Polynomials Of First Degree Research Articles

The Effect of Graded Running Protocols On Peak Oxygen Consumption and Intramuscular Oxygen Saturation

Aim: The aim of this investigation was to examine the impact of two different running protocols on peak oxygen consumption (VO2max) and regional intramuscular oxygen saturation (SmO2) levels in one highly trained runner. Methods: VO2 and SmO2 were measured simultaneously during a gradually ascending running protocol and a Bruce protocol. VO2 was measured by breath-by-breath spiroergometer gas analysis system (ZAN®, Germany), and SmO2 was measured from both gastrocnemius muscles using a wireless near-infrared spectroscopy device (BSXinsight®, USA). The correlation between VO2 and SmO2 data was determined by Pearson correlation coefficients over the test stage mean values. The change of VO2 and SmO2 within each stage was determined by first-degree polynomials. Results: Peak VO2 in the Bruce protocol (4390 ml/min) was higher than peak VO2 in the running protocol (4640 ml/min), but no difference was observed in end-test SmO2 decreases. There were highly significant negative correlations between VO2 values and SmO2 values (r=-0.960-0.990, p

Percentile Bootstrap Interval on Univariate Local Polynomial Regression Prediction

This study offers a new technique for constructing percentile bootstrap intervals to predict the regression of univariate local polynomials. Bootstrap regression uses resampling derived from paired and residual bootstrap methods. The main objective of this study is to perform a comparative analysis between the two resampling methods by considering the nominal coverage probability. Resampling uses a nonparametric bootstrap technique with the return method, where each sample point has an equal chance of being selected. The principle of nonparametric bootstrapping uses the original sample data as a source of diversity in contrast to parametric bootstrapping, where the variety comes from generating a particular distribution. The simulation results show that the paired and residual bootstrap interval coverage probabilities are close to nominal coverage. The results showed no significant difference between paired bootstrap interval and percentile residual. Increasing the bootstrap sample size sufficiently large gives the scatterplot smoothness of the confidence interval. Applying the smoothing parameter by choice gives a second-order polynomial regression with a smoother distribution than the first-order polynomial regression. The scatterplot shows that the second-degree polynomial regression can capture the data curvature feature compared to the first-degree polynomial. The bands made from second-degree polynomials give a narrower width than first-degree polynomials. In contrast, applying optimal smoothing parameters to the model provides different conclusions by using smoothing parameters based on choice. In addition to the differences based on the scatterplot, the bootstrap estimates of the coverage probability are also other. Selecting smoothing parameters based on a particular value provides probability coverage with the paired bootstrap method for the first-degree local polynomial regression is 0.93, while the second-degree local polynomial is 0.96. The probability of coverage based on the residual bootstrap method for the first-degree local polynomial regression is 0.95, while the second-degree local polynomial is 0.96. The probability coverage based on the optimal parameters of the paired bootstrap method for the first-degree local polynomial regression is 0.945, while the second-degree local polynomial is 0.93. The residual bootstrap method gives the first-degree local polynomial regression of 0.95, while the second-degree local polynomial is 0.93. In general, both bootstrap methods work well for estimating prediction confidence intervals.

Modification artificial neural networks for solving singular perturbation problems

In this paper, we introduce a modified method for solving singular perturbation problem for ordinary differential equations by using artificial neural networks. This modified method is called modified artificial neural network method (MANN). This new method is based on replacing each in the input vector training set) by a polynomial of first degree which will be as . Using MANN makes that training points should be selected over the open interval (a, b) without training the neural network in the range of first and end points. We illustrate this modified method by solving some numerical examples and provide a comparison between our results and those calculated by other numerical methods, this method demonstrates that neural networks produce solutions with good generalization and accuracy. Several examples are provided to demonstrate the suggested strategy.

Piecewise acceleration orbital modeling: a GOCE satellite case study

In this work, the Precise Science Orbit (PSO) of the Gravity Field and Steady-State Ocean Circulation Explorer Mission (GOCE) satellite, acquired via the European Space Agency, served as the reference orbit for the testing various variants of the GOCE satellite orbit modeling. The GOCE satellite positions along the reduced-dynamic PSO orbit were treated as pseudo-observations in the satellite orbit improvement process in the least squares sense. This process was realized using dedicated extended Toruń Orbit Processor software which enabled determining corrections to the orbital initial conditions and the set of parameters, necessary to determine additional empirical accelerations of the satellite in the radial-track, along-track and cross-track directions. These piecewise accelerations were determined in successive equal intervals (pieces) into which the processed orbital arcs were divided. For modeling the accelerations, polynomials of different degrees were used. The obtained RMS differences between the improved orbits and the reference PSO orbit were determined for various orbital arc lengths up to 1-day. The best RMS of the fits for the 1-day arcs was in the range from 2.0 to 3.2 mm with significantly worst results for along-track direction. Based on the set of solutions determined, the number of orbital parameters for adopted accuracy thresholds and the upper limit of their number, that can be estimated depending on the length of the orbital arc, under given numerical conditions were obtained. The optimal form of the polynomial modeling of the estimated accelerations also depends partly on the length of the processed orbital arc. For shorter arcs (45 min and less), the second- third- and fourth-order polynomial gives the best results, while for longer arcs (90, 180, 360, 720 and 1440 min), zero- and first-degree polynomials are the most effective. A very promising solution with the RMS of the fit of 0.1 mm for a 1-min arc using the fourth-order polynomial was obtained in a perspective of the future use of the short arc approach to the fit of 1-day arcs. Additionally, solution variants with non-equal numbers of orbital pieces for the radial-along-and cross-track directions were obtained. In the case of the solution for an arc length of 1-day, the distribution of residuals and their statistics in the aforementioned radial-, along-and cross-track directions are presented.

On Calculating Method of the Kelly Criterion for Financial Investment in Single Risky Asset with Various Distributions of Returns

In this paper, the expectation of the reciprocal of first-degree polynomials of non-negative valued random variables is calculated. This is motivated to compute the Kelly criterion, which is the optimal solution of the maximization of the expected logarithm of the investment return. As soon as the expectation of the reciprocal of first-degree polynomials of asset returns is calculated, which is our main interest, the Kelly criterion can be obtained by using the ordinary optimization technique or applying the appropriate algorithm.

An Affine Integral Inequality of an Arbitrary Degree for Stability Analysis of Linear Systems With Time-Varying Delays

This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional’s derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.

Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems

Abstract Schemes with the second-order approximation in time are considered for numerically solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Padé polynomial approximation is unconditionally stable. It demonstrates good asymptotic properties in time and provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic (SM) stability). In fact, the only drawback of this scheme is the necessity to solve an equation with an operator polynomial of second degree at each time level. We consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree. Such computational implementations occur, for example, if we apply the fully implicit two-level scheme (the backward Euler scheme). A three-level modification of the SM-stable scheme is proposed. Its unconditional stability is established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm involves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage and the following correction of the approximate solution using a factorized operator. The SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work, a model problem is solved numerically.

Numerical Solution of Fuzzy Partial Differential Equations by Using Modified Fuzzy Neural Networks

The aim of this work is to present a modified method for finding the numerical solutions of fuzzy partial differential equations by using fuzzy artificial neural networks. Using a fuzzy trial neural solution depending on the fuzzy initial values and the fuzzy boundary conditions of the problem. Using modified fuzzy neural network makes that training points should be selected over an open interval without training the network in the range of first and end points. In fact, This new method based on replacing each element in the training set by a polynomial of first degree. The fuzzy trial solution of fuzzy partial differential equation is written as a sum of two parts. The first part satisfies the fuzzy conditions, it contains no fuzzy adjustable parameters. The second part involves a feed-forward fuzzy neural network containing fuzzy adjustable parameters. In comparison with existing similar fuzzy neural networks, the proposed method provides solutions with high accuracy. Finally, we illustrate our approach by two problems.

Asymptotic form of level density distributions for a class of inhomogeneous 1D vertex models

Level density distributions for some Haldane-Shastry like spin chains associated with AN−1 root system have been computed recently by Enciso et al., using the connection of these spin chains with inhomogeneous one-dimensional vertex models. The energy functions of such vertex models are given by specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions are given by arbitrary polynomials of any possible degree and show that the level densities for this class of vertex models asymptotically follow the Gaussian distribution for large number of vertices.

Level density distribution for one-dimensional vertex models related to Haldane-Shastry like spin chains

The energy level density distributions of some Haldane-Shastry like spin chains associated with the AN − 1 root system have been computed recently by Enciso et al., exploiting the connection of these spin systems with inhomogeneous one-dimensional vertex models whose energy functions depend on the vertices through specific polynomials of first or second degree. Here we consider a much broader class of one-dimensional vertex models whose energy functions depend on the vertices through arbitrary polynomials of any possible degree. We estimate the order of mean and variance for such energy functions and show that the level density distribution of all vertex models belonging to this class asymptotically follow the Gaussian pattern for large number of vertices. We also present some numerical evidence in support of this analytical result.

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.

Poisson Yang-Baxter maps with binomial Lax matrices

A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first-degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on C4×C4 with 3 × 3 Lax matrices is also presented.

Nonlocal transformations and linearization of second-order ordinary differential equations

The class of nonlinear second-order equations that are linearizable by means of generalized Sundman transformations (S-transformations) is identified as the class of equations admitting first integrals that are polynomials of first degree in the first-order derivative. This class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing S-transformations are presented. Only the equations of a well-defined subclass can also be linearized by invertible point transformations. These invertible point transformations can be constructed by using the algorithms for the calculation of linearizing S-transformations. Several examples illustrate that both types of linearization are strictly different.

Yang–Baxter maps with first-degree-polynomial 2 × 2 Lax matrices

A family of non-parametric Yang–Baxter (YB) maps is constructed by re-factorization of 2 × 2 matrix polynomials of first degree. These maps are Poisson with respect to the Sklyanin bracket. For each Casimir function a parametric Poisson YB map is generated by reduction on the corresponding level set. By considering a complete set of Casimir functions symplectic multi-parametric YB maps are derived. These maps are quadrirational with explicit formulae in terms of matrix operations. Their Lax matrices are, by construction, 2 × 2 first-degree-polynomial in the spectral parameter and are classified by THE Jordan normal form of the leading term. Degenerate parametric YB maps constructed as limits of the non-degenerate ones are connected to known integrable systems on quadgraphs.

Highly Efficient Designs to Handle the Incorrect Specification of Linear Mixed Models

We apply a maximin criterion to examine the relative efficiency of several D q -optimal designs for a family of linear mixed models. Incorrect specifications of the order of the polynomial, size of the autocorrelation parameter, number of random parameters, and the correlation between random intercept and random slope are investigated. The results of our study allow us to draw the following conclusions: (1) the maximin D q -optimal design encountered appears to be highly efficient; (2) the variation of the minimum relative efficiencies of D q -optimal designs of the family of linear mixed models that were studied decreases as the order of the polynomial increases; (3) the effect of the autocorrelation parameter on the relative efficiencies of D q -optimal designs is the largest for first-degree polynomials; and (4) the relative efficiency of the equidistant design is lower than that of the maximin value and also lower than the reference value 0.85.

Psychotropic drugs interfere with the tight coupling of polyphosphoinositide cycle metabolites in human platelets: A result of receptor-independent drug intercalation in the plasma membrane?

Incubation of platelets with increasing concentrations of thrombin produced large amounts of phosphatidic acid (PA) and distinct changes in phosphatidylinositol-4-phosphate (PIP) and phosphatidylinositol-4,5-bisphosphate (PIP 2), prominent metabolites in the polyphosphoinositide (PPI) cycle. The relation between normalized PA and PIP or PIP 2 levels in such thrombin-treated platelets from 22 normal donors gave a very similar pattern, suggesting tight control of the metabolites in the polyphosphoinositide (PPI) cycle. Prochlorperazine (PCP), trifluoperazine (TFP), haloperidol (HPD), quetiapine (QTP), pimozide (PMZ) and clozapine (CLO) interfered with this tight coupling produced by treating platelets with increasing thrombin concentrations. All drugs decreased the formation of PA at a given thrombin concentration, a decrease that varied greatly among platelets from different donors. This made it difficult to treat the PIP/PA and PIP 2/PA relationships with ordinary, descriptive statistics. The data were therefore subjected to regression analysis using polynomials of second or first degree and gave the interference ranking order: PCP > TFP ≫ PMZ = HPD > CLO > QTP. All six drugs increased the mean molecular area of monolayers of dipalmitoyl phosphatidylserine on pure water at 37 °C by 20–50%, while they had little effect on monolayers of dipalmitoyl phosphatidylcholine. These results suggest that the drugs are membrane-active and may intercalate in biomembranes containing negatively charged phospholipids. Since human platelets do not contain D 2 receptors, the interference with the tight coupling of PPI cycle metabolites was not receptor-mediated. We suggest that the drugs are intercalated in the plasma membrane and alter the relative, spatial positioning of phospholipid-consuming enzymes and thereby alter the velocities of the enzyme-catalyzed reactions. Such intercalation could be part of the side effects of the drugs and may explain their psychotropic action(s).

Haematological and biochemical indices in young African children: in search of reference intervals

The reference intervals of haematological and biochemical indices currently used in Africa are derived from data collected from populations living in industrialized countries. Few studies have been performed in Africa questioning the validity of these values when applied to local African populations. To provide reference intervals of haematological [haemoglobin (Hb), white blood cells (WBC), haematocrit (Htc) and platelets] and biochemical indices (ALT, creatinine and bilirubin) for children aged 1-4 from a rural area of southern Mozambique. Reference intervals were developed using the 2.5 and 97.5 centiles. Partition tests were performed to evaluate age and gender differences. Quantile regression models were estimated for those variables in which age partition was recommended. Deviances from linearity in the estimated models were evaluated using fractional polynomials of first or second degree. Agreement to classify normality, using the estimated reference intervals or values in use in a western paediatric hospital, was made using the kappa statistic. Reference intervals for Hb, WBC, Htc, platelets, ALT and creatinine show significant differences by age. Gender differences were observed for creatinine values, while for bilirubin there were no significant differences for age or gender. Estimated Hb and Htc reference intervals in African children were lower than the accepted western ones, while ALT values were higher in the former. Agreement between normal classification, using the estimated intervals or the western values, was from slight to fair. Reference intervals of haematological and biochemical indices based on results from western individuals of the same age are not in agreement with the estimated values for African children. These observed values should not be used as a rule to define normality, but are the ones usually found in this population where anaemia, malaria and high mortality rates are also common.

Classification of bivariate configurations with simple Lagrange interpolation formulae

In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes than the degree is used for classifying these configurations.

Recurrence equations and their classical orthogonal polynomial solutions

The classical orthogonal polynomials are given as the polynomial solutions p n ( x) of the differential equation σ(x)y″(x)+τ(x)y ′(x)+λ ny(x)=0, where σ( x) is a polynomial of at most second degree and τ( x) is a polynomial of first degree. In this paper a general method to express the coefficients A n, B n and C n of the recurrence equation p n+1(x)=(A nx+B n)p n(x)−C np n−1(x) in terms of the given polynomials σ( x) and τ( x) is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three-term recurrence equation, i.e., a homogeneous linear three-term recurrence equation with polynomial coefficients. In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equation σ(x)Δ∇y(x)+τ(x)Δy(x)+λ ny(x)=0, where Δy( x)= y( x+1)− y( x) and ∇ y( x)= y( x)− y( x−1) denote the forward and backward difference operators, respectively, and a similar approach applies to classical q-orthogonal polynomials, being solutions of the q-difference equation σ(x)D qD 1/qy(x)+τ(x)D qy(x)+λ q,ny(x)=0, where D qf(x)= f(qx)−f(x) (q−1)x , q≠1, denotes the q-difference operator.

On a structure formula for classical q-orthogonal polynomials

The classical orthogonal polynomials are given as the polynomial solutions P n ( x) of the differential equation σ(x)y″(x)+τ(x)y′(x)+λ ny(x)=0, where σ( x) turns out to be a polynomial of at most second degree and τ( x) is a polynomial of first degree. In a similar way, the classical discrete orthogonal polynomials are the polynomial solutions of the difference equation σ(x) Δ∇y(x)+τ(x) Δy(x)+λ ny(x)=0, where Δ y( x)= y( x+1)− y( x) and ∇ y( x)= y( x)− y( x−1) denote the forward and backward difference operators, respectively. Finally, the classical q-orthogonal polynomials of the Hahn tableau are the polynomial solutions of the q-difference equation σ(x)D qD 1/qy(x)+τ(x)D qy(x)+λ q,ny(x)=0, where D qf(x)= f(qx)−f(x) (q−1)x , q≠1, denotes the q-difference operator. We show by a purely algebraic deduction — without using the orthogonality of the families considered — that a structure formula of the type σ(x)D 1/qP n(x)=α nP n+1(x)+β nP n(x)+γ nP n−1(x) (n∈ N≔{1,2,3,…}) is valid. Moreover, our approach does not only prove this assertion, but generates the form of this structure formula. A similar argument applies to the discrete and continuous cases and yields σ(x)∇P n(x)=α nP n+1(x)+β nP n(x)+γ nP n−1(x) (n∈ N) and σ(x)P n′(x)=α nP n+1(x)+β nP n(x)+γ nP n−1(x) (n∈ N). Whereas the latter formulas are well-known, their previous deduction used the orthogonality property. Hence our approach is also of interest in these cases.