Higher Derivatives Research Articles

Deriving interaction vertices in higher derivative theories

We derive cubic interaction vertices for a class of higher-derivative theories involving three arbitrary integer spin fields. This derivation uses the requirement of closure of the Poincaré algebra in four-dimensional flat spacetime. We find two varieties of permitted structures at the cubic level and eliminate one variety, which is proportional to the equations of motion, using suitable field redefinitions. We then consider soft theorems for field theories with these higher-derivative interactions and construct amplitudes in these theories using the inverse-soft approach.

On the gauge invariance of the higher-derivative Yang–Mills–Chern–Simons action

In this work, we study the perturbative generation of the gauge invariant effective action for the non-Abelian gauge field in a (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional spacetime. We present a detailed analysis of the two, three and four-point functions in order to determine the non-Abelian Chern–Simons terms (parity odd) and Yang–Mills terms (parity even). Moreover, these terms are supplemented by the higher-derivative corrections which resulted in the Alekseev–Arbuzov–Baikov effective action (parity even) plus the higher-derivative (HD) corrections to the Chern–Simons terms (parity odd). In order to highlight some features about the perturbative generation of the effective action, we present a discussion based on the dimensional analysis, which allows us to establish the general structure of the permissible terms to guarantee the gauge invariance of the higher-derivative parts.

Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection

The inner transition layer evolution for two-dimensional quasi-linear initial-boundary value problem for the reaction-advection-diffusion equation in an inhomogeneous media with a small parameter within the high order derivatives is considered. Within the framework of the main (zero order) sum of the asymptotic series, the position of the inner transition layer is described by the Hamilton–Jacobi equation. The potential is calculated as an integrated density function of the reaction sources within the limits of the equilibrium levels. The front line of the transition layer evolves in the same way as the constant-eikonal line (the wavefront line in the other words) for the problem of wave propagation in an inhomogeneous medium in short-wave (geometro-optical) asymptotics. The sum of the asymptotic series of zero order and first order is found, the existence gap of a smooth front line, the time of destruction of the contrasting structure are calculated.

Higher derivatives of functions with given critical points and values

Let f:Bn→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: B^n \\rightarrow {{\\mathbb {R}}}$$\\end{document} be a d+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d+1$$\\end{document} times continuously differentiable function on the unit ball Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^n$$\\end{document}, with maxz∈Bn‖f(z)‖=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm{max\\,}_{z\\in B^n} \\Vert f(z) \\Vert =1$$\\end{document}. A well-known fact is that if f vanishes on a set Z⊂Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z\\subset B^n$$\\end{document} with a non-empty interior, then for each k=1,…,d+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=1,\\ldots ,d+1$$\\end{document} the norm of the k-th derivative ||f(k)||\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$||f^{(k)}||$$\\end{document} is at least M=M(n,k)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M=M(n,k)>0$$\\end{document}. A natural question to ask is “what happens for other sets Z?”. This question was partially answered in Goldman and Yomdin (Lower bounds for high derivatives of smooth functions with given zeros. arXiv:2402.01388), Yomdin (Anal Math Phys 11:89, 2021), Yomdin (J Singul 25:443–455, 2022) and Yomdin (Higher derivatives of functions vanishing on a given set. arXiv:2108.02459v1). In the present paper, we ask a similar (and closely related) question: what happens with the high-order derivatives of f, if its gradient vanishes on a given set Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}? And what conclusions for the high-order derivatives of f can be obtained from the analysis of the metric geometry of the “critical values set” f(Σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(\\Sigma )$$\\end{document}? In the present paper, we provide some initial answers to these questions.

Solving the cohomological equation for locally Hamiltonian flows, part I - local obstructions

We study the cohomological equation Xu=f for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution u when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimal components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf. [9,11]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of f around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) to be shown in [15].The main contribution of this article is to define the aforementioned new families of invariant distributions dσ,jk, Cσ,lk and analyze their effect on the regularity of u and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of f near degenerate singularities inspired by tools from [14] and [17]. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs.

High affinity and low PARP-trapping benzimidazole derivatives as a potential warhead for PARP1 degraders

PARPi have been explored and applied in the treatment of various cancers with remarkable efficacy, especially BRCA1/2 mutated ovarian, breast, prostate, and pancreatic cancers. However, PARPi renders inevitable drug resistance and showed high toxicity because of PARP-Trapping with long-term clinic tracking. To overcome the drug resistance and the high toxicity of PARPi, many novel methods have been developed including PROTACs. Being an event-driven technology, PROTACs needs a high affinity, low toxicity warhead with no steric hindrance in binding process. Veliparib shows the lowest PARP-Trapping effect but could hardly to be the warhead of PROTACs because of the strong steric hindrance. Other PARP1 inhibitors showed less steric hindrance but owns high PARP-Trapping effect. Thus, the development of novel warhead with high PARP1 affinity, low PARP1-Trapping, and no steric hindrance would be valuable. In this work, we reserved benzimidazole as the motif to reserve the low PARP1-Trapping effect and substituted the pyrrole by aromatic ring to avoiding the steric hindrance in PARP1 binding cave. Thus, a series of benzimidazole derivates were designed and synthesized, and some biological activities in vitro were evaluated including the inhibition for PARP1 enzyme and the PARP-Trapping effect using MDA-MB-436 cell line. Results showed that the compound 19A10 has higher PARP1 affinity(IC50 = 4.62 nM)) and similar low PARP-Trapping effect compared with Veliparib(IC50 (MDA-MB-436) >100 μM). Docking study showed that the compound 19A10 could avoiding the steric hindrance which was much better than Veliparib. So, the compound 19A10 could potentially be a perfect warhead for PARP1 degraders. Besides, because of the depletion of the PARP1 and the decreasing of the binding capability, we suppose that the PROTACs using 19A10 as the warhead would be no-PARP-Trapping effect. Furthermore, QSAR study showed that to develop novel compounds with high PARP1 binding affinity and low PARP-Trapping, we can choose the skeleton with substituent R1H, R2 = piperiazine, and R3 with large tPSA. And, if we want to develop the compounds with high PARP1 binding affinity and high PARP-Trapping which can possibly improve the lethality against tumor cells, we can choose the skeleton with substituent R1F, R2 = 3-methy-piperiazine, and R3 with large tPSA.

Tunable products of 5-hydroxymethylfurfural oxidation by Zn/Co-ZIF derived catalysts

As a bio-based platform compound, 5-hydroxymethylfurfural (HMF) can be effectively converted into other high value-added derivatives. Here we report two non-noble metal catalysts for the selective oxidation of HMF, NC and Co@NC are easily obtained by pyrolysis of ZIF-8 and ZIF-67, respectively. The NC and Co@NC catalysts oxidized HMF to FFCA (Select. 95 %) and FDCA (Select. 98 %), respectively, using O2 as oxidant. Experimental and theoretical studies show that pyridinic N and graphitic N in NC has high catalytic activity for the oxidation of carbonyl groups in HMF to FFCA, while the synergistic effect of Co-N in Co@NC can completely oxidize HMF to FDCA. DFT calculations show that, compared with NC, the Co@NC containing Co-N exhibits a lower energy barrier during the aerobic oxidation of HMF to FDCA. And Co-N leads to the conversion of electrons from N to Co and promotes the activation of O2 to form O2-, and then oxidize HMF to FDCA. Interestingly, our reaction system is universal, and could achieve almost 100 % conversion of various furfural substrates to corresponding carboxylic acid. This study may provide a new/highly efficient catalysts for biomass conversion.

High order asymptotic inequalities for some dissipative systems

We obtain some important fundamental inequalities concerning the long time behavior of high order derivatives for solutions of some dissipative systems in terms of their algebraic L2 decay. Some of these inequalities have not been observed in the literature even for the fundamental Navier–Stokes equations. To illustrate this new approach, we derive bounds for the asymmetric incompressible fluids equations, where improved inequalities were established for the micro-rotational field w. Finally, we show how this technique can be applied to other dissipative systems.

Self-similar solutions to a multidimensional singular heat equation with power nonlinearity

The paper deals with the construction of exact solutions to a singular heat equation with power nonlinearity in the case of numerous independent variables with spatial (e.g. axial or central) symmetry. A new class of self-similar solutions is proposed, which reduce to solving the Cauchy problem for a second-order nonlinear ordinary differential equation having singularities at the higher derivative with respect to the required function and/or the independent variable. The ordinary differential equation is studied in two ways: analytically and numerically. The analytical study uses a truncated Taylor series with recurrently computed coefficients, for which explicit formulas are obtained. The numerical solution to the problem uses an iteration algorithm based on the collocation method and radial basis functions. The numerical analysis shows the convergence of the proposed numerical algorithm and its sufficient accuracy enabling one to use the found self-similar solutions to verify approximate solutions to the original heat equation. Besides, the numerical analysis has allowed the radius of convergence of the constructed Taylor series to be evaluated. The form of the constructed self-similar solutions, namely their unboundedness near the symmetry center (axis), enables us to study the behavior and exactness of the numerical solutions to the nonlinear singular parabolic-type equation that have been obtained by the stepwise solution method proposed by us earlier and possess the same property.

Development and Validation of HPTLC and First Derivative Spectrophotometric Methods for Determination of Levetiracetam in Bulk and Tablet Dosage Form

A simple, sensitive high performance thin layer chromatographic (HPTLC) method and first order derivative spectrophotometric methods were developed for estimation of US FDA drug levetiracetam in bulk and its tablet dosage form. Chromatographic estimation was carried out on silica gel 60 F245 TLC plate as stationary phase and toluene: methanol: Ethyl acetate (4:4:2v/v/v) as mobile phase. Densitometric analysis was carried out at wavelength 218nm. The Rf values for Levetiracetam was found to be 0.47±0.03. Linearity was observed over range of 400-3200ng/spot for Levetiracetam (R2 0.998). The %recovery of Levetiracetam was found to be 99.25-100.7%. The LOD and LOQ were found to be 0.0917ng/ml and 0.278ng/ml respectively indicating sensitivity of the method. The first order derivative spectra were obtained at n=4, Δλ=3nm and determinations was made at 215 nm using methanol as solvent. Linearity was observed in the range of 80μg/ml - 280μg/ml with good regression values. Percent mean recovery of Levetiracetam was obtained as 101.5%. Both proposed methods were simple, fast, accurate, precise and reproducible and hence can be applied for routine quality control analysis of levetiracetam in bulk and its tablet dosage form.

Advances in succinic acid production: the enhancement of CO2 fixation for the carbon sequestration benefits.

Succinic acid (SA), one of the 12 top platform chemicals produced from biomass, is a precursor of various high value-added derivatives. Specially, 1mol CO2 is assimilated in 1mol SA biosynthetic route under anaerobic conditions, which helps to achieve carbon reduction goals. In this review, methods for enhanced CO2 fixation in SA production and utilization of waste biomass for SA production are reviewed. Bioelectrochemical and bioreactor coupling systems constructed with off-gas reutilization to capture CO2 more efficiently were highlighted. In addition, the techno-economic analysis and carbon sequestration benefits for the synthesis of bio-based SA from CO2 and waste biomass are analyzed. Finally, a droplet microfluidics-based high-throughput screening technique applied to the future bioproduction of SA is proposed as a promising approach.

Cluster perturbation theory IX: Perturbation series for the coupled cluster singles and doubles ground state energy.

In this paper, we develop and analyze a number of perturbation series that target the coupled cluster singles and doubles (CCSD) ground state energy. We show how classical Møller-Plesset perturbation theory series can be restructured to target the CCSD energy based on a reference CCS calculation and how the corresponding cluster perturbation series differs from the classical Møller-Plesset perturbation series. Subsequently, we reformulate these series using the coupled cluster Lagrangian framework to obtain series, where fourth and fifth order energies are determined only using parameters through second order. To test the methods, we perform a series of test calculations on molecular photoswitches of both total energies and reaction energies. We find that the fifth order reaction energies are of CCSD quality and that they are of comparable accuracy to state-of-the-art approximations to the CCSD energy based on local pair natural orbitals. The advantage of the present approach over local correlation methods is the absence of user defined threshold parameters for neglecting or approximating contributions to the correlation energy. Fixed threshold parameters lead to discontinuous energy surfaces, although this effect is often small enough to be ignored, but the present approach has a differentiable energy that will facilitate derivation and implementation of gradients and higher derivatives. A further advantage is that the calculation of the perturbation correction is non-iterative and can, therefore, be calculated in parallel, leading to a short time-to-solution.

The four-loop β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-function from vacuum supergraphs and the NSVZ relation for N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {N}}}}=1$$\\end{document} SQED regularized by higher derivatives

In N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {N}}}}=1$$\\end{document} SQED with Nf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_f$$\\end{document} flavors regularized by higher derivatives we obtain the four-loop beta function using a method based on calculating vacuum supergraphs. For this purpose we use a special C++ program which obtain contributions to the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-function from supergraphs without external legs in the form of integrals of double total derivatives. After that the result was compared with the three-loop anomalous dimension calculated earlier. We explicitly check the NSVZ relation in this order.

Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a method is generated using m steps (m a natural number) of convergence order 2 m. This method avoids a large increase in the number of operator evaluations. However, there are several problems with the conditions used to show the convergence: the existence of high order derivatives is assumed, which are not in the method; there are no a priori results for the error distances or information on the uniqueness of the solutions. Therefore, the earlier studies cannot guarantee the convergence of the method to solve nondifferentiable equations. However, the method may converge to the solution. Thus, the convergence conditions can be weakened. These problems arise since the convergence order is determined using the Taylor series which requires the existence of high-order derivatives which are not present in the method, and they may not even exist. These concerns are our motivation for authoring this article. Moreover, the novelty of this article is that all the aforementioned problems are addressed positively, and by using conditions only related to the divided differences in the method. Furthermore, a more challenging and important semi-local analysis of convergence is presented utilizing majorizing sequences in combination with the concept of the generalized continuity of the divided difference involved. The convergence is also extended from the Euclidean to the Banach space. We have chosen to demonstrate our technique in the present method. But it can be used in other studies using the Taylor series to show the convergence of the method. The applicability of other single- or multi-step methods using the inverses of linear operators with or without derivatives can also be extended with the same methodology along the same lines. Several examples are provided to test the theoretical results and validate the performance of the method.

Synthesis and comprehensive characterization with anticancer activity assessment of salicylaldehyde and 2-acetylphenol based chitosan thiosemicarbazones and their copper(II) complexes

Functionalization of chitosan as salicylaldehyde and 2-acetylphenol-based chitosan thiosemicarbazone involving amino group at C2 position of pyranose ring was confirmed by comprehensive characterization by spectroscopic and analytical ways. Both the low molecular weight (Mw) chitosan oligosaccharide (CS) (Mw < 3000 Da, degree of deacetylation (DDA) = 87 %) and the crab shell chitosan (CCS) (Mw = 350 kDa, DDA = 67 %) functionalized thiosemicarbazones behaved as ONS tridentate ligand to form the square planar copper(II) complexes with a chloride ion as an additional ligand to satisfy the co-ordination sphere. All the derivatives were found thermally stable till the commencement of chitosan backbone disruption at 200 °C. The MTT assay profiles showed a conspicuous enhancement in the in vitro antitumorigenic and anticancer activity of native chitosan upon its functionalization as phenolic carbaldehyde chitosan thiosemicarbazones. Further, the assays more often revealed a further lowering of cell viability with chitosan oligosaccharide derivatives in comparison to high molecular weight chitosan derivatives, and a lower viability of the MCF-7 cancer cells than the tumorigenic MDCK cells. Both the low Mw chitosan with high DDA and high Mw chitosan with low DDA, the chitosan-functionalized salicylaldehyde based thiosemicarbazones, and their copper(II) chitosan thiosemicarbazone complexes were found to have minimal toxicity with IC50 >600 μgmL−1 against the mouse embryonic normal fibroblast NIH3T3 cells. These results were indicatives of more protonation and enhanced anticancer activity of chitosan oligosaccharide derivatives, and preferential cytotoxicity of chitosan biomaterials against the tumorigenic and cancer cells with more negatively charged cell surfaces.

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

A Taylor series method which is taught at undergraduate levels and which has been hitherto used for the solution of initial value problem is successfully used in this work for the solutions of eighth-order two point boundary-value problems. The method is based on successive differentiation of the governing equation to obtain high order derivatives and then evaluated at the boundary point x=a. The solution is expressed inform a Taylor series with the unknown coefficients at a point x=a. By applying boundary conditions at x=b in the Taylor series solution, the system of unknown coefficient is obtained. After solving the system, then unknown coefficient are determined. The procedure is applied on both linear and nonlinear boundary-value problems. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is simple, effective and also is in good agreement with the previous result and exact solution as showing in the tables and figures.

Black hole thermodynamics in natural variables: quadrophenia

It was recently observed in [1] for thermal Kerr-Newman black holes in 4d flat space that one can rewrite the conventional thermodynamics on the inner and outer horizons in terms of left- and right-moving variables with a remarkable simplification of the corresponding expressions. With the goal of illustrating the wide applicability of these newly proposed natural variables, we extend the original observation in four independent directions that can be further superimposed on each other. These four generalizations can be thought of as different deformations of the original 4d Einstein-Maxwell theory, all within the framework of supergravity: higher derivative (HD) corrections in minimal 4d supergravity; additional scalar and vector couplings in matter-coupled 4d supergravity; higher dimensions, in particular 5d minimal supergravity; and a cosmological constant in 4d minimal gauged supergravity with Anti-de Sitter (AdS) vacuum. Each of these generalizations offers a different lesson about the novel thermodynamics, and we pay special attention to the respective BPS limits that can be understood from fixed point formulae, demonstrating the power of the natural variables to capture the full phase space.

Higher Derivations Satisfying Certain Identities in Rings

Let n and m be fixed positive integers. In this paper, we establish some structural properties of prime rings equipped with higher derivations. Motivated by the works of Herstein and Bell-Daif, we characterize rings with higher derivations D=dii∈N satisfying (i) dnx,dmy∈ZR for all x,y∈R and (ii) dnx,y∈ZR for all x,y∈R.

Uniqueness of meromorphic functions and their powers in set sharing

Abstract The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing.

Advancing Convergence Rates in Nonlinear Equation Solvers

It is crucial to achieve high-order convergence for nonlinear equations in numerical analysis and scientific computing. Newton-Raphson and other conventional techniques frequently converge slowly. High-order algorithms offer quicker convergence and less expensive processing.This abstract outlines important ideas and innovations, such as improved Newtonian methods for better convergence. To hasten convergence, these techniques make use of enhanced Jacobian matrix approximations and higher-order derivatives. Additionally, it covers useful topics like dealing with stability problems and computing higher-order derivatives, which could lead to efficiency and accuracy gains in scientific and engineering applications. Keywords: Non- linear Equations, High Order Convergence, Numerical Methods, Jacobian Matrix, High Order Derivatives