The author wishes to correct the errors. Sections 5, 6, 7, and 8 are supplemented here.

Stability and exact periodic solutions of indefinite equations arising from the Kepler problem on the sphere

Stability and exact periodic solutions of indefinite equations arising from the Kepler problem on the sphere

ИССЛЕДОВАТЕЛЬСКИЕ ЗАДАЧИ В ОБУЧЕНИИ МАТЕМАТИКЕ СТУДЕНТОВ ПЕДВУЗОВ

The development of society requires modern graduates to be ready for continuous self-education, updating their knowledge and skills, the ability to solve research problems and to scientific research. One of the most important goals of the implemented educational programs that meet the requirements of the federal state educational standards of higher education is the formation of research competencies. This is especially relevant for future teachers, as they will be able to apply the acquired skills in their future professional activities. The article actualizes the issue of using research problems in the study of mathematics as a forming means of research competencies, since in the process of solving them, all components of research activity are mastered. The content features of various mathematical disciplines make it possible to do this especially effectively. The article describes the method of using research problems in the study of the discipline «Algebra and number theory», which is one of the components of mathematical and professional pedagogical education. The listing of the material is presented by a detailed description of the study of indefinite equations solved by the method of searching the remainders of dividing an algebraic expression which is containing a variable by a certain number. It is presented how a change in some parameters of the problem leads to a change in the result. The solution of such an equation is already a full-fledged research task. In addition, experience has shown that a positive cognitive result is obtained by a method that allows, based on the analysis of ready-made solutions, to learn how to construct new equations with predefined properties, which brings students to a new level of creative work. In conclusion, it is noted that, despite a number of objective difficulties associated with the low level of training of some students, the need for additional time expenditures, this is significant, not only from the point of view of forming the necessary research competencies, but also contributes to the activation of educational and cognitive activities, the motivation growth to study the discipline.

The Collatz Conjecture and Linear Indefinite Equation

For the collatz conjecture, we define an iterative formula of odd integers according to the basic theorem of arithmetic, and give the concept of iterative exponent. On this basis, a continuous iterative general formula for odd numbers is derived. With the formula, the equation of cyclic iteration is deduced and get the result of the equation without a positive integer solution except 1. On the other hand, the general formula can be converted to linear indefinite equation. The solution process of this equation reveals that odd numbers are impossible to tend to infinity through iterative operations. Extending the result to even numbers, it can be determined that all positive integers can return 1 by a limited number iterations.

The floating-body problem: an integro-differential equation without irregular frequencies

The linear boundary value problem under consideration describes time-harmonic motion of water in a horizontal three-dimensional layer of constant depth in the presence of an obstacle adjacent to the upper side of the layer (floating body). This problem for a complex-valued harmonic function involves mixed boundary conditions and a radiation condition at infinity. Under rather general geometric assumptions the existence of a unique solution is proved for all values of the nonnegative problem's parameter related to the frequency of oscillations. The proof is based on the representation of solution as a sum of simple- and double-layer potentials with densities distributed over the obstacle's surface, thus reducing the problem to an indefinite integro-differential equation. The latter is shown to be soluble for all continuous right-hand side terms for which purpose S.~G. Krein's theorem about indefinite equations is used.

Solutions of Indefinite Equations

Indefinite equation is an unsolved problem in number theory. Through explo-ration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations. In this paper, we will introduce and prove the solutions of Pythagorean equation, Fermat’s the-orem, Bill equation and so on.

О МЕТОДИКЕ ОБУЧЕНИЯ СОСТАВЛЕНИЮ НЕКОТОРОГО ВИДА АРИФМЕТИЧЕСКИХ ЗАДАЧ

The article is devoted to a current topic related to the development of methods for composing problems in teaching students of pedagogical universities of mathematical faculties. This problem becomes especially important in the context of the need to involve students in independent creative activities to acquire and apply knowledge. The material is presented in relation to the section of the discipline «Algebra and number theory», dedicated to solving Diophantine equations, the main objectives of which are not only mastering the theory and algorithms for solving basic problems, but also obtaining the necessary knowledge and skills for further professional activity. Solving a problem, the student must not only solve it correctly and quickly enough, but also show the creative component of the activity, using it as much as possible for their mathematical development. In this regard, the process of composing problems by students is undoubtedly useful, which reflects the systematic application of the material and elements of mathematical actions based on the laws and methods of mathematics. In addition, the ability to compose problems will be required in future activities related to teaching mathematics. The processes of solving and composing tasks are interconnected and this allows you to increase the efficiency and effectiveness of composing and solving tasks. Therefore, the teacher can give a task to the student with the requirement to compose (fully or partially) and solve the problem. In this paper, examples of tasks for the compilation of indefinite equations solvable in integers are considered, for the solution of which the methods of number theory are used: the study of possible residuals from dividing an algebraic integer expression by a specific integer; finding integer solutions to a linear equation with two variables. The stages of composing Diophantine equations are described in detail, the ways of obtaining equations solvable on a given set of integers or natural numbers are analyzed, and the application of various theoretical propositions used for their solution is shown.

Non-real eigenvalues of nonlocal indefinite Sturm\u2013Liouville problems

The present paper deals with non-real eigenvalues of regular nonlocal indefinite Sturm–Liouville problems. The existence of non-real eigenvalues of indefinite Sturm–Liouville differential equation with nonlocal potential K(x,t) associated with self-adjoint boundary conditions is studied. Furthermore, a priori upper bounds of non-real eigenvalues for a class of indefinite differential equation involving nonlocal point interference potential function is obtained.

Fast algorithm for 3D wideband LFM source localisation based on time delay under a uniform circular array

Traditional localisation technologies for wideband linear frequency modulation (LFM) source are always confined to the far-field situation and suffer from high computational cost for dealing with multiple sub-narrowband components. In this study, by performing temporal delay-and-correlation to wideband LFM signals, a fast and closed-form algorithm based on phase differences is proposed for three-dimensional (3D) source localisation (azimuth angle, elevation angle, and range) under a uniform circular array. First, the cross-correlation of the array outputs with their temporally delayed version is calculated to develop a new single-frequency signal model. Then, the phase differences of adjacent sensors’ transformed outputs are exploited to construct indefinite equations. After that, the least-square method is employed to estimate the source's 3D parameters simultaneously. Moreover, theoretical analysis and numerical examples show the aperture restriction in avoiding ambiguity can be relaxed under the new signal model. Finally, simulations demonstrate the effectiveness and advantages of the proposed algorithm.

高等学校数学における数学的活動を生かした 授業づくりに関する実践研究（I） ─ オープンエンドアプローチによる「二元一次不定方程式」の学習指導を通して ─

高等学校数学における数学的活動を生かした 授業づくりに関する実践研究（I） ─ オープンエンドアプローチによる「二元一次不定方程式」の学習指導を通して ─

A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity

Efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equation $$\begin{aligned} u''=\frac{h(t)}{u^{\lambda }},\quad \lambda \in (0,1), \end{aligned}$$ are established. Here, $$h\in L(\mathbb {R}/T\mathbb {Z})$$ is a rather general sign-changing function with $$\overline{h}<0$$ . In contrast with the results in Godoy and Zamora (Proc R Soc Edinb Sect A Math) and Hakl and Zamora (J Differ Equ 263:451–469, 2017), the key ingredient to solve the aforementioned problem seems to be connected more with the oscillation and the symmetry aspects of the weight function h than with the multiplicity of its zeroes. Roughly speaking, the solvability for the above-mentioned problem can be guaranteed when $$H_+\approx H_-$$ and $$H_+$$ is large enough.

Mixed Incoherent Far-Field and Near-Field Source Localization under Uniform Circular Array.

A high-accuracy algorithm is presented for the localization of mixed incoherent near-field and far-field narrow-band sources under uniform circular array (UCA). Herein, considering that it is difficult to classify the mixed sources, we first decouple mixed sources’ angles and ranges by calculating centro-symmetric sensors’ phase differences. Then, as the phase differences including only sources’ angles can be transformed as indefinite equations, each source’s azimuth angle and elevation angle are obtained by performing the least squares method. After that, on the basis of the estimated angles of the mixed sources, one-dimensional (1-D) multiple signal classification (MUSIC) method and corresponding spatial spectrum are utilized to identify the mixed sources and estimate the ranges of the near-field sources. Finally, simulation and comparison results verify the superior performance of our proposed algorithm.

Closed-Form Algorithm for 3-D Near-Field OFDM Signal Localization under Uniform Circular Array.

Due to its widespread application in communications, radar, etc., the orthogonal frequency division multiplexing (OFDM) signal has become increasingly urgent in the field of localization. Under uniform circular array (UCA) and near-field conditions, this paper presents a closed-form algorithm based on phase difference for estimating the three-dimensional (3-D) location (azimuth angle, elevation angle, and range) of the OFDM signal. In the algorithm, considering that it is difficult to distinguish the frequency of the OFDM signal’s subcarriers and the phase-based method is always affected by errors of the frequency estimation, this paper employs sparse representation (SR) to obtain the super-resolution frequencies and the corresponding phases of subcarriers. Further, as the phase differences of the adjacent sensors including azimuth angle, elevation angle and range parameters can be expressed as indefinite equations, the near-field OFDM signal’s 3-D location is obtained by employing the least square method, where the phase differences are based on the average of the estimated subcarriers. Finally, the performance of the proposed algorithm is demonstrated by several simulations.

Fast FRFT-Based Algorithm for 3-D LFM Source Localization With Uniform Circular Array

Localization of linear frequency modulation (LFM) source based on combination of time–frequency analysis and a spatial spectrum estimation technique has received extensive research. However, this scheme is always confined to far-field sources and suffers from high computational cost. In this paper, by performing the fractional Fourier transform to multiple incoherent LFM signals, a fast and closed-form algorithm based on phase difference is proposed for 3-D source localization under a uniform circular array. First, the phase relationship among sensors’ maximum outputs in fractional Fourier domain is exploited to construct indefinite equations, and then the least square method is employed to estimate the source’s azimuth angle, elevation angle, and range simultaneously. Finally, comparing with current approaches, the satisfactory performance of our proposed algorithm is demonstrated, and the effectiveness for dealing with mixed far-field and near-field sources is also confirmed.

Positive subharmonic solutions to nonlinear ODEs with indefinite weight

We prove that the superlinear indefinite equation [Formula: see text] where [Formula: see text] and [Formula: see text] is a [Formula: see text]-periodic sign-changing function satisfying the (sharp) mean value condition [Formula: see text], has positive subharmonic solutions of order [Formula: see text] for any large integer [Formula: see text], thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

Pixel-wise partial volume effects correction on arterial spin labeling magnetic resonance images

Arterial spin labeling is a recently emerging imaging modality in functional magnetic resonance, and it is widely acknowledged to be effective in directly measuring the cerebral blood flow of patients while being scanned, which makes it a promising indicator in contemporary dementia disease diagnosis studies. However, partial volume effects mainly caused by signal cross-contamination due to pixel heterogeneity and limited spatial resolution of the arterial spin labeling scanning protocol often prevent the cerebral blood flow from being accurately measured. In order to correct the partial volume effects, contemporary studies usually rely on neighboring pixles to solve indefinite equations of partial volume correction, which makes shortcomings of blurring and brain tissue details loss inevitable in their correction outcomes. In this study, a novel pixel-wise correction method is proposed to tackle partial volume effects in arterial spin labeling images. The main idea is to formalize the correction problem as a series of quadratic programming sub-problems using split-Bregman iterations, then formulate each sub-problem via the regularization of least absolute shrinkage and selection operator, and finally solve the regularization sub-problem via the fast proximal gradient descent approach. A real-patients database composed of 360 demented patients is incorporated for experimental evaluation of the pixel-wise method. Extensive experiments and comprehensive statistical analysis are carried out to demonstrate the superiority of the pixel-wise method with comparisons towards the popular region-based method. Promising results are reported from the statistical point of view.

Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles

We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter $$\lambda $$ , which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from infinity occur at $$\lambda =0$$ . Our approach combines variational and bifurcation techniques.