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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]

Let c=(Cn)n≥0$$ c={\left({C}_n\right)}_{n\ge 0} $$ be the Catalan sequence and T$$ T $$ a linear and bounded operator on a Banach space X$$ X $$ such 4T$$ 4T $$ is a power‐bounded operator. The Catalan generating function is defined by the following Taylor series: C(T):=∑n=0∞CnTn.$$ C(T):= \sum \limits_{n=0}^{\infty }{C}_n{T}^n. $$Note that the operator C(T)$$ C(T) $$ is a solution of the quadratic equation TY2−Y+I=0$$ T{Y}^2-Y+I=0 $$. In this paper, we define powers of the Catalan generating function C(T)$$ C(T) $$ in terms of the Catalan triangle numbers. We obtain new formulae that involve Catalan triangle numbers: the spectrum of c∗j$$ {c}^{\ast j} $$ and the expression of c−∗j$$ {c}^{-\ast j} $$ for j≥1$$ j\ge 1 $$ in terms of Catalan polynomials (∗$$ \ast $$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future. [ABSTRACT FROM AUTHOR]

A method is proposed with which the locations of the roots of the monic symbolic quintic polynomial x5+a4x4+a3x3+a2x2+a1x+a0$x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ can be determined using the roots of two resolvent quadratic polynomials: q1(x)=x2+a4x+a3$q_1(x) = x^2 + a_4 x + a_3$ and q2(x)=a2x2+a1x+a0$q_2(x) = a_2 x^2 + a_1 x + a_0$, whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of q1(x)$q_1(x)$ and q2(x)$q_2(x)$ and on some specific relationships between them. The method is illustrated with the full analysis of one of the possible cases. Some of the roots of the symbolic quintic equation for this case have their isolation intervals determined and, as this cannot be done for all roots with the help of quadratic equations only, finite intervals containing 1 or 3 roots, or 0 or 2 roots, or, rarely, 0, or 2, or 4 roots of the quintic are identified. Knowledge of the stationary points of the quintic lifts this indeterminacy and allows finding the isolation interval of each root. Separately, using the complete root classification of the quintic, one can also lift this indeterminacy. The method also allows to see how variation of the individual coefficients of the quintic affect its roots. No root finding iterations or any numerical approximations are used and no equations of degree higher than two are solved. [ABSTRACT FROM AUTHOR]

We consider a map f from one abelian group into another that satisfies either an additive or quadratic functional equation on any given pair of elements of its domain. Particular emphasis is placed on the possibility that f itself is neither additive nor quadratic and a complete description of all those cases is obtained. [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]

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP -complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time. [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]

The aim of this work is to investigate the alternative quadratic functional equation f(x+y)+f(x-y)-2f(x)-2f(y)∈{0,1,2},where f:G→R, G is an Abelian group, and provide a procedure for the construction of the solutions. [ABSTRACT FROM AUTHOR]

The asymptotic stability behavior of Drygas, quadratic and Jensen functional equations is investigated. Indeed, we show that if these equations hold approximately for large arguments with an upper bound ε, then they are also valid approximately everywhere with a new upper bound which is a constant multiple of ε. These results will be applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. We also obtain some results of hyperstability character for these functional equations. [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: The study of differential growth in various animals suggests that a similar growth pattern occurs throughout the bilateral animals. This growth pattern is based on the assumption that a quadratic equation describes the relationship between two body measurements, yielding a quadratic parabola in a graphic presentation. Aim: Can human growth be studied by comparing body measurements? If the growth of one body part has a definite relation to the growth of another body part and if this relation can be expressed in a quadratic formula, then important conclusions can be reached. Subjects and methods: An official database of the mean measurements of the Belgian population has been used. Results: (1) The growth in human length is, from the beginning, constantly and negatively influenced by the growth in girth, so it is evident that growth has to stop; (2) The influence of the growth in girth is stronger in females, resulting in shorter females; (3) The growth of the human head is, from the beginning, constantly and negatively influenced by the growth in body length (both sexes show a very similar pattern); (4) Not all comparisons resulted in a quadratic parabola: the nipple distance is constantly at ∼24% of the thoracic girth in males and young females. Conclusion: The study of differential growth by using a quadratic parabola gives the answers to the questions “Why do we stop growing?” and “Why are women shorter than men?”. [ABSTRACT FROM AUTHOR]

The generalized Gaussian distribution with location parameter μ, scale parameter σ, and shape parameterpcontains the Laplace, normal, and uniform distributions as particular cases forp= 1, 2, +∞, respectively. Derivations of the true maximum-likelihood estimators of μ and σ for these special cases are popular exercises in many university courses. Here, we show how the true maximum-likelihood estimators of μ and σ can be derived forp= 3, 4, 5. The derivations involve solving of quadratic, cubic, and quartic equations. [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]

We prove some stability results for the equation in the class of functions mapping a groupoid ( X, ∗) into a Banach space Y , where $${p, q, r, s: X \rightarrow X}$$ are endomorphisms of the groupoid, and A, B, C, D are fixed scalars. Particular cases of the equation are the equation of the p-Wright affine functions, the additive Cauchy equation, the Jensen equation, the quadratic equation and the general linear equation (in two variables). [ABSTRACT FROM AUTHOR]

We prove that the Diophantine problem for spherical quadratic equations in free metabelian groups is solvable and, moreover, NP-complete. [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]

Julian Huxley (1924) came to the conclusion that intra-specific growth usually follows a sequence of power curves. So Huxley claimed that during growth sudden changes in the growth rate can occur. The restudy of his material, however, reveals that his observations closely follow single quadratic curves. As a result the intra-specific allometry studied by Huxley is comparable to ontogenetic allometry. The quadratic factor of the quadratic equations obtained, represents the growth rate; it shows the constant increase (positive factor) or decrease (minus factor) of one of the measurements for a constant increase in the other measurement with which it is compared. The quadratic factor explains the entire growth process and is the same for the smaller (younger) and larger (older) specimens. It could probably permit the prediction of the shape of larger and/or smaller animals not yet found, or give a clue to some evolutionary changes. By using the quadratic parabola there is no need to postulate "sudden changes in the growth curve" and so it appears that Huxley's power curve can be abandoned. [ABSTRACT FROM AUTHOR]

Solving an autoconvolution equation is a nonlinear ill-posed inverse problem. Besides standard methods for general nonlinear problems several customized methods for deautoconvolution are available. Recently, a new decomposition approach for solving ill-posed quadratic equations, e.g. autoconvolutions, has been proposed. In this article we compare the new approach to the TIGRA method of Ramlau and to the local regularization method of Dai and Lamm. Numerical tests show that the new method yields better approximations to the unknown true solution than existing methods in comparable computation time. [ABSTRACT FROM AUTHOR]

In this paper, we present a solution for any standard quaternion quadratic equation, i.e. an equation of the form z2 + μz + ν = 0 where μ and ν belong to some quaternion division algebra Q over some field F, assuming the characteristic of F is 2. [ABSTRACT FROM AUTHOR]

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations. [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]

Standard methods for regularizing ill-posed nonlinear equations rely on derivatives of the nonlinear forward mapping. Thereby stronger structural properties of the concrete problem are neglected and the derived algorithms only show mediocre efficiency. We concentrate on nonlinear mappings with quadratic structure and develop a derivative-free regularization method that allows us to apply classical techniques known from linear inverse problems to quadratic equations. In fact, regularization of a quadratic problem can be reduced to regularization of one linear problem and a downstream inversion of a well-posed quadratic mapping. The motivation for considering problems with quadratic structure in more detail comes from applications in laser optics where kernel-based autoconvolution-type equations have to be solved. [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]

The square roots of the bicomplex number A = a + ib + jc + ijd, where a, b, c, d are real numbers and i, j ij are the bicomplex units, are found. The solutions of the quadratic equation X² + pX + q = 0 of the bicomplex variable X and bicomplex parameters p, q, are given [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present a method that allows to study the Hyers–Ulam stability of some systems of functional equations connected with the Cauchy, Jensen and quadratic equations. In particular we generalize and extend some already known results. [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]

A necessary and sufficient solvable condition for diagonal quadratic equation with prime variables in arithmetic progressions is given, and the best qualitative bound for small solutions of the equation is obtained. [ABSTRACT FROM AUTHOR]

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 paper, we present an approach to the solution of the Hamilton–Jacobi–Isaacs equation (HJIE) arising in the H∞ control problem for nonlinear systems. We show that the HJIE can be solved analogously to a scalar quadratic equation with some additional side conditions, and present a computational procedure for determining symmetric solutions. Examples are given for second-order affine nonlinear systems to illustrate the procedure, and the method can be extended to higher-order systems. [Copyright &y& Elsevier]

• Two novel oxadiazole ring derivatives were synthesized. • The solvent effects were studied by Infrared spectroscopy. • The theoretical results assigned using PED contributions. • The ν (C O), ν (C N) and ν (C S) were correlated with the Swain, AN, KBM and LSER. • The quadratic and interactive effects of solvent parameters were also reported. In the present study novel 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazol-5(4 H)-one (compound (4)) and 4-(4-ethyl-phenyl)-3-(4-methyl-phenyl)-1,2,4-oxadiazole-5(4 H)-thione (compound (5)) were synthesized. These oxadiazole ring derivatives were characterized by IR, 1H NMR, 13C NMR and HRMS analyses. The solvent effects on C O, C N and C S stretching vibrational frequencies (ν (C O), ν (C N) and ν (C S)) of synthesized compounds were investigated experimentally using attenuated total reflection (ATR) infrared spectroscopy and compared with the theoretical results assigned using the potential energy distribution (PED) contributions. Furthermore, the ν (C O), ν (C N) and ν (C S) of compound (4) and compound (5) were correlated with empirical solvent parameters such as the solvent acceptor numbers, the Swain equation, the Kirkwood-Bauer-Magat equation, and the linear solvation energy relationships. Apart from the linear effects investigated in similar studies, solvent-induced vibrational shifts were investigated using the quadratic equation. The prediction capabilities of empirical solvent parameters were statistically compared. It was found that the linear solvation energy relationships show better correlation than the other empirical solvent parameters. Additionally, the quadratic equation provided more accurate predictions for the vibrational frequency locations than the Swain and the linear solvation energy relationships equations. [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]