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This work deals with fracture mechanisms in quasi-brittle materials, focusing on the characterization of the Fracture Process Zone (FPZ) of specimens under tensile load. Particularly, paper was used as model material. Experiments were conducted on notched and unnotched specimens. Based on an image analysis of these observations, a stochastic finite element model was developed, using both a nonlocal stress-based approach and a discretized random field modeling of the Young's modulus. The proposed methodology allowed characterizing the damage zone and the size of the FPZ, analyzing the influence of the mesostructure, composed of flocs (fiber aggregates where the basis weight is larger than the average one) and antiflocs (complement of flocs). The area of the active FPZ and the normalized stress drop were linked using a surface energy dissipated in the active FPZ. The stress drop, until limiting value, increased with the width of the active FPZ. Finally, a relationship between the surface energy and the nonlocal internal length was established. [ABSTRACT FROM AUTHOR]

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Coal pillar is a crucial element formed during underground coal mining, playing a significant role in ensuring coal mining operations are conducted safely. Historically, research on coal pillars has primarily focused on regular-shaped pillars, with limited attention given to special-shaped pillars. Therefore, this paper aims to investigate the special-shaped coal pillar formed after the excavation of the newly constructed 3307 headgate in Dayang Coal Mine, China. The primary object of this paper is to elucidate the mechanism behind the instability failure of special-shaped coal pillars and verify the influence of various geological conditions on their stability. To accomplish these goals, numerical simulation techniques were employed to analyze the stress distribution patterns and plastic zone failure characteristics of special-shaped coal pillar. The investigation revealed the presence of stress concentration phenomena within special-shaped coal pillar, where the stress peak values were directly correlated with the width of the pillar. Moreover, it was observed that as the width of coal pillar decreased, the stress curve gradually transitioned from a bimodal shape to a unimodal shape. Additionally, the ratio of the plastic zone within coal pillar increased as the width decreased, with pronounced tensile failures occurring at sharp corners. Building upon theoretical analysis and numerical simulation results, the interior of special-shaped coal pillar was classified into four distinct zones: damaged zone, progressive zone, stable zone, and special zone. Based on this zoning approach, a stability control technology was developed, employing a combination of "pier pillar support + flexible formwork concrete + grouting reinforcement + high-strength cable and bolt + U-steel." Subsequently, an industrial experiment was conducted to validate the efficacy of this control technology. Field monitoring results demonstrated that the newly excavated roadway exhibited minimal deformation and damage, thereby satisfying the requirements for safe and efficient mine production over the anticipated service life. Highlights: The failure characteristics and bearing mechanism of special-shaped coal pillars were analyzed, and a mechanical model was established, allowing exploration of the influence of different geological conditions on pillar stability. The stress evolution and plastic zone development characteristics of special-shaped coal pillar during the whole cycle were studied. A zoning criterion for special-shaped coal pillar was proposed, along with corresponding control measures, which were successfully applied to the site, effectively controlled the deformation of roadway. [ABSTRACT FROM AUTHOR]

Positive semidefinite kernels on spheres cross time are on demand in spatial statistics, machine learning and numerical analysis, motivated by applications in the climate, environmental and earth sciences. Most of the recent literature has focused on kernels that are isotropic on the sphere and stationary in time, but this assumption is often overly limited in quantifying spatial dependence properly. A broader class of kernels can be obtained by trading isotropy for axial symmetry, i.e., stationarity with respect to longitudes only. This paper challenges theoretical aspects by providing a spectral representation that uniquely characterizes space-time positive semidefinite kernels that are axially symmetric on the sphere and stationary in time. We also address the problem of partial Fourier inversion and show that it is feasible under mild technical conditions. The second part of the paper focuses on constructive methods to build nonseparable kernels having axial symmetry on the sphere and stationarity over time. [ABSTRACT FROM AUTHOR]

The rock mechanics parameters are important indicators for risk assessment in tunnel construction. However, rock mechanical parameters estimated by traditional mechanical testing in the field are subject to considerable uncertainties due to limitations testing technology, which can directly lead to unreliable risk assessment results. Deformation data for parametric back analysis is a reliable and efficient method for estimating mechanical parameters of rock masses. Deformation monitoring data used for parameter back analysis is an efficient and reliable method for estimating rock mechanical parameters, but the current research generally suffers from inaccurate constitutive models, confusing selection of input data, and failure to consider the intrinsic correlation of mechanical parameters. In this paper, a deformation-based back analysis method for rock mechanical parameters was proposed by XGBoost with RC. Taking the Changning Tunnel in Yunnan Province, China, as an example, the construction process of the back analysis model was described in detail. To improve the reliability of the sample data, the Ubiquitous-Joint constitutive model was used and a more characteristic deformed data selection method was proposed. Furthermore, the performance of the back analysis model was improved through Bayesian Optimization of hyperparameters and order filtering of Regressor Chain, and the intrinsic correlation of the parameters was effectively considered through the Regressor Chain. The results showed that applying the rock mechanics parameters obtained from the back analysis model to numerical simulation can effectively predict deformation during tunnel construction, and the error rate of deformation prediction was within 10%, which demonstrates the reliability of the method in this paper. This method can be extended to other heavily fractured tunnels. Highlights: Summarized the technical process of back analysis of mechanics parameters based on machine learning, analyzed the current research status, and put forward the existing problems. Through the spatial–temporal effect analysis of deformation of rock mass, a reasonable selection method of the surrounding rock deformation index in the sample data is proposed. The performance of the back analysis model of rock mechanics parameters is effectively improved by hyperparameter optimization of Bayesian and order filtering of Regressor Chains. [ABSTRACT FROM AUTHOR]

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both "superelements" are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis. [ABSTRACT FROM AUTHOR]

This paper describes the experiments carried out on a mediaeval masonry tower in the historic centre of Lucca and some finite element numerical simulations of the tower's experimental response. The Guinigi Tower, one of the most iconic monuments in Lucca, has been continuously monitored by high-sensitivity seismic stations that recorded the structure's response to the dynamic actions of the surrounding environment. The monitoring campaign results have been analysed to show the effectiveness of dynamic monitoring as a valuable source of information on the structural properties of the tower. The dynamic analyses of the tower and the surrounding palace subjected to some seismic events recorded during the experiments have highlighted the capabilities of experiment-based finite element modelling. The calibration of the finite element model and the numerical analysis have been carried out by resorting to procedures developed at ISTI-CNR and able to consider the nonlinear behaviour of masonry materials. [ABSTRACT FROM AUTHOR]

In soft soil layers, piles are frequently loaded laterally by horizontal soil movements. In many cases, the lateral pressure acting on piles due to horizontal soil movements is calculated with empirically or analytically based approaches, respectively. However, most of these design approaches do not consider possible influences on the resulting pile loads. This paper presents the results of model tests and numerical simulations on single piles and pile groups in cohesive soil subjected to lateral loads which were carried out to overcome limitations of available design approaches. Based on extensive small-scale 1 × g-model tests and numerical investigations with the finite element method, influencing factors on the lateral pressure, such as the roughness of the pile–soil interface, the pile size, the pile shape and the pile spacing were identified. A parametric study with the numerical model quantified the most relevant factors influencing the development of lateral pressure on piles due to adjacent surface loads and lead to the development of a simplified analysis approach. [ABSTRACT FROM AUTHOR]

The paper presents findings from a parametric study analysing geometric (e.g., shape ratio, edge inclination) and stratigraphic factors (e.g. impedance ratio) influencing ground motion in trapezoidal valleys. The study involved 2160 visco-elastic analyses, considering 180 2D models with diverse shapes and soil properties, undergoing 12 synthetic input motions. Analyses results showed that the motion at the valley centre increases with both shape and impedance ratios, while it is independent of the edge slope; on the other hand, the maximum amplification at the edges depends on their inclination and on the impedance ratio, while it is independent of the valley shape. The position and size of the zone of maximum amplification at the edges depend on all the previous parameters. A valley amplification factor (VAF) is introduced to quantify spectral acceleration increase due to 2D effects. Closed-form equations are proposed to evaluate VAF based on valley properties. The proposed VAF is then applied to predict seismic amplification in two central Italian valleys, providing results well-comparable to those obtained from 2D numerical analyses. The described approach can be easily implemented into codes of practice as a conservative design tool to estimate 2D amplification along the surface of 'shallow valleys' subjected to moderate seismic actions. [ABSTRACT FROM AUTHOR]

This contribution discusses the influence of different barrier update strategies on the performance and robustness of an interior point algorithm for single-crystal plasticity at small strains. To this end, single-crystal plasticity is first briefly presented in the framework of a primal-dual interior point algorithm to outline the general algorithmic structure. The manner in which the barrier parameter is modified within the interior point method, steering the penalization of constraints, plays a crucial role for the robustness and efficiency of the overall algorithm. In this paper, we compare and analyze different strategies in the framework of crystal plasticity. In a thorough analysis of a numerical example covering a broad range of settings in monocrystals, we investigate robust hyperparameter ranges and identify the most efficient and robust barrier parameter update strategies. [ABSTRACT FROM AUTHOR]

Nonsymmetric algebraic Riccati equations are a class of nonlinear matrix equations with wide applications. In this paper, numerical solution of nonsymmetric algebraic Riccati equations is studied. A generalized ALI iteration method is proposed to solve the equation, which is an extension of the ALI iteration method and the MALI iteration method. Theoretical analysis and numerical experiments show that the proposed method is feasible and is more effective than the ALI iteration method and the MALI iteration method. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a finite volume approximation of a convective diffusive elliptic problem with Neumann boundary conditions and L 1 data. To deal with the non-coercive character of the equation and the low regularity of the right hand-side we mix the finite volume tools and the renormalized techniques. To handle the Neumann boundary conditions we choose solutions having a null median and we prove a convergence result. We present also some numerical experiments in dimension 2 to illustrate the rate of convergence. [ABSTRACT FROM AUTHOR]

Rapid flow-like landslides frequently occur in mountainous regions. To mitigate the disasters caused by these landslides, it is crucial to develop robust numerical models that can accurately predict their run-out processes. Models based on the material point method (MPM) offer significant advantages in simulating large deformation issues in geomaterials, including landslides. However, applying these models to accurately simulate real-world rapid flow-like landslides remains a challenge, primarily due to the complexities involved in handling the three-dimensional (3D) sliding bed boundary. This paper introduces a novel 3D MPM model specifically designed for simulating rapid flow-like landslides that propagate across curved beds. The constraints of the sliding bed on the landslide are imposed by the boundary nodes close to the bed. These boundary nodes carry information about the normal vector of the sliding bed, derived directly from the digital terrain model (DTM). Furthermore, the model integrates a hybrid formulation that combines the Full Lagrangian Implicit Particle (FLIP) method with the Particle In Cell (PIC) method, facilitating a stable solution for the velocity and position of material points. The effectiveness of the proposed model is confirmed through a numerical analysis of a rigid block sliding down an inclined plane and an experiment of sand flow on a curved bed. The simulation results from these two benchmark scenarios align closely with both analytical and experimental data, attesting to the validity of the model. The model is then applied to analyze a rapid flow-like landslide that occurred in Gansu Province, China, characterized by a curved sliding bed. The outcomes illustrate the model's capability to efficiently capture the landslide's climbing and turning motions induced by the meandering topography. Moreover, it successfully reproduces the main deposition characteristics observed in the field, demonstrating the model's strong suitability for simulating the propagation of rapid flow-like landslides on naturally curved beds. [ABSTRACT FROM AUTHOR]

Core-shell droplets exhibit advantageous physical and chemical properties attributed to their unique structural composition, thereby presenting broader prospects for application in diverse fields including biomedicine, drug delivery, food science, and cosmetics industry. Controllable preparation of monodisperse, small-volume, and effectively encapsulated core-shell droplets is crucial for advancing various applications. In this paper, coaxial electrohydrodynamic (EHD) printing technology is used to generate uniform and stable arrays of water-in-oil core-shell droplets. By using theoretical analysis and numerical simulation to determine the conditions for stable generation of water-in-oil droplets, this study successfully addresses the issue of unstable coaxial charged jets and the generation of satellite droplets by monitoring the jet recognition. The simulation results show that too high voltage and inner axis solution supply speed can lead to the formation of satellite droplets or spray. When Qout/Qin≥1, it is less likely to produce satellite droplets. Combining the results of numerical simulation and the jet recognition module, the coaxial jet mode can be effectively monitored. The jet is controlled in a stable printing mode (pulsating droplet mode) and complete in one injection within 5 ms to achieve uniform droplet arrays with diameters around 50 μm and a droplet diameter coefficient of variation (CV) below 6%. This work provides significant help on the injection morphology control of water-in-oil droplets, and provides a feasible technical means for the application of coaxial EHD printing technology in various fields such as bioassay, biosensing, and fluorescence display. [ABSTRACT FROM AUTHOR]

The segment misalignment is a common defect in the segment installation of a shield tunnel construction, due to the complexity and uncertainty of its construction conditions. This raises a concerning about the safety of the segment lining for the shield tunnel. In this paper, the integrated numerical models with surrounding rock, peastone grouting and segment lining were built using three-dimensional finite element method, by introducing two kinds of misaligned defects of segments. One was the misalignment that elongated the horizontal axis of the transversal ring section of segment lining, another was the misalignment between adjacent rings of segments. To eliminate the impact of boundary on numerical results, seven rings of segments are built for the three-dimensional finite element models, and the middle ring is dealt with for the results analysis under V-class surrounding rock, including the circumferential stress of outer and inner layers of the segment, the contact stress on joint surface between segments, and the stress of locating pins between adjacent rings. Results indicate that the two kinds of misaligned defects create pronounced impact on tensile stress of segment at the bottom and the crown segments, respectively, which produces a greater tensile stress to increase cracking risk even abrupt fracture of segment. However, the misaligned defects have a slight impact on the contact stress and the locating pins stress. Therefore, attention should be paid to the cracking of bottom or crown segments in the segment lining of shield tunnel. [ABSTRACT FROM AUTHOR]

Reservoir simulation is crucial for understanding the flow response in underground reservoirs, and it significantly helps reduce uncertainties in geological characterization and optimize methodologies for field development strategies. However, providing efficient and accurate solutions for the strong heterogeneity remains challenging, as most of the discretization methods cannot handle this complexity. In this work, we perform a comprehensive assessment of various numerical linearization techniques employed in reservoir simulation, particularly focusing on the performance of the nonlinear solver for problem dealing with fluid flow in porous media. The primary linearization methods examined are finite difference central (FDC), finite forward difference (FDF), and operator-based linearization (OBL). These methods are rigorously analyzed and compared in terms of their accuracy, computational efficiency, and adaptability to changing reservoir conditions. The results demonstrate that each method has distinct strengths and limitations. The FDC method is more accurate particularly in complex simulations where strong heterogeneity are introduced but is generally slower in convergence. The OBL on the other hand, is more efficient and converges quickly, which makes it suitable for scenarios with limited computational resources and simple physics, while the FDF method provides a balanced combination of precision and computational speed, contingent upon careful step size management of the derivative estimations. This paper aims to guide the selection of appropriate linearization techniques for enhancing nonlinear solvers' accuracy and efficiency in reservoir simulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the complex flows when all three regimes pre-Darcy, Darcy and post-Darcy may be present in different portions of a same domain. We unify all three flow regimes under mathematics formulation. We describe the flow of a single-phase fluid in by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stabilit1y of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters. [ABSTRACT FROM AUTHOR]

Micro-mixers and piezoelectric pumps have been studied as independent parts, which achieves only single mixing or pumping functions. A new valveless piezoelectric mixing pump (VPMP) driven by a sinusoidal signal is proposed in this paper to achieve a structure with both pumping and mixing functions. It accomplishes this by placing obstacles in the pump chamber and in the flow channel then adding active mixing. Because it is small in size, has a simple structure and is easy to control, it can be used in micro mixing applications. In this study, numerical simulation analysis using the Fluent software tests if the new VPMP has good mixing and pumping performance, and analyzes the effect of the obstacles. Simultaneously, the results of numerical simulation are verified with experiments. It was proven that the flow rate of the VPMP can reach 12.26 mL/min, and that the liquid can achieve uniform mixing. Moreover, the obstacle placed in the pump chamber has a more obvious impact on the mixing capability, and the obstacle placed in the channel has a more obvious impact on the flow rate. In addition, with the increase of the amplitude of driving signal, the flow rate of the VPMP increased correspondingly. With the increase of the frequency, the flow rate first increased and then decreased. The capability of mixing increased with an increase of working frequency. The experimental results showed that the VPMP can achieve good pumping and mixing performance, and that one structure can possess both mixing and pumping functions. [ABSTRACT FROM AUTHOR]

This paper presents simple numerical solutions for electromagnetic plane waves in spatially homogenous time varying medium. The solution is based on converting the resulting second order differential equation into two combined ordinary differential equations which are solved numerically by using the built-in ode113 function in Matlab. By using this method, the time domain responses of the electric and magnetic fields at fixed point in space are obtained. The proposed method is applied on two cases: linearly time varying medium and sinusoidally time varying medium. The corresponding frequency domain response is obtained by using inverse Fourier transformation of the obtained time domain response. The proposed method is compared with FDTD solution. It is found that the proposed method has the same accuracy of FDTD with much less computational time. [ABSTRACT FROM AUTHOR]

This paper investigates the performance of a water sprinkler cooling system for a centrifugal Babbitt lining machine, used to renovate industrial bearings. A three-dimensional and transient numerical simulation was conducted to study the effects of various parameters on the solidification rate, including mold rotational speed, sprinkler flow rate and angle, and initial temperatures of the mold and Babbitt alloy. Results demonstrate that the temperature drop in the Babbitt layer and mold is directly proportional to the rotational speed, with a 15% change in rotational speed from the reference case resulting in a 5 °C change in the average temperature of the Babbitt alloy, while clockwise (CW) rotation of the mold leads to greater temperature reduction. An optimal rotational speed of 550 rpm CW was determined. The study also identified an optimal total water flow rate of 8.7e−5 m3/s, with minimal additional cooling benefits beyond that point. Furthermore, vertical water sprinkler angles were found to provide superior cooling performance. The initial mold temperature was found to have a more significant influence on the final average Babbitt temperature than the initial Babbitt temperature, due to the mold's larger mass and thermal inertia. The qualitative and quantitative results of this research offer valuable insights for enhancing the design of water sprinkler cooling systems in centrifugal Babbitt lining machines, providing a pathway for transitioning from manual to automated cooling processes. This innovative approach has the potential to transform industries in underdeveloped countries, improving their efficiency, quality, and safety. [ABSTRACT FROM AUTHOR]

A hollow-core photonic crystal fiber based Refractive Index sensor in the terahertz frequency regime for bio-sensing application is proposed in this paper. Refractive index variation is used for the identification of different analytes with refractive indices ranging from n = 1.35 to 1.39. Zeonex is chosen as the fiber material. The proposed photonic crystal fiber (PCF) sensor features a solitary rectangular core enclosed by an arrangement of 16 rectangular air holes within the cladding area with different heights and widths. Simulation and numerical analysis are conducted using Finite Element Method-based based Comsol Multiphysics Software. The proposed PCF sensor exhibits relative sensitivity of 98.8%, an Effective Mode Area of 1.06 × 10−7 m2 with a very low confinement loss of 7.751 × 10–14 dB/cm are achieved at a frequency of 1.4 THz for an analyte of refractive index 1.39 in X Polarization direction of electric field. Moreover, birefringence, non-linearity, effective refractive index and power fraction are also analyzed. Further simple PCF structure based on rectangle ensures that fabrication is feasible with current fabrication techniques.Please check and confirm that the authors and their respective affiliations have been correctly identified and amend if necessary.I have checked the authors name and affiliations and found its correct.Kindly check and confirm whether the corresponding author and corresponding mail ID is correctly identified.I have checked about corresponding authors name and email id. I found it correct. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified single-mode rate equations of semiconductor lasers (SCLs) wherefrom the high dimension of complex dynamics of hyperchaotic islands is studied. Detailed analytical and numerical analysis are investigated under the conjugated impact of three decisive parameters of SCL-systems most specifically, the impact of a new hyperchaotic parameter. Thereby, various quantifying dynamic methods are applied and underscore high-dimensional hyperchaotic states as well as fractal characteristics of new hyperchaotic attractors featuring an entangled central part. The high dimension of the said hyperchaos is also supported by a constructive interference of specific control parameters used which give arise to a complex dynamics of the maximum Lyapunov exponent by means of various two-parameter bifurcation diagrams. Additionally, several cross-sections of Poincaré map bring out odd folded dynamical structures and irregular truncated torus pertaining thereto. In this light, the electronic circuit of the modified model is constructed and simulated in the PSpice environment so as to further support analytical and numerical findings. Finally, controlling hyperchaos is structurally investigated, resulting in the suppression of hyperchaotic states with the generation of continuous wave emission, periodic and quasi-periodic attractors. [ABSTRACT FROM AUTHOR]

Layered rock slope exists widely. Because of its special slope structure, it is prone to bending deformation and toppling failure, which is a serious threat to engineering construction and safety operation. At present, the research of layered rock slope still has great innovation potential. During the construction of Wudongde Hydropower Station on Jinsha River, safety and stability problems such as slope geological structure development, face rock unloading and relaxation, and even slip and large deformation were encountered. Through field exploration, it is found that the rock and soil stratification of the slope on both sides of Wudongde Hydropower Station is highly obvious. At present, there is a lack of research on-site long-term displacement monitoring of layered rock high-steep slope, especially for layered slope in complex hydrogeology and construction environment. In order to strengthen the research on the deformation and stability of layered rock slope, this paper analyzes the measured displacement data of Wudongde hydropower station slope, and establishes three-dimensional geological finite element model with the help of numerical simulation software. The stability of the slope is calculated by combining the finite difference method and the strength reduction method. Finally, the evolution mechanism of the deformation of the layered rock slope is explained according to the geological structure characteristics. The main conclusions of this paper are as follows: the layered slope in the dam reservoir area is prone to deformation under the combined action of long-term construction disturbance and fissure water seepage, and the construction disturbance has a strong influence on the artificial excavation area below 1070 m, and the maximum rock mass deformation and surface displacement in the artificial excavation area of the slope reach 92.2 mm and 312.5 mm, respectively. However, the influence of construction disturbance on the natural mountain above 1070 m is limited, the valley deformation of the natural mountain on the left bank of the reservoir area is higher than that on the right bank, and the cumulative deformation is still less than 20 mm. The influence of seepage on the displacement of the area with higher elevation at the top of the slope is more obvious, and the influence of excavation and other disturbances on the displacement of the artificial excavation area with lower elevation is more obvious. The deformation of the river valley in the water cushion pond behind the dam increases slowly, and the change trend of the field deformation data is mostly consistent with that of the numerical calculation. The horizontal shrinkage of the mountains on both sides shows a contraction trend on the whole, and the maximum horizontal shrinkage calculated by numerical simulation is close to 20 mm, which is located at the elevation of 990 m. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach for analyzing efficient numerical solution of a class of singularly perturbed parabolic convection-diffusion PDEs having a finite jump discontinuity mostly in the source function at the interior of the spatial domain. These PDEs often appear in mathematical modeling of the semiconductor devices; and solutions of such problems usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. The prime objectives of this paper are to overcome the theoretical challenge related to the monotonocity of the finite difference operator which utilizes one-sided second-order difference operators at the interface point; and also to establish higher-order numerical approximation in space, regardless of smaller and larger values of the parameter ε. Proving discrete maximum principle of the proposed difference operator is found to be challenging task on the standard Shishkin mesh adapted to both boundary and weak interior layers. We overcome this difficulty by constructing a special non-uniform mesh, called modified layer-adapted mesh; and hereby establish the stability as well as the parameter-uniform error estimate in the discrete supremum norm. Finally, the theoretical estimate is verified with numerical results for test examples with and without the exact solution. Moreover, numerical results are obtained for the semilinear PDEs and also compared with the implicit upwind scheme to exhibit the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The paper deals with numerical analysis of the effect of stress state and loading direction with respect to the rolling direction on damage and fracture behavior of anisotropic metals. The continuum damage model has been enhanced to take into account the influence of production-induced anisotropies and loading direction on damage criteria and on evolution equations of damage strains. Constitutive parameters are determined using experimental results taken from tests with uni- and biaxially loaded specimens. The focus of the paper is on three-dimensional micro-mechanical numerical analyses of micro-defect-containing representative volume elements covering a wide range of stress states. These calculations lead to more insight in the different damage and failure processes on the micro-scale and their influence on the macroscopic damage laws. With the obtained numerical results it is possible to detect general trends, to propose governing equations for the damage criteria, to develop evolution equations for the damage strains, and to identify constitutive parameters of the anisotropic material model. It is shown that the anisotropic behavior and the loading direction with respect to the principal axes of anisotropy affect the evolution of damage mechanisms on the micro-level as well as the corresponding damage strains. [ABSTRACT FROM AUTHOR]

The aim of this Authors’ Reply is to examine critically certain assertions by Logo in a Discussion and in two conference papers about a publication by Rozvany and Maute on a reliability based benchmark example [ABSTRACT FROM AUTHOR]

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect. [ABSTRACT FROM AUTHOR]

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data. [ABSTRACT FROM AUTHOR]

This paper deals with the mathematical analysis of Tuberculosis by using fractal fractional operator. Mycobacterium TB is the bacteria that causes tuberculosis. This airborne illness mostly impacts the lungs but may extend to other body organs. When the infected individual coughs, sneezes or speaks, the bacterium gets released into the air and travels from one person to another. Five classes have been formulated to study the dynamics of this disease: susceptible class, infected of DS, infected of MDR, isolated class, and recovered class. To study the suggested fractal fractional model’s wellposedness associated with existence results, and boundedness of solutions. Further, the invariant region of the considered model, positive solutions, equilibrium point, and reproduction number. One would typically employ a fractional calculus approach to obtain numerical solutions for the fractional order Tuberculosis model using the Adams-Bashforth-Moulton method. The fractional order derivatives in the model can be approximated using appropriate numerical schemes designed for fractional order differential equations. [ABSTRACT FROM AUTHOR]

This paper investigates an adapted loss function for the estimation of a linear regression with right censored responses. The adapted loss function could be used in composite quantile regression, which is a good method to handle the responses with high censored rate. Under some regular conditions, we establish the consistency and asymptotic normality of the resulting estimator. For estimation of regression parameters, we propose the MMCD algorithm, which generates satisfactory results for the proposed estimator. In addition, the algorithm can also be extended to the fused adaptive lasso penalized method to identify the interquantile commonality. The finite sample performances of the methods are further illustrated by numerical results and the analysis of two real datasets. [ABSTRACT FROM AUTHOR]

Selectively thickening core wall is an effective approach to improve the birefringence of hollow core photonic bandgap fiber (HC-PBGF). However, the variation of core wall thickness has a significant influence on the transmission loss and transmission bandwidth. In this paper, the effects of selectively thickening core wall on birefringence and loss performance of the HC-PBGF are systematically investigated. Numerical analysis results indicate that when the core wall thickness in the y-direction is 2.4 to 3.3 times as that in the x-direction, the birefringence of a 19-cell HC-PBGF can be improved to 10− 4, meanwhile the transmission loss can be controlled below 10 dB/km. The high birefringence (≥ 10− 4), low transmission loss (< 10 dB/km) performance can be maintained in a wide waveband from 1460 to 1610 nm, even under a tight bending radius of 6 mm. The applicability of selectively thickening core wall on improving birefringence is also confirmed in the 7-cell HC-PBGF and 37-cell HC-PBGF, where the birefringence of 7-cell HC-PBGF and 37-cell HC-PBGF can be enhanced to 5.5 × 10− 4 and 5.0 × 10− 5, respectively. Moreover, to further reduce the transmission loss, two modified structures (adding nodes or anti-resonant layer) are discussed. The numerical analysis results demonstrate that the structure by adding nodes on the thickened core wall is more advantageous in terms of reducing transmission loss and expanding transmission bandwidth. [ABSTRACT FROM AUTHOR]

Particle morphology is a crucial determining factor in the physicochemical properties of geological materials. Currently, there are still substantial challenges with the characterization and reconstruction of the morphology of irregular particles. Most currently used characterization and reconstruction methods were developed separately, and lack cohesive integration of parameters between the two processes. This limitation hinders the comprehensive characterization of the multiscale geometric features and impedes a clear relation between geometrical features and reconstruction parameters. Therefore, a unified 2D and 3D morphological characterization method for irregular particles is developed in this paper, enabling a systematic morphology quantification across three relative scales. Furthermore, a geometric parameters-driven particle reconstruction approach is proposed, which is the reverse of the quantification process, achieving precise and controllable 2D and 3D irregular morphology reconstruction. The proposed methods achieve, for the first time, an integrated quantification and reconstruction framework of irregular particle morphology, as well as a unified 2D and 3D morphology characterization. Notably, the proposed approach exhibits strong robustness and computational efficiency when performing 2D and 3D particle morphology quantification and precisely controllable virtual particle reconstruction, which are the bases for accurate mechanical analyses and numerical simulations. [ABSTRACT FROM AUTHOR]

With the development of cloud computing, users are more inclined to outsource complex computing tasks to cloud servers with strong computing capacity, and the cloud returns the final calculation results. However, the cloud is not completely trustworthy, which may leak the data of user and even return incorrect calculations on purpose. Therefore, it is important to verify the results of computing tasks without revealing the privacy of the users. Among all the computing tasks, the polynomial calculation is widely used in information security, linear algebra, signal processing and other fields. Most existing polynomial-based verifiable computation schemes require that the input of the polynomial function must come from a single data source, which means that the data must be signed by a single user. However, the input of the polynomial may come from multiple users in the practical application. In order to solve this problem, the researchers have proposed some schemes for multi-source outsourced data, but these schemes have the common problem of low efficiency. To improve the efficiency, this paper proposes an efficient polynomial-based verifiable computation scheme on multi-source outsourced data. We optimize the polynomials using Horner's method to increase the speed of verification, in which the addition gate and the multiplication gate can be interleaved to represent the polynomial function. In order to adapt to this structure, we design the corresponding homomorphic verification tag, so that the input of the polynomial can come from multiple data sources. We prove the correctness and rationality of the scheme, and carry out numerical analysis and evaluation research to verify the efficiency of the scheme. The experimental indicate that data contributors can sign 1000 new data in merely 2 s, while the verification of a delegated polynomial function with a power of 100 requires only 18 ms. These results confirm that the proposed scheme is better than the existing scheme. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to analyzing the dynamics of COVID-19 using nonstandard finite difference (NSFD) schemes. Our model incorporates both asymptomatic and symptomatic infected individuals, allowing for a more comprehensive understanding of the epidemic's spread. We introduce an unconditionally stable NSFD system that eliminates the need for traditional Runge–Kutta methods, ensuring dynamical consistency and numerical accuracy. Through rigorous numerical analysis, we evaluate the performance of different NSFD strategies and validate our analytical findings. Our work demonstrates the benefits of using NSFD schemes for modeling infectious diseases, offering advantages in terms of stability and efficiency. We further illustrate the dynamic behavior of COVID-19 under various conditions using numerical simulations. The results from these simulations demonstrate the effectiveness of the proposed approach in capturing the epidemic's complex dynamics. [ABSTRACT FROM AUTHOR]

Enhancing information security has become increasingly significant in the digital age. This paper investigates the concept of physical layer security (PLS) within a relay-aided power line communication (PLC) system operating over a multiple-input multiple-output (MIMO) channel based on MK model. Specifically, we examine the transmission of confidential signals between a source and a distant destination while accounting for the presence of multiple eavesdroppers, both colluding and non-colluding. We propose a two-phase jamming scheme that leverages a full-duplex (FD) amplify-and-forward (AF) relay to address this challenge. Our primary objective is to maximize the secrecy rate, which necessitates the optimization of the jamming precoding and transmitting precoding matrices at both the source and the relay while adhering to transmit power constraints. We present a formulation of this problem and demonstrate that it can be efficiently solved using an effective block coordinate descent (BCD) algorithm. Simulation results are conducted to validate the convergence and performance of the proposed algorithm. These findings confirm the effectiveness of our approach. Furthermore, the numerical analysis reveals that our proposed algorithm surpasses traditional schemes that lack jamming to achieve higher secrecy rates. As a result, the proposed algorithm offers the benefit of guaranteeing secure communications in a realistic channel model, even in scenarios involving colluding eavesdroppers. [ABSTRACT FROM AUTHOR]

The Double-Yield (D-Y) model is a widely used method for predicting the mechanical properties of caved rock. However, determining the numerical simulation parameters for this model typically requires extensive trial and error. In this paper, a new approach for parameter inversion of the D-Y model is proposed and applied to a real case study involving repeated mining in multiple coal seams. The proposed framework combines FLAC3D, a finite element numerical simulation software, with a new computational tool developed using MATLAB® App Designer. By applying this framework to the case study, the accuracy and efficiency of parameter determination in the mining area are improved. In addition, a 3D finite element model of multiple coal seams is generated based on the geological information of the case study, and the stress evolution is analyzed. The simulation results are then validated through field measurements, providing valuable reference information for safe and efficient production. Overall, this improved method for parameter inversion of the D-Y model, along with the combined numerical simulation and field measurement approach, enhances the understanding of stress evolution in multiple coal seams and contributes to safer and more efficient mining operations. Highlights: An in-depth analysis was conducted to examine how variations in Double-Yield parameters impact the numerical simulation curve. A breakthrough approach for parameter inversion of the Double-Yield (D-Y) model is introduced, and the computational tool is developed. A 3D finite element model is crafted to analyse the dynamic stress evolution during repeated mining of multiple coal seams. The accuracy of simulation outcomes is bolstered by validation against field measurements of mining-induced stress. [ABSTRACT FROM AUTHOR]

The occurrence of rockfalls is a prevalent geological hazard, especially in the Three Gorges Reservoir area of the Yangtze River. Analyzing the pre-rockfall evolution process of hazardous rock mass is crucial for stability assessment, risk monitoring, and evaluation. Generally, numerical analysis is conducted to study the rock stability and failure; however, it is primarily based on the two-dimensional calculation of typical cross-sections without considering the shape of the three-dimensional space of the hazardous rock mass, thus leading to distorted results. To address this issue, this paper proposes a methodological approach for the analysis of the stability conditions of a hazardous rock mass. The approach starts with field investigations and an Unmanned aerial vehicle (UAV) photogrammetric survey to gather data. These data are then used to construct a three-dimensional (3D) geological model of rock mass. Finally, a 3D numerical simulation is performed to analyse the potential failure processes of hazardous rock mass. In this study, we focused on the hazardous Diaozui rock mass of the Qutang Gorge in the Three Gorges Reservoir area (China). We constructed a 3D geological model of the rock mass based on the data from the field survey and UAV photogrammetric survey. Using this 3D geological model, we established a 3D numerical analysis model to assess the rockfall hazard. By utilizing the Strength reduction method (SRM) and simulating the collapse process of the hazardous rock mass, we analyzed the instability mechanism and failure evolution process of the Hazardous Diaozui rock mass. The results illustrated the potential failure mode, the key-oriented wedges of the rock mass, and the potential critical failure point for the hazardous rock mass. These findings provide valuable insights for stability assessment, risk monitoring, and evaluation of the hazardous rock mass in the Three Gorges Reservoir area. [ABSTRACT FROM AUTHOR]

The large deviations of linear control system trajectories from the equilibrium position during the stabilization process, known as peak-effect, represent a serious obstacle to the design of cascade control systems and to guidance stabilization. Evaluation of such deviations is an important problem not only for control theory but also for other branches of mathematics where similar phenomenon appears as, for example, in numerical analysis, in the study of hydrodynamic stability and transition to turbulence, in the dynamics of vehicular platoons, and many others. The aim of this paper is to improve one of the main results concerning upper bounds for the peak-effect in linear control systems in the work by B.T. Polyak and myself in 2016. As the lower bound for overshoot proved by R.N. Izmailov in 1987 shows, the asymptotic estimate for the upper bound obtained here cannot be improved. [ABSTRACT FROM AUTHOR]

This paper investigates a 2-bit all-optical analog-to-digital converter (AO-ADC) using the photonic crystal (PhC) structure. The proposed structure comprises a 27 × 27 grid of silicon rods in air with a refractive index of 3.6. In the design of this converter, a linear demultiplexer and an optical encoder are used. The optical demultiplexer has one input and three outputs. Optical coding has three inputs and two outputs. In general, this converter consists of one input and two outputs, and according to the output light power from the structure and the threshold level of the converter, two logical 0 and 1 states are created in the output. The footprint of the proposed AO-ADC is as small as 118.5 µm2 making it more suitable for use in the optical integrated circuits (OICs) compared to the previous structures. The proposed AO-ADC works at an operating wavelength of 1550 nm with a response time of about 0.6 ps and also works with lower optical intensity. [ABSTRACT FROM AUTHOR]

Because of the increased mobility of vehicle users, it might be difficult to keep communication services in vehicle networks effective and dependable. Huge hurdles have been presented to vehicular networks as a result of the meteoric rise in the amount of data, which comes with the needs of high dependability and low latency. The deployment of access point servers at geographic locations that are closer to the vehicles in order to provide real-time service to applications that are based on the vehicles is one possible option. However, there is a limited amount of cache store space, and there is also a lack of a tractable access mode allocation method. As a result of these factors, it is very difficult to strike a compromise between the network transmission performance and fronthaul savings. Because the signal-to-interference-ratio (SIR) can be enhanced with switching mode in vehicular infrastructure, it may be possible to achieve higher levels of dependability. To serve all of the vehicles, the conventional allocation in vehicular network may not be sufficient on its own for two reasons: (1) the number of vehicles exceeds the number of paths, and (2) a vehicle may be located outside of the coverage path. Therefore, the implementation of switching mode allocation in vehicular communication is very necessary in order to increase the number of vehicles that can be supplied. In this paper, allocation using V2I, V2V, and V2X modes have been analyzed to provide dependable coverage for vehicles. These methods are used for communicating with other vehicles. In this paper, the numerical analysis has been performed such that SIR is optimized. In switching mode allocation, it has been shown that establishing a variable SIR threshold is helpful in achieving a path coverage that can be relied upon. It has been shown beyond a reasonable doubt that the coverage probability is likewise directly dependent on SIR thresholds. The theoretical analysis is verified, and it is confirmed that the suggested method is capable of achieving significant performance improvement in terms of coverage probability and data rate. [ABSTRACT FROM AUTHOR]

Because of the increased mobility of vehicle users, it might be difficult to keep communication services in vehicle networks effective and dependable. Huge hurdles have been presented to vehicular networks as a result of the meteoric rise in the amount of data, which comes with the needs of high dependability and low latency. The deployment of access point servers at geographic locations that are closer to the vehicles in order to provide real-time service to applications that are based on the vehicles is one possible option. However, there is a limited amount of cache store space, and there is also a lack of a tractable access mode allocation method. As a result of these factors, it is very difficult to strike a compromise between the network transmission performance and fronthaul savings. Because the signal-to-interference-ratio (SIR) can be enhanced with switching mode in vehicular infrastructure, it may be possible to achieve higher levels of dependability. To serve all of the vehicles, the conventional allocation in vehicular network may not be sufficient on its own for two reasons: (1) the number of vehicles exceeds the number of paths, and (2) a vehicle may be located outside of the coverage path. Therefore, the implementation of switching mode allocation in vehicular communication is very necessary in order to increase the number of vehicles that can be supplied. In this paper, allocation using V2I, V2V, and V2X modes have been analyzed to provide dependable coverage for vehicles. These methods are used for communicating with other vehicles. In this paper, the numerical analysis has been performed such that SIR is optimized. In switching mode allocation, it has been shown that establishing a variable SIR threshold is helpful in achieving a path coverage that can be relied upon. It has been shown beyond a reasonable doubt that the coverage probability is likewise directly dependent on SIR thresholds. The theoretical analysis is verified, and it is confirmed that the suggested method is capable of achieving significant performance improvement in terms of coverage probability and data rate. [ABSTRACT FROM AUTHOR]

The performance of a permissioned blockchain, namely, Hyperledger Fabric, is modeled and analyzed in a quantitative manner in this paper. Various types of nodes contribute to the performance of Hyperledger Fabric, and each of those is modeled and tracked along the operational flow of the permissioned blockchain. There are nodes for endorsement, ordering, and commitment to realize the due decentralized network operations. A quantitative model for each type of nodes has been proposed in Ke and Park (Performance study on various Hyperledger Fabric node types and transaction flow, IEEE BCCA, 2021) along with numerical analysis. Each type of the nodes is characterized in terms of transaction/block queue size and waiting time, and the transaction/block arrival rates and the transaction/block service rates are considered for simulation purposes. The analysis is extended beyond the analysis in Ke and Park (Performance study on various Hyperledger Fabric node types and transaction flow, IEEE BCCA, 2021) in this paper to particularly demonstrate how the arrival rates and the service rates co-influence the performance and how the number of channels impacts the performance, in order to ultimately facilitate a more dynamic way of optimization, taking the co-relation across different types of nodes into account. The major contribution of this paper to the field of computer science by creating a series of queuing models to evaluate the performance of different types of nodes: Chaincode Execution and Endorsement, Block Creation and Delivery, Transaction Validation and Block Committing, and Transaction Processing. [ABSTRACT FROM AUTHOR]

Data from natural sources show counter-intuitive distribution patterns for the leading digits to the left of the decimal point and the digit 1 is observed more frequently than all other numbers. This pattern, which was first described by Newcomb and later confirmed by Benford, is used in financial and tax auditing to detect fraud. Deviations from the pattern indicate possible falsifications. Anesthesiology journals are affected not only by ghostwriting and plagiarism but also by counterfeiting. In the present study 20 publications in anesthesiology known to be falsified by an author were investigated for irregularities with respect to Benford's law using the χ-test and the Z-test. In the 20 retracted publications an average first-digit frequency of 243.1 (standard deviation SD ±118.2, range: 30-592) and an average second-digit frequency of 132.3 (SD ±72.2, range: 15-383) were found. The observed distribution of the first and second digits to the left of the decimal point differed significantly (p< 0.01) from the expected distribution described by Benford. Only the observed absolute frequencies for digits 3, 4 and 5 did not differ significantly from the expected values. In an analysis of each paper 17 out of 20 studies differed significantly from the expected value for the first digit and 18 out of 20 studies varied significantly from the expected value of the second digit. Only one paper did not vary significantly from expected values for the digits to the left of the decimal. For comparison, a meta-analysis using complex mathematical procedures was chosen as a control. The analysis showed a first-digit distribution consistent with the Benford distribution. Thus, the method used in the present study seems to be sensitive for detecting fraud. Additional statements of specificity cannot yet be made as this requires further analysis of data that is definitely not falsified. Future studies exploring conformity might help prevent falsified studies from being published. [ABSTRACT FROM AUTHOR]

Abstract  A web is continuous, flexible materials such as paper, plastic film, metal foil, textiles, etc. Web wrinkle is a problem which plagues the process industries. Most webs are very thin and prone to buckle or wrinkle. The effects of wrinkling can range from compromised product quality to down time of a processing operation, leading to less profitability in the manufacturing operation. The web wrinkles which will be treated in this paper are those that can be affected by roller misalignment and friction characteristics between web and roller. In this paper, observation method is presented for paper-web wrinkling and the prediction model of wrinkling is proposed with experimental verifications. [ABSTRACT FROM AUTHOR]

In the recent paper [1], a new scaling theory of electron localization was proposed. We show that numerical data for the quasi-one-dimensional Anderson model do not support predictions of this theory. [ABSTRACT FROM AUTHOR]

Recently, Al-Odat and Al-Hussien presented consideration of the critical energy of a composite superconductor in this journal. Various numerical and other deficiencies have been identified in this paper. These problems are discussed and clarified in this Comment. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method to transform the results of qualitative risk analysis into a numeric optimal control problem for autonomous ship navigation. Today, making autonomous high-level decisions replacing a crew onboard is considered difficult, in some part due to the complexity of managing the operational risks involved. Although human supervisors, e.g., located in remote operating control centers are still needed for safety and liability reasons, there is a growing demand for complex decisions to be made by the onboard control system itself, both during normal operations and in emergencies. This paper suggests general principles for how the results from systems-theoretic process analysis (STPA) can be transformed into a quantitative and computationally tractable optimization problem, solved by a MPC-based decision-making algorithm for autonomous navigation. The proposed method is demonstrated and evaluated by simulating an autonomous ship navigating in a coastal environment. It is concluded that the proposed method may serve as a reasonable and valuable bridge between the realms of qualitative risk analysis and numerical optimal control for risk-aware autonomous control and decision-making. [ABSTRACT FROM AUTHOR]

The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by Korteweg (1901) that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set of observations. (The implicit generalization developed in this paper is not a sub-class of the implicit generalization of the Navier-Stokes fluid developed by Rajagopal (2003), (2006) or the generalization due to Průša and Rajagopal (2012), as spatial gradients of the density appear in the constitutive relation developed by Korteweg (1901).) Third, we introduce a challenging set of partial differential equations that would lead to new techniques in both analysis and numerical analysis to study such equations. [ABSTRACT FROM AUTHOR]

This paper discusses the main results of a full-scale shaking table test campaign carried out under the auspices of the EU funded research project SERA, whose objective is to investigate the seismic performance of three-leaf masonry walls with weak lime-mortar joints. These masonry walls are widely found in seismic prone regions in the Mediterranean area, thus assessing their behaviour under dynamic actions is an important pre-requisite for the seismic vulnerability evaluation of a plethora of historical centres. The first part of the paper presents a preliminary study on the mechanical properties of the wall component materials that was carried out through an ad-hoc experimental campaign. The outcomes are of particular interest for the characterization of the mortar and of the infill materials, that were designed to reproduce the low strength that is typically found in old masonry buildings. The design of the masonry wall that was tested and the test set-up are presented next. The applied loading protocol consisted of the horizontal component of a ground motion record that is repeatedly applied to the shaking table with increasing intensity. Finally, the main results of the experimental test are discussed. The damage patterns, drift ratios and base shear are presented for the ground motion sequence. The results are also discussed through a dynamic capacity curve that shows the attainment of different limit states with increasing ground motion intensity. A set of nonlinear numerical simulations, both static and dynamic, using a 3D FE model of the wall verify the experimental study as they report good agreement with the experimental tests and exhibit stable numerical behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we discuss properties of the singular value decomposition (SVD-decomposition) within the framework of the analysis of the numerical rank in ill-posed problems for determining frequency properties of discrete sequences consisting of trigonometric binomials. The properties of the numerical rank of the SVD-decomposition are indicated. We propose an algorithm for determining the frequencies of trigonometric binomials involved in the original function that forms a discrete sequence; this algorithm is based on estimates of the numerical rank. We obtain a stable criterion for estimating the numerical rank based on the arithmetic-mean estimators of the pseudo-null-space of the trajectory matrix. Also, we present the results of numerical experiments that demonstrate the consistency of the arithmetic-mean estimator of the pseudo-zero-space for the analysis of the spectrum of noisy sequences. [ABSTRACT FROM AUTHOR]