Simple Roots Research Articles

Fourth-order iterative method without calculating the higher derivatives for nonlinear equation

In this paper, we present firstly several one-step iterative methods including classical Newton’s method and Halley’s method for root-finding problem based on Thiele’s continued fraction. Secondly, by using approximants of the second derivative and the third derivative, we obtain an improved iterative method, which is not only two-step iterative method but also avoids calculating the higher derivatives of the function. Analysis of its convergence shows that the order of convergence of the modified iterative method is four for a simple root of the equation. Finally, to illustrate the efficiency and performance of the proposed method, we give some numerical experiments and comparison.

Extending ROOT through Modules

The ROOT software framework is foundational for the HEP ecosystem, providing multiple capabilities such as I/O, a C++ interpreter, GUI, and math libraries. It uses object-oriented concepts and build-time components to layer between them. We believe that a new layering formalism will benefit the ROOT user community. We present the modularization strategy for ROOT which aims to build upon the existing source components, making available the dependencies and other metadata outside of the build system, and allow post-install additions on top of existing installation as well as in the ROOT runtime environment. Components can be grouped into packages and made available from repositories in order to provide a post-install step of missing packages. This feature implements a mechanism for the more comprehensive software ecosystem and makes it available even from a minimal ROOT installation. As part of this work, we have reduced inter-component dependencies in order to improve maintainability. The modularization effort draws inspiration from similar efforts in the Java, Python, and Swift ecosystems. Keeping aligned with modern C++, this strategy relies on forthcoming features such as C++ modules. We hope formalizing the component layer provides simpler ROOT installs, improves extensibility, and decreases the complexity of embedding ROOT in other ecosystems.

Nonlinear boundary value problems for nondegenerate differential-algebraic systems

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

On good deformations of <inline-formula><tex-math id="M1">\begin{document}$ A_m $\end{document}</tex-math></inline-formula>-singularities

The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an $ n $-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.

Root extension as a linguistic problem

This paper provides an overview of different approaches to the linguistic problem that the origins of PIE root extension present. Accepted definitions of root extension, like the one given by B. W. Fortson IV, “extra phonetic material (one or two sounds) added on to [roots], generally without any discernable change to the meaning of the root”1, shirk the question of why the simple root gets extended, and whether this process is governed by any definable principles.

Approximation of peak-to-average power ratio of generalized frequency division multiplexing

Generalized frequency division multiplexing (GFDM) is a generalized form of the multi-carrier system with a flexibility to vary the number of subcarriers, the number of time slots and the associated pulse shaping filter for achieving desired performance. Peak to average power ratio (PAPR) is a crucial parameter to be considered while designing systems for low power consumption devices. It has been found in the literature that the PAPR of a GFDM system is higher as compared to the versatile orthogonal frequency division multiplexing (OFDM) systems for the same number of subcarriers.This paper presents a simple intuitive methodology based on order-statistics to derive an expression for the PAPR of a GFDM system. The expression of the PAPR for a simple square root companded GFDM system is also derived based on the same principle. The derived PAPR expressions for both these cases are validated through extensive Monte-Carlo simulations. The derived expressions can be useful to the research community to evaluate the PAPR of GFDM system for a given number of subcarriers K, the number of time slots M, and the considered pulse shaping filter. The PAPR results also conform to the simulation results presented by other researchers in the literature.

Contrast Structures with a Multizone Internal Layer

A boundary value problem for a singularly perturbed second-order differential equation is considered in two cases, in each of which one of the roots of the degenerate equation is double. In the first case, a narrow internal layer is proven to be formed where there is a rapid transition of the solution from the double root of the degenerate equation to a simple root; in the second case, the solution is proven to have a “spike” in the internal layer. Such solutions are called step-like contrast structures and spike-like contrast structures, respectively. In each case, the asymptotic expansion of the contrast structure is constructed. It significantly differs from the known expansion when all roots of the degenerate equation are simple; in particular, the internal layer turns out to be multizone.

Existence, Asymptotics, Stability and Region of Attraction of a Periodic Boundary Layer Solution in Case of a Double Root of the Degenerate Equation

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.

$$L^p$$ L p -solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory $$L^p$$ -solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in $$L^p([t_0,\infty [)$$ . These results are applied to study a fourth order linear differential equation of Poincare type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

Leveraging the genetic basis of Rett syndrome to ascertain pathophysiology.

Mutations in the methyl-CpG binding protein 2 (MECP2) gene cause Rett syndrome (RTT), a progressive X-linked neurological disorder characterized by loss of developmental milestones, intellectual disability and breathing abnormality. Despite being a monogenic disorder, the pathogenic mechanisms by which mutations in MeCP2 impair neuronal function and underlie the RTT symptoms have been challenging to elucidate. The seemingly simple genetic root and the availability of genetic data from RTT patients have led to the generation and characterization of a series of mouse models recapitulating RTT-associated genetic mutations. This review focuses on the studies of RTT mouse models and describe newly obtained pathogenic insights from these studies. We also highlight the potential of studying pathophysiology using genetics-based modeling approaches in rodents and suggest a future direction to tackle the pathophysiology of intellectual disability with known or complex genetic causes.

Development and application of long-term root zone salt balance model for predicting soil salinity in arid shallow water table area

A simple root zone salt balance model was developed for long-term soil salt prediction. A groundwater balance module was integrated to obtain the water flux at the bottom of the root zone, since groundwater level is easily and reliably measured data. An assumption that the net percolation at the bottom of root zone equals to the net recharge from root zone to unconfined groundwater system was used, which makes capillary rise and infiltration water leaching of root zone obtainable. The model accuracy and limitation were evaluated by comparing the simulation results with those from HYDRUS-1D under various soil texture and bottom boundary conditions. The mean relative error (MRE) of soil salt content ranged from −6.25% to 9.95%, and the root mean square error (RMSE) from 0.02 kg/100 kg to 0.05 kg/100 kg, which suggest the model accuracy. The determination coefficient (R2) of the cumulative percolation at the bottom of root zone and recharge to groundwater at the end of each calculation period was 0.99, which validates the model assumption. The model was then applied to predict soil salinity in the Longsheng irrigation district of Hetao Irrigation District, China. The soil salinity and water table depth data from 1999 to 2016 were used to calibrate and validate the model parameters. The MRE values of water table depth and average soil salt content were 10.08% and 10.84%, and RMSE values 0.37 m and 0.04 kg/100 kg in the validation period. Subsequently, the validated model was used to simulate the soil salinity in future 100 years under various water saving conditions. Water table depth and autumn irrigation for salt leaching were the two major factors to control soil salinity. The required autumn irrigation to keep root zone in mild salinity level (0.3 kg/100 kg) can decrease by 25.1 mm with 0.1 m water table depth increase, while it should increase by 64 mm with 1.0 kg/m3 groundwater salinity increase in the crop growing season. Moderate groundwater exploitation to increase water table depth is recommended to save autumn irrigation water and control soil salinity.

A family of optimal fourth‐order methods for multiple roots of nonlinear equations

Newton‐Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton‐type method at the first step. In this paper, we present a new two‐step optimal fourth‐order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.

A simple root selection method for univariate finite normal mixture models

–It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.

Geometric Realization of Dynkin Quiver Type Quantum Affine Schur–Weyl Duality

For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $\beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.

Deciphering conical spouted bed hydrodynamics using high intensity microphone

Pressure fluctuations emanating from three dimensional (3D) conical spouted bed using dense zirconia particles (6200 kg/m3) were investigated via high intensity microphone. The effects of static bed height and gas velocity on the spouted bed hydrodynamics were studied. Studies reveal that unprocessed pressure signals are adequate to decipher the various hydrodynamic regimes of the spouted bed. Simple root mean square (RMS) of pressure fluctuation signals can be used to identify the spouted bed hydrodynamic regimes. Pressure fluctuation signals were further analyzed in the frequency domain and the analysis shows drastic lowering of peak intensity of power spectral density at minimum spouting velocity. The differentiating factor of the present study being that the pressure signals exhibit distinctly identifiable general trend associated with various spouted bed regimes and such trend is further confirmed by the frequency analysis. This very feature could be useful tool in fabricating TRISO particles in single batch process. Experimental data on minimum spouting velocity (Ums) was compared with relevant correlations and the results were discussed.

Stability analysis of a family of optimal fourth-order methods for multiple roots

Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m > 1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.

EFFECTIVE DIVISORS ON BOTT–SAMELSON VARIETIES

We compute the cone of effective divisors on a Bott–Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain B-equivariant ℙ1 bundles. As an application, we compute the cone of effective codimension-two cycles on Bott–Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott–Samelson variety to contain a dense B-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott–Samelson variety is log Fano.

The strategy and results of percutaneous endoscopic surgery for cervical disc herniation

Objective To evaluate the strategy and clinical effects of percutaneous endoscopic surgery for cervical disc herniation. Methods Fifty-one patients with cervical disc herniation were treated with percutaneous endoscopic surgery from June 2015 to March 2017, including 32 men and 19 women, with an average age of 52.2 years (range, 28-66 years). Radicular symptoms were present in all patients, while 23 patients had mild myelopathy (Nurick Grade: 0-3) and 3 patients of multilevel stenosis had severe myelopathy (Nurick Grade: 4-5). According to axial image of preoperative magnetic resonance imaging (MRI), 31 patients had lateral herniation that was located lateral to the edge of spinal cord, 20 patients had central herniation that was located within the lateral edge of spinal cord. Among them, 48 patients had soft herniation and 3 patients had ossified lateral herniation combined with foraminal stenosis. All surgery was carried out under general anesthesia, while posterior and anterior percutaneous endoscopic surgeries were performed for lateral herniation and central herniation respectively. Posterior endoscopic surgery was performed with keyhole fenestration at V point (the junction of lateral edge of lamina space and inner edge of facet). Lateral edge of thecal sac and nerve root were exposed and decompressed, soft herniation was explored and removed. Anterior endoscopic surgery was performed through puncture and 4mm tube between the visceral sheath and vascular sheath. The tube was inserted through disc to the base of herniation under fluoroscopy. The herniation was removed until the dura sac was exposed and relaxed. One stage open-door laminoplasty was performed for 3 patients with severe multiple segmental stenosis and huge central herniation. The operative time and blood loss were recorded, and patients were followed-up (range, 6-18 months, average 12.1 months) to evaluate the clinical efficacy. Results The mean operative time of posterior endoscopic surgery was 90 min (range, 45-150 min). The nerve root was not well exposed, and the fenestration was too lateral in 1 patient, with partial relieve of symptoms; and simple nerve root decompression was performed for 3 patients of ossified herniation combined with foraminal stenosis. Herniated or sequestered nucleus pulposus was removed for 27 patients, one of them had transient paralysis ipsilateral limb and 2 of them had linkage of cerebrospinal fluid. The Visual Analogue Score (VAS) score improved form preoperative 8.9±1.6 to 0.5±0.4, and the Oswestry Disability Index (ODI) score improved form 32.8±4.2 to 2.3±1.9 at final follow-up. For anterior percutaneous endoscopic surgery, the mean operative time was 80 min (range, 45–120 min). Herniated or free nucleus was successfully removed for all patients. The thecal sac was lacerated due to unclear exposure in 1 case. The VAS score improved form preoperative 6.9±2.3 to 0.9±0.8, and the ODI score improved form 40.1±8.6 to 5.6±3.0 at final follow-up, with improvement of myelopathy at least one Nurick Grade. During follow-up, the alignment of cervical spine was well preserved without kyphosis for two groups, while the height of intervertebral space decreased with 0.4±0.3 mm and 0.9±0.6 mm in posterior and anterior surgery respectively. Conclusion Percutaneous endoscopic surgery provides minimally invasive alternatives for some cervical disc herniation with predominant radicular pain. Posterior endoscopic surgery is suitable for lateral herniation, and anterior endoscopic discectomy is suitable for some central soft herniation without obvious collapse and instability. However, the long-term results of disc space collapsed after anterior approach remains unclear. Key words: Cervical vertebrae; Intervertebral disc displacement; Endoscopy; Surgical procedures, minimally invasive

Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Solving nonlinear equations is one of the most important problems in numerical analysis, and has a wide range of application in various aspects, as well as many branches of science, engineering, physics, computing, astronomy, finance, .. . Generally, it is difficult to find the exact root of the nonlinear equations, and so iterative methods become the efficient way to obtain approximate solutions. Recently Kou et. al. presented a class of new variants of Ostrowski's method with order of convergence equals seven (OSM7) for solving simple roots of nonlinear equations proposed in [7]. Ostrowski's method (OSM7) has efficiency index equals to ∜7≈1.626, it has four functions per iteration but its order of convergence is seven that means it's not optimal method. In this paper, we have proposed new two improvements of Ostrowski's method (OSM7) to make it an optimal eight family and to increase its efficiency index. The first improvement has obtained by multiplied the third step of (OSM7) by product of two weight functions with some special conditions and the second improvement has obtained by multiplied the third step of (OSM7) by summation of two others weigh functions with special conditions, too. Using weight functions in every improvement helped us to improve the order of convergence of (OSM7) from seven to eight without changing the number of function evaluations to be an optimal family. New two optimal families have efficiency index equals to ∜8≈1.682. Some numerical examples are provided to show the good performance of the new methods.

On the Cohomology of Certain Rank 2 Vector Bundles on G/B

Let G be a semisimple, simply connected, linear algebraic group over an algebraically closed field k. Donkin (In J. Algebra, 307, 570–613 2007), Donkin gave a recursive description for the characters of the cohomology of line bundles on the three dimensional flag variety in prime characteristic. The recursion involves not only line bundles but also certain natural rank 2 bundles associated to two dimensional B −modules Nα(λ), where λ in an integral weight and α is a simple root. In this paper we compute the cohomology of these rank 2 bundles and simplify the recursion in Donkin (In J. Algebra, 307, 570–613 2007). We also compute the socle of Nα(λ) and give a rank 2 version of Kempf’s vanishing theorem.