Absolute Convergence Research Articles

Analysis of shield tunnel response to bilateral pit excavation with a focus on perimeter pressure and deformation mechanisms

This study aims to investigate the responses of shield tunnel structures subjected to disturbances caused by bilateral pit excavation, and it systematically reveals for the first time the impact mechanism of bilateral pit excavation on the distribution of perimeter pressure and deformation patterns of shield tunnels. Using a bilateral pit excavation project in Nanjing as a case study, this research establishes methods for calculating longitudinal displacement and circumferential pressure of tunnels under bilateral pit excavation conditions, employing the image source method for analysis. A refined three-ring segment model is developed, and the load structure method is used to analyze the impact of deep foundation excavation on the tunnel located between the two excavation sites. The results indicate that, compared to unilateral excavation, bilateral excavation significantly increases the perimeter pressure at the top and bottom of the tunnel, with a smaller increase in pressure at the arch waist. The deformation pattern is characterized by contraction at the top and bottom and expansion at the waist, forming a transverse elliptical deformation. The maximum vertical convergence values of the middle segment ring are 25.00 mm at the top and 25.88 mm at the bottom, with a vertical absolute convergence value of 44.5 mm and a convergence ratio (ΔDt/Dt) of 0.72%. As the foundation coefficient increases, the perimeter pressure at the top and bottom of the tunnel also increases. When the tunnel is closer to the foundation pits (Sp decreases), the perimeter pressure at the bottom of the tunnel increases. Conversely, as the distance between the two foundation pits (S) increases, the impact of excavation on the tunnel shifts from the upper part to the lower part, resulting in decreased upper perimeter pressure and increased lower perimeter pressure. The research findings provide important references for similar engineering projects.

ANALYSIS OF CONVERGENCE BETWEEN PROVINCES IN INDONESIA

<p>Indonesia is one of the developing countries that is always trying to increase its economic growth. Indonesia has a large number of people with various ethnic backgrounds and diverse customs, so this situation has given rise to disparities between regions. This study aims to determine whether or not sigma convergence, to find out whether absolute and conditional convergence occurs or not, and to determine the speed of convergence. This study uses secondary data, namely Gross Regional Domestic Product (GRDP) per capita, poverty, and inequality for each province in Indonesia from 2017-2021. The method used in this study is panel data with a fixed effect approach. The results of the analysis using the sigma convergence (σ) approach show the occurrence of convergence. The results of the absolute convergence analysis explain the divergence of GRDP per capita in Indonesia with a divergence speed of 13.76% per year. The results of the conditional convergence analysis show that there is a divergence of GRDP per capita in Indonesia with a divergence speed of 13.62% per year.</p>

The SOR and AOR Methods with Stepwise Optimized Values of Parameters for the Iterative Solutions of Linear Systems

We modify the successive overrelaxation (SOR) method and accelerated overrelaxation (AOR) method for solving linear equations systems. The optimal value of the acceleration parameter is determined, using the maximal reduction method of the residual vector’s length, or equivalently an orthogonality condition. Rather than the constant value, the MAOR method endows a step-by-step varying acceleration parameter to possess the property of absolute convergence and the orthogonality of consecutive residual vector. In SOR, the relaxation parameter is also optimized by using the orthogonality condition. Numerical examples ensure that the MSOR and MAOR iterative schemes converge faster than the original SOR and AOR iterative schemes. They are easily implemented with low computational cost, and without needing of a detailed spectral analysis to determine the optimal values of parameters has a great advantage.

On formal series solutions to 4th-order quadratic homogeneous differential equations and their convergence

Abstract It is known that all τ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions combinatorially by using conformal blocks. In this paper, we show the convergence of the formal series, including the solutions of more general equations. The convergence of the conformal block function also follows in the case c = 1 by the absolute convergence of τ series, since it is a partial sum of the τ series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlevé equations.

A connection between tests for absolute convergence of infinite series, or how to be fair

The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers ∑ n = 0 ∞ a n are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for more series) than the Ratio Test. This relation inspires introduction of some intermediate strength tests (stronger than the Ratio Test and weaker than the Root Test) that we call Power Mean Tests (they, in particular, include the Arithmetic Mean Test). We show the connection between the Root, the Power Mean and the Ratio Tests. We also note that all these tests are related to the test that we formulate and call Generalised f-mean Test (or Kolmogorov–Nagumo–de Finetti mean Test). We provide an example of an infinite series, where the Arithmetic Mean test is the test that should be used for convergence verification, because the Root and the Ratio Tests are not easy to apply. We conclude this work with the statement emphasising why it is ‘fair’ to include the summarising Corollary 3.1 in our Calculus course.

Assessment of Green Innovation Efficiency in Chinese Industrial Enterprises Based on an Improved Relational Two-Stage DEA Approach: Regional Disparities and Convergence Analysis

Industrial enterprises are characterized by significant energy consumption and high emissions. Therefore, the implementation of green innovation by these enterprises is beneficial for promoting sustainable economic development and safeguarding the ecological environment. In this study, a relational two-stage DEA model containing shared inputs and undesired outputs is constructed to evaluate and decompose the green innovation efficiency (GIE) of Chinese industrial enterprises across 30 provinces from 2012 to 2021. Since the objective function of this model is nonlinear, a heuristic search method is employed for its resolution. On the basis of efficiency evaluation, the Gini coefficient, kernel density estimation, and convergence analysis are further employed to investigate the regional disparities and convergence properties in the two-stage efficiency of green innovation. Our findings are as follows: (1) The average GIE of Chinese industrial enterprises demonstrates a fluctuating upward trajectory, with significant regional disparities observed between provinces. (2) Regional disparities in R&D efficiency (RDE) and achievement conversion efficiency (ACE) have diminished in all regions. Super-variable density and interregional differences serve as the primary sources of regional disparities in RDE and ACE, respectively. (3) The presence of absolute and conditional convergence in RDE and ACE is observed across all regions. To improve the GIE of Chinese industrial enterprises, it is imperative to emphasize the heterogeneous impact of economic levels, industrial structure, and the degree of openness across various regions and stages of green innovation.

Regional Difference and Spatial Convergence of Land Use Carbon Emissions in Three Urban Agglomerations of Yangtze River Economic Belt

It is of great significance to explore the regional differences of land use carbon emission （LUCE） in the Yangtze River Economic Belt and the path of coordinated emission reduction for regional sustainable development. Based on the LUCE estimation method, this study scientifically calculated the LUCE of the three major urban agglomerations of the Yangtze River Economic Belt （Yangtze River Delta, middle reaches of the Yangtze River, and Chengdu-Chongqing urban agglomeration） from 2010 to 2020. Kernel density estimation and the spatial convergence model were used to study the dynamic evolution, regional differences, and convergence characteristics of LUCE. The results showed that： ① The carbon absorption of forest land, water areas, grassland, and unused land were relatively small in terms of carbon emissions from cultivated land and construction land. The carbon emission of construction land increased gradually, whereas the carbon absorption of four carbon sinks fluctuated little during the study period. ② The core density curves of different urban agglomerations showed different distribution patterns, extensibility, and polarization characteristics but generally tended to be balanced. ③ From 2010 to 2020, the LUCE of the Yangtze River Economic Belt as a whole showed the spatio-temporal characteristics of increasing first and then decreasing and high in the east and low in the west. The LUCE of the central cities of the three urban agglomerations were at the highest level steadily, and stable coupling mechanisms had not been established between the economic development level and the ecological environment. ④ The LUCE of the three urban agglomerations in the Yangtze River Economic Belt all had absolute β convergence and also had conditional β convergence under the model control variables such as economic development level, urbanization level, industrial structure, population density, and environmental regulation, etc., and the conditional convergence speed was greater than the absolute convergence speed in each region. The convergence speed of the Yangtze River Delta urban agglomeration was the slowest. The above conclusions provide support for the coordinated emission reduction path of the three urban agglomerations in the Yangtze River Economic Belt and are also conducive to actively and steadily promoting the realization of the goals of carbon peak and carbon neutralization.

Characterizations of some new classes of four-dimensional matrices and dual spaces on the double spaces of Cesàro means

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces |C_{1,1}|_{k} defined by combining the first order Cesàro means with the concept of absolute summability k≥1. Beside this, we determine the α-dual of the space |C_{1,1}|₁ and the β(bp)- and γ-duals of the spaces |C_{1,1}|_{k} for k≥1. Finally, we characterize some new four-dimensional matrix classes (|C_{1,1}|_{k},υ), (|C_{1,1}|₁,υ), (|C_{1,1}|₁,L_{k}), (|C_{1,1}|_{k},L_{u}), (L_{u},|C_{1,1}|_{k}) and (L_{k},|C_{1,1}|₁), where υ∈{M_{u},C_{bp}} for 1≤k

Norm of a generalized Hilbert linear and bilinear form on the Hardy spaces

In this paper, the precise norm of a generalized Hilbert linear form HLλ on the Hardy space H1 and that of the corresponding bilinear form HBλ on the Hardy space pair (Hp,Hq) are given for all λ∈{x∈R:x≠0,−1,−2,⋯}. Furthermore, the classical Hardy's inequality on the absolute convergence of HL1 is generalized to all HLλ with λ>0.

Impact of Covid-19 on the Economic Growth of ASEAN Countries: Convergence or Divergence?

This contribution addresses the question of whether economic growth among ASEAN member countries is converging or diverging within the context of the COVID-19 pandemic. ASEAN has significantly developed economic and social mechanisms aimed at achieving greater integration among member countries, with the goal of enhancing economic growth and especially accelerating the catch-up of the poorer member countries. How resilient have the member countries been? Has the COVID-19 pandemic more severely affected the poorer member countries? To answer these questions, our contribution addresses this issue based on economic growth theories, particularly drawing on Solow’s theory regarding beta and gamma convergence. The study covers the period from 2000 to 2022. In the pre-pandemic period from 2000 to 2019, there is a general phenomenon of absolute convergence, demonstrating a reduction in the disparity in absolute values between less advanced and more modern countries, which also leads to a reduction in inequality as shown by the confirmation of the σ-convergence hypothesis. This is true for both GDPC and GDPW variables. The catch-up effect of absolute β-convergence is not evident when looking at the period from 2020 to 2022. During this timeframe, both the absolute β-convergence and σ-convergence hypotheses are rejected, which slows down the economic growth of poorer economies relative to the more advanced ASEAN countries. Overall, the entire 2000–2022 period confirms the unconditional β-convergence and σ-convergence assumptions of the pre-pandemic period.

Transition towards a low-carbon global economy: An integrated analysis of club convergence, catch-up and the agenda 2030

This study employs the Biennial Malmquist-Luenberger (BML) productivity approach to measure Green Total Factor Productivity (GTFP) in 120 countries from 1990 to 2021. Next, the Phillips and Sul methodology is applied to investigate whether these countries have witnessed an absolute convergence process, or whether there has been a club convergence process. We also use an Ordered Logit method to identify the factors affecting the formation of clubs. Finally, the fuzzy regression discontinuity method is used to examine the exogenous shock of the 2030 agenda on the global GTFP. The key findings are presented as: (1) during the study period, the global GTFP on average has shown an upward trajectory. The European region performed better than the other regions, whereas Sub-Saharan Africa lagged in GTFP. Developed economies led in technological progress, while developing ones have achieved slightly higher carbon efficiency; (2) although the full sample divergence was supported, three clubs were identified with significant disparities in GTFP. We found that per capita GDP, industrial structure, energy structure, and particularly, indigenous and foreign innovations play major roles in explaining the formation of convergence clubs; (3) there was an apparent positive jump at the cut-off, indicating agenda 2030′s favorable treatment effect on the global GTFP. Therefore, all countries, poorer and richer alike, should use the half-way momentum to self-critically review and revise their national SDG strategies and commit to strengthening multilateralism.

Cross-sectional and spatial panel data analysis of territorial economic cohesion in the European Union regions based on convergence approach: From 2 to 8 per cent?

From an economic, political and social standpoint, one of the most evident and visible features of today's European Union as a supranational regional organization is its heterogeneity, where disparity seems to be the common denominator. This leads to the interest for measuring the territorial economic cohesion of the EU. From an eminently economic perspective, and working with the GDP per capita of the EU NUTS-2 regions for the period 2003–2021, this paper aims to provide evidence of a lack of territorial economic cohesion through a beta and sigma convergence methodology by applying cross-sectional and spatial panel data analysis.The findings show that the speed of convergence depends mainly on the level of economic development, its cycles and the heterogeneity of the, which implies conditional convergence. Less developed regions show higher convergence speeds, which are also accentuated during recession periods. Greater heterogeneity among the regions also increases the convergence speed, while accentuating in the less developed regions. In general terms, the results reveal convergence speeds of the entire NUTS-2 regions between 7 and 11 per cent (much higher than 2 per cent under absolute convergence). Likewise, when considering spatial dependence, a reduction in convergence speeds between approximately 3 and 8 per cent is detected. Finally, the 29 vulnerable regions have been identified, with economic development and growth below the EU average mean, emphasizing the need to take the concerns of territorial economic cohesion into account.

About Absolute Convergence of Fourier Series of Almost Periodic Functions

The current stage of development of the theory of almost periodic functions is characterized by a desire for analysis and processing of a huge amount of accumulated scientific and practical material. The theory of almost periodic functions arose in the 20-30 s of the twentieth century; currently, extensive literature has accumulated on various issues of this theory. Long before the creation of the general theory of almost periodic functions, the outstanding Riga mathematician P. Bol drew attention to such functions. For functions of many variables f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>), Bol considered the corresponding multiple Fourier series and, in p-dimensional Euclidean space, a straight line passing through the origin: x<sub>1</sub>=a<sub>1</sub> t, x<sub>2</sub>=a<sub>2</sub> t,..., x<sub>p</sub>=a<sub>p</sub>t, where a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - some real, non-zero numbers. Considering the value of the function f(x<sub>1</sub>, x<sub>2,</sub>...x<sub>p</sub>) on this line, he obtains a function of one variable φ(t) = f(a<sub>1</sub> t, a<sub>2</sub> t,...a<sub>p</sub> t) and proves that this function is almost periodic. With some choice of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> - it may happen that this function is periodic. However, if the numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> are linearly independent, then you can easily make sure that the function will not be a periodic function. Further development of the problem was carried out by the French mathematician E. Escalangon. However, the main significant drawback of the results of Bol and Escalangon was that from the very beginning, starting with the definition of almost-periodic functions, they introduced into consideration a fixed system of numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub> associated with the almost-period (τ). This drawback was eliminated by the Danish mathematician G. Bohr, who developed in general terms the theory of continuous almost-periodic functions. Bohr's research in its methods was closely related to Bohl's research. However, Bohr did not impose restrictions such as Bohl’s inequality in advance for the almost period. The results obtained by Bol and Bohr were based on the deep connection between almost periodic functions and periodic functions of many variables. The article examines the question of sufficient conditions for the absolute and uniform convergence of Fourier series of uniform almost periodic functions in the case when the Fourier exponents have a single limit point at zero, i.e. λ<sub>k</sub>→0 (k→∞). In this case, the Laplace transform is used for the first time as a structural characteristic.

On generalization of some theorems with absolute summability factors of infinite series

AbstractThe generalizations of some results of H. Bor, L. Leindler, and H. Sevli pertaining to absolute summability are examined.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

Regional disparities in the European Union. A machine learning approach

We investigate the hypothesis of regional convergence in the per-capita GDP in 242 European regions (NUTS2) during the 2000–2021 period. The literature shows mixed results, from absolute convergence towards a joint long-run distribution to multiple regimes (convergence club). Our results show a broad convergence to an unimodal distribution. Although the GDP distribution was characterized by a twin-peak property in 2000, it tends to disappear over time, bringing, in 2021, to an unimodal distribution. Physical and human capital is the most responsible for the convergence process and the EU cohesion funds. To empirically investigate the question, we first apply alternative techniques of cluster identification. Later, we assess whether clusters and covariates affect the per-capita GDP. We use a novel machine learning algorithm (GPBoost) instead of the more traditional techniques used in the current literature. The results show that a convergence process is at work; physical and human capital are mainly responsible for the gdp explanation. but eu funds play a relevant role as well. moreover, complementarities do exist among these variables.

Analysing the impact of digital technology diffusion on the efficiency and convergence process of the commercial banking industry of Pakistan

In a dynamic financial ecosystem, digitalisation is supporting banks in revamping their business processes to be more efficient, reducing costs, coping with customers’ evolving demands, and keeping them abreast of market competition. This two-stage research aims to investigate three issues in Pakistan’s banking industry from 2006 to 2020: (i) banking efficiency; (ii) the impact of digitalisation on banking efficiency; and (iii) banking efficiency’s absolute and conditional convergence. In our first-stage analysis, bootstrap data envelopment analysis has been applied, which exhibits bias-corrected overall, pure, and scale efficiencies of 74, 77, and 96%, respectively. In the second-stage analysis, we executed Tobit and two-step dynamic panel data system generalised method of moments (DPDSYS-GMM) models, and the results uncover that digitalisation has a positive influence on banking efficiency. Findings confirm that return on assets, bank size, interest rate, and gross domestic product growth rate have a positive association with banking efficiency. Our research reveals that state-controlled banks outperform their private sector and special-purpose counterparts. Our DPDSYS-GMM findings validate β and σ-convergence, implying that initially, low-efficient banks caught up (converged) to the more efficient opponents, reducing cross-sectional efficiency dispersion and attaining common equilibrium. The findings of banking efficiency conditional β-convergence assert that the adoption of digital technology has played a critical role in the convergence process and that digitalisation has acted as a catalyst for less efficient banks to catch up significantly faster to their more efficient rivals. This study displays digitalisation’s disruptive influence on the overall transformation of Pakistan’s banking system in the modern era, resulting in significant efficiency gains.

Re-Orthogonalized/Affine GMRES and Orthogonalized Maximal Projection Algorithm for Solving Linear Systems

GMRES is one of the most powerful and popular methods to solve linear systems in the Krylov subspace; we examine it from two viewpoints: to maximize the decreasing length of the residual vector, and to maintain the orthogonality of the consecutive residual vector. A stabilization factor, η, to measure the deviation from the orthogonality of the residual vector is inserted into GMRES to preserve the orthogonality automatically. The re-orthogonalized GMRES (ROGMRES) method guarantees the absolute convergence; even the orthogonality is lost gradually in the GMRES iteration. When η<1/2, the residuals’ lengths of GMRES and GMRES(m) no longer decrease; hence, η<1/2 can be adopted as a stopping criterion to terminate the iterations. We prove η=1 for the ROGMRES method; it automatically keeps the orthogonality, and maintains the maximality for reducing the length of the residual vector. We improve GMRES by seeking the descent vector to minimize the residual in a larger space of the affine Krylov subspace. The resulting orthogonalized maximal projection algorithm (OMPA) is identified as having good performance. We further derive the iterative formulas by extending the GMRES method to the affine Krylov subspace; these equations are slightly different from the equations derived by Saad and Schultz (1986). The affine GMRES method is combined with the orthogonalization technique to generate a powerful affine GMRES (A-GMRES) method with high performance.

An investigation into the convergence of economic growth among Indian States and the path ahead

This study offers a comprehensive examination of economic growth and convergence in the Indian States and Union Territories (U.T.) over the period from 1991 to 2020. It investigates absolute, sigma, and conditional convergence within this diverse set of states and utilizes the augmented Solow and extended Solow models to explore conditional convergence dynamics. The empirical findings reveal several significant insights. First, there is no substantial negative correlation between the initial per capita GDP ratio and the average annual growth rate, indicating the absence of absolute convergence across the Indian States and U.T. economies during the study period. These results align with those obtained from sigma convergence analysis, reinforcing the absence of widespread convergence. However, conditional convergence is observed, as evidenced by the rate of conditional convergence (coefficient of initial GDP per capita) estimated at 0.038 among the Indian States and U.T. The presence of conditional convergence implies that while initial conditions matter, other factors, including physical and human capital, population growth, and additional variables, significantly contribute to the growth and convergence of Indian regions. The study concludes that policies aimed at promoting economic growth in the Indian States should prioritize the expansion of the labor force, investments in physical and human capital, and prudent government consumption. Furthermore, fostering an environment that encourages access to new technologies and ideas, maintaining sound macroeconomic management, and increasing investments in human capital formation are essential for sustained growth. Effective resource allocation through prudent budgetary policies and heightened investments in the health sector are recommended. Incentives to reduce fertility rates and adept monetary policy management are also identified as crucial elements for ensuring stability and sustainable growth. In summary, this research underscores the importance of adopting a holistic approach to foster economic growth and convergence in the Indian States. The suggested policy measures create a conducive environment for sustained development and prosperity in this diverse and dynamic region.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).