Polynomial Equation Research Articles

PHB Production by Bacillus megaterium LSRB 0103 Using Cornstarch and Urea.

Polyhydroxybutyrates (PHBs) are biopolymers that are good green alternative for synthetic carbon-based polymers, and are also one of the most researched members of the Polyhydroxyalkanoates (PHA) family. In this study, a gram-positive bacterial strain Bacillus megaterium LSRB 0103 was isolated from Pallikaranai Marshland, Chennai, India. Primary screening using Sudan Black dye revealed the presence of intracellular PHB granules. Minimal Davis Media (MDM) which was used or PHB production gave a yield of 0.7107g/L. Subsequently, using response surface methodology (RSM), a central composite design (CCD) model was designed for media optimization having cornstarch, urea, and pH as independent variables. The findings of the CCD model were fitted into a second-order polynomial equation. The RSM model predicted the maximum PHB yield of 0.93g/L, at these independent variable levels, cornstarch, 5g/L; urea, 2.1g/L; and pH 7.0; while the experimental PHB yield was 0.94g/L, with a percentage error of 1.1%. This study is the first-time report of production of PHB by Bacillus megaterium using cornstarch and urea as substrate.

ENHANCED SURFACE QUALITY AND STRENGTH OF FDMed SPECIMENS USING BBD AND BIO-INSPIRED ALGORITHMS

This research investigated and optimized the parameters of the FDM process by employing bio-inspired algorithms for determining the optimal parameter settings in terms of surface quality and mechanical performance. Four important process parameters including layer thickness (0.11–0.33[Formula: see text]mm), part orientation (0–90∘), raster width (0.2–0.56[Formula: see text]mm), and the raster angle (0–60∘) at three variation levels were selected for fabricating the specimens (ABS material P430) using the statistical Box–Behnken design. ANOVA analysis and multiple regression analysis were used to fit the experimental data to a second-order polynomial equation. Through, the RSM analysis, the layer thickness is the key important factor that accounts for all of the responses. The fracture behavior of specimens was examined using a scanning electron microscope (SEM). From the SEM analysis, a substantial amount of plastic deformation on the fracture surface indicative of craze cracking is visible from a 0∘ orientation, indicating a totally ductile fracture mechanism. Then, three swarm intelligence algorithms such as Tasmanian Devil Optimization (TDO), Remora Optimization Algorithm (ROA), Tuna Swarm Optimization (TSO) were implemented to optimize the input parameters that would lead to minimum surface roughness and maximum tensile strength. Experimental data and predicted values varied between 1.64% and 1.84%, as shown by verification experiments.

Nonlinear Regression Approach as a Correction Factor of Measurements of Low-Cost Electrochemical Air Quality Sensors

Air quality directly affects the health of humans. The health implications of poor air quality are recognized by professionals and the public alike and these concerns have driven both the proliferation of formal sensor networks, but also low-cost sensors which can be used in the home. The advancement of technology in recent years has also led to the rapid development of low-cost sensors. Given that citizens are concerned about the air quality of the environment in which they live, they are turning to the supply of low-cost sensors, as they are affordable. The question of the reliability of measurements from low-cost sensors remains an area of research. In this research work, the optimization of ozone (O3) and nitrogen dioxide (NO2) measurements of low-cost electrochemical air quality sensors is investigated by applying nonlinear regression, using a second-order polynomial equation as a correction factor. The proposed correction method is implementable in IoT devices, as it does not require high computational resources. The results show that the measurements are susceptible to correction, with the effect that the corrected values are close to the actual values obtained by the reference instruments of the Department of Environmental Pollution Control Project of Athens (PERPA), a service of the Greek Ministry of the Environment and Energy.

Symmetry and symmetric transformations in mathematical imaging

The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetriesreflection, rotational, and translationalexemplified through concrete mathematical expressions. Evariste Galoiss Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrots fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetrys practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetrys foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.

Study of the intermolecular interaction in hydrophobic deep eutectic solvent of (2-n-octyl-4-isothiazolin-3-one + 1-decanol) from the density, viscosity, thermodynamic, spectroscopic and quantum chemical calculations

Hydrophobic deep eutectic solvents (HDESs) are considered as green solvents and have been developed rapidly in recent years, which are widely used in diverse applications. In this study, we introduce a binary system composed of 2-n-octyl-4-isothiazolin-3-one (OIT) as a novel hydrogen bond acceptor (HBA) and 1-decanol as a hydrogen bond donor (HBD) under atmospheric pressure. The fundamental physicochemical properties, including density and dynamic viscosity, of the binary systems with various mole ratios were measured at temperatures of 293.15 K, 303.15 K, 313.15 K, 323.15 K, and 333.15 K. The excess molar volume (VmE), viscosity deviation (Δη) and excess Gibbs free energy (ΔG∗E) for activation of viscous flow were calculated and fitted according to the Redlich-Kister type polynomial equation. Furthermore, the spectral analysis (FTIR, UV–vis and 1H NMR) confirmed the presence of intermolecular hydrogen bonding in the binary systems. In addition, quantum chemical calculations revealed that there may be multiple modes of hydrogen bond formation between OIT molecule and 1-decanol molecule. However, selecting the most energetically favorable point, the most prominent hydrogen bond was found to be between the amide O atom of OIT and the hydroxyl H atom of 1-decanol.

Hydrogen storage properties of magnesium based alloys Mg67Ni(32-x)Nb1Alx (x = 1, 3, and 5) by using response surface methodology

Nano metals and hydrogen storage have attracted significant attention in recent years due to their numerous unique properties and wide range of applications. This study explores the synthesis of nanostructured Al and Nb-substituted Mg2Ni intermetallic compounds through high-energy ball milling and investigates their electrochemical performance for energy-related applications. The research emphasizes the critical influence of crystallinity and crystallite size on electrode material performance. Employing Response Surface Methodology (RSM), the study identifies key factors affecting discharge capacity. Notably, current density emerges as the most significant factor, contributing 73% to discharge capacity, as confirmed by perturbation plots. Interaction effects among the factors were found to be relatively insignificant concerning the chemical kinetics of the electrode material. Furthermore, a second-order polynomial equation was developed through RSM to quantitatively relate discharge capacity to composition, milling time, and current density, with a high R2 value of 98.3%. To optimize discharge capacity, a fuzzy parameter setting was generated based on the mathematical model, resulting in a predicted discharge capacity of 398.209 mAh g−1, closely aligned with actual experimental results (394.203 mAh g−1). This work showcases the significance of advanced statistical techniques in elucidating the intricate relationships governing electrochemical performance, particularly in the context of nanocrystalline materials.

Excess Thermodynamic Parameters of Binary Liquid Mixtures of Dimethyl Malonate with Isomeric Xylenes at Different Temperatures

Over the complete composition range of mole fractions, densities and ultrasonic velocities of binary liquid mixtures of dimethyl malonate with o-xylene, p-xylene and m-xylene were investigated at 303.15-318.15 K. These parameters were chosen to investigate the  impact of substituents and specific locations on several thermodynamic characteristics of binary mixtures. The deviation in adiabatic  compressibility (∆βad), excess molar volume (VE) and excess intermolecular free length (LfE) were estimated using the experimental measurements. These extraneous parameters are fitted to a polynomial equation of the Redlich-Kister type. Along with the obtained parameter values, the standard deviation (σ) was also determined.

Estimation of Forest Canopy Fuel Moisture Content in Dali Prefecture by Combining Vegetation Indices and Canopy Radiative Transfer Models from MODIS Data

Forest canopy fuel moisture content (FMC) is a critical factor in assessing the vulnerability of a specific area to forest fires. The conventional FMC estimation method, which relies on look-up tables and loss functions, cannot to elucidate the relationship between FMC and simulated data from look-up tables. This study proposes a novel approach for estimating FMC by combining enhanced vegetation index (EVI) and normalized difference moisture index (NDMI). The method employs the PROSAIL + PROGeoSAIL two-layer coupled radiation transfer model to simulate the vegetation index, the water index, and the FMC value, targeting the prevalent double-layer structure in the study area’s vegetation distribution. Additionally, a look-up table is constructed through numerical analysis to investigate the relationships among vegetation indices, water indices, and FMC. The results reveal that the polynomial equations incorporating vegetation and water indices as independent variables exhibit a strong correlation with FMC. Utilizing the EVI–NDMI joint FMC estimation method enables the direct estimation of FMC. The collected samples from Dali were compared with the estimated values, revealing that the proposed method exhibits superior accuracy (R2 = 0.79) in comparison with conventional FMC estimation methods. In addition, we applied this method to estimate the FMC in the Chongqing region one week before the 2022 forest fire event, revealing a significant decreasing trend in regional FMC leading up to the fire outbreak, highlighting its effectiveness in facilitating pre-disaster warnings.

Decomposition and factorisation of transients in functional graphs

Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be quite large, it is interesting to decompose and factorise them into several subgraphs acting together. Polynomial equations over functional graphs provide a formal way to represent this decomposition and factorisation mechanism, and solving them validates or invalidates hypotheses on their decomposability. The current solution method breaks down a single equation into a series of basic equations of the form A×X=B (with A, X, and B being FGs) to identify the possible solutions. However, it is able to consider just FGs made of cycles only. This work proposes an algorithm for solving these basic equations for general connected FGs. By exploiting a connection with the cancellation problem, we prove that the upper bound to the number of solutions is closely related to the size of the cycle in the coefficient A of the equation. The cancellation problem is also involved in the main algorithms provided by the paper. We introduce a polynomial-time semi-decision algorithm able to provide constraints that a potential solution will have to satisfy if it exists. Then, exploiting the ideas introduced in the first algorithm, we introduce a second exponential-time algorithm capable of finding all solutions by integrating several ‘hacks’ that try to keep the exponential as tight as possible.

GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations

Traditional machine learning is generally treated as a black-box optimization problem and does not typically produce interpretable functions that connect inputs and outputs. However, the ability to discover such interpretable functions is desirable. In this work, we propose GINN-LP, an interpretable neural network to discover the form and coefficients of the underlying equation of a dataset, when the equation is assumed to take the form of a multivariate Laurent Polynomial. This is facilitated by a new type of interpretable neural network block, named the “power-term approximator block”, consisting of logarithmic and exponential activation functions. GINN-LP is end-to-end differentiable, making it possible to use backpropagation for training. We propose a neural network growth strategy that will enable finding the suitable number of terms in the Laurent polynomial that represents the data, along with sparsity regularization to promote the discovery of concise equations. To the best of our knowledge, this is the first model that can discover arbitrary multivariate Laurent polynomial terms without any prior information on the order. Our approach is first evaluated on a subset of data used in SRBench, a benchmark for symbolic regression. We first show that GINN-LP outperforms the state-of-the-art symbolic regression methods on datasets generated using 48 real-world equations in the form of multivariate Laurent polynomials. Next, we propose an ensemble method that combines our method with a high-performing symbolic regression method, enabling us to discover non-Laurent polynomial equations. We achieve state-of-the-art results in equation discovery, showing an absolute improvement of 7.1% over the best contender, by applying this ensemble method to 113 datasets within SRBench with known ground-truth equations.

Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix

AbstractMany computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.

Transforming waste perlite into super lightweight ceramsite: Ratios optimization via uniform design, and investigating calcium fluoride and silicon carbide effects on foaming

The utilization of waste perlite powder generated from mining operations in the construction industry represents a significant step towards enhancing environmental sustainability. This study applies a uniform design of experiments to delve into the apparent density characteristics of super lightweight ceramsite concerning the proportions of solid waste materials, with a particular focus on waste perlite powder. To gain insights into the apparent density, first- and second-order polynomial equations were developed. These equations are founded on the utilization of five core raw materials, including waste perlite powder, fly ash, bentonite, calcium fluoride, and silicon carbide, and were meticulously analyzed using stepwise regression techniques. This approach led to the successful optimization of test ratios, achieving the envisioned outcomes. Additionally, the research explores the influence of calcium fluoride and silicon carbide on the foaming properties of materials under varying maximum sintering temperature conditions. Ultimately, the optimized conditions yielded super lightweight ceramsite with impressive properties, boasting a bulk density of 281 kg/m³, water absorption efficiency of 1.2%, and a compressive strength of 2.72 MPa. This research showcases the potential for utilizing waste perlite powder in the construction industry, not only as a sustainable alternative but also as a means to reduce environmental impact. The development of super lightweight ceramsite underlines the promise of eco-friendly construction materials, driving forward the agenda of environmentally responsible building practices.

Adsorption isotherm of building materials at different temperatures: A review and measurement

The temperature dependency of the adsorption isotherm renders its significance as one of the primary input parameters in models dealing with moisture transport. The present study involved the determination of the adsorption isotherm of sixteen building materials from various categories, including cladding, building paper, wood, sheathing boards, and insulations. The measurements were conducted under nine climatic conditions, a combination of three temperatures (3 °C (37.4 °F), 21 °C (69.8 °F), 45 °C (113 °F)) and three relative humidity levels (50%, 70%, 90%). The study indicated that relative humidity increased the building materials’ equilibrium moisture content (EMC). In opposition, an increase in temperature decreased the EMC of the tested materials. It was discovered that cellulose fiber and wood-based materials had a greater capacity to store moisture than others. Conversely, the minimum moisture content was observed in brick, Densglass gold gypsum, and EPS products. Furthermore, we used the ABC model as an adaptable mathematical equation for a novel nonlinear surface fitting of adsorption isotherms of tested materials to make optimal use of the experimental data and effectively analyze the simultaneous impact of all temperatures and relative humidities. Tested against the measured experimental data, we obtained the best R-squared values from the second-order polynomial Equation.

Fundamental properties and intermolecular interaction of triethylene glycol + ethylenediamine aqueous solution for CO2 capture

Solvent absorption as a significant method for carbon dioxide capture, exploring the mechanism of CO2 capture and regeneration solvents is essential for industrial applications. In this work, water (H2O) + (ethylenediamine (EDA) + triethylene glycol (TEG)) system with the constant mole ratio (TEG: EDA = 1: 1) was chosen as the liquid absorbent. In order to gain a deeper comprehension of solution properties of H2O + (EDA + TEG) system, the density (ρ) and viscosity (η) of fundamental properties were measured at P = 1005 hPa and T = (298.15 K to 318.15 K) with the temperature step of 5 K. The excess properties, including excess molar volume (VmE), viscosity deviation (Δη), and excess Gibbs energy of activation of viscous flow (ΔG*E) of H2O + (EDA + TEG) system were calculated from the experimental ρ and η values, and fitted well with the Redlich-Kister polynomial equation. The above results combined with the spectra proved the existence of intermolecular interaction in H2O + (EDA + TEG) system, which favorably fixed CO2. Subsequently, the aqueous solution with mass fraction of 15 wt% EDA and molar ratio EDA: TEG = 1: 1 was used for absorbing CO2 (room temperature) and desorbing (378.15 K) for 5-time cycles, and the cycling absorption performance decreased. So as to explore the reason of degradation, Fourier transform infrared spectroscopy (FTIR), Raman spectroscopy, and nuclear magnetic resonance carbon spectroscopy (13C NMR) were used to analyze the circulating absorption mechanism of CO2 in H2O + (EDA + TEG) system, which revealed that alkyl carbonates and partially indecomposable substance 2-imidazolidinones generated in the cyclic process, and provided some implications for the industrial application.

On the connectedness of multistationarity regions of small reaction networks

A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected; indeed, a better understanding of the topology and geometry of such regions may help elucidate how robust multistationarity is to perturbations. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks – those with one species and up to three reactions, and those with two species and up to two reactions – we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.

Modulating Leaching Behavior of Boscalid Fungicide in Soil Through Organic Amendments

ABSTRACT The leaching loss and movement of pesticides need to be minimized to safeguard the aquatic environment and human health. In a laboratory study, the changes in the leaching and downward movement behavior of boscalid fungicide were examined in soil (Typic Hapludoll) columns (0–30 cm) receiving no amendment (control) or with the application of farmyard manure (FYM) and pine needle biochar (PNB) in surface (0–15 cm) soil @ 5 t ha−1. No boscalid residues (<0.01 µg mL−1) were detected in the leachate under any treatment. Both FYM and PNB were effective in limiting the movement of boscalid in soil compared to control. FYM maintained the highest boscalid concentration (1.41 µg g−1) in the top (0–2.5 cm) soil while in PNB treated soil the highest sorbed boscalid (0.58 µg g−1) could accumulate at 2.5–5.0 cm soil depth. The depth-wise distribution of the boscalid concentration in soil could be well predicted by third order polynomial equations (R2 value = 0.929 to 1.000, all significant at p ≤ 0.01). The FTIR spectra show that the boscalid retention onto these amendments was through H-bonding and π–π interactions. As compared to FYM, the PNB had a relatively lower proportion of soluble- and desorbed-boscalids but a higher proportion of irreversibly sorbed boscalid and appeared to be an effective organic amendment for the areas prone to high surface run-off.

Threshold amounts of nutrients and the relationship with chlorophyll a during eutrophication phenomenon in small-scale artificial reservoirs

This study aimed at determining the relationship of chlorophyll a (Chl a) and the nutrients in two reservoirs of different sizes, namely Bueng Si Than and Bueng Thung Sang, which are located in Khon Kaen Province in Thailand. The evaluation of the trophic state index and the estimated threshold amounts of nutrients was consequently considered. Water samples were collected during both the dry and rainy seasons with twenty-seven sampling points in each reservoir. Parameters indicating water quality and nutrients were based on the APHA method. The results showed a significant difference in the average physicochemical properties and nutritional parameters of the water in a smaller proportion, with Bueng Si Than having more than Bueng Thung Sang (p < 0.05). Nutrient factors (TP, TKN) showed the impact on the occurrence of eutrophication during both the dry season and the rainy season. Average ratios of nutrients and Chl a together with the trophic state index revealed the occurrence of hypertrophic conditions in both reservoirs. The high value of Chl a:TP signaled that the algal growth in these reservoirs could be controlled by TP. The Pearson correlation analysis indicated a strong positive relationship of Chl a and TP in Bueng Si Than. Moreover, the obtained results from the simulation were achieved using MATLAB R2022a, and were performed with the fourth order polynomial equation, which indicated the boundary values of 28.25–29.42 μg/L and 4.15-4.19 mg/L for the threshold amounts of TP and TKN prior to Eutrophic occurrence, ascertain further eutrophication management in small-scale artificial reservoirs.

Effect of contact length of bias ply tractor tire on its tractive performance

In order to study the effect of contact length of tire on tractive performance of tractor, experiments were conducted using a single wheel tester fitted with 13.6–28 bias ply tire in a soil bin in soft soil condition. Contact length and contact width were measured at different normal loads (9.8 kN and 13.72 kN) and inflation pressures (83, 103, 124 and 138 kPa). Results showed that the contact length had higher influence on tire pulling ability and tractive efficiency as compared to contact width of the tire. An equation for predicting contact length was developed using XLSTAT software, with normal load and inflation pressure as an independent variables and contact length as a dependent variable. The model demonstrated high efficiency with a coefficient of determination (R2) 0.96, a percentage of variation 0.76 %, a root mean square error 10.841, and an adjusted R2 0.95. Additionally, a second-order polynomial equation was developed using curve fitter app to estimate drawbar pull of a tractor by keeping wheel slip and contact length as independent parameters. Validation with another set of data obtained for 14.6–28 tire yielded R2 0.93 and less than 4 % variation, thus indicated the model’s accuracy in predicting drawbar pull.

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.