Polynomial Expression Research Articles

Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

On the enumeration of bipartite simple games

This paper provides a classification of all monotonic bipartite simple games. The problem we deal with is very versatile since simple games are inequivalent monotonic Boolean functions, functions that are used in many fields such as game theory, neural networks, artificial intelligence, reliability or multiple-criteria decision-making. The obtained classification can be implemented in an algorithm able to enumerate bipartite simple games. These numbers provide some light on enumerations of several subclasses of bipartite simple games, for which we find formulas.Complete simple games, a subclass of all simple games for which the desirability relation is a complete preordering, were already classified by means of two parameters: a vector and a matrix fulfilling some conditions. Complete simple games are inequivalent monotonic regular Boolean functions. In this paper, we deduce a procedure for bipartite non-complete games, which allows enumerating the number of bipartite simple games. Several formulas are obtained, in particular polynomial expressions for the number of bicameral meet games and the number of bicameral join games, two types of voting systems widely used in practice.

Analytical and numerical simplified modeling of a single-lap joint

This paper is devoted to the theoretical modeling of the mechanical response of bonded structures and particularly focuses on the single-lap joint which corresponds to the simplest and most fundamental bonding configuration. Such a geometry brings into play all the features of any bonded assembly in terms of mechanical response (stress/strain heterogeneities, singularities, adhesion at the interfaces,...). It can especially be used to characterize the mechanical behavior of new adhesives within an assembly. In all cases, an accurate description of the stress/strain distribution (specifically in the adhesive layer) is required both for the calibration of an adhesive behavior and for dimensioning purposes. Accordingly, the present study aims at developing a specific 1D enriched finite element devoted to the numerical modeling of single-lap joints, especially of the overlap region. First, analytical solutions for a single-lap joint under tensile forces are investigated in the framework of elasticity. They are particularly based on the choice of a 2D representation of the adhesive layer with polynomial displacement fields in terms of the thickness coordinate. Such a preliminary study allows one to identify optimally the appropriate kinematics for each layer (adhesive but also substrates) and highlights the importance of non-linear terms in the polynomial expressions of both longitudinal and transverse displacements within the adhesive. A three-layer finite element model is then formulated for the overlap region, based on the retained kinematics, and involving an elastoplastic constitutive law for the adhesive material. The numerical integration through the adhesive thickness and the assembly of the three layers lead to the definition of a very low-cost 1D finite element, which provides nevertheless a complete and accurate description of the stress fields, especially within the adhesive layer. This new finite element (used simultaneously with a more classical beam finite element for the unbonded parts of the adherends) is finally validated by comparison with 2D reference results computed using Abaqus software.

Feature-based characterization and extraction of ripple errors over the large square aperture.

Freeform surfaces play an important role in modern optical systems with compactness and better performance. The fabrication tools tend to impart a structured signature on optical surfaces, called ripple errors, during the freeform surface manufacturing process. The description and extraction of ripple errors for freeform surface fabrication and testing have attracted extensive attention. In this paper, we develop a fast and accurate method to describe ripple errors for the large aperture based on Fourier model coupling. The polynomial expression is transformed into Fourier series form and surface errors are reconstructed by frequency feature extraction combining with the least square method. The high accuracy and efficiency of the proposed method for representing and filtering ripple errors consuming little computer memory are demonstrated using real experimental data. The proposed method offers a robust and powerful tool not only suitable for surface error characterization but also for image filtering and analysis.

Riveted joints in composites, a practical tool to estimate stresses around the rivet hole

Riveted joints have different failure mechanisms, some associated to the failure of the rivet itself, and others associated to the materials to be joined. The failures associated to the material are mainly due to the stress concentrations at the rivet holes. In joints with metallic (ductile) materials, these stress concentrations might not play a relevant role in the failure initiation, but with unidirectional composite (brittle) materials, they do control the failure. Both the orthotropic behavior of joined materials and the non-linear contact conditions between the rivet and the hole, make that the problem does not have a general solution. In the present work, several finite element analyses have been carried out for basic load cases, different materials, and orientations of the fiber. Furthermore, a two-steps least squares adjustment has been performed to obtain a polynomial expression of the stress distributions for all cases. As a result, a tool is provided, allowing the users to easily estimate the stress distribution (σrr, σθθ) along the hole contour, and σxy along the tangential lines to the hole circumference. Such stress ditribution can be used for failure predictions. All input and output data are available to readers, by means of a spreadsheet, to estimate mentioned stress distributions.

Influence of dynamic loads in joints on dimensional stability

In small arms manufacturing the lock dimensional chain final link precision is secured with manual file fitting and removing the tolerance intentionally left on the recoil lugs. The manual process reduces the contact stiffness of the lock components, and the final link dimension (between the cylindrical gauge end that simulates a cartridge, and the breeching lock face) set during the assembly is distorted when the breech lock components are exposed to shot impact loads. For this reason, the cylindrical cartridge dummy used for the initial assembly is sized to allow for subsequent plastic deformation at the breech lock component interfaces. The deformation value is determined experimentally on an ad hoc basis. Polynomial expressions have been obtained experimentally for the interface plastic deformation vs. relative positioning errors and surface finish relation for parts made of typical military steel grades. To study the relation between the interface surface machining methods and contact stiffness flat specimens, 50RA steel quenched to HRC 40…45 (the roughness Rz is 10…15 μm) were machined as follows: peripheral pregrinding; flank milling; file crosswise (at 45°) movements. The results generated after processing the measurements made with a HOMMEL TESTER W55 instrument show that finish flank milling yields the best results.

Characterization and Analysis of Extensile Fluidic Artificial Muscles

Extensile fluidic artificial muscles (EFAMs) are soft actuators known for their large ranges of extension, low weight, and blocked forces comparable to those of pneumatic cylinders. EFAMs have yet to be studied in a way that thoroughly focuses on their manufacturing, experimental characterization, and modeling. A fabrication method was developed for production of two EFAMs. The quasi-static axial force response of EFAMs to varying displacement was measured by testing two specimens under isobaric conditions over a pressure range of 103.4–517.1 kPa (15–75 psi) with 103.4 kPa (15 psi) increments. The muscles were characterized by a blocked force of 280 N and a maximum stroke of 98% at 517.1 kPa (75 psi). A force-balance model was used to analyze EFAM response. Prior work employing the force-balance approach used hyper-elastic constitutive models based on polynomial expressions. In this study, these models are validated for EFAMs, and new constitutive models are proposed that better represent the measured stress values of rubber as a function of strain. These constitutive models are compared in terms of accuracy when estimating pressure-dependent stress–strain relationships of the bladder material. The analysis demonstrates that the new hyper-elastic stress models have an error 5% smaller than models previously employed for EFAMs for the same number of coefficients. Finally, the analysis suggests that the new stress functions have smaller errors than the polynomial stress model with the same number of coefficients, guarantee material stability, and are more conservative about the stress values for strains outside of the testing range.

Evaluation of Accelerometer-Derived Data in the Context of Cycling Cadence and Saddle Height Changes in Triathlon.

In the multisport of triathlon cycling is the longest of the three sequential disciplines. Triathlon bicycles differ from road bicycles with steeper seat tube angles with a change to saddle height altering the seat tube angle. This study evaluated the effectiveness of a tri axial accelerometer to determine acceleration magnitudes of the trunk in outdoor cycling in two saddle positions. Interpretation of data was evaluated based on cadence changes whilst triathletes cycled in an aerodynamic position in two saddle positions. The evaluation of accelerometer derived data within a characteristic overground setting suggests a significant reduction in mediolateral acceleration of the trunk, yielding a 25.1% decrease when saddle height was altered alongside reduced rate of perceived exertion (3.9%). Minimal differences were observed in anteroposterior and longitudinal acceleration. Evaluation of sensor data revealed a polynomial expression of the subtle changes between both saddle positions. This study shows that a triaxial accelerometer has capability to continuously measure acceleration magnitude of trunk movements during an in-the-field, varied cadence cycle protocol. Accessible and practical sensor technology could be relevant for postural considerations when exploring saddle position in dynamic settings.

Toward the Reverse Decomposition of Unipotents. II. The Relative Case

Recently, Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In 2018, the author discussed various generalizations of these results to exceptional groups, specifically those of types E6 and E7. Here, a further variation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4, similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1 by relative elementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J ⊴ R, are obtained. Again, in particular, this allows to give very short proofs for the description of subgroups normalized by E(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

Method for determining load magnitude and location from the plastic deformation of fixed beams using a neural network.

Fixed beam structures are widely used in engineering, and a common problem is determining the load conditions of these structures resulting from impact loads. In this study, a method for accurately identifying the location and magnitude of the load causing plastic deformation of a fixed beam using a backpropagation artificial neural network (BP-ANN). First, a load of known location and magnitude is applied to the finite element model of a fixed beam to create plastic deformation, and a polynomial expression is used to fit the resulting deformed shape. A basic data set was established through this method for a series of calculations, and it consists of the location and magnitude of the applied load and polynomial coefficients. Then, a BP-ANN model for expanding the sample data is established and the sample set is expanded to solve the common problem of insufficient samples. Finally, using the extended sample set as training data, the coefficients of the polynomial function describing the plastic deformation of the fixed beam are used as input data, the position and magnitude of the load are used as output data, a BP-ANN prediction model is established. The prediction results are compared with the results of finite element analysis to verify the effectiveness of the method.

Fractional polynomial function in stochastic response surface method for reliability analysis

The explicit polynomial function can replace the actual limit state function for failure probability. In this study, the fractional order is introduced to an extended family of polynomial functions to guarantee a smooth relationship between the sample data and polynomial expression. The fractional polynomial method is proposed to establish the function and analyze the system’s reliability accurately and flexibly. Integer, fractional, positive, and negative orders are considered. The scope of orders, constraint conditions, and sampling method are investigated under different conditions of the proposed method. Furthermore, the calculation process for the fractional polynomial method when the approach is applied to a multivariable system is examined, and engineering examples are provided to illustrate the application of the proposed method. The fractional polynomial function can describe system performance and reliability with few terms and high accuracy. The results calculated by the proposed method agree with those from Monte Carlo simulation.

Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

In this paper, applying the Fa\`a di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.

ВЫБОР МЕСТ РАЗМЕЩЕНИЯ НАЗЕМНЫХ ОПТИЧЕСКИХ СРЕДСТВ НАБЛЮДЕНИЯ НА ОСНОВЕ НЕЧЕТКОВОЗМОЖНОСТНОЙ МОДЕЛИ

When choosing the location of groundbased optical surveillance devices (NOSN), the problem arises of evaluating the suitability of a particular area of terrain or choosing the best of a number of considered ones, provided that the ability to perform professional tasks with specified characteristics is quantified. A method for solving this problem is proposed by constructing a fuzzyprobability model of a generalized (integral) indicator of the quality of NOS placements based on the knowledge and experience of experts. Such an indicator is determined by a systemically confirmed set of seven linguistic variables included in the factor space for constructing a fuzzy probability model in the form of a nonlinear polynomial expression. At the same time, the dependent variablethe ability to perform the task with the specified characteristicsindirectly determines the degree of the best placement of the NOS, taking into account its astroclimate. Practical recommendations on the choice and justification of the factor space are given, the main stages of constructing a fuzzy probability model are shown, and the degree of adequacy of calculations based on it to the actual ability to perform tasks with specified characteristics, evaluated by independent experts, for the locations of NOS in various geographical regions of the Russian Federation.

Form deviations caused by lateral displacement errors in annular subaperture stitching interferometry

Annular subaperture stitching interferometry (ASSI) is a common approach for the measurement of aspherical surfaces. A common obstacle of ASSI is the occurrence of lateral displacement errors when the sensor or specimen is repositioned between the subaperture measurements. Our contribution focuses on modeling of the statistical displacement errors. A virtual experiment is presented simulating the propagation of the displacement errors through a cumulative and a global stitching algorithm to the retrieved surface form. For the considered experimental setup, the uncertainty in lateral position depends on the positioning uncertainties of the employed motion system and the uncertainty in the absolute distance measurement between the sensor and specimen. The lateral displacement uncertainty is determined experimentally employing a calibratable lateral grating. Thus, it is traceable to the SI unit of the length (meter). The experimental results show that the lateral displacement errors may be modeled by a normal distribution, and the results of the virtual experiment indicate that the statistical lateral displacement errors transfer linear through the stitching procedure and also cause a normal distributed topography error. This enables the assignment of an expanded uncertainty to each individual sample point employing the Zernike polynomial expression of the topography measurement.

Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots

Operator square-roots are ubiquitous in theoretical physics. They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation. Often the use of a perturbative expansion is the only recourse when dealing with them. In this work we show that under certain conditions differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square-roots. Specifically, for the number operator $\hat n=\hat a^\dag a$ we show that the square-root $\sqrt{\hat n}$ near $\hat n=0$ can be approximated by a polynomial in $\hat n$. This result is unexpected because a Taylor expansion fails. A polynomial expression in $\hat n$ is possible because $\hat n$ is an operator, and its constituents $a$ and $a^\dag$ have a non-trivial commutator $[a,a^\dag]=1$ and do not behave as scalars. We apply our approach to the zero mass Klein-Gordon Hamiltonian in a constant magnetic field, and as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators. We prove that these new expressions exactly reproduce spin operators. Our expressions are manifestly Hermitian, which offer an advantage over other methods, such as the Dyson-Maleev representation.

Numerical research of elastic-plastic fracture toughness of aged ferritic-martensitic steel

Experimental test methods are used to determine material properties and fracture parameters. However, due to cost, lack of test material, etc., experimental tests cannot always be conducted. Therefore, there is a need for alternative methods to determine material properties and fracture parameters. In this paper, a numerical methodology for determination of the elastic-plastic fracture toughness of steel P91 is suggested. For this purpose the universal finite element software ABAQUS was used. J-integral resistance curves (J-R) were constructed by numerical simulation of tension experiment of compact tension (CT) specimen and by using the suggested quadratic polynomial expression for the calculation of the crack length extension of steel P91. The suggested method was validated by experimental tests.The secondary aim of the work is to evaluate the ageing effect on the elastic-plastic fracture toughness of steel P91. Thermal ageing of the steel was carried out in an electric furnace at 600, 650 and 700 °C for up to 20000 h. Research has revealed that JQ values decrease by up to 20%, determined experimentally, and by up to 22%, determined numerically, when the steel is aged up to ~10900 h at 650 °C.

Calculation of Astrophysical S-factor and Thermonuclear Reaction Rates for (p,n) Medium Elements Reactions

The cross-sections of (p,n) medium elements reactions as a function of proton energies such as 45Sc(p,n)45Ti, 48Ti(p,n)48V, 51V(p,n)51Cr, 52Cr(p,n)52Mn, 55Mn(p,n)55Fe, 56Fe(p,n)56Co, 59Co(p,n)59Ni, 62Ni(p,n)62Cu, 63Cu(p,n)63Zn, and 66Zn(p,n)66Ga have been interpolated near threshold up to 10 MeV in step of 0.05 MeV using MATLAB program. Weighted averages of cross-sections have been used to calculate the astrophysical S-factor and thermonuclear reaction rates as a function of the center of mass energy, Ec.m. and T9 respectively. Polynomial expressions have been used to fit the calculated astrophysical S-factor and thermonuclear reaction rates to determine the astrophysical S-factor at various Ec.m. and thermonuclear reaction rates at various T9 from best fitting equations with minimum Chi-Square. Empirical formulae of reactions set 45Sc(p,n)45Ti, 48Ti(p,n)48V, 55Mn(p,n)55Fe, 59Co(p,n)59Ni, 66Zn(p,n)66Ga and reactions set 48Ti(p,n)48V, 51V(p,n)51Cr, 59Co(p,n)59Ni, 63Cu(p,n)63Zn, 66Zn(p,n)66Ga have been used to calculate the astrophysical S-factor as a function of Ec.m. and Z and thermonuclear reaction rates as a function of T9 and Z of target nucleus. The results have been compared with the adopted data that have been calculated from the fitting equations which have a good agreement.

Higher-order rogue wave solutions of the Kadomtsev Petviashvili—Benjanim Bona Mahony (KP-BBM) model via the Hirota-bilinear approach

We study the higher-order rogue wave solutions of the Kadomtsev Petviashvili—Benjanim Bona Mahony (KP-BBM) model with and without controllable center via the Hirota bilinear approach. To construct higher-order rogue wave solutions of the model, we apply cross-product terms in a polynomial expression as a test function. We construct three kinds of rogue waves with center at the origin and three kinds of higher-order rogue waves with a controllable center by choosing different test functions. We show how the center can control the shapes and orders of the rogue waves by choosing the values of parameters of the center. In particular, the 3-rogue wave solutions exhibit double rogue waves for lower values and distinct triple rogue waves in a triangular structure for a sufficiently large value parameters of the center. Moreover, the 6-rogue wave solutions of the model present lower-order rogue waves for small values and higher-order rogue waves for large value parameters of the center. It is shown that the order of the rogue gradually increases for rising value of parameters, and it will be maximum six distinct rogues for a sufficiently large values. Finally, we explain the solutions of the model in 3D and in density plots graphically.

Accurate Chebyshev Approximations for the Width of the Voigt Profile, Differential Peaks, and Deconvolution of the Lorentzian Width

Accurate formulas have been developed using rational Chebyshev approximations for the spectral features/quantities of the Voigt profile and its first two derivatives. The rational polynomial expression for the half width at half maximum (HWHM) of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (7) and provides 12-digit accuracy. The correction expression for the deconvolution of the Gaussian broadening from the Lorentzian broadening in the Voigt profile is presented in Eq. (11) and provides 13-digit accuracy. The expression for the position of the peak in the first derivative of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (15) and provides 14-digit accuracy. The expression for the position of the side peak in the second derivative of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (17) and provides 12-digit accuracy. The expressions for the peak amplitude ratios for the first and second derivatives of the Voigt profile in terms of the Lorentzian and Gaussian widths are presented in Eq. (19) and Eq. (21), respectively, and provide 13-digit accuracy. The reported accuracies are higher than those of earlier work, and the presented expressions provide a reference for the direct calculation of spectral quantities for Voigt profiles. The accuracy of the presented formulas establishes a general foundation for the deconvolution of instrument-based broadening and other broadening mechanisms in the HWHW and the positions of the first- and second-derivative peaks.

Macrohomogeneity condition for strain gradient homogenization of periodic heterogeneous media with interfacial strong discontinuities

The Hill macrohomogeneity condition is revisited in the context of strain gradient homogenization for heterogeneous materials prone to interfacial displacement jumps. The consideration of strain gradient effects is motivated by their use as a regularization method for strain-softening constitutive damage models leading to strain localization and displacement discontinuity. Starting from the weak form of the boundary value problem formulated at the microscopic level, a polynomial expression of the virtual velocity is adopted as the minimum microscopic kinematics consistent with the selected macroscopic kinematics of the strain gradient effective continuum. The effective volumetric and interfacial mesoscopic strains and stress measures for the effective substitution are obtained versus the microscopic strains and stresses. The Hill macrohomogeneity condition is successively formulated for continuous interfaces and discontinuous interfaces witnessing strong discontinuities. It highlights the expressions of the effective stress measures associated to the volumetric and interfacial behavior for both classical and higher-order effects.