Higher-order Iteration Research Articles

Metabolic flux optimization of iterative pathways through orthogonal gene expression control: Application to the β-oxidation reversal

Balancing relative expression of pathway genes to minimize flux bottlenecks and metabolic burden is one of the key challenges in metabolic engineering. This is especially relevant for iterative pathways, such as reverse β-oxidation (rBOX) pathway, which require control of flux partition at multiple nodes to achieve efficient synthesis of target products. Here, we develop a plasmid-based inducible system for orthogonal control of gene expression (referred to as the TriO system) and demonstrate its utility in the rBOX pathway. Leveraging effortless construction of TriO vectors in a plug-and-play manner, we simultaneously explored the solution space for enzyme choice and relative expression levels. Remarkably, varying individual expression levels led to substantial change in product specificity ranging from no production to optimal performance of about 90% of the theoretical yield of the desired products. We obtained titers of 6.3 g/L butyrate, 2.2 g/L butanol and 4.0 g/L hexanoate from glycerol in E. coli, which exceed the best titers previously reported using equivalent enzyme combinations. Since a similar system behavior was observed with alternative termination routes and higher-order iterations, we envision our approach to be broadly applicable to other iterative pathways besides the rBOX. Considering that high throughput, automated strain construction using combinatorial promoter and RBS libraries remain out of reach for many researchers, especially in academia, tools like the TriO system could democratize the testing and evaluation of pathway designs by reducing cost, time and infrastructure requirements.

Hadronic structure on the light front. V. Diquarks, nucleons, and multiquark Fock components

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front, motivated by the QCD vacuum structure and lattice results. In this paper we focus on the importance of diquark correlations, which we describe by a quasi-local four-fermion effective 't Hooft interaction induced by instantons. The same interaction is also shown to generate extra quark-antiquark pair of the "sea". Its higher order iteration can be included via "pion-mediation": both taken together yield a quantitative description of the observed flavor asymmetry of antiquarks sea. Finally we discuss the final step needed to bridge the gap between hadronic spectroscopy and parton observables, by forward DGLAP evolution towards the chiral upper scale of $\sim 1\, {\rm GeV}^2$.

Вдосконалення ітераційного алгоритму з реалізації способу опису екологічного стану території

З використанням способу дослідження екологічного стану території при техногенному забрудненні реалізовано програмний додаток «Комп’ютерна програма з реалізації способу опису забрудненої території «Випадкова точка». Цей програмний додаток призначений для створення моделей пересічених територій, які зазнали впливу небезпечних речовин, з урахуванням ступеня забруднення ділянки. В основу покладено підхід, що дозволяє описати територію відрізками. Проте це викликає нев’язку апроксимації, яка може призвести до наближеного або помилкового результату. Задача роботи — вдосконалення алгоритму ітерації при реалізації способу опису екологічного стану забрудненої небезпечними речовинами території для побудови її моделі. Особливістю вирішення поставленої задачі є те, що при використанні ітераційних формул, заснованих на розкладанні за нев’язками, є ще одна додаткова можливість для скорочення числа дій — обчислення зі змінною розрядністю. Це пов’язано з тим, що при використанні ітераційної формули p-го порядку на кожному кроці збільшується точність приблизно в p разів. Представлений алгоритм реалізовано за допомогою знаходження базових послідовностей ітераційних формул з модифікацією методу Чебишова для побудови ітерацій вищих порядків. Зокрема, застосування різних методів апроксимації функцій для отримання коефіцієнтів базових послідовностей на основі відомих розкладань. Наведене може використовуватися в комп’ютерній реалізації програм для побудови моделей пересіченої території, наприклад водних об’єктів, та прогнозів забруднення техногенно-навантажених промислових територій, відстійників, русла річок, шахтних виробіток. Вдосконалений алгоритм дозволяє виконувати задачі опису з більшою точністю шляхом мінімізації помилок та наближення їх значення до нуля.

On a high-order iteration technique for a wave equation with nonlinear viscoelastic term

In this paper, a high-order iterative technique for solving a wave equation with strong damping and nonlinear viscoelastic term is constructed. For this purpose, we adapt the high-order iterative method used in the earlier works and establish the existence theorem of a recurrent sequence associated with the proposed problem. Thereafter, we prove a N-order convergence of the obtained sequence to a unique solution of the problem.

О теореме Кенига для целых функций конечного порядка

Показано, что теорема Кенига о нулях аналитической функции, примененная к логарифмической производной целой функции конечного порядка, приводит к алгоритму отыскания нулей, для которого областями сходимости являются многоугольники Вороного искомых нулей. Так как диаграмма Вороного последовательности нулей составляет множество меры нуль, то алгоритм имеет глобальную сходимость. Дана оценка скорости сходимости. Для итераций высших порядков, которые строятся с помощью теоремы Кенига, рассмотрено влияние кратности корня на область сходимости и приводится оценка скорости сходимости. It is shown that Koenig's theorem on zeros of analytic functions applied to the logarithmic derivative of an integer function of finite order leads to an algorithm of finding zeros whose convergence domains are the Voronoi polygons of the zeros to be found. Since the Voronoi diagram of a sequence of zeros is a set of measure zero, this algorithm is globally convergent. The rate of convergence is estimated. For higher-order iterations that are constructed using Koenig's theorem, the effect of root multiplicity on the convergence domain is considered and the convergence rate is estimated for this case.

Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations

We present a three-step family of iterative methods to solve systems of nonlinear equations. This family is a generalization of the well-known fourth-order King’s family to the multidimensional case. The convergence analysis of the methods is provided under mild conditions. The analytical discussion of the work is upheld by performing numerical experiments on some application oriented problems. Finally, numerical results demonstrate the validity and reliability of the suggested methods.

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme

The goal of this research is to extend and investigate an improved approach for calculating the weighted Moore–Penrose (WMP) inverses of singular or rectangular matrices. The scheme is constructed based on a hyperpower method of order ten. It is shown that the improved scheme converges with this rate using only six matrix products per cycle. Several tests are conducted to reveal the applicability and efficiency of the discussed method, in contrast with its well-known competitors.

Bar recursion is not computable via iteration

We show that the bar recursion operators of Spector and Kohlenbach, considered as third-order functionals acting on total arguments, are not computable in Gödel’s System T plus minimization, which we show to be equivalent to a programming language with a higher-order iteration construct. The main r esult is formulated so as to imply the non-definability of bar recursion in T+min within a variety of partial and total models, for instance the Kleene–Kreisel continuous functionals. The paper thus supplies proofs of some results stated in the book by Longley and Normann. The proof of the main theorem makes serious use of the theory of nested sequential procedures (also known as PCF Böhm trees), and proceeds by showing that bar recursion cannot be represented by any sequential procedure within which the tree of nested function applications is well-founded.

Some higher-order iteration functions for solving nonlinear models

In this paper, we present a new efficient sixth-order family of Jarratt type methods for solving scalar equations. Then, we extend this family to the multidimensional case preserving the same order of convergence. We also discuss the theoretical convergence properties of the proposed scheme in the case of scalar as well as multidimensional case. The derivation of these schemes are based on weight function approach and free disposable parameters. We also demonstrate the applicability of them on total six number of problems: first five are real life problems namely, continuous stirred tank reactor (CSTR), chemical engineering, the trajectory of an electron in the air gap between two parallel plates, Hammerstein integration and boundary value problems; last one is the standard academic test problem. In addition, numerical comparisons are made to show the performance of the proposed iterative techniques with the existing techniques of the same order in the scalar as well as multi-dimensional case. Finally on the basis of numerical results, we conclude that our techniques perform better in terms of residual error, error between the two consecutive iterations, asymptotic error constant term and approximated root as compared to the existing ones of same order in scalar as well as multidimensional case.

An eighth-order family of optimal multiple root finders and its dynamics

There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.

Ptychographic iterative engine with partially coherent illumination for weakly scattering samples

As an expansion of coherent diffraction imaging, ptychographic iterative engine (PIE) not only inherits advantages such as ultra-high resolution and compact optical system, but also expands the field of view in quantitative imaging, thus PIE is widely used in short wavelength imaging such as X-ray and electron beam imaging, and then extended to visible light field. However, PIE requires coherent illumination for both phase and amplitude retrieval, while traditional X-ray or electron beam sources often cannot satisfy this strict coherent condition, which leads to poor-quality information retrieval with low signal-to-noise ratio. Though several proposed methods such as multiple wavelength and multi-mode algorithms can eliminate incoherency influence to some extent, various details such as quantitative spectrum of illuminating source should be obtained before information retrieval, which complicates computing procedures. In addition, it is hard to acquire the spectrum of the illuminating source in most cases. In order to acquire high-quality information based on PIE with partially coherent illumination, a newly designed enhanced phase retrieval method for weakly scattering samples in PIE with partially coherent illumination is presented in this paper, in which only the bright field of the diffraction patterns is used in the iterative procedures mimicking the coherent cases especially for weakly scattering samples without any prior illuminating details. The bright field area can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. While the dark-field area generated by interference of diffracted beams cannot satisfy the requirement for coherence, therefore, dark-field diffraction patterns should be excluded in sample information extraction and only the bright field is used for phase retrieval via iterative process. Compared with the proposed multiple wavelength and multi-mode algorithms, the proposed method can simplify sample reconstructing procedures due to needing no prior knowledge. Moreover, in order to enhance the information of weakly scattering samples in retrieval, high order iteration method is also introduced in the paper. Since the bright field can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. For weakly scattering sample, the weakened diffracted beam is much lower than zeroth order beam, thus it is difficult to acquire high-contrast information with classical PIE algorithms. Introducing high order iterative tactic, the contrast of weakly scattering sample is obviously improved and the details of weakly scattering sample can be retrieved clearly. Both theoretical analysis and numerical simulations are illustrated in detail, proving the robustness and availability of the designed method: high-contrast phase information can be obtained with the proposed method, while traditional phase retrieval algorithm almost loses all details of the sample. In order to mimic the real experimental situation, a 30 dB white noise is added into the simulation, the details of weakly scattering sample phase information can also be retrieved clearly by using the bright field of the diffraction patterns with high order iteration method. With the newly designed enhanced phase retrieval method for weakly scattering samples with partially coherent illumination, sample retrieval via PIE can not only use ordinary X-ray source or electron beam as illumination source, thereby avoiding the dependence on complete coherent source, but also obviously improve the retrieval quality of the sample characteristics, which widely expands the application fields of the PIE.

Effect of Equivalence Ratio on Composition and performance of Biogas and Gasoline Exhaust from Spark Ignition Engine by Mathematical Modeling

This paper presents the numerical computationnal of pressure, temperature and exhaust characteristics of spark ignition engine with biogas as fuel. The solution of non-linear combustion equation systems have been computed, that based on a quasi-one-dimensional engine model, high order iteration method with the equilibrium constants method. Computer program was used to calculate the mole fractions of 10 combustion products when biogas and gasoline fuel are burnt along with variable equivalence ratios. In cylinder chamber model is based on the classical two-zone approach, wherein parameters like heat transfer from the cylinder, blow by energy loss and heat release rate are also considered and calculated. Biogas is defined as fuel produced from using anaerobic digestion of biodegradable or waste materials and the constituents are C 5 H 7 O 2 N, CH 4 , CO 2 N 2 H 2 O of biogas and C 7 H1 7 of gosoline. Which general fuel model is specified by way of its C a H b O c N d values. The curve-fitted coefficients of energy were then employed to simulate air and fuels data along with frozen composition and practical chemical equilibrium routines from Gill data. The calculated data were used to plot the various pressure and temperature with the crank angle of each step of four stroke engine cycle and combustion products versus equivalence ratio. All results were compared with gasoline as reference fuel in the spark ignition engine according to the same numerical method.

Computing outer inverses by scaled matrix iterations

Convergence analysis for a class of iterative methods for computing outer inverses with prescribed range and null space is studied. Furthermore, several heuristics for accelerating iterative methods via scaling are proposed. In fact, we are motivated by the fact that scaling of iterative methods for computing generalized inverses is investigated so far only on low-order methods. Our intention is to test the scaling on higher-order iterative methods. Here, we also propose a new higher-order iteration. Although the introduced method is efficient in terms of computational efficiency index, we test its acceleration through several experiments.

The Nonlinear Multi-Fuel Combustion Equation: Comparison between the Model of Heat in SIE for Ethanol-Gasoline Blended and High Order Iteration Method

A quasi one-dimensional engine cycle, the simulation program was developed to predict the simple heat transfer during combustion stroke in a spark ignition engine (SIE) of ethanol-gasoline blended fuels. A two-zone heat release model was utilized to model the combustion process inside the combustion cylinder by using high order iteration method (HIM) in combustion equation. The fuel, air and burned gas properties throughout the engine cycle were calculated by using variable specific heats. The transient heat flux inside the combustion chamber due to the change in the cylinder gas temperature and pressure during combustion was determined by using the constant of heat rate from enthalpy heat rate coefficient. The program was written in MATLAB. The results of this paper such as the average heat flux cycles of the burned gas during the combustion and showing the emissions of at during the combustion process by high iteration method. Based on these results, it was concluded that the fundamental thermodynamic properties of the fuel, air, and combustion product species were in agreement with previously published results to within the scatter of those results typically less than 1% form references 1%. The combustion products were calculated by the high order iteration methods by around 1% from the old methods.

Approximate Analytic Expression of Surface Charge Density/Surface Potential Relationship for a Spherical Colloidal Particle Immersed in a General Electrolyte Solution

We implement a variational iteration method for solving a Poisson–Boltzmann equation (PBE) describing a spherical colloid immersed in a general electrolyte solution. In this method, a general Lagrange multiplier is introduced to construct correction functional for the problem, and the multiplier is identified optimally via a variational method; a linearization solution of the original PBE is chosen as an initial approximation for the iteration. To proceed with the iteration analytically, we approximate the highly nonlinear term of the PBE by a polynomial of proper order. Due to presence of an exponent integral function in the first-order iteration solution, higher-order iteration solutions are analytically unavailable currently. Based on the first-order iteration solution, analytic expression for surface charge density/surface potential relationship is acquired. The present analytic solution contrasts sharply with previous ones by two striking features: (1) the present one surpasses well beyond previous approximate solutions in that general electrolyte type can be dealt with in a unified way, and (2) valid application scope of the present expression is in small κa domain and thus is complementary with those of previous ones.

A comparison between the iteration methods and Adomian decomposition method for solving nonlinear combustion equations

In this paper, we present a comparative study between Newton-Raphson method, Householder‘s iteration method, Adomian decomposition method and High order iteration method applied to the solution of nonlinear hydrocarbon combustion equations for investigation of equilibrium composition in combustion products. For this purpose, the paper produces a step of numerical for calculating the composition of the hydrocarbon combustion, which is based on the development 6236 Paramust Juntarakod and Thanakom Soontornchainacksaeng of a model using 11 species chemical equilibrium considerations. The 11×11 non-linear system, which is obtained from the seven non-linear equilibrium equations and the four linear atom balance equations, is converted to a 4×4 non-linear system. This system is solved using the all iteration Method for non-linear systems, with no conversion problems and small computational cost. The program is applied successfully for hydrocarbon fuel, which typically represents the gasoline fuel. The comparison gives similar results and supplies quantitatively reliable results.

Iteration functions for pth roots of complex numbers

A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.

Comparison of different BEM+Born series modeling schemes for wave propagation in complex geologic structures

Complex geologic structures generally consist of irregular subregions with piecewise constant properties. Two different boundary-element method (BEM) plus Born-series schemes have been formulated for wave-propagation simulation in such piecewise homogeneous media by incorporating a Born series and boundary integral equations. Both schemes decompose the resulting boundary integral equation matrix into two parts: (1) the self-interaction operator, handled with a fully implicit BEM, and (2) the extrapolation operator, approximated by a Born series. The first scheme associates the self-interaction operator with each boundary itself and interprets the extrapolation operator as cross-interactions between different boundaries in a subregion. The second scheme relates the self-interaction operator toeach subregion itself and expresses the extrapolation operator as cross-interactions between different subregions in a whole model. In the second scheme, the matrix dimension of the resulting boundary integral equation is reduced by eliminating the traction field. Both numerical schemes have been validated by dimensionless frequency responses to a semicircular alluvial valley, compared with the full-waveform BEM numerical solution. We then extended these schemes to a complex fault model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments indicate that the [Formula: see text] series modeling schemes significantly improve computational efficiency, especially for high frequencies and with multisource seismic surveys. The tests also confirmed that the second modeling scheme has a faster convergence but may cause more noise in higher-order iterations than the first scheme.

A combinatorial construction of high order algorithms for finding polynomial roots of known multiplicity

We construct a family of high order iteration functions for finding polynomial roots of a known multiplicity $s$. This family is a generalization of a fundamental family of high order algorithms for simple roots that dates back to Schröder’s 1870 paper. It starts with the well known variant of Newton’s method $\hat {B}_{2}(x) = x-s \cdot p(x)/p’(x)$ and the multiple root counterpart of Halley’s method derived by Hansen and Patrick. Our approach demonstrates the relevance and power of algebraic combinatorial techniques in studying rational root-finding iteration functions.

A high order iteration formula for the simultaneous inclusion of polynomial zeros

In this paper, using Lagrange interpolation with special nodes, we construct an iteration for simultaneously finding all zeros of an algebraic polynomial with high order convergence. The convergence analysis, initial conditions and numerical examples are given.