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The growth of meromorphic solutions of linear difference equations containing Askey—Wilson divided difference operators is estimated. The ϕ -order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order ρ log ( f ) and the classical order ρ ( f ) of a meromorphic function f .

In this research we introduce the concept of -convex function of higher order by means of the so called -divided difference; elementary properties of this type of functions are exhibited and some examples are provided.

A method and system for measuring multiple temperature points using two wire configuration Resistance Temperature Detectors (RTD) for temperature range −100°C to +100°C is mainly composed of the Micro Controller Unit (MCU), analog-digital converter (ADC), reference Voltage source, Multiplexers (MUX) and resistance network including the RTD temperature sensors. The RTDs are sequentially scanned using a reference/constant voltage source, and resistance values of the RTDs are calculated based on the voltage measured across the RTDs and the current sensing resistor. Improving the precision of temperature measurement is based on a line fitting algorithm-Newton's divided differences interpolation polynomial based on three reference resistor values representing two extreme end points and the center point of the temperature range for which the system is made. The test results show that the device temperature measurement precision can reach ±0.02°C, has the advantage of using a single quartic function for the entire temperature range −100°C to +100°C with dynamic calibration to address errors related to in-circuit resistance, tolerance, temperature coefficient, linearity and offset errors in the circuit, and also supports detection of degradation of linearity and offset in the circuit because of the aging factor.

Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain regions. While this idea, known as the tensor product case, has been around for over a century, this exposition uses modern multi-index notation to provide generality, concise algorithms, and rigor, appropriate as a topic in introductory numerical analysis. Node configurations where the tensor product case applies are called lower sets, and they include n-dimensional rectangles (boxes) and triangles (corners), the latter having unique interpolation over polynomials of fixed degree. Arbitrary distinct nodes do not ensure unique interpolating polynomials but, when possible, a different basis generalization, which we call Newton--Sauer, results in a nice triangular linear system for the coefficients. For the right number of distinct nodes, an algorithm based on Gaussian elimination produces either this Newton-Sauer basis or a polynomial that is zero on all nodes, showing that unique interpolation is impossible. The algorithm may also be used on any distinct nodes to produce a polynomial subspace of minimal degree where unique interpolation is possible. A variation of this algorithm using matrix block operations produces another basis generalization that we call Newton-Olver, which exactly agrees with the classic one for a corner of nodes and computes an interpolating polynomial using formulas similar to divided differences.

We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order systems of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and nonautonomous dynamical systems with conserved quantities of arbitrary forms, such as time-dependent conserved quantities. Sufficient conditions to construct conservative schemes of arbitrary order are derived using the multiplier method. General formulas for first-order conservative schemes are constructed using divided difference calculus. New conservative schemes are found for various dynamical systems such as Eider's equation of rigid body rotation, Lotka-Volterra systems, the planar restricted three-body problem, and the damped harmonic oscillator.

For r, s ∈ ℝ and ρ = min {r, s}, let D[x + r, x + s; ψ n−1] ≡ −ϕn (x) be the divided difference of the functions ψ n−1 = (−1)ⁿ ψ(n−1) (n ∈ N) on (−ρ,∞), where ψ (n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions x ↦ ∏ i = 1 k ϕ m i ( x ) − λ k ∏ i = 1 k ϕ n i ( x ) , x ↦ ∏ i = 1 k ϕ n i ( x ) − μ k ϕ s n k ( x ) to be completely monotonic on (−ρ,∞), where mi, ni ∈ N for i = 1, .., k with k ≥ 2 and s n k = ∑ i = 1 k   n i . These generalize known results and gives an answer to a problem.

We consider a sequence of positive linear operators Ln of Bernstein-Schnabl type. It was studied in the literature from various points of view; we provide new properties of it. The eigenstructure of these operators is described. We investigate the kernel of Ln which is related with the set of solutions of a difference equation. Several algorithms are proposed in order to solve the involved problems.

Interpolating non-uniform rational B-spline (NURBS) curves often leads to undesirable feedrate fluctuations due to the non-analytic relationship between the spline parameter and arc length. In this paper, we introduce a feedback method that effectively reduces feedrate fluctuations while maintaining simplicity in the interpolation process. Our approach begins by defining the chord length-parameter ratio (CPR), which we then predict for the current interpolation point using Newton’s divided difference interpolation polynomial and historical CPR data. Subsequently, we calculate the compensation length based on tangent and chord directions, and we correct the initial parameter using the second-order Runge–Kutta method. Importantly, our method involves only two calculations of the first derivative of the NURBS curve, making it computationally efficient compared to most approximation methods. To validate our proposed method, we conduct numerical simulations and compare its performance against several approximation methods and a recent feedback method. The results demonstrate that our approach excels in terms of both efficiency and accuracy, making it a promising solution for improving NURBS curve interpolation.

The problem of numerical recovery of intermediate derivatives or a differential operator is considered. With this aim, the norm of the leading derivative is estimated, from which optimal difference schemes for intermediate derivatives are obtained. An example of a smooth model function is considered for which the measurement error is simulated using real-world experimental data. The trusted probability of the obtained estimate for the norm of the leading derivative is estimated. Intervals of trusted recovery of intermediate derivatives are estimated.

The ephemeris of a pulsar is stable for a long time, which allows navigation based on pulsar orientation to be vector feasible. The formation of a satellite navigation model using the orientation vector of an X-ray pulsar signal is presented in this paper. To obtain the time difference of arrival (TDOA), a new estimation method is constructed, which can measure the photon sequence of an X-ray pulsar signal and is based on the fast Fourier transform (FFT). Next, three new observation variables are constructed. The variables are satellite phase incrementation; the angle between the satellite baseline and the pulsar direction vector; and the angle between the plane spanned by three satellite baselines and the pulsar direction vector. All three variables, along with the TDOA of the X-ray pulsar signal, are utilized to determine the orbit. The position and velocity of the satellite formations are estimated by using the adaptive divided difference filter (ADDF) to eliminate nonlinearity. Several simulation cases are designed to verify the proposed method.