Search

This article introduces a new equation of the form 945(x2 + y2) − 1889xy + 10(x + y) + 100 = pzq and explores its solution by gradually changing the parameter q from 1 to 10 with generalized value q = m. A transformation x = u + v and y = u − v simplifies the equation which results (u + 10)2 + 3779v2 = pzq. We have employed multiple techniques to discover integral solutions for the equation and conducted a thorough analysis to examine the properties and characteristics of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution of the quadratic equation T Y 2 - Y + I = 0 where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C (T) : = ∑ n = 0 ∞ C n T n , where the sequence (C n) n ≥ 0 is the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (λ T) - 1 . Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers. [ABSTRACT FROM AUTHOR]

Estimation of the traditional transverse electric (TE) and transverse magnetic (TM) impedances of the magnetotelluric tensor for two-dimensional structures can be decoupled from the estimation of the strike direction with significant implications when dealing with galvanic distortions. Distortion-free data are obtainable by combining a quadratic equation with the phase tensor. In the terminology of Groom–Bailey, the quadratic equation provides amplitudes and phases that are immune to twist, and the phase tensor provides phases immune to both, twist and shear. On the other hand, distortion-free strike directions can be obtained using Bahr's approach or the phase tensor. In principle, this is all that is needed to proceed to a two-dimensional (2D) interpretation. However, the resulting impedances are strike ignorant because they are invariant under coordinate system rotation, and if they are to be related to a geological strike, they must be linked to a particular direction. This is an additional ambiguity to the one of 90° arising in classic strike-determination methods, which must be resolved independently. In this work, we use the distortion model of Groom–Bailey to resolve the ambiguity by bringing back the coupling between impedances and strike in the presence of galvanic distortions. Our approach is a hybrid between existing numerical and analytical methods that reduces the problem to a binary decision, which involves associating the invariant impedances with the correct TE and TM modes. To determine the appropriate association, we present three algorithms. Two of them require optimizing the fit to the data, and the third one requires a comparison of phases. All three keep track of possible crossings of the phase curves providing a clear-cut solution. Synthetic and field data illustrate the performance of the three schemes. [ABSTRACT FROM AUTHOR]

The solvent effects on C=O stretching vibrational frequency, ν(C=O), of ketoprofen (KETO) were studied experimentally using attenuated total reflection infrared spectroscopy (ATR-IR). The experimental ν(C=O) of KETO were correlated with empirical solvent parameters, including the Kirkwood–Bauer–Magat (KBM) equation, the acceptor numbers (ANs) of the solvents, the Swain equation, linear solvation energy relationships (LSERs), and the quadratic equation (QE). The solvent-induced ν(C=O) shifts of KETO displayed a better correlation with the LSER equation than with the KBM equation, ANs of the solvents, and the Swain equation. The linear effect of the solvent hydrogen-bond donor acidity (Aj) on ν(C=O) of KETO was found to be highly significant, whereas the hydrogen-bond acceptor basicity (Bj) and the interaction effect of Aj and Bj were not significant. It was also observed that the quadratic effects of Aj and Bj were slightly significant. Additionally, the linear effect of LSER parameters (π*, δ, α, and β) and the interaction effect of π*β on the ν(C=O) of KETO were highly significant. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann–Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, let H be an abelian group which is uniquely 2-divisible, and let σ be an involutive automorphism of S. We express the solutions f: S → H of the following variant of the quadratic functional equation f(xy) + f(σ(y)x) = 2f(x) + 2f(y), x,y ∈ S, in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. [ABSTRACT FROM AUTHOR]

Background: Automated calculations by laboratory information system (LIS) are efficient and accurate ways of providing calculated laboratory test results. Due to the lack of established advanced mathematical functions and equation logic in LIS software, calculations beyond simple arithmetic functions require a tedious workaround. Free and bioavailable testosterone (BT) calculations require a quadratic solver currently unavailable as ready to use the function on most commercial LIS platforms. We aimed to develop a module within the Epic Beaker LIS to enable automatic quadratic equation solving capability and real‑time reporting of calculated free and BT values. Materials and Methods: We developed and implemented an advanced calculation module from the ground up using existing basic calculation programming functions in the Epic Beaker LIS. A set of calculation variables were created, and mathematical logic and functions were used to link the variables and perform the actual quadratic equation based calculations. Calculations were performed in real‑time during result entry events, and calculated results populated the result components in LIS automatically. Results: Free and BT were calculated using instrument measured results of total testosterone, sex hormone binding globulin, and/or serum albumin, by applying equations widely adopted in laboratory medicine for endocrine diseases and disorders. Calculated results in Epic Beaker LIS were then compared and confirmed by manual calculations using Microsoft Excel spreadsheets and scientific calculators to have no discrepancies. Conclusions: Automated calculations of free and BT were successfully implemented and validated, the first of such implementation for the Epic Beaker LIS platform, eliminating the need of offline manual calculations, potential transcription error, and with improved turnaround time. It may serve as a model to build similarly complex equations when the clinical need arises. [ABSTRACT FROM AUTHOR]

In this paper, some classes of permutation polynomials of the form ( x p m − x + δ ) s + L ( x ) over the finite field F p 2 m are investigated by determining the number of solutions of certain equations, where L ( x ) = x or x p m + x . More precisely, for an integer s satisfying s ( p m + 1 ) ≡ p m + 1 (mod p 2 m − 1 ), we give four classes of permutation polynomials of the form ( x 2 m + x + δ ) s + x over F 2 2 m , and five classes of permutation polynomials of the form ( x 3 m − x + δ ) s + x 3 m + x over F 3 2 m , respectively. [ABSTRACT FROM AUTHOR]

Background: Automated calculations by laboratory information system (LIS) are efficient and accurate ways of providing calculated laboratory test results. Due to the lack of established advanced mathematical functions and equation logic in LIS software, calculations beyond simple arithmetic functions require a tedious workaround. Free and bioavailable testosterone (BT) calculations require a quadratic solver currently unavailable as ready to use the function on most commercial LIS platforms. We aimed to develop a module within the Epic Beaker LIS to enable automatic quadratic equation solving capability and real-time reporting of calculated free and BT values. Materials and Methods: We developed and implemented an advanced calculation module from the ground up using existing basic calculation programming functions in the Epic Beaker LIS. A set of calculation variables were created, and mathematical logic and functions were used to link the variables and perform the actual quadratic equation based calculations. Calculations were performed in real-time during result entry events, and calculated results populated the result components in LIS automatically. Results: Free and BT were calculated using instrument measured results of total testosterone, sex hormone binding globulin, and/or serum albumin, by applying equations widely adopted in laboratory medicine for endocrine diseases and disorders. Calculated results in Epic Beaker LIS were then compared and confirmed by manual calculations using Microsoft Excel spreadsheets and scientific calculators to have no discrepancies. Conclusions: Automated calculations of free and BT were successfully implemented and validated, the first of such implementation for the Epic Beaker LIS platform, eliminating the need of offline manual calculations, potential transcription error, and with improved turnaround time. It may serve as a model to build similarly complex equations when the clinical need arises. [ABSTRACT FROM AUTHOR]

The study of stability problems of functional equations was motivated by a question of S. M. Ulam asked in 1940. The first result giving answer to this question is due to D.H. Hyers. Subsequently, his result was extended and generalized in several ways. In this paper we prove some hyperstability results for the equation g(ax + by) + g(cx + dy) = Ag(x) + Bg(y) on restricted domain. Namely, we show, under some weak natural assumptions, functions satisfying the above equation approximately (in some sense) must be actually solutions to it. [ABSTRACT FROM AUTHOR]

Energy loss represents a traditional key objective in the optimal operation and planning of electrical networks, and various estimation methods have been studied. In this study, two formulas are proposed for calculation of the loss factor (LSF) to improve the classical LSF method based on the minimum load factor (MLF) and the LF. The former is an approximate formula, whose accuracy is good enough for engineering calculations. While the latter is an empirical quadratic equation determined by the statistical analysis, which is more accurate to estimate losses. To conduct a complete feasibility study for project practices, a large amount of measurement data is used to calculate energy losses in a district of Guangdong using the classical LSF method and the improved LSF (ILSF) method. Results of statistical analyses indicate that the real data fall in the proposed three‐dimensional region and the use of MLF can help improve accuracy in the energy losses estimation. The classical and ILSF methods are used to estimate the effect in loss reduction by inserting a distributed generation in a 43‐bus distribution network, and the candidate bus can be identified effectively. [ABSTRACT FROM AUTHOR]

Kung-Traub's conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2d-1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached by a method with three function evaluations is four, even for quadratic functions. In this paper, we show that the conjecture fails for quadratic functions. In fact, we can find a 2-point method with three function evaluations reaching fifth order convergence. We also develop 2-point 3rd to 8th order methods with one function and two first derivative evaluations using weight functions. Furthermore, we show that with the same number of function evaluations we can develop higher order 2-point methods of order r + 2, where r is a positive integer, ≥ 1. We also show that we can develop a higher order method with the same number of function evaluations if we know the asymptotic error constant of the previous method. We prove the local convergence of these methods which we term as Babajee's Quadratic Iterative Methods and we extend these methods to systems involving quadratic equations. We test our methods with some numerical experiments including an application to Chandrasekhar's integral equation arising in radiative heat transfer theory. [ABSTRACT FROM AUTHOR]

In the paper, the equivalence of the functional inequality ∥2f(x) + f(y) + f(-y) - f(x - y)∥ ≤ ∥f(x + y)∥ (x,y ∈ G) and the Drygas functional equation f(x + y) + f(x - y) = 2f(x) + f(y) + f(-y) (x,y ∈ G) is proved for functions f : G → E where (G, +) is an abelian group, (E,<.,.>) is an inner product space, and the norm is derived from the inner product in the usual way. [ABSTRACT FROM AUTHOR]

When we talk about quadratic equation, it is immediate to associate with the general formula erroneously attributed to Bhaskara in Brazil. This is due to overuse and often unique to this formula. The objective of this paper is to encourage and discuss the relevance and scope of other methods for solving quadratic equations that currently has been left aside by most Brazilian teachers, such as a geometric method present in *The Elements* of Euclid, the completion of squares (which demonstrate the general formula by geometric reasoning - adding areas) and the method of sum and product (which we will present in a device that extends its use for fractional roots). [ABSTRACT FROM AUTHOR]

A study was conducted to determine dietary energy levels for optimum productivity and carcass characteristics of indigenous Venda chickens raised in closed confinement. Four dietary treatments were considered in the first phase (1 to 7 weeks) on two hundred day-old unsexed indigenous Venda chicks indicated as EVS1, EVS2, EVS3 and EVS4 (11, 12, 13 and 14 MJ ME/kg DM, respectively) and each treatment was replicated five times. In the second phase (8 - 13 weeks), 160 indigenous Venda female chickens, aged eight weeks, were randomly allocated to four dietary treatments and each treatment was replicated five times in a completely randomized design. The diets used in both trials were isonitrogenous but with different energy levels. A quadratic equation was used to determine dietary energy levels for optimum feed intake, growth rate, FCR and ME intake at both the starter and grower phases and the carcass characteristics of the birds at 91 days. Dietary energy levels of 12.91, 12.42, 12.34 and 12.62 MJ ME/kg DM feed supported optimum feed intake, growth rate, FCR and ME intake, respectively, for the starter phase. At the grower phase, dietary energy levels of 12.56, 12.66, 12.62 and 12.71 MJ ME/kg DM feed supported optimum feed intake, growth rate, FCR and ME intake, respectively. Carcass, drumstick, thigh and wing had optimum weights at dietary energy levels of 13.81, 13.23, 13.43 and 13.18 MJ ME/ kg DM, respectively. Thus, dietary energy level for optimization depended on the particular production parameter in question. [ABSTRACT FROM AUTHOR]

Abstract: The scalar–vector representation is used to derive a simple algorithm to obtain the roots of a quadratic quaternion polynomial. Apart from the familiar vector dot and cross products, this algorithm requires only the determination of the unique positive real root of a cubic equation, and special cases (e.g., double roots) are easily identified through the satisfaction of algebraic constraints on the scalar/vector parts of the coefficients. The algorithm is illustrated by computed examples, and used to analyze the root structure of quadratic quaternion polynomials that generate quintic curves with rational rotation-minimizing frames (RRMF curves). The degenerate (i.e., linear or planar) quintic RRMF curves correspond to the case of a double root. For polynomials with distinct roots, generating non-planar RRMF curves, the cubic always factors into linear and quadratic terms, and a closed-form expression for the quaternion roots in terms of a real variable, a unit vector, a uniform scale factor, and a real parameter is derived. [Copyright &y& Elsevier]

We establish a sufficient condition for an m-dependent system to be a Sidon system. It is proved that this condition cannot be strengthened for any natural m. The proof is based on the consistency of a certain system of quadratic equations over the field ℝ. [ABSTRACT FROM AUTHOR]

Abstract: In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with -order of convergence at least four for quadratic equations. [Copyright &y& Elsevier]

Abstract: In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter . These methods reach the order of convergence when they are applied to quadratic polynomials with different roots. Newton’s and Chebyshev’s methods appear as particular choices of the family appear for and , respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods. [Copyright &y& Elsevier]

Abstract: The plan of this paper is to obtain one-point iterative methods with any R-order of convergence, when they are applied to approximate solutions of quadratic equations in Banach spaces. To do this, we consider real Cauchy''s method and, under certain natural modifications, it is extended to Banach spaces. Some applications are also provided. [Copyright &y& Elsevier]

We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form x1x2 · x3x4 = x5x6 · x7x8 with xi ∊ {x, y, u, v} (1 ⩽ i ⩽ 8) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some recent papers of Föorg-Rob and Krapež, and Polonijo. [ABSTRACT FROM AUTHOR]

An SMT machine has many working coordinate frames—the fiducial mark camera frame, component camera frame, machine table frame, PCB frame, and reference frame. Because of many influences such as mechanical dimension errors, machine assembling errors, and camera lens distortions, all frames on the SMT machine must be calibrated to compensate for these machine errors. This paper applies machine vision techniques to auto-calibrate an SMT machine, including frame transformations and nozzle compensation, bringing the accuracy of this system to within ±0.1 mm. The coordinate mapping from the camera frame to machine table frame is a quadratic form, while the other frame mappings use linear forms. The merits for this machine’s vision-based, auto-calibration methods are: (1) It has little calibration time, (2) It does not need expensive calibration instruments, and (3) Its expense is very low. [ABSTRACT FROM AUTHOR]

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results. [ABSTRACT FROM AUTHOR]