Approximate Linear Equation Research Articles

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Making individualized men's basic body block patterns with a measuring garment incorporating stretchable capacitance sensors

PurposeTo make individualized men's basic upper garment patterns without sleeves, the authors developed a measuring garment that measures necessary body dimensions and angles all together. Additionally, the authors proposed a method for making individualized basic body block patterns using the obtained dimensions and angles.Design/methodology/approachThe authors decided on the locations of the dimensions of the body required for making the individualized garments. The authors then sewed multiple stretchable capacitance sensors to corresponding locations on a stretchable T-shirt. To obtain the dimensions with sensors of short length, the authors indirectly obtained each length from the relationship between the actual body length and the capacitance of the sensor. Beforehand, the authors obtained linear-approximation equations for the relationship between actual body dimensions and the capacitance of sensors for five participants and a dummy. The authors then used the measuring garment and the equations to obtain the body dimensions of another six participants. The authors compared the obtained and actual body dimensions to verify the equations. The authors made individualized upper-garment patterns without sleeves and garments for the 11 participants with the obtained dimensions and angles. The authors verified the proposed method in wearing tests comparing garments designed using the proposed method with conventionally designed garments.FindingsUsing the measuring garment, the authors obtained body dimensions close to actual body dimensions. A pattern of the individualized basic upper garment using the obtained dimensions and angles could be drawn. Compared with the conventional patterns, the individualized patterns had notable differences in the locations of the shoulder point and side neck point and directions of the shoulder line, which relate to the shoulder shape (i.e. square, sloping, forward, or backward). In wearing tests, all participants declared that the individualized garment better fitted their shoulders than the conventional garment without tightness around the shoulders, neck, and armpits. The proposed method with the developed measuring garment was thus found to be effective in designing individualized garments.Originality/valueThis paper presents the possibility of not only measuring body dimensions but also designing individualized basic upper garments using the proposed measuring garment. The proposed measuring garment will assist the efficient manufacture of individualized upper garments without a three-dimensional scanner or special skills.

Photocatalytic Hydrogen Generation from Aqueous Methanol Solution over n-Butylamine-Intercalated Layered Titanate H2La2Ti3O10: Activity and Stability of the Hybrid Photocatalyst

The stability of platinized n-butylamine-intercalated layered titanate H2La2Ti3O10 during the process of photocatalytic hydrogen production from aqueous methanol under UV irradiation has been thoroughly investigated by means of XRD, CHN, TG, 13C NMR, BET, SEM and GC-MS analysis. It was revealed that n-butylamine completely abandons the interlayer space and transforms into n-butyraldehyde within 3 h of the reaction, while the particle morphology and specific surface area of the photocatalyst are preserved. The resulting solid phase contains carbon in at least two different oxidation states, which are attributed to the intermediate products of methanol oxidation bound to the perovskite matrix. The activity of the photocatalyst formed in this way is stable in time and strongly depends on the medium pH, which is not typical of either the parent H2La2Ti3O10 or TiO2. An approximate linear equation φ ≈ 29−2∙pH holds for the apparent quantum efficiency of hydrogen production in the 220–340 nm range at 1 mol. % methanol concentration. In the acidic medium, the photocatalyst under study outperforms the platinized H2La2Ti3O10 by more than one order of magnitude. The variation in methanol concentration allowed a maximum quantum efficiency of hydrogen production of 44% at 10 mol. % to be reached.

Choice of DLVO approximation method for quantifying the affinity between latex particles and membranes

The Derjaguin-Landau-Verwey-Overbeek (DLVO) model, as well as the extended model (XDLVO), is popularly employed to quantify the interfacial interactions underlying membrane fouling. However, disagreements between membrane fouling extent and the interaction energies derived from DLVO or XDLVO models have been reported, which suggests gaps in the understanding. This study demonstrates that the DLVO approximation methods for predicting the interfacial foulant-membrane interaction are sensitive to the boundary condition assumptions (e.g., constant charge versus constant potential). In particular, while both the Poisson-Boltzmann (P–B) and linear superposition approximation (LSA) equations can quantify the electrostatics (EL) interaction energy component, the former assumes constant potential, while the latter additionally considers constant charge scenarios. The relative accuracy of these two equations were evaluated here. For dead-end filtration tests, flux decline trends, OCT analysis results and fouling model parameters were obtained. Regarding fouling of pristine PCTE membrane by latex particles of opposite charge signs, both the P–B and LSA equations contribute to predict relative fouling extents correctly. However, for the case of the BPEI-coated membrane, the P–B equation failed whereas the LSA equation gave good agreements. For cross-flow filtration tests in organic solvents, LSA also out-performed the P–B equation in providing more accurate predictions of membrane fouling. The results here highlight the shortcomings in the commonly used P–B equation and are expected to be potentially valuable in the development of better approximations for quantifying interfacial interaction energies.

Robust stable iterated unscented Kalman filter based on maximum correntropy criterion

The Unscented Kalman filter (UKF) based on maximum correntropy criterion (MCC) is robust to heavy-tailed non-Gaussian noise. However, the approximate linear measurement equation obtained by statistical linearization technique may be not accurate enough since it only uses prior information. In this paper, a robust stable iterative maximum correntropy criterion UKF (RS-IMCC-UKF) is proposed by using nonlinear measurement function directly and numerical stability methods. Different from the existing UKF algorithms, we only need to perform one-time unscented transformation in each filtering cycle, reducing the execution time of algorithm. Then a nonlinear enhancement model is constructed to handle predictions and observations simultaneously, which will be included in the cost function of robust MCC. In the process of iterative solution, thanks to the latest iteration values which are used to update the measurement information, RS-IMCC-UKF has more accurate results compared with traditional filters. In order to avoid non-positive definite characteristic in covariance matrix and the inverse operation in ill-conditioned matrix, the hyperbolic QR decomposition and Moore–Penrose pseudo-inversion are introduced to improve the numerical stability of the algorithm. Finally, a target tracking is modeled to verify the effectiveness of the algorithm.

H0 tension, phantom dark energy, and cosmological parameter degeneracies

Phantom dark energy can produce amplified cosmic acceleration at late times, thus increasing the value of $H_0$ favored by CMB data and releasing the tension with local measurements of $H_0$. We show that the best fit value of $H_0$ in the context of the CMB power spectrum is degenerate with a constant equation of state parameter $w$, in accordance with the approximate effective linear equation $H_0 + 30.93\; w - 36.47 = 0$ ($H_0$ in $km \; sec^{-1} \; Mpc^{-1}$). This equation is derived by assuming that both $\Omega_{0 \rm m}h^2$ and $d_A=\int_0^{z_{rec}}\frac{dz}{H(z)}$ remain constant (for invariant CMB spectrum) and equal to their best fit Planck/$\Lambda$CDM values as $H_0$, $\Omega_{0 \rm m}$ and $w$ vary. For $w=-1$, this linear degeneracy equation leads to the best fit $H_0=67.4 \; km \; sec^{-1} \; Mpc^{-1}$ as expected. For $w=-1.22$ the corresponding predicted CMB best fit Hubble constant is $H_0=74 \; km \; sec^{-1} \; Mpc^{-1}$ which is identical with the value obtained by local distance ladder measurements while the best fit matter density parameter is predicted to decrease since $\Omega_{0 \rm m}h^2$ is fixed. We verify the above $H_0-w$ degeneracy equation by fitting a $w$CDM model with fixed values of $w$ to the Planck TT spectrum showing also that the quality of fit ($\chi^2$) is similar to that of $\Lambda$CDM. However, when including SnIa, BAO or growth data the quality of fit becomes worse than $\Lambda$CDM when $w< -1$. Finally, we generalize the $H_0-w(z)$ degeneracy equation for $w(z)=w_0+w_1\; z/(1+z)$ and identify analytically the full $w_0-w_1$ parameter region that leads to a best fit $H_0=74\; km \; sec^{-1} \; Mpc^{-1}$ in the context of the Planck CMB spectrum. This exploitation of $H_0-w(z)$ degeneracy can lead to immediate identification of all parameter values of a given $w(z)$ parametrization that can potentially resolve the $H_0$ tension.

An Efficient Three-Term Iterative Method for Estimating Linear Approximation Models in Regression Analysis

This study employs exact line search iterative algorithms for solving large scale unconstrained optimization problems in which the direction is a three-term modification of iterative method with two different scaled parameters. The objective of this research is to identify the effectiveness of the new directions both theoretically and numerically. Sufficient descent property and global convergence analysis of the suggested methods are established. For numerical experiment purposes, the methods are compared with the previous well-known three-term iterative method and each method is evaluated over the same set of test problems with different initial points. Numerical results show that the performances of the proposed three-term methods are more efficient and superior to the existing method. These methods could also produce an approximate linear regression equation to solve the regression model. The findings of this study can help better understanding of the applicability of numerical algorithms that can be used in estimating the regression model.

The New Analytical Algorithm for Determining the Strapdown INS Orientation

The analytical solution of an approximate (truncated) equation for the vector of a rigid body finite rotation has made it possible to solve the problem of determining the quaternion of orientation of a rigid body for an arbitrary angular velocity and small angle of rotation of a rigid body with the help of quadratures. Proceeding from this solution, the following approach to the construction of the new analytical algorithm for computation of a rigid body orientation with the use of strapdown INS is proposed: 1) By the set components of the angular velocity of a rigid body on the basis of mutually — unambiguous changes of the variables at each time point, a new angular velocity of a rigid body is calculated; 2) Using the new angular velocity and the initial position of a rigid body, with the help of the quadratures we find the exact solution of an approximate linear equation for the vector of a rigid body finite rotation with a zero initial condition; 3) The value of the quaternion orientation of a rigid body (strapdown INS) is determined by the vector of finite rotation. During construction of the algorithm for strapdown INS orientation at each subsequent step the change of the variables takes into account the previous step of the algorithm in such a way that each time the initial value of the vector of finite rotation of a rigid body will be equal to zero. Since the proposed algorithm for the analytical solution of the approximate linear equation for the vector of finite rotation is exact, it has a regular character for all angular motions of a rigid body).

An Approximation Technique for Fractional Order Delay Differential Equations

In this research article, an Iterative Decomposition Method is applied to approximate linear and non-linear fractional delay differential equation. The method was used to express the solution of a Fractional delay differential equation in the form of a convergent series of infinite terms which can be effortlessly computable.The method requires neither discretization nor linearization. Solutions obtained for some test problems using the proposed method were compared with those obtained from some methods and the exact solutions. The outcomes showed the proposed approach is more efficient and correct.

Grain Size Effect on Optimum Clearance Determination in Blanking Non-Oriented Electrical Steel Sheet

Non-oriented electrical steel, as the core magnetic material, is firstly blanked into lamination in motor manufacturing. As for the newly developed steel, there is a general tendency toward thinner and coarser-grained. Due to blanking clearance and thickness are both down to the sub-millimeter scale, grain size becomes an important role in formation of blanked edge quality, which mainly determines the deterioration level of magnetic properties. This paper aims to systemically investigate the influence of blanking clearance and grain size on blanked edge quality. In this research, non-oriented electrical steel sheets of the same chemical composition, 3 thicknesses and 3 grain sizes are prepared for blanking tests over the conventional relative blanking clearance range. The blanking edges are quantitatively examined by means of optical microscopy to visualize the distribution of plastic deformation. The results show that there exists an optimum clearance that leads to a fine blanked edge. In further study, an approximate linear equation of the ratio of clearance/grain size (c/D) vs. D is found for optimizing the blanked edge quality. This research thus provides an in-depth understanding and guidance for optimum blanking clearance determination influenced by size effect.

Third-order AVO inversion for lamé parameter based on inverse operator estimation algorithm

Various linearized Zoeppritz approximations are often used in the forward modeling of amplitude variation with offset (AVO). However, they are usually limited by the elastic parameters with weak properties contrast. The high contrast situations between layers will cause big errors when linear approximate equations are used. Given this problem, the more precise third-order Zoeppritz approximate equation for velocity and density is derived in this study. Parameter sensitivity analysis illustrates that the third-order approximate equation for velocity and density is not favorable for nonlinear AVO inversion because the equation is extremely insensitive to S-wave velocity. In consideration of the sensitivity of elastic parameters and the poor inversion results, we derived the third-order approximate equation for Lamé parameter. A direct inversion method (Named as inverse operator estimation algorithm) is further used for nonlinear AVO inversion. The L1 norm constraint is considered to improve stability and efficiency during the nonlinear inversion process. The suitability and feasibility of the equation and inversion method can be demonstrated by the synthetic examples. Field data examples show that the inversion results have a good agreement with logging curves.

A Gauss–Newton iteration for Total Least Squares problems

The Total Least Squares solution of an overdetermined, approximate linear equation $Ax \approx b$ minimizes a nonlinear function which characterizes the backward error. We show that a globally convergent variant of the Gauss--Newton iteration can be tailored to compute that solution. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix $A$ is perturbed by a rank-one term.

Relationship between dexmedetomidine dose and plasma dexmedetomidine concentration in critically ill infants: a prospective observational cohort study.

BackgroundDexmedetomidine is a highly selective central α2-agonist used as a sedative in pediatric intensive care unit (PICU). However, little is known about the relationship between dexmedetomidine dose and its plasma concentration during long-term infusion. We have previously demonstrated that the sedative plasma dexmedetomidine concentration is moderately correlated with the administered dose in adults (r = 0.653, P = 0.001). We hypothesized that there would be a similar relationship between the sedative dexmedetomidine concentration and administered dose in infants.MethodsAll patients admitted to the PICU at Nagoya City University Hospital, Japan, between November 2012 and March 2013 were eligible for inclusion in the study. Plasma dexmedetomidine concentration was measured by ultra-performance liquid chromatography coupled with tandem mass spectrometry.ResultsWe measured the plasma dexmedetomidine concentration in 203 samples from 45 patients. Of these, 96 samples collected from 27 patients < 2 years old were included in this study. All patients received dexmedetomidine at 0.12–1.40 µg/kg/h. The median administration duration was 87.6 hours (range: 6–540 hours). Plasma dexmedetomidine concentration ranged from 0.07 to 3.17 ng/ml. Plasma dexmedetomidine concentration was not correlated with the administered dose (r = 0.273, P = 0.007). The approximate linear equation was y = 0.690x + 0.423.ConclusionsIn infants, plasma dexmedetomidine concentration did not exhibit any correlation with administered dose, which is not a reliable means of obtaining optimal plasma concentration.

He is variational iteration method to approximate time fractional wave non linear like equation with variable coefficient

In this paper, we will consider variationaliteration method (VIM) for solvingtime fractional non linear wave like with variable coeffcient. In this method we usean approximate value for time fractional derivativewhen find Lagrange multiplier.Three examples show theefficiency and the importance of the method

Extracellular Linkers Completely Transplant the Voltage Dependence from Kv1.2 Ion Channels to Kv2.1

The transmembrane voltage needed to open different voltage-gated K (Kv) channels differs by up to 50 mV from each other. In this study we test the hypothesis that the channels’ voltage dependences to a large extent are set by charged amino-acid residues of the extracellular linkers of the Kv channels, which electrostatically affect the charged amino-acid residues of the voltage sensor S4. Extracellular cations shift the conductance-versus-voltage curve, G(V), by interfering with these extracellular charges. We have explored these issues by analyzing the effects of the divalent strontium ion (Sr2+) on the voltage dependence of the G(V) curves of wild-type and chimeric Kv channels expressed in Xenopus oocytes, using the voltage-clamp technique. Out of seven Kv channels, Kv1.2 was found to be most sensitive to Sr2+ (50 mM shifted G(V) by +21.7 mV), and Kv2.1 to be the least sensitive (+7.8 mV). Experiments on 25 chimeras, constructed from Kv1.2 and Kv2.1, showed that the large Sr2+-induced G(V) shift of Kv1.2 can be transferred to Kv2.1 by exchanging the extracellular linker between S3 and S4 (L3/4) in combination with either the extracellular linker between S5 and the pore (L5/P) or that between the pore and S6 (LP/6). The effects of the linker substitutions were nonadditive, suggesting specific structural interactions. The free energy of these interactions was ∼20 kJ/mol, suggesting involvement of hydrophobic interactions and/or hydrogen bonds. Using principles from double-layer theory we derived an approximate linear equation (relating the voltage shifts to altered ionic strength), which proved to well match experimental data, suggesting that Sr2+ acts on these channels mainly by screening surface charges. Taken together, these results highlight the extracellular surface potential at the voltage sensor as an important determinant of the channels’ voltage dependence, making the extracellular linkers essential targets for evolutionary selection.

A time-integration method for stable simulation of extremely deformable hyperelastic objects

This paper presents a time integration method for realtime simulation of extremely deformable objects subject to geometrically nonlinear hyperelasticity. In the presented method, the equation of motion of the system is discretized by the backward Euler method, and linearly approximated through the first-order Taylor expansion. The approximate linear equation is solved with the quasi-minimal residual method (QMR), which is an iterative linear equation solver for non-symmetric or indefinite matrices. The solution is then corrected considering the nonlinear term that is omitted at the Taylor expansion. The method does not demand the constitutive law to guarantee the positive definiteness of the stiffness matrix. Experimental results show that the presented method realizes stable behavior of the simulated model under such deformation that the tetrahedral elements are almost flattened. It is also shown that QMR outperforms the biconjugate gradient stabilized method (BiCGStab) in this application.

Gravito-electromagnetic effects of massive rings

The Einstein field equations in linear post-Newtonian approximation can be written in analogy with electromagnetism, in the so-called gravito-electromagnetic (GEM) formalism. We use this analogy to study the gravitational field of a massive ring: In particular, we consider a continuous mass distribution on Keplerian orbit around a central body, and we work out the gravitational field generated by this mass distribution in the intermediate zone between the central body and the ring, focusing on the gravitomagnetic (GM) component that originates from the rotation of the ring. In doing so, we generalize and complement some previous results that focused on the purely Newtonian effects of the ring (thus neglecting its rotation) or that were applied to the case, of rotating spherical shells. Eventually, we study in some simple cases, the effect of the rotation of the ring, and suggest that, in principle, this approach could be used to infer information about the angular momentum of the ring.

Partition coefficient vs. binding constant: How best to assess molecular lipophilicity

Partition coefficient, P, is the preferred descriptor of molecular lipo- or hydrophilicity, and thus of relationships between a solute (S, e.g., a drug), a polar medium (W, e.g., an aqueous buffer), and an essentially apolar, organic, medium or a drug carrier (O). The coefficient is commonly identified with the linear ratio of solute quantities in the two media, P=nSO/nSW, even to characterise solute association with or binding to a surface (e.g., of a HPLC column or a drug carrier). To check the latter practice correctness—and credibility of the prevailing P definition—this paper compares an ideal solute distribution between two separate homogeneous fluid media (i.e., partitioning) to solute association with a uniform surface immersed in one such medium (i.e., binding). This reveals that solute partitioning and binding fundamentally differ and can only exceptionally be described, or analysed, with similar equations. Nonlinearised formulae that describe partitioning (Eq. (9)) and binding (Eq. (11)) can yield similar lipophilicity descriptor values only if solute preparation is relatively dilute; employing a large organic medium fraction is helpful in this respect. Additional prerequisites for partitioning and binding models match are: 1:1 stoichiometry at the association maximum and identical interfacial area of solute and organic medium molecules. If these requirements are not met, binding model yields different, potentially somewhat higher, but more often up to >10 times lower results than partitioning model. The main reason is saturation of organic medium with solute molecules. Partitioning model excludes this phenomenon, which cannot always be prevented by focussing on dilute solute preparations. The current practice of using a linear model and approximate equations to study partitioning or binding can cause large errors (>103×), and is one possible reason for the notoriously high experimental logP values scattering. The latter makes logP predictions more difficult and less reliable than they could be if the measured data were evaluated with non-linearised partitioning or binding equations, as appropriate.

The Effect of Molybdenum on the Tribological Behavior of Nfgrey (8) Cast Iron

The effect of molybdenum on the morphological and wear behaviour of Nigeria Foundries grey cast iron (8) [NFgrey(8)] has been studied. The microstructural and mechanical properties of the cast samples for the control and the NFGrey(8)/Molybdenum alloy were characterised using SEM, XRD, Brinell Hardness and impact energy testing Machine respectively. MATLAB linear curve fitting technique was used to obtain the approximate linear equation via first differential approach. The results show that molybdenum addition improved significantly the wear resistance of the NFGrey(8) cast iron. However, the toughness of the alloy was sacrificed. The strong and hard intermetallic compounds within the matrix are attributable to wear resistance of the NFGrey(8)/Mo alloy.

A comparison between dosages and plasma concentrations of dexmedetomidine in clinically ill patients: a prospective, observational, cohort study in Japan.

BackgroundDexmedetomidine is a highly selective central α2-agonist with anesthetic and analgesic properties for patients in intensive care units. There is little information about the relationship between dosage and plasma concentration during long drug infusions of dexmedetomidine in critically ill patients, especially in Asians. In addition, the administration of dexmedetomidine with a dosage of 0.2–0.7 μg/kg/h in Japan is different from that with a dosage of 0.2–1.4 μg/kg/h in European countries and the USA. There has been concern about obtaining an effective concentration with a small dosage and estimating the relationship between dosage and plasma concentration. We conducted a prospective, observational, cohort study measuring plasma dexmedetomidine concentrations.MethodsPlasma dexmedetomidine concentrations of 67 samples from 34 patients in an intensive care unit for 2 months were measured by ultra performance liquid chromatography coupled with tandem mass spectrometry using single-blind method, and the correlation coefficient between dosages and plasma concentrations was estimated. Exclusion criteria included young patients (<16 years) and samples obtained from patients in which the dosage of dexmedetomidine was changed within 3 h.ResultsAmong the patients, 20 (58.8%) of the 34 received dexmedetomidine at 0.20–0.83 μg/kg/h, and in 40 of the 67 samples for which dexmedetomidine had been administered, this occurred for a median duration of 18.5 h (range, 3–87 h). The range of the dexmedetomidine plasma concentration was 0.22–2.50 ng/ml. By comparison with other studies, with a dosage of 0.2–0.7 μg/kg/h, the patients in this setting could obtain an effective dexmedetomidine concentration. The plasma dexmedetomidine concentration was moderately correlated with the administered dosage (r = 0.653, P < 0.01). The approximate linear equation was y = 0.171x + 0.254. The range of Richmond Agitation-Sedation Scale was 0 to -5.ConclusionsWe concluded that, with a dosage of 0.2–0.83 μg/kg/h, the patients in this setting could obtain an effective dexmedetomidine concentration of 0.22–2.50 ng/ml. In addition, the plasma dexmedetomidine concentration was moderately correlated with the administered dosage (r = 0.653, P < 0.01).Trial registrationUniversity Hospital Medical Information Network Clinical Trials Registry (UMIN-CTR) UMIN000009115.