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Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed "NP-completeness" as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm's potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts. [ABSTRACT FROM AUTHOR]

We study a system of equations, involving a large parameter, arising from the study of stellar pulsation for which a combination of procedures is used to approximate a fun damental solution. We present a combination of singular and non-singular perturbation methods which, aided by symbolic computation, may be of multi-disciplinary interest for the analysis as well as a astrophysics application. Example software is presented in the Wolfram Language (Mathematica version 13.2). [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a system of mappings on the finite product of metric spaces. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Song-il Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results.

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Heat capacity data are unavailable or incomplete for many minerals at geologically relevant temperatures. Despite the availability of entropy and enthalpy values in numerous thermodynamic tables (even sometimes at elevated temperatures), there remains need for extrapolation beyond, or interpolation between, temperatures. This approach inevitably results in estimates for entropy and enthalpy values because the heat capacity coefficients required for optimal thermodynamic treatment are less frequently available. Here we propose a simple method for obtaining heat capacity coeficients of minerals. This method requires only the empirically measured temperature-specific heat capacity for calculation via a matrix algorithm. The system of equations solver is written in the Python computing language and has been made accessible in an online repository. Thermodynamically, the solution to a system of equations represents the heat capacity coefficients that satisfy the mineral-specific polynomial. Direct coefficient calculation will result in more robust thermodynamic data, which are not subject to fitting uncertainties. Using hematite as an example, this method provides results that are comparable to conventional means and is applicable to any solid material. Coeficients vary within the traditional large 950 K temperature interval, indicating that best results should instead utilize a smaller 400 K temperature interval. Examples of large-scale implications include the refinement of geothermal gradient estimation in rapidly subsiding sedimentary basins or metamorphic and hydrothermal evolution. [ABSTRACT FROM AUTHOR]

Systems of whole‐stand yield models facilitate projections of forest attributes, but their inputs may be difficult to measure accurately. This study conducted sensitivity analyses to examine the effect of systematic and stochastic measurement errors on outputs from a representative system of equations. Simulated error was added to explanatory variables stand age, site index, or both. Results showed that large systematic error in one variable tended to produce moderate to large percent changes in all models, particularly the height and volume equations (often >50% change). Systematic error in both variables amplified this effect, especially for young, less productive stands. Stochastic error dramatically increased estimate variability (some relative standard errors >50%), particularly in the height and volume models at young ages and low site indices. These results suggest that measurement error may considerably alter projections and increase uncertainty when using whole‐stand yield models, highlighting the need for careful crew training. Recommendations for Resource Managers: Measurement error in input variables has the potential to significantly affect estimates from systems of whole‐stand yield models.Effects can be over‐ or underestimation of volume yields and financial returns or other forest attributes of interest.Careful crew training is paramount to ensure forest measurements remain consistent with the quality of the data used to fit the models. [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Virtual Element Method for a system of time-dependent nonlinear convection-diffusion-reaction equations on polygonal meshes. We present a fully nonlinear discrete scheme with a proof of existence and uniqueness of the numerical solution. In order to overcome the complexity of solving nonlinear discrete system, we propose an equivalent iterative linearized system of equations. We perform the stability analysis and then derive the optimal convergence estimates in the H 1 semi-norm. Finally, we investigate the performance of the fully discrete scheme through a set of numerical tests. To improve computational efficiency, we have performed the simulations using a two-grid method. [ABSTRACT FROM AUTHOR]

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with m < n ) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter ε → 0 , proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results. [ABSTRACT FROM AUTHOR]

A new theory of solvability of linear equations with an infinite number of unknowns has been developed. [ABSTRACT FROM AUTHOR]

The demand for digital technologies has been growing due to a shift in the technological and economic paradigm. The need for online services has increased since the beginning of the COVID pandemic. There are significant disparities between Russian regions in the digital technology accessibility and the development of computer skills. In 2020, the Internet diffused rapidly in most regions, although previously, there had been a slowdown. As markets got saturated with digital services, the digital divide between Russian regions narrowed. Overall, the Internet use patterns are consistent with those of the spatial diffusion of innovations. Amongst the leaders, there are regions home to the largest agglomerations and northern territories of Russia, whereas those having a high proportion of rural population lag behind. Coastal and border regions (St. Petersburg, the Kaliningrad region, Karelia, Primorsky Krai, etc.) have better access to the Internet due to their proximity to the centres of technological innovations as well as the high intensity of external relations. Leading regions have an impact on their neighbours through spatial diffusion. Econometrically, access to the Internet depends on income, the average age and level of education, and its use depends on the business climate and Internet accessibility factors. Regional markets are gradually getting more saturated with digital services and technologies. The difference between regions in terms of access to the Internet is twofold, whereas, in terms of digital technology use, the gap is manifold. In many regions, the share of online commerce, which became the driver of economic development during the lockdown, is minimal. Based on the results of the study, several recommendations have been formulated.

Under study are the finite systems of Diophantine equations over finite configurations. We propose some consistency verification procedure for such a system and use the output of the procedure to constructing the complete solution set. We estimate the running time of the procedure in general and distinguish the class of systems for which the consistency problem is decidable in polynomial time. [ABSTRACT FROM AUTHOR]

Pakistan, a developing country, desperately looked for foreign investment in order to promote growth and investment. Pakistan introduced a number of policy measures in the last few decades and attracted substantial foreign investments. But it is not clear whether foreign investments inflows in Pakistan played any role in growth and development in Pakistan? This study employs a system of equations to investigate the nexus between FDI, portfolio and domestic investment in Pakistan. The main objective of this research is to measure the impact of foreign direct, port-folio investment and other forms of capital inflows on domestic, especially on private investment, in Pakistan. The results show that FDI inflows promoted domestic investment in Pakistan, where FDI and domestic investment reported a one-to-one relationship. The findings of the study also confirm that domestic investments contribute to FDI inflows; thus, FDI and domestic investment mutually cause each other. On the other hand, portfolio investments play an insignificant role in determining domestic investment in Pakistan. This shows that compared to portfolio investment, Pakistan needs to focus on foreign direct investment inflows for promoting growth and private investment at home.

Free-surface seepage problems are one of the most challenging issues in the practice of geotechnical engineering since the determination of the free surface commonly demands sophisticated numerical techniques. This paper introduces an implicit numerical model for determining the free surface of seepage accurately. The present model is based upon the fractional step and finite volume method, which solves the governing equation on an unstructured triangular mesh. In order to reduce the computational complexity and cost, a robust algorithm for solving a system of equations has been presented, which greatly increases the convergence rate. An iterative method has been deployed in an efficient way, through which the non-zero values are extracted from the coefficient matrix of the resultant equation system, whereby the problem unknowns are achieved swiftly. To verify the developed model, several test cases have been utilized. Comparisons between the numerical outcomes, analytical solutions, experimental data, and those of other numerical methods indicate the efficiency and accuracy of the developed scheme and its perfect convergence characteristic. Comparisons between the developed model and other numerical models indicate that the present method is highly robust. The proposed technique provides sufficient accuracy and reliability; besides, it increases the convergence rate through decreasing computational efforts. [ABSTRACT FROM AUTHOR]

A system of nonlinear convolution-type integral equations on the entire axis is considered in this work. This system is met in many areas of mathematical physics and mathematical biology. The issues of existence and absence of nontrivial bounded solutions are investigated. The theorem about the uniqueness of the solution in a certain class of bounded functions is proved. In the end of the work, applied examples of this system are provided. [ABSTRACT FROM AUTHOR]

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with m

This manuscript is related to compute approximate solutions for a class of fractional distributed order differential equations (FDODEs). The corresponding derivative of fractional order is taken in Caputo sense. The adopted numerical scheme is based on collocation method together with Haar wavelet. This is an efficient numerical algorithm which convert the consider problem to a some system of equations of algebraic type. Upon utilizing the Broyden tools the obtain nonlinear system is solved for the intended numerical results. Further in case of linear equations the obtained system is solved by Gauss elimination procedure. To strengthen our results we testify several problems corresponding to different collocation points. We present also maximum absolute and root mean square errors. The graphical presentations are also given.

This paper provides an extension of results connected with the problem of the optimization of a linear objective function subject to max -* fuzzy relational equations and an inequality constraint, where * is an operation. This research is important because the knowledge and the algorithms presented in the paper can be used in various optimization processes. Previous articles describe an important problem of minimizing a linear objective function under a fuzzy max -* relational equation and an inequality constraint, where * is the t-norm or mean. The authors present results that generalize this outcome, so the linear optimization problem can be used with any continuous increasing operation with a zero element where * includes in particular the previously studied operations. Moreover, operation * does not need to be a t-norm nor a pseudo-t-norm. Due to the fact that optimal solutions are constructed from the greatest and minimal solutions of a max -* relational equation or inequalities, this article presents a method to compute them. We note that the linear optimization problem is valid for both minimization and maximization problems. Therefore, for the optimization problem, we present results to find the largest and the smallest value of the objective function. To illustrate this problem a numerical example is provided. [ABSTRACT FROM AUTHOR]

Pakistan, a developing country, desperately looked for foreign investment in order to promote growth and investment. Pakistan introduced a number of policy measures in the last few decades and attracted substantial foreign investments. But it is not clear whether foreign investments inflows in Pakistan played any role in growth and development in Pakistan? This study employs a system of equations to investigate the nexus between FDI, portfolio and domestic investment in Pakistan. The main objective of this research is to measure the impact of foreign direct, portfolio investment and other forms of capital inflows on domestic, especially on private investment, in Pakistan. The results show that FDI inflows promoted domestic investment in Pakistan, where FDI and domestic investment reported a one-to-one relationship. The findings of the study also confirm that domestic investments contribute to FDI inflows; thus, FDI and domestic investment mutually cause each other. On the other hand, portfolio investments play an insignificant role in determining domestic investment in Pakistan. This shows that compared to portfolio investment, Pakistan needs to focus on foreign direct investment inflows for promoting growth and private investment at home. [ABSTRACT FROM AUTHOR]

Incompliance of the settings of the system to control actual values of the parameters of a variable frequency induction electric drive may sometimes result in complete non-operability of a variable frequency electric drive as well as in the considerable reduction of the dynamic quality parameters. Such parameters as active rotor resistance, rotor inductance, and inductance of the magnetization circuit are available for the immediate measuring. They are not identified in terms of the acceptance tests, and the values presented in catalogues and reference books are calculated ones that may differ considerably from the real values of a certain machine. Despite constant studies by the researchers, a task to identify electromagnetic parameters of the equivalent circuit of an induction motor is still important and topical. The objective of the paper is to develop a method of online-identification of the electromagnetic parameters of an induction motor making it possible to implement accurate regulator adjustment of the frequency control system in terms of operational changes in the driving motor parameters. For the first time, the paper analyzes a steady mode of induction motor operation which does not apply T-network of the equivalent circuit of an induction motor. An approach has been proposed relying on the equation of an induction motor in three-phase fixed coordinate system obtained on the basis of the theory of generalized electromechanical converter.

This paper reports a solution to the problem of determining the configuration parameters of PID controllers when arranging the poles of the transfer function of a linear single-circuit automated control system for a predefined set of control objects. Unlike known methods in which the task to find the optimal settings of a PID controller is formed as a problem of nonlinear programming, in this work a similar problem is reduced to solving a system of linear algebraic equations. The method devised is based on the generalized Viète theorem, which establishes the relationship between the parameters and roots of the characteristic equation of the automatic control system. It is shown that for control objects with transfer functions of the first and second orders, the problem of determining the configuration parameters of PID controllers has an unambiguous solution. For control objects with transfer functions of the third and higher orders, the generated problem is reduced to solving the redefined system of linear algebraic equations that has an unambiguous solution when the Rouché– Capelli theorem condition is met. Such a condition can be met by arranging one of the roots of the characteristic equation of the system on a complex plane. At the same time, the requirements for the qualitative indicators of the system would not always be met. Therefore, alternative techniques have been proposed for determining the configuration parameters of PID controllers. The first of these defines configuration parameters as a pseudo solution to the redefined system of linear algebraic equations while the second produces a solution for which the value of the maximum residual for the system of equations is minimal. For each case, which was used to determine the settings of PID controllers, such indicators of the control process as overshooting and control time have been determined. [ABSTRACT FROM AUTHOR]

This manuscript is related to compute approximate solutions for a class of fractional distributed order differential equations (FDODEs). The corresponding derivative of fractional order is taken in Caputo sense. The adopted numerical scheme is based on collocation method together with Haar wavelet. This is an efficient numerical algorithm which convert the consider problem to a some system of equations of algebraic type. Upon utilizing the Broyden tools the obtain nonlinear system is solved for the intended numerical results. Further in case of linear equations the obtained system is solved by Gauss elimination procedure. To strengthen our results we testify several problems corresponding to different collocation points. We present also maximum absolute and root mean square errors. The graphical presentations are also given. [ABSTRACT FROM AUTHOR]

We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.

In this paper, we consider the solvability of the system of two operator equations A X = C and X B = D in the framework of Hilbert C ∗ -modules as well as the existence of Hermitian and positive solutions. We also provide formulae for the general forms of such solutions. [ABSTRACT FROM AUTHOR]

Methodological approaches to investing in companies and reducing the negative impact of risks that are formed at the macro and micro levels are considered in the article. The algorithm for expressing investment risks through related risks and conducting an investment risk assessment as a group process is defined. It has been determined that the defining features of investment risks are the environment, duration, and scope of the project, risk position, profile, risk appetite, consequences, capacity, and results of the impact on the investment project. An investment risk accounting system is formed, which is represented by a set of organized structural elements that perform functions related to planning and implementation of a set of measures that identify, assess, monitor, and control risks to minimize negative consequences and enhance opportunities. A method of forming a real portfolio of investment projects considering the dynamic risk factor has been developed. [ABSTRACT FROM AUTHOR]

Biomass additivity is a desirable characteristic of a system of equations for predicting components and total biomass, since equations independently adjusted generate biologically inconsistent results. The aim of this study was to fit and compare three methods for modelling biomass: (i) total biomass individual regression, (ii) total biomass regression function calculated as the sum of separate biomass components, and (iii) simultaneous equations of biomass components based on Nonlinear Seemingly Unrelated Regression. A total of 208 trees of Eucalyptus dunnii and Eucalyptus grandis were harvested and destructively sampled to record above-ground biomass. Results indicate that a system of equations adjusted by simultaneous equations provides accurate biomass estimations, guaranteeing additivity. This model system showed good fit and good prediction performance, given that the correlation coefficient was higher than 97 % for total above-ground biomass, for both species; whereas root mean square error was 23.9 kg and 30.2 kg for E. grandis and E. dunnii, respectively. A system of biomass equations was developed for each eucalyptus species, such that the sum of the estimations of the biomass components equaled the estimate of above-ground biomass. Results showed that the systems of equations have high potential for improving the accuracy of individual tree above-ground biomass estimates for both species. [ABSTRACT FROM AUTHOR]

The local convergence analysis of iterative methods is important since it indicates the degree of difficulty for choosing initial points. In the present study we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study. [ABSTRACT FROM AUTHOR]

Minimal linear codes as a special class of linear codes attract much interest for their application in secret sharing schemes and secure two-party computation. It is interesting to characterize minimal linear codes with kinds of mathematical tools and construct more minimal linear codes. This letter considers minimal linear codes constructed from the defining sets formed by vectors with given weights, follows the method in (“Minimal linear codes from characteristic functions, IEEE Trans. Inf. Theory. DOI: 10.1109/TIT.2020.2978387), and fully characterizes these minimal linear codes by weights. [ABSTRACT FROM AUTHOR]

Purpose. The measurements of the resistance of grounding devices for various purposes using a three-electrode installation under real conditions are considered. Methodology. On the basis of the use of a three-electrode installation, a technique for full-scale measurements of the resistance of a grounding device of any design in an arbitrary soil structure is presented. Results. Based on the measurement results, a system of sixth-order equations is solved which allows to determine the own and mutual resistances in a three-electrode installation with a sufficiently high accuracy. Originality. It is not necessary to find a point of zero potential. Practical value. The proposed technique allows to perform measurements in conditions of dense urban and industrial development.

A distribution system upgrade moving towards Smart Grid implementation is necessary to face the proliferation of distributed generators and electric vehicles, in order to satisfy the increasing demand for high quality, efficient, secure, reliable energy supply. This perspective requires taking into account system vulnerability to cyber attacks. An effective attack could destroy stored information about network structure, historical data and so on. Countermeasures and network applications could be made impracticable since most of them are based on the knowledge of network topology. Usually, the location of each link between nodes in a network is known. Therefore, the methods used for topology identification determine if a link is open or closed. When no information on the location of the network links is available, these methods become totally unfeasible. This paper presents a method to identify the network topology using only nodal measures obtained by means of phasor measurement units.

This article presents a one-stage efficiency frontier analysis based on the scaling property. This type of analysis is a not very often applied in empirical work in spite of its nice features. Due to the scaling property the influence of exogenous (managerial) variables on efficiency can be modelled and estimated in one stage. It also opens the possibility of estimating a system of equations, consisting of a cost function and the corresponding cost share equations. The model is applied to a unique data set of Dutch secondary education school boards in the period 2007–10, not only consisting of regular data on cost, inputs and outputs, but also of specific data on operational management. The model provides reliable and plausible estimates for the cost efficiency, scale elasticity, and technical change. Average cost efficiency is about 96%. Economies of scale prevail for school boards with size less than 0.8 times average size, whereas annual productivity growth is 2.2% on average. [ABSTRACT FROM AUTHOR]

The use of a system of basic hydrogasdynamic equations, which was written for the portion of contraction of the flow at entry into a cascade of axial-flow turbine profiles at large angles of reattachment to the walls of a blade passage, has enabled the authors to obtain analytical dependences (characteristics) of the loss on the angles of entry of the flow into the cascade. The obtained procedure of calculating the loss permits taking account of the radius of rounding of the profiles at entry, on which the region of insensitivity to the angle of attack is dependent, and also determining relevant flow-entry angles limiting the insensitivity region. The authors have proposed variants of the procedure for both an incompressible-fluid model and a compressible gas, where use is made of the gasdynamic functions of the reduced velocity. The calculated characteristics of the turbine cascades are in good agreement with the existing experimental data in a wide range of angles of attack and radii of rounding of the profiles′ leading edges. The influence of the flow compressibility on the loss has been shown. To test the procedure under the conditions of pulsed flow through the turbine where the angles of attack are particularly large, the authors carried out investigations whose results are also in good agreement with the results of corresponding calculations. [ABSTRACT FROM AUTHOR]

This article proposes a novel robustness test method which combines binary sample splitting and system of equations techniques. First, we build a structural model to analyze relationship between the conditional exogeneity assumption and the explanations of structural parameters, thus establish principles of construction of alternative models. Then, we propose two construction methods: Binary sample splitting and confounding variables adjusting. Finally, we propose a robustness test statistic which based on system of equations method. Furthermore, we adjust computation process of the test to avoid singularity issue. Our Monte Carlo experiments verify the finite sample properties of this robustness test. As an application, we analyze the model settings of an article by Professor Acemoglu, which studied the influence of aging on economic growth. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a unit root test for a system of equations using a spectral variance decomposition method based on the Maximal Overlap Discrete Wavelet Transform. We obtain the limiting distribution of the test statistic and study its small sample properties using Monte Carlo simulations. We find that, for multiple time series of small lengths, the wavelet-based method is robust to size distortions in the presence of cross-sectional dependence. The wavelet-based test is also more powerful than the Cross-sectionally Augmented Im et al. unit root test (Pesaran, M. H. 2007. "A Simple Panel Unit Root Test in the Presence of Cross-section Dependence." Journal of Applied Econometrics 22 (2): 265–312.) for time series with between 20 and 100 observations, using systems of 5 and 10 equations. We demonstrate the usefulness of the test through an application on evaluating the Purchasing Power Parity theory for the Group of 7 countries and find support for the theory, whereas the test by Pesaran (Pesaran, M. H. 2007. "A Simple Panel Unit Root Test in the Presence of Cross-section Dependence." Journal of Applied Econometrics 22 (2): 265–312.) finds no such support. [ABSTRACT FROM AUTHOR]

Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property (SDC) . Our main result is the description of finite lattices and semilattices with Property (SDC) : we prove that a finite lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite semilattice has Property (SDC) if and only if it is distributive. [ABSTRACT FROM AUTHOR]