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Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most do(1) where the degree of the input polynomial in every variable is less than d. They also stated the question of designing fast algorithms for the large variable case (i.e., n x do(1)) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field Fq of characteristic p, which evaluates an n-variate polynomial of degree less than d in each variable on N inputs in time provided that p is at most do(1), and q is at most (exp(exp(exp(· · · (exp(d))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n x do(1)), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: --We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program [26]. More precisely, for every fixed prime p, we show that for every constant e > 0, and large enough n, the rank of any n × n Vandermonde matrix V over the field Fpa can be reduced to (n/exp(O(poly(e)log0.53 n))) by changing at most nT(e) entries in every row of V, provided a = poly(logn). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [9]. [ABSTRACT FROM AUTHOR]

Recently, some new sequence spaces ℓ p (A α) (0 < p < ∞) , c 0 (A α) , c (A α) , and ℓ ∞ (A α) have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of A α = (a n , v α) , where a m , v α = { v α ρ (α) (m) , v ∣ m , 0 , v ∤ m , and ρ (α) (m) : = sum of the α th power of the positive divisors of m ∈ N . They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes. This article deals with some geometric properties of these sequence spaces. [ABSTRACT FROM AUTHOR]

Flow field regulation plays an important role in the optimization of column flotation units with agitation function. This paper aims to discuss the regulating effect of baffle design on the flow field in stirred column flotation units by numerical simulation. A series of geometric models of changing baffle structure and layout were established, and detailed flow field simulation analysis was carried out by using COMSOL software. The results show that the internal flow field of the traditional standard cylindrical tank will cause the solid particles to adhere to the wall movement and the bubbles to the middle area, resulting in the uneven distribution of particles and bubbles, and the internal flow field is not conducive to the mineralization process and the stability of the foam layer. This limitation can be significantly improved by introducing baffles. The results show different baffle shapes and layouts are as follows: wide and narrow on top, narrow on the bottom and wide on top, and only arranged at the bottom of the column body, can increase the average turbulent kinetic energy in the collision zone by 22%, 46% and 43%, and the average turbulent dissipation rate by 18%, 43% and 36%, respectively, while maintaining the turbulence intensity in the transport zone. Among them, the baffle plate with wide top and narrow bottom has the best effect. Arranging baffles with a length shorter than 2/3 of the height of the cylinder at the upper part inside the cylinder can also enhance the turbulence intensity in the collision zone, but at the same time, it also increases the turbulence intensity in the transport zone. Moreover, as the length of the baffle decreases, the enhancement effect becomes stronger. Since this method significantly affects the transport zone, it is more suitable for columnar mixing devices. The study also reveals that by adjusting the proportion of wide and narrow baffles, the center of the turbulent region can be laterally shifted to the short baffles, and the flow field distribution can be adjusted. Meanwhile, with the increase in the number of long baffles, the intensity of the flow field in the device will gradually decrease. By moving the center of the turbulent region to the side inlet, the particle dispersion can be accelerated, but in general, the particles will accumulate toward the region with lower turbulence intensity. These findings have important guiding significance for the design and optimization of the flow field of the stirred column flotation unit, which is helpful in improving the agitation uniformity and flotation efficiency. [ABSTRACT FROM AUTHOR]

Zero inflation, zero deflation, overdispersion, and underdispersion are commonly encountered in count time series. To better describe these characteristics of counts, this article introduces a zero‐modified geometric first‐order integer‐valued autoregressive (INAR(1)) model based on the generalized negative binomial thinning operator, which contains dependent zero‐inflated geometric counting series. The new model contains the NGINAR(1) model, ZMGINAR(1) model, and GNBINAR(1) model with geometric marginals as special cases. Some statistical properties are studied, and estimates of the model parameters are derived by the Yule–Walker, conditional least squares, and maximum likelihood methods. Asymptotic properties and numerical results of the estimators are also studied. In addition, some test and forecasting problems are addressed. Three real‐data examples are given to show the flexibility and practicability of the new model. [ABSTRACT FROM AUTHOR]

The worldwide pesticide marketplace was approximate 85 billion dollars in 2019, growing at a geometric progression rate of 4.2% from 2015, and by 2023 it is projected to grow at a rate of 11.5% to approximately 130.7 billion dollars. Pesticides' uniqueness is defined by the distinctiveness of their chemistry and their relationship with the environment. Pesticide mobility and bioavailability in water bodies are determined by their desorption and absorption processes from soil particles. Waterbodies are severely affected with the build-up of these poisonous impurities with its imbalance in pH and accumulation of heavy metals, which adversely distress the aquatic ecology in the waterbodies. Pesticides are not only damaging to the ecology of the waterbodies; it is also consequentially harmful to humans. It causes gastrointestinal disorders, cardiac and respiratory issues, and even affects the biotic molecules and organs subsequently leading to acute and/or chronic illnesses. This article reviews 200 scientific literatures for existence of pesticides in drinking water and the various remediation technologies available for the treatment to render the water potable. Upon review of more than 20 technologies, suggestions have been made for the best probable technologies for water with presence of pesticide. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

The given text discusses the concept of representative points in a graph and how they can be used to define a function. It explains that by choosing three rational points, one inside each region of the graph, a function can be created. The text aims to show that this function is injective, meaning that it is one-to-one. It provides a diagram and references a related problem about the possibility of having uncountably many disjoint figure 8s in the plane. The text concludes by mentioning that the result discussed is due to R. L. Moore. [Extracted from the article]

Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple. [ABSTRACT FROM AUTHOR]

Data assimilation (DA) plays a pivotal role in diverse applications, ranging from weather forecasting to trajectory planning for autonomous vehicles. A prime example is the widely used ensemble Kalman filter (EnKF), which relies on the Kalman filter's linear update equation to correct each of the ensemble forecast member's state with incoming observations. Recent advancements have witnessed the emergence of deep learning approaches in this domain, primarily within a supervised learning framework. However, the adaptability of such models to untrained scenarios remains a challenge. In this study, we introduce a new DA strategy that utilizes reinforcement learning (RL) to apply state corrections using full or partial observations of the state variables. Our investigation focuses on demonstrating this approach to the chaotic Lorenz 63 and 96 systems, where the agent's objective is to maximize the geometric series with terms that are proportional to the negative root‐mean‐squared error (RMSE) between the observations and corresponding forecast states. Consequently, the agent develops a correction strategy, enhancing model forecasts based on available observations. Our strategy employs a stochastic action policy, enabling a Monte Carlo‐based DA framework that relies on randomly sampling the policy to generate an ensemble of assimilated realizations. Numerical results demonstrate that the developed RL algorithm performs favorably when compared to the EnKF. Additionally, we illustrate the agent's capability to assimilate non‐Gaussian observations, addressing one of the limitations of the EnKF. Plain Language Summary: Reliable forecasts of the state of chaotic systems, such as environmental flows, require combining observational data and dynamical model outputs through a process called data assimilation. The ensemble Kalman filter (EnKF) is the most commonly adopted algorithm for this task, however, is subject to some limitations when applied to nonlinear/non‐Gaussian systems. Recently, there has been interest in using deep learning (DL), particularly within a supervised learning setup, for DA. However, making DL models work well in new situations that differ from those experienced during training is challenging. In this work, we propose a new DA approach that leverages reinforcement learning (RL). RL helps the system make corrections to its predictions based on incoming observations, even when the model has not been trained for those specific scenarios. Compared to the EnKF framework, RL offers a novel algorithm for nonlinear corrections of the forecasts. Numerical results show that the proposed RL algorithm outperforms the EnKF and demonstrates the RL agent's ability at addressing some shortcomings of the EnKF. Key Points: Deep reinforcement learning (RL) is introduced for data assimilation in application to the Lorenz 63 and 96RL generalizes to new situations unseen during training through actively learning from the data and system dynamicsThe RL agent allows for nonlinear correction of the forecast using the observationsThe performance of the proposed RL algorithm generally surpasses that of the standard ensemble Kalman filter (EnKF) [ABSTRACT FROM AUTHOR]

Calculations about compound interest serve as the basis of most financial decisions; therefore, it is imperative to explore what mathematical knowledge people need to correctly calculate simple and compound interest. The aim of this study is to analyze the relationship between college students' competence in calculating simple and compound interest and their understanding of the arithmetic and geometric progressions. It is also pointed out whether the results vary according to gender. Population proportion tests are carried out, and gender proportion differences are considered for inferential analysis. The dichotomous Probit model was used for correlation analysis. Results demonstrate that 59.8% of students know how to formulate a whole-number succession, and only 30.9% in the case of fractional numbers. Less than 50% of students can calculate compound interest, but 76.7% can calculate simple interest. There is no significant difference between men and women. The results show a positive relationship between male students' competence in calculating compound interest with the possibility to correctly formulate a geometric succession. Findings can be useful for mathematics teaching strategy design and its applications in finance contexts with the purpose of training students to be better at finance decision making. [ABSTRACT FROM AUTHOR]

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol–Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler's identity, we give relations among this characteristic function, the Apostol–Bernoulli polynomials and numbers, and also trigonometric functions including cosine and sine. A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol–Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas. [ABSTRACT FROM AUTHOR]

We establish lower estimates for the characteristic determinant of a problem with multipoint conditions in the selected variable with simple interpolation nodes and periodicity conditions with respect to the remaining variables for an Euler-type partial differential equation. We consider particular cases in which either the interpolation nodes form a geometric progression or the real parts of the roots of characteristic equation behave as power functions at infinity. [ABSTRACT FROM AUTHOR]

Ensuring the alignment of perspectives between unmanned ground vehicles (UGVs) and Building Information Modeling (BIM) is crucial for the precise retrieval and analysis of BIM-stored information during inspection tasks. However, accumulative localization errors often result in deviations between the viewpoints of UGV cameras and their corresponding representations in BIM at specific waypoints. Therefore, this study introduces a sequential rectification method to correct the UGV's pose within the BIM environment to ensure seamless alignment of perspectives. By leveraging visual features and geometric strategies in sequence, this method correlates the UGV-captured point cloud data with the BIM, thereby deriving accurate and robust pose rectification. Experimental validation in a featureless indoor environment demonstrated that this method reduced the angle and distance error of reference lines in two-dimensional (2D) views to approximately 2° and 7 pixels, respectively, and the root mean square error (RMSE) of three-dimensional (3D) lines to approximately 17 cm. The validation also demonstrated that the proposed method was particularly robust in correcting the pose and improving the alignment of perspectives between BIM and the UGV, even in cases of significant misalignment. Hence, this study improves the reliability of decisions made by UGVs for indoor inspection when cross-referencing with BIM data. Practical Applications: UGVs have been leveraged to automate inspection tasks, thereby reducing human participation. Generally, these UGVs are programmed to perform analyses by comparing the as-built condition with the as-designed BIM. However, for these tasks to accurately reference the data stored in BIM, it is crucial to have a seamless alignment between the UGV's perspectives and the as-designed BIM. Unfortunately, due to the accumulation of localization errors, discrepancies often arise between the perspectives in BIM and UGV cameras at waypoints. Such misalignments can compromise the reliability of decisions made for assigned tasks compared with BIM, such as verifying measurements of elements or confirming equipment installation during the project handover phase. Hence, this study, having demonstrated superior performance and robustness, could provide an effective solution to this alignment challenge. By ensuring seamlessly aligned perspectives, this method improves the reliability of UGV inspection assessments when accessing corresponding data stored in BIM. [ABSTRACT FROM AUTHOR]

We present an investigation of the infinite sequences of numbers formed by calculating the pairwise averages of three given numbers. The problem has an interesting geometric interpretation related to the sequence of triangles with equal perimeters which tend to an equilateral triangle. Investigative activities of the problem are carried out in several stages and involve various technological tools and various fields of mathematics. The activities can be useful for the instruction of pre-service and experienced mathematics teachers. [ABSTRACT FROM AUTHOR]

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

Liquid hydrogen (LH2), a key clean energy carrier, requires precise thermal management, especially for long-term storage and long-distance transport. Accurate prediction of thermal stratification, self-pressurization, and Boil-Off Gas (BOG) generation is considered pivotal for optimizing low-temperature cryogenic storage systems. Most of the literature uses the two-zone model, where the vapor-liquid interface is treated as a simple saturation condition. The influence of interfacial thermal behavior on the calculation of thermal stratification, self-pressurization, and Boil-Off Gas (BOG) generation has not been sufficiently analyzed. To overcome such limitations, in this study, the liquid-vapor interface is modeled by establishing a separate governing equation and optimal geometric progression grid discretization instead of treating it as a simple condition. The Diffusion-based Interface Model (DIM) of the present study transcends traditional methods by eliminating the typical saturation condition at the vapor-liquid interface, providing a deeper insight into the thermal dynamics of the interface. The DIM's findings include a daily BOG rate of 0.14%, aligning with literature, and temperature deviations of 0.008% in liquid and 0.13% in vapor near the interface, indicating a greater vapor phase influence on interfacial thermal behavior. Furthermore, a 0.35% deviation is observed in vapor pressure towards the simulation's end. The DIM can be a complementary tool for engineering works. It enables the design of safe storage of liquid hydrogen. • Novel Diffusion-based Interface Model (DIM) for LH2 storage. • Analyze the interfacial thermal behavior of LH2 storage tank. • DIM improved prediction accuracy of boil-off gas (BOG) generation. [ABSTRACT FROM AUTHOR]

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba's conjecture. [ABSTRACT FROM AUTHOR]

A finite zero-sum game defined on a subset of the unit square is considered. The game is a generalized progressive discrete silent duel, in which the kernel is skew-symmetric, and the players, referred to as duelists, have identical linear accuracy functions featured with an accuracy proportionality factor. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end time moments, every following time moment is the partial sum of the respective geometric series. Due to the skew-symmetry, both the duelists have the same optimal strategies and the game optimal value is 0. If the accuracy factor is not less than 1, the duelist’s optimal strategy is the middle of the duel time span. If the factor is less than 1, the duel solution is not always a pure strategy saddle point. In a boundary case, when the accuracy factor is equal to the inverse numerator of the ratio that is the time moment preceding the duel end moment, the duel has four pure strategy saddle points which are of the mentioned time moments. For a trivial game, where the duelist possesses just one moment of possible shooting between the duel beginning and end moments, and the accuracy factor is 1, any pure strategy situation, not containing the duel beginning moment, is optimal. [ABSTRACT FROM AUTHOR]

Modern giraffids are nowadays represented by few species; however, this family was quite diverse during the Neogene. Several lineages radiated, showing high morphological disparity in metapodial shapes. During the last few years, not many scholars have focused on this subject. Furthermore, modern statistical approaches have been totally neglected. Herein, we present the first attempt to fill this gap of knowledge adopting a plethora of phylogenetic comparative approaches alongside multivariate statistics. Namely, we performed a series of traditional and geometric morphometric analyses for a total of 207 metapodial specimens of extant and fossil giraffid species. Our results recognized four different clusters, grouped in different morphospaces. Moreover, two of these clusters are clearly outliers in respect to an overall homogeneous distribution. The two outlying clusters include short and robust metapodials observed in Sivatherium, and long and slender metapodials observed in Giraffa and Bohlinia, respectively. Similarities between the modern Okapia and the extinct genus Decennatherium were also detected. This study explores the relations between the morphological adaptations and the ecological roles of different members of the Giraffidae. It also provides insights into how different environmental factors might have influenced the evolution of this group and the adaptation of their members to diverse environments. Furthermore, locomotory adaptations of the most hyper-specialized Giraffidae groups are briefly discussed and investigated. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

In this paper, we introduce dual numbers with components including Leonardo number sequences. This novel approach facilitates our understanding of dual numbers and properties of Leonardo sequences. We also investigate fundamental properties and identities associated with Leonardo number sequences, such as Binet's formula and Catalan's, Cassini's and D'ocagne's identities. Furthermore, we also introduce a dual vector with components including Leonardo number sequences and dual angles. This extension not only deepens our understanding of dual numbers, it also highlights the interconnectedness between numerical sequences and geometric concepts. In the future it would be valuable to replicate a similar exploration and development of our findings on dual numbers with Leonardo number sequences. [ABSTRACT FROM AUTHOR]

Integrating vendors' and buyers' production-inventory systems can significantly reduce the total costs incurred by both parties. In this paper, we study the synchronised multi-vendor multiple-buyer integrated production-inventory system with a production flow where the lot sizes follow a geometric sequence. We model this new problem with the objective to minimise the total cost incurred by the vendors and the buyers and determine the optimal production shipment policy. The computational experiments realised on real case study show the efficiency of the developed model for optimising production shipment and synchronisation of vendor-buyer inventories. [ABSTRACT FROM AUTHOR]

A generalized class of the discrete game of timing is solved as a finite zero-sum game defined on a symmetric lattice of the unit square. The game is a progressive discrete silent duel whose kernel is skew-symmetric, and the players, referred to as duelists, have identical shooting accuracy functions featured with an accuracy proportionality factor a and a power constant β describing nonlinearity of the shooting accuracy. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end time moments, every following time moment is the partial sum of the respective geometric series. Due to the skew-symmetry, both the duelists have the same optimal strategies and the game optimal value is 0. As the duelist has a single bullet, there is no reason for considering solutions in mixed strategies, if any, with non-singleton supports. If a ⩾ 2 β − 1 , the duelist's optimal strategy is the middle of the duel time span; otherwise, the duel solution may be not a pure strategy solution. The case of quadratic accuracy for β = 2 is further considered, which is reasoned by that linearly developing real-time processes are quite rare, and the quadratic accuracy is a non-linearity pattern that makes an interaction model more reliable by slightly diminishing the duelist's confidence. Thus, the case of a < 2 β − 1 is thoroughly studied for pure strategy solutions. A boundary case of a is found, by which the duel has four pure strategy solutions which are of the time moment preceding the duel end moment and the duel end moment itself. • Discrete game of timing is solved as a finite zero-sum game on a symmetric lattice. • Duelist is featured with shooting accuracy proportionality and nonlinearity factors. • The breaking point of the accuracy is found above which pure strategy solutions exist. • Pure strategy solutions are found below the breaking point for the quadratic accuracy. [ABSTRACT FROM AUTHOR]

Fracture expansion in rock masses can be observed by monitoring the break of contacts between the bounding particles via the discrete element method. The latter's realization in this study via the PFC2D program tracked the evolution process of the zonal disintegration in an exemplary roadway-surrounding rock affected by mining. Besides, the damage evolution pattern in a high-stress soft rock roadway was simulated by the FLAC2D program using a strain-softening plastic model, revealing the effects of rock mass strength, stress state, and anchor support on the zonal disintegration of the roadway. Numerical simulation results show that in a roadway with high-level stress, the obvious fractures spread from the roadway surface to the depth of the surrounding rock along a series of geometric planes and cut the surrounding rock into rock mass blocks. Under high crustal stress, conjugate shear fractures occur near the roadway surfaces and form a closed-loop fractured zone after intersecting the conjugate fracture faces. The closed fractured zone becomes a free face, from which conjugate shear fractures develop, forming new closed fractured zones in the deep surrounding rock. By repeatedly generating the closed fracture zones, a fracture network appears in the roadway-surrounding rock. The development of zonal disintegration of roadway-surrounding rock mainly depends on the rock mass strength and its stress state. Zonal disintegration only occurs when the crustal stress of the roadway-surrounding rock exceeds its strength. When the horizontal stress is low and the vertical stress exceeds the rock mass strength, zonal disintegration only occurs on two sides of the roadway. When the vertical stress is low and the horizontal stress exceeds the rock's mass strength, it only appears on the roof and floor. When the values of cohesion, internal friction angle, and tensile strength are reduced in the same proportion, cohesion has the greatest impact on the expansion of the zonal disintegration zone, followed by the internal friction angle, while the tensile strength effect is the least. In anchor-supported roadways undergoing zonal disintegration processes, the intact zone blocks slide relatively along the fracture surface during the process of loosening and deformation of the surrounding rock, making the anchor rods susceptible to tensile, shear, and bending actions. [ABSTRACT FROM AUTHOR]

Laser peen forming (LPF) is promising to form large-scale metal sheets in various applications. Understanding the bending deformation of the LPF process is of particular importance for accurate shape prediction and process planning elaboration. This study established a simplified eigenstrain-based finite element (FEM) model by incorporating loading sequence and geometric non-linearity to explore the bending deformation and shape prediction of the LPF process. The simplified eigenstrain was inversely determined based on the arc heights of plates subjected to full-covered LPF. The above model incorporating loading sequence and geometric non-linearity was verified by the comparison of experimental results, and it improved shape prediction accuracy considerably compared to the common beam bending analytical model. Moreover, the bending deformation processes and mechanisms of LPF plates were elucidated. It was found that biaxially curved surfaces were formed for thick plates with 5 mm or larger thickness, in which the geometric non-linearity and flattening effects played significant roles on their formation. On the other side, monoaxially curved surfaces were obtained for thin plates with 4 mm or smaller thickness after the LPF process. Three stages, i.e., forward bending, reverse flattening, and buckling, occurred and contributed to the formation of the monoaxially curved surface. With the increase in the eigenstrain amplitude and side length, and the decrease in plate thickness, the LPF plates transit from a biaxially curved shape to a monoaxially curved one. [ABSTRACT FROM AUTHOR]

When the rotating machinery fails, the signal generated by the faulty component often no longer maintains the original symmetry, which makes the vibration signal with nonlinear and non-stationary characteristics, and is easily affected by background noise and other equipment excitation sources. In the early stage of fault occurrence, the fault signal is weak and difficult to extract. Traditional fault diagnosis methods are not able to easily diagnose fault information. To address this issue, this paper proposes an early fault diagnosis method for symplectic geometry mode decomposition (SGMD) based on the optimal weight spectrum index (OWSI). Firstly, using normal and fault signals, the optimal weight spectrum is derived through convex optimization. Secondly, SGMD is used to decompose the fault signal, obtaining a series of symplectic geometric modal components (SGCs) and calculating the optimal weight index of each component signal. Finally, using the principle of maximizing the OWSI, sensitive components reflecting fault characteristics are selected, and the signal is reconstructed based on this index. Then, envelope analysis is performed on the sensitive components to extract early fault characteristics of rolling bearings. OWSI can effectively distinguish the interference components in fault signals using normal signals, while SGMD has the characteristic of unchanged phase space structure, which can effectively ensure the integrity of internal features in data. Using actual fault data of rolling bearings for verification, the results show that the proposed method can effectively extract sensitive components that reflect fault characteristics. Compared with existing methods such as Variational Mode Decomposition (VMD), Feature Mode Decomposition (FMD), and Spectral Kurtosis (SK), this method has better performance. [ABSTRACT FROM AUTHOR]

To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously continued. The resulting problems are solved using the developed method of iterative extensions. To expedite the convergence of this method, the relationships between physical quantities on the extension of systems and additional parameters of the iterative method are employed. The formulations of sufficient convergence conditions for the iterative process utilize interdisciplinary connections with functional analysis, applying discrete analogs of the principles of function extensions while preserving norm and class. In the algorithmic implementation of the iterative extensions method, automation is applied to control the selection of the optimal iterative parameter value during information processing. In accordance with the fictitious domain methodology, solvable problems from domains with a complex geometry are reduced to problems in a rectangle in the two-dimensional case and in a rectangular parallelepiped in the three-dimensional case. But now, in the problems being solved, the minimization of the error of the iterative processes is carried out with a norm stronger than the energy norm. Then, all relative errors are estimated from above in the used norms by terms of infinitely decreasing geometric progressions. A generalization of the developed methodology to boundary value problems for polyharmonic equations is possible. [ABSTRACT FROM AUTHOR]

In the 3rd century B.C.E., Archimedes wrote his treatise on the quadrature of the parabola, in which he laid out his solution to the problem of finding the area of a parabolic segment. Archimedes attacks the problem from two sides; first, he uses classical mechanics to derive an area formula, and then he uses pure geometry to prove his formula is correct. In this paper, we will present two modern approaches to this problem using calculus. The first approach can be presented to a calculus class that has learned about optimization methods and convergent geometric sequences, and the second can be presented after they have learned the Fundamental Theorem of Calculus. [ABSTRACT FROM AUTHOR]

In this study, we investigated whether the cognitive system, known to be able to register regular visual event sequences and the violation of these sequences automatically, had the capacity of processing two sequences simultaneously. To this end, we measured the visual mismatch negativity (vMMN) component of ERPs as interwoven event sequences simultaneously presented to the left and right side of the screen. One of the sequences consisted of geometric patterns (diamonds); the other, photographs of human faces. In successive cycles, parts of the stimuli vanished and then re-appeared (the OFF/ON method). The vanishing parts served as either standard (frequently vanishing parts) or infrequent (deviant) events, but these events were task-irrelevant. The 20 adult participants (age 21.40 ± 2.72 years) performed a visual tracking task, with the OFF/ON task being a passive oddball paradigm. According to the results, both OFF and ON events, and both diamond and face stimuli elicited the vMMN component, showing that the system underlying this activity is capable of processing two event sequences if the sequences consist of fairly different kind of objects as stimuli. The sLORETA analysis showed that the source of vMMN was more frequent contralaterally to the deviant event, and the sources comprised loci from ventral and dorsal structures, as well as some anterior loci. [ABSTRACT FROM AUTHOR]

Estimating the execution time of assembly task sequences of a new geometric model is an essential prerequisite for task allocation and resource planning decisions in smart multi-robotic assembly cells. This paper discusses an optimization-based method to estimate the execution time of a full sequence of assembly tasks, including picking, aligning, and insertion. Task descriptions, joint space trajectories, assembly task-related motions, and process conditions are used as inputs for the method. The required descriptions of assembly tasks are parametrized, and the resulting function is then combined with the process conditions, assembly task-relevant motions, and the capabilities of the resources in the robotic assembly cell to estimate the execution times. The advantage of the method is the possibility of determining the execution time of an assembly task, taking into account the capabilities of the robot, before assigning it to the multi-robot assembly station. The proposed method is demonstrated experimentally by using a CAD model with the information of Fanuc robot CR7iA/L. The method is implemented in MATLAB using AMPL-API and tested. To determine its effectiveness, the outcomes of the proposed method are compared to the values of the 3-D Fanuc Roboguide (RG). Finally, further research steps to improve the accuracy of the total time for the entire assembly sequence are also discussed. [ABSTRACT FROM AUTHOR]

Switching power converters bring a desired output magnitude using semiconductor devices. These switching converters are generally classified as voltage lift and super lift. The conventional lift converters such as buck, boost and buck–boost come under the category of voltage lift technique, in which the output voltage increases in arithmetic progression. The other lift approach that increases the output voltage in the geometric progression is the super-lift method. Nowadays super-lift converters find their role in various applications, because of their high gain and low ripple. The converter taken for analysis is a modified negative output super-lift Luo converter (MNOSLC). Current mode control (CMC) is employed to maintain a stable DC voltage at the load side. This is investigated using different types of controllers such as PI, fuzzy, sliding mode controller (SMC) and fuzzy SMC. The performance of these controllers is compared and it is found that fuzzy SMC results in better line and load regulation. A prototype is developed that validates the simulation results. [ABSTRACT FROM AUTHOR]

The increasing number of objects in Earth orbit has encouraged the development of space surveillance and tracking (SST) applications. A critical aspect of SST is the identification and characterization of close encounters between pairs of space objects. The present work introduces a tool for the analysis of conjunctions, consisting of several modules. The first module, which has been shown to greatly speed up the process, employs a series of geometric and temporal filters to shorten the list of potential colliding pairs. The remaining objects are then propagated to compute important parameters such as time of closest approach (TCA), miss distance (MD), and probability of collision (PoC), the latter using three different methods. When a conjunction assessment returns an MD or a PoC that exceeds predefined alert thresholds, the algorithm enables the planning of an impulsive collision avoidance maneuver (CAM) at specific maneuver epochs. CAM candidates are determined using an analytical Keplerian approach, with the goal of achieving the desired PoC or MD. The user can then verify the performance of a specific candidate through perturbed propagation, and the MD and PoC are recalculated after the maneuver to ensure that they meet the desired thresholds. In conclusion, this paper evaluates the performance of the tool using synthetic and real data, providing valuable insights into its effectiveness. [ABSTRACT FROM AUTHOR]

The selection of an appropriate third-party logistics (3PL) provider has become an inescapable option for shippers in today's business landscape, as the outsourcing of logistics activities continues to increase. Choosing the 3PL supplier that best meets their requirements is one of the most difficult difficulties that logistics consumers face. Effective decision-making (DM) is critical in dealing with such scenarios, allowing shippers to make well-informed decisions within a restricted timeframe. The importance of DM arises from the possible financial repercussions of poor decisions, which can result in significant financial losses. In this regard, we introduce p, q-spherical fuzzy set (p, q-SFS), a novel concept that extends the concept of T-spherical fuzzy sets (T-SFSs). p, q-SFS is a comprehensive representation tool for capturing imprecise information. The main contribution of this article is to define the basic operations and a series of averaging and geometric AOs under p, q-spherical fuzzy (p, q-SF) environment. In addition, we establish several fundamental properties of the proposed aggregation operators (AOs). Based on these AOs, we propose a stepwise algorithm for multi-criteria DM (MCDM) problems. Finally, a real-life case study involving the selection of a 3PL provider is shown to validate the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

Exponential smoothing is one of the most widely used forecasting methods for univariate time series data. Based on the difference between protected and confidential time series data, we derive theoretical bounds for the absolute change to forecasts generated from additive exponential smoothing models. Given time series data up to time t, we discover a functional form of robust bounds for the absolute change to forecasts for any T ≥ t + 1 , which can be represented as a compact form of geometric series. We also find robust bounds for the Change in Mean Absolute Error ( Δ MAE ) and Measured Mean Absolute Error (MMAE). [ABSTRACT FROM AUTHOR]

We show that a sequence of graphs with uniformly bounded vertex degrees, number of vertices going to infinity, and with geometric property (T) has many small cycles. We also show that when a small part of such a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls. [ABSTRACT FROM AUTHOR]

Species-abundance distribution (SAD) is essential in understanding the formation and maintenance mechanisms of species diversity. In alpine meadows, aspect affects almost all biological and ecological processes. Therefore, the soil ciliate data obtained by Illumina MiSeq sequencing were used to explore the characteristics of SADs to the soil ciliate communities on different aspects in alpine meadows (south aspect, southwest aspect, west aspect, northwest aspect, north aspect; SA, SWA, WA, NWA, NA, respectively). Statistical models: Log-series model (LSM) and Log-normal model (LNORM), Niche models: Broken stick model (BSM) and Geometric series model (GSM), and a neutral theory model: Meta-community zero-sum multinomial distribution model (MZSM) were selected to fit the soil ciliate data for analysis. The results showed that: (1) the number of common species increased and then decreased from SA to NA, the number of rare species changed in the opposite trend; (2) BSM and GSM were able to fit the SAD of overall soil ciliates species on different aspects, while MZSM was rejected on all aspects, indicating that the niche process dominated the formation of soil ciliate communities in the alpine meadows, and the resource allocation pattern was dominated by fixed allocation; (3) best-fit models for common species were GSM and BSM which were consistent with best-fit models for all species, while the best-fit model for rare species was LNORM, common species dominated the formation of SAD on different aspects and had an irreplaceable role in maintaining community productivity and stability. [ABSTRACT FROM AUTHOR]

The importance of renewable energy resources in the energy province necessitates the use of photovoltaic (PV) and wind systems to power brushless DC (BLDC) motors for applications such as water pumping and automobiles. In this research, an optimized MPPT-based positive output super-lift Luo converter (PO-SLLC) is presented for a hybrid power system with a BLDC motor to overcome the constraints of ordinary DC–DC converters. The Luo converter produces a positive output voltage which aids in achieving a high power density with fewer ripples in the voltage and current profile. The output voltage grows in geometric progression, increasing the converter's efficiency, according to its mathematical model. Adjustment parameter enhanced Aquila optimization (APEAO) is used to optimize the duty cycle of the PO-SLLC from the MPPT controller and the controller parameters of the PI controller of the wind system. The proposed system's behavior for variations in irradiation patterns with constant load torque is simulated in MATLAB/Simulink. The settling time and overshoot obtained with the proposed optimized controller are compared to those obtained with other optimization controllers, such as ant colony optimization (ACO), genetic algorithm (GA), Aquila optimization (AO), advanced version of harmony search algorithm (2N + 1-HSA) and opposition based quantum bat algorithm (OQBA) and it is demonstrated that the proposed APEAO optimized controller has good dynamic performance. [ABSTRACT FROM AUTHOR]

The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized. [ABSTRACT FROM AUTHOR]

The distribution of emergency shelter materials in emergency cases around the world is a hard task, the goal of this research is to offer a Complex Non-linear Diophantine Fuzzy (C-NLDF) decision-making model for earthquake shelter construction. Essentially, the article is divided into three sections to acquire acceptable and precise measures in emergency decision-making situations. First, we present the Complex Non-Linear Diophantine Fuzzy Set (CN-LDFS), a new generalization of the complex linear Diophantine fuzzy set (CLDFS) and q-linear Diophantine fuzzy set (q-LDFS), as well as explore its key aspects. Furthermore, aggregation operators are useful for aggregating uncertainty in decision-making issues. As a result, algebraic norms for CN-LDFSs are produced based on certain operational laws. In the second section of the work, we offer a series of averaging and geometric aggregation operators under CN-LDFS that are based on defined operating laws. In the final section of the work, under complex Non-linear Diophantine fuzzy information, the ranking algorithms based on suggested aggregation operators are present to address the case study regarding emergency situation of earthquakes. In comparison section, results of existing and proposed operators explore the effectiveness of proposed methodologies and provide accurate emergency measures to address the global uncertainty about the construction of emergency shelters in earthquakes. [ABSTRACT FROM AUTHOR]