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Research has shown the importance of helping students, especially those with mild-to-moderate learning disabilities, to offload information during problem-solving. When students can get their thoughts onto paper, number line strategies can help them develop a firm foundation in mathematical problem-solving while understanding the relationships between mathematical operations. These strategies are helpful for the development of addition, subtraction, multiplication, division, and later, fractional mathematics. In this article, we describe the progression of number lines as a supportive strategy for elementary students and those with developmental delays in mathematics to improve mathematical understanding. This strategy is based on students being able to show their work and think about what they have written on paper or how they have used manipulatives.

Background: The Mental Starters Assessment Project (MSAP) seeks to address poor performance in Grade 3 mathematics. The programme focusses on eliminating inefficient counting methods and promoting strategic mathematical skills, including numerical sense, mental calculation and rapid recall skills. Additionally, MSAP supports teachers' professional growth by providing them with a toolkit of effective calculation strategies to bridge the performance gap and enhance mathematical education. Aim: This paper explores the insight gained from reflections of final year Bachelor of Foundation Phase (FP) initial teacher educators (ITE) students in South Africa. Setting: Grade 3 classrooms. Methods: The ITE students were given training and materials to implement the MSAP, and this occurred over a 4-week teaching practicum, after which they completed a reflective task on the implementation. A total of 20 students were selected from a cohort of 138 based on their academic performance. Results: The analysis of the reflections showed that ITE students benefitted from reflecting on their practice and highlighted important elements of their professional learning. The reflections raised issues related to challenges in their pedagogical content knowledge (PCK) as well as their confidence and competence to teach mathematics and manage the classroom context. Conclusion: With a multi-dimensional model of reflection, ITE students can achieve a deeper understanding of mathematics teaching and learning when building learners' mental strategies, fostering professional growth and elevating the overall quality of mathematics education. Contribution: Overall, the findings provide insight into the benefits of reflective practices for ITE students' professional development and the improvement of mathematics education.

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

In this paper, a stability analysis and the controller improvement of T-S fuzzy Descriptor system are studied. Firstly, by making full use of the related theory of fuzzy affiliation function and combining the design method of fuzzy Lyapunov function with the method of inequality deflation, a stability condition with wider admissibility and less system conservatism is proposed. The advantage of this method is that it is not necessary to ensure that each fuzzy subsystem is progressively stable. We also maximise the boundary of the derivatives of the affiliation function mined. Secondly, a PDC controller and a Non-PDC controller are designed, and the deflation conditions for the linear matrix inequalities of the two controllers are constructed. Finally, some arithmetic simulations and practical examples are given to demonstrate the effectiveness of the method studied in this paper, and the results obtained are less conservative and have larger feasible domains than previous methods. [ABSTRACT FROM AUTHOR]

A binary operation # on Z + is said to be an associative arithmetic if both # and its iteration — the binary operation ∗ defined recursively by: x∗1 = x and x∗y = [x ∗ (y −1)]#x — are associative. E. Rosinger [6] showed that under reasonable conditions an associative arithmetic must be ordinary addition. However, in the general case, there are associative arithmetics that are not ordinary addition. This paper gives examples of these as well as results towards a structure theorem for associative arithmetics. The paper also describes the role that this particular math problem has played in my mathematical life. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present three related studies that build on each other to demonstrate first the need and then the efficacy of the Blended Arithmetic Curriculum (BAC) to help students overcome both slow language processing and the environmental effects of being a student in an urban school district. The author's underlying theory is that K–3 students with slow language processing may be good at complex reasoning, but still struggle with retrieving basic computational facts. Nonetheless, if they did not learn their facts, these students would struggle with K–3 multi-digit arithmetic computation, and ultimately struggle with their hypothesized strength: seeing numeric patterns as would be needed in university level computation. To teach arithmetic facts conceptually, the author developed a paper and pencil curriculum that first teaches complex multi-digit addition with regrouping using a limited number of facts such as 5 + 5, 9 + 1, 1 + 9; 7 + 7, 7 + 8, 8 + 7, and 8 + 8 in problems such as 197 + 108 = 305 so that fact retrieval and computation are fast and accurate. At the end of 2nd grade, urban students with learning disabilities solved 42 two-digit by two-digit problems with 92 percent accuracy in an average of 7 min. The results matched those of suburban students and were significantly faster and more accurate than general education students in the same urban school. [ABSTRACT FROM AUTHOR]

This research paper develops an algorithm to solve a neutrosophic linear fractional optimization problem, where the cost of the objective functions, the technology coefficients, and the resources are neutrosophic numbers (triangular and trapezoidal numbers). Two defuzzification methods are used here to convert the neutrosophic linear fractional optimization problem into a multi-objective linear fractional optimization problem (MOLFOP). The fractional objective function is defuzzified using the component-wise optimization method, and the linear constraints are defuzzified using the aggregation ranking function strategy. Hence, the MOLFOP is then solved using the fuzzy goal programming method. In order to further explain our suggested algorithm, two numerical examples are solved at the end, and the outcomes obtained by our method are compared with the outcomes obtained using the other algorithm. [ABSTRACT FROM AUTHOR]

To model fuzzy numbers with the confidence degree and better account for information uncertainty, Zadeh came up with the notion of Z-numbers, which can effectively combine the objective information of things with subjective human interpretation of perceptive information, thereby improving the human comprehension of natural language. Although many numbers are in fact Z-numbers, their higher computational complexity often prevents their recognition as such. In order to reduce computational complexity, this paper reviews the development and research direction of Z-numbers and deduces the operational rules for symmetric triangular Z-numbers. We first transform them into classical fuzzy numbers. Using linear programming, the extension principle of Zadeh, the convolution formula, and fuzzy number algorithms, we determine the operational rules for the basic operations of symmetric triangular Z-numbers, which are number-multiplication, addition, subtraction, multiplication, power, and division. Our operational rules reduce the complexity of calculation, improve computational efficiency, and effectively reduce the information difference while being applicable to other complex operations. This paper innovatively combines Z-numbers with classical fuzzy numbers in Z-number operations, and as such represents a continuation and innovation of the research on the operational laws of Z-numbers. [ABSTRACT FROM AUTHOR]

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C , including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. [ABSTRACT FROM AUTHOR]

This paper mainly studied the distribution of values of Hardy sums involving Chebyshev polynomials. By using the method of analysis and the arithmetic properties of Hardy sums and Chebyshev polynomials of the first kind, we obtained a sharp asymptotic formula for the hybrid mean value of Hardy sums S5(h, q) involving Chebyshev polynomials of the first kind. In addition, we also gave the value of Hardy sums S (h, q) and S 3(h, q) involving Chebyshev polynomials. Finally, we found the reciprocal formulas of S3(h, q) and S 4(h, q) involving Chebyshev polynomials of the first kind. [ABSTRACT FROM AUTHOR]

With the rapid proliferation of the Internet of Things (IoT) in recent years, the number of IoT devices has surged exponentially. These devices collect and transmit vast amounts of data, including sensitive information. Encrypting data is a crucial means to prevent unauthorized access and potential misuse. However, the traditional cryptographic schemes offering robust security demand substantial device resources and are unsuitable for lightweight deployments, particularly in resource-constrained IoT devices. On the other hand, with the automotive industry making strides in autonomous driving, self-driving vehicles are beginning to integrate into people's daily lives. Ensuring the security of autonomous driving systems, particularly in preventing hacker infiltrations, is a paramount challenge currently facing the industry. An emerging lightweight sequence cipher—aiming to strike a balance between security and resource efficiency—has been proposed in this paper based on S-box substitution and arithmetic addition. The designed security threshold is 280. It has been verified that with a slight performance disadvantage, it can reduce memory usage while ensuring the security threshold. The key stream generated by this structure exhibits excellent pseudo-randomness. [ABSTRACT FROM AUTHOR]

The Tree-Seed Algorithm (TSA) has been effective in addressing a multitude of optimization issues. However, it has faced challenges with early convergence and difficulties in managing high-dimensional, intricate optimization problems. To tackle these shortcomings, this paper introduces a TSA variant (DTSA). DTSA incorporates a suite of methodological enhancements that significantly bolster TSA's capabilities. It introduces the PSO-inspired seed generation mechanism, which draws inspiration from Particle Swarm Optimization (PSO) to integrate velocity vectors, thereby enhancing the algorithm's ability to explore and exploit solution spaces. Moreover, DTSA's adaptive velocity adaptation mechanism based on count parameters employs a counter to dynamically adjust these velocity vectors, effectively curbing the risk of premature convergence and strategically reversing vectors to evade local optima. DTSA also integrates the trees population integrated evolutionary strategy, which leverages arithmetic crossover and natural selection to bolster population diversity, accelerate convergence, and improve solution accuracy. Through experimental validation on the IEEE CEC 2014 benchmark functions, DTSA has demonstrated its enhanced performance, outperforming recent TSA variants like STSA, EST-TSA, fb-TSA, and MTSA, as well as established benchmark algorithms such as GWO, PSO, BOA, GA, and RSA. In addition, the study analyzed the best value, mean, and standard deviation to demonstrate the algorithm's efficiency and stability in handling complex optimization issues, and DTSA's robustness and efficiency are proven through its successful application in five complex, constrained engineering scenarios, demonstrating its superiority over the traditional TSA by dynamically optimizing solutions and overcoming inherent limitations. [ABSTRACT FROM AUTHOR]

The extensions of ordinary fuzzy sets are problematic because they require decimal numbers for membership, non-membership and indecision degrees of an element from the experts, which cannot be easily determined. This will be more difficult when three or more digits' membership degrees have to be assigned. Instead, proportional relations between the degrees of parameters of a fuzzy set extension will make it easier to determine the membership, non-membership, and indecision degrees. The objective of this paper is to present a simple but effective technique for determining these degrees with several decimal digits and to enable the expert to assign more stable values when asked at different time points. Some proportion-based models for the fuzzy sets extensions, intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and spherical fuzzy sets are proposed, including their arithmetic operations and aggregation operators. Proportional fuzzy sets require only the proportional relations between the parameters of the extensions of fuzzy sets. Their contribution is that these models will ease the use of fuzzy set extensions with the data better representing expert judgments. The imprecise definition of proportions is also incorporated into the given models. The application and comparative analyses result in that proportional fuzzy sets are easily applied to any problem and produce valid outcomes. Furthermore, proportional fuzzy sets clearly showed the role of the degree of indecision in the ranking of alternatives in binomial and trinomial fuzzy sets. In the considered car selection problem, it has been observed that there are minor changes in the ordering of intuitionistic and spherical fuzzy sets. [ABSTRACT FROM AUTHOR]

The architectonic is key for situating Kant's understanding of science in the coming century. For Kant the faculty of reason turns to ideas to form a complete system. The coherence of the system rests on these ideas. In contrast to technical unity which can be abstracted a posteriori, architectonic ideas are the source of a priori unity for the system of reason because they connect our reasonable pursuit to essential human ends. Given Kant's focus on mathematics, in the architectonic and his critical philosophy more generally, we must have some sense of the architectonic idea of mathematics. In this paper, I argue for the key principles of the architectonic idea of mathematics: 1) because mathematics is grounded in a priori intuition, it is a peculiarly human activity; 2) the method of mathematics is one of a priori construction, a method only mathematics can employ and: 3) the objects of mathematics are extensive magnitudes. Given these principles, we can use the architectonic idea to have some clarity about how mathematics has dealt with historical development. [ABSTRACT FROM AUTHOR]

Deep neural network (DNN) inference demands substantial computing power, resulting in significant energy consumption. A large number of negative output activations in convolution layers are rendered zero due to the invocation of the ReLU activation function. This results in a substantial number of unnecessary computations that consume significant amounts of energy. This paper presents ECHO, an accelerator for DNN inference designed for computation pruning, utilizing an unconventional arithmetic paradigm known as online/most significant digit first (MSDF) arithmetic, which performs computations in a digit-serial manner. The MSDF digit-serial computation of online arithmetic enables overlapped computation of successive operations, leading to substantial performance improvements. The online arithmetic, coupled with a negative output detection scheme, facilitates early and precise recognition of negative outputs. This, in turn, allows for the timely termination of unnecessary computations, resulting in a reduction in energy consumption. The implemented design has been realized on the Xilinx Virtex-7 VU3P FPGA and subjected to a comprehensive evaluation through a rigorous comparative analysis involving widely used performance metrics. The experimental results demonstrate promising power and performance improvements compared to contemporary methods. In particular, the proposed design achieved average improvements in power consumption of up to 81 % , 82.9 % , and 40.6 % for VGG-16, ResNet-18, and ResNet-50 workloads compared to the conventional bit-serial design, respectively. Furthermore, significant average speedups of 2.39 × , 2.6 × , and 2.42 × were observed when comparing the proposed design to conventional bit-serial designs for the VGG-16, ResNet-18, and ResNet-50 models, respectively. [ABSTRACT FROM AUTHOR]

In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form α (v) → β (v) . Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by T + E ) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by T + E – Rule ). The theory T + E will be stronger than T + E – Rule , but how much stronger can T + E be? More specifically, is T + E conservative over T + E – Rule for theorems of some fixed syntactical complexity Γ ? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas Π and in the applications of E – Rule side formulas from the class Π and can be eliminated (in a certain precise sense), then T + E is ∀ B (Π) -conservative over T + E – Rule . Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of E – Rule of a depth at most of m suffice to obtain ∀ B (Π) conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework. [ABSTRACT FROM AUTHOR]

In order to improve the efficiency of the "last-mile" distribution in urban logistics and solve the problem of the difficult charging of electric logistics vehicles (ELVs), this paper proposes a charging station location optimization scheme for ELVs that takes into account time-window and load constraints (TW-LCs). Taking the optimal transportation path as the objective function and considering the time-window and vehicle load constraints, a charging station siting model was established. For the TW-LC problem, an improved genetic algorithm combining the farthest-insertion heuristic idea and local search operation was designed. Three different types of standardized arithmetic examples, C type, R type, and RC type, were used to test the proposed algorithm and compare it with the traditional genetic algorithm. The results indicate that, under the same conditions, compared to the traditional genetic algorithm, the improved genetic algorithm reduced the optimal path length by an average of 11.12%. It also decreased the number of charging stations selected, the number of vehicles in use, and the algorithm complexity by 22.97%, 13.71%, and 46.81%. Building on this, a case study was conducted on the TW-LC problem in a specific area of Chongqing, China. It resulted in a 50% reduction in the number of charging stations and a 25% reduction in the number of vehicles selected. In terms of economic indicators, the proposed algorithm improves unit electricity sales by 73.88% and reduces the total annualized cost of the logistics company by 18.81%, providing a theoretical basis for the subsequent promotion of ELVs. [ABSTRACT FROM AUTHOR]

This study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, according to the convergence analysis carried out using symbolic computation in the computer algebra system CAS-Maple 18. Without further evaluations of a given nonlinear function and its derivatives, a very rapid convergence rate is achieved, demonstrating the remarkable computing efficiency of the novel technique. To determine the simple roots of nonlinear equations, this paper discusses the dynamic analysis of one-parameter families using symbolic computation, computer animation, and multi-precision arithmetic. To choose the best parametric value used in iterative schemes, it implements the parametric and dynamical plane technique using CAS-MATLAB@R2011b. The dynamic evaluation of the methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this method illustrates not only iterative processes but also gives useful information regarding the convergence of the numerical scheme based on initial guessed values. Some nonlinear problems that arise in science and engineering are used to demonstrate the performance and efficiency of the newly developed method compared to the existing method in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we derive novel families of inclusion sets for the spectra and pseudo- spectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687-704]. [ABSTRACT FROM AUTHOR]

The uncertainties of distribution generations (DGs) and loads lead to severe voltage fluctuations in active distribution networks (ADNs). Meanwhile, energy storage systems (ESSs) and static var compensators (SVCs) can mitigate the uncertainties of power injections by regulating the active and reactive power. Considering the variations of multiple uncertain factors, this paper proposes a complex affine arithmetic (CAA) based uncertain sensitivity analysis method of voltage fluctuations in ADNs. First, affine models of active and reactive power injections are established. The correlations of noisy symbols are used to reflect the mitigation effects of ESSs and SVCs on the uncertainties introduced by DGs and loads. Next, sensitivity indicators of voltage fluctuations are defined based on the transitivity of noisy symbols. Then, a calculation method for sensitivity indicators based on the micro-increments of coefficients is proposed. Combined with the obtained indicators, a fast sensitivity method for calculating interval values of voltages is further proposed. The modified IEEE 33-bus system is tested to validate the accuracy and efficiency of the proposed method by comparison with the continuous utilization of power flow method. Moreover, the 292-bus system is tested to validate its applicability in a large distribution system. Facts have proved that this method improves the efficiency and reliability of calculations, and in different scenarios, it can achieve fast calculation of nodes and online analysis of the voltage fluctuation range in uncertain environments, provides an effective tool for voltage quality management in active distribution networks. [ABSTRACT FROM AUTHOR]

High-dimensional datasets present significant challenges for classification tasks. Dimensionality reduction, a crucial aspect of data preprocessing, has gained substantial attention due to its ability to improve classification performance. However, identifying the optimal features within high-dimensional datasets remains a computationally demanding task, necessitating the use of efficient algorithms. This paper introduces the Arithmetic Optimization Algorithm (AOA), a novel approach for finding the optimal feature subset. AOA is specifically modified to address feature selection problems based on a transfer function. Additionally, two enhancements are incorporated into the AOA algorithm to overcome limitations such as limited precision, slow convergence, and susceptibility to local optima. The first enhancement proposes a new method for selecting solutions to be improved during the search process. This method effectively improves the original algorithm's accuracy and convergence speed. The second enhancement introduces a local search with neighborhood strategies (AOA_NBH) during the AOA exploitation phase. AOA_NBH explores the vast search space, aiding the algorithm in escaping local optima. Our results demonstrate that incorporating neighborhood methods enhances the output and achieves significant improvement over state-of-the-art methods. [ABSTRACT FROM AUTHOR]

First-grade students often encounter challenges in understanding and solving arithmetic word problems due to their limited reading comprehension abilities. Despite these difficulties, students may employ arbitrary strategies, such as combining numbers based on specific keywords, even if they lack a full understanding of the problems. Research suggests that effective mathematical reasoning involves the use of visual mental representations during the problem-solving process. To address this, some studies have explored methods to enhance students' comprehension of word problems. Building on this, the current study explores the impact of first-grade pupils creating visual representations of problem situations on their comprehension and the number of correct solutions. In a typical math class, 45 first graders received a paper-and-pencil task, and, in a visual context, they solved similar problems after reading and illustrating the situation. The findings reveal that while most participants correctly represented the problem situations through drawing, about half struggled to determine the numeric solutions. Nevertheless, the visual context led to an increase in the number of correct problem solutions compared to the normal context, suggesting the potential benefits of incorporating visual representations in enhancing comprehension and problem-solving skills. [ABSTRACT FROM AUTHOR]

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh's extension principle may not satisfy the condition of becoming a discrete fuzzy number. In order to solve these problems, special discrete fuzzy numbers on countable sets are investigated in this paper. Since the representation theorem of fuzzy numbers is the basic tool of fuzzy analysis, two kinds of representation theorems of special discrete fuzzy numbers on countable sets are studied first. Then, the metrics of special discrete fuzzy numbers on countable sets are defined, and the relationship between these metrics and the uniform Hausdorff metric (i.e., supremum metric) of general fuzzy numbers is discussed. In addition, the triangular norm and triangular conorm operations (t-norm and t-conorm for short) of special discrete fuzzy numbers on countable sets are presented, and the properties of these two operators are proven. We also prove that these two operators satisfy the basic conditions for closure of operation and present some examples. Finally, the applications of special discrete fuzzy numbers on countable sets in image fusion and aggregation of subjective evaluation are proposed. [ABSTRACT FROM AUTHOR]

After the Hindu-Arabic decimal positional system was introduced in Europe, throughout many centuries textbooks on elementary arithmetic, intended for beginners, had a more or less fixed organization of content, usually starting with chapters on numeration. These chapters, as a rule, contained one or more examples of large integers the purpose of which was simply to be named (read out loud), sometimes also vice versa. This tradition apparently began with the two first texts that significantly contributed to the spread of the decimal system in Europe--the Latin translations of al-Khwarizmi's treatise on decimal arithmetic, and Leonardo's Liber Abaci, containing examples of reading a 16-digit and a 15-digit number respectively. Throughout the centuries, the order of magnitude of these introductory numbers increased, in general up to some 30 digits, but in some cases to over 60 digits. In this paper we examine the development and extent of this characteristic of introductory arithmetic textbooks from the period 13th-19th century, and the conditions which lead to this, now extinct, practice. [ABSTRACT FROM AUTHOR]

In this paper, an optimization approach based on an arithmetic optimization algorithm (AOA) is proposed for specifying the optimal allocation of distribution generations/generators (DGs) and capacitor banks (CBs) in radial distribution systems. The AOA is a new population-based meta-heuristic algorithm that is essentially based on using basic arithmetic operators in mathematics. The proposed approach is employed to specify the optimum placement, capacity, and power factor of DGs and CBs to decrease the distribution systems' total power loss and voltage deviation. To state the performance of the proposed approach, DGs and CBs are placed in IEEE 33-bus and 69-bus systems separately or together. When only DGs are used and the parameters of location, capacity, and power factor of DGs are determined simultaneously, the total active power loss reductions in the IEEE 33-bus and 69-bus systems are achieved at 94.42% and 98.03%, respectively. When the results of other optimization algorithms are examined, it is seen that better results are obtained with AOA. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel concept termed a semi B-algebra, which acts as an extension of the existing B-algebra. Our investigation delves into the derivation of essential arithmetical properties unique to semi B-algebras, coupled with diverse characterizations shedding light on various aspects of this particular algebraic structure. [ABSTRACT FROM AUTHOR]

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set Z. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most 1016 . In this paper we answer another question of Erdős, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly n elements is exp ((4√τ/3+o(1)n3/2/(logn)1/2) as n→∞, where τ=∑∞ t=1(logt+1/t)².En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense. [ABSTRACT FROM AUTHOR]

In recent years, a variety of work has been done in the field of single, double and triple sequences. Study on n-tuple sequence is new in this field. The main interest of this paper is to explore the idea of n-tuple sequences x = (xi1,i2,...,in) in metric spaces. We introduce the concept of n-sequence space of interval number and discussed its arithmetic properties. Furthermore, we combined the concept of Orlicz function, statistical convergence, interval number and n-sequence to construct some new n-sequence spaces and discussed their properties. Some suitable examples for these spaces have been constructed. [ABSTRACT FROM AUTHOR]

The D-S evidence theory is extensively applied to manage uncertain information. However, the theory encounters challenges related to conflicts during the fusion process, impeding the precise identification of multi-subset focal elements. This paper introduces a novel method for conflicting evidence fusion that incorporates the Bray–Curtis dissimilarity, cosine distance of the included angle, and belief entropy. The method comprehensively evaluates three aspects—evidence similarity, evidence distance, and the amount of information—while considering factors like the credibility and uncertainty of evidence. Initially, the evidence undergoes conversion into single-subset focal element evidence through the improved Pignistic probability function. Subsequently, the credibility between pieces of evidence is established using the Bray–Curtis dissimilarity and angle cosine distance, while the uncertainty of the evidence is computed using belief entropy. The weighted correction coefficient of the evidence is determined by integrating the credibility and uncertainty of the evidence. Subsequently, the corrected evidence is fused using the D-S evidence theory to derive the final judgment. An analysis of two sets of arithmetic examples, considering both single-subset and multi-subset focal elements, demonstrates the faster convergence and enhanced accuracy and reliability of the proposed method in comparison to existing approaches. [ABSTRACT FROM AUTHOR]

In the era of low-power very large-scale integration design, approximate computing is a soaring design paradigm that assures energy-efficient circuit design at the cost of some accuracy. Multiplication is a very important operation of an arithmetic unit and compressor is the inherent part of the multiplier. In this paper, an energy-efficient design of a 4:2 approximate compressor is proposed. The inclusion of the extra leakage control transistors in the logic has led to leakage power reduction, which ultimately reduces the power dissipation in the circuits. The proposed 4:2 compressor is simulated at 10 nm fin-field effect transistor (FinFET) technology. The proposed compressor has been verified for its functionality and robustness. The proposed design shows a power dissipation of 154.48 nW and a delay of 9.16 ps. In comparison to other 4:2 approximate compressors, the proposed compressor shows the least power dissipation, the lowest power delay product (PDP) and the highest robustness. The power dissipation of the proposed compressor is 92.84% less than that of the base compressor. The PDP of the designed compressor is 80.58% less than the minimum PDP among other compressors. The aging analysis for the negative bias temperature instability (NBTI) effect is also checked for the proposed compressor and compared with the existing designs. [ABSTRACT FROM AUTHOR]

The design of a Ternary Logic Processor using CNTFETs (Carbon-Nanotube-Field-Effect-Transistor) is a challenging task, but it also has the potential to offer significant advantages over the traditional binary logic processors based on CMOS (Complementary-Metal-Oxide-Semiconductor) technology. This paper presents the design and implementation of a Ternary Logic Processor (TLP) using CNTFETs. The TLP is a single-cycle processor that operates on three-trit data. An Instruction Set Architecture (ISA) is defined, at first, for this TLP that consists of instructions of the Register type, Load-store type, Immediate type, and branch type. Based on the ISA, the architecture of the CNTFET-based TLP is proposed and the transistor level designs of the TLPs' fundamental blocks like the Ternary Instruction Fetch (TIF), Ternary Register File (TRF), Ternary Arithmetic and Logic Unit (TALU) and Ternary Data Memory (TDM) are presented. HSPICE simulations using a standard CNTFET model, are performed for the TLP and the TLPs' individual blocks and the performance parameters like the power consumption, propagation delay, and the number of CNTFETs required are calculated. In addition to this, the functionality of the processor is verified using a few of the standard programs. [ABSTRACT FROM AUTHOR]

Fix an integer d ≥ 2 . The parameters c 0 ∈ Q ¯ for which the unicritical polynomial f d , c (z) = z d + c ∈ C [ z ] has finite postcritical orbit, also known as Misiurewicz parameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work from a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we call p-special, which may be of independent number theoretic interest. [ABSTRACT FROM AUTHOR]

A universal number (Unum) is a number representation format that can reduce the memory contention issues in multicore processors and parallel computing systems by optimizing the bit storage in the arithmetic operations. Given the safety-critical nature of applications of Unum format, there is a dire need to rigorously assess the correctness of the Unum based arithmetic operations. Unums are of three types, namely, Unum-I, Unum-II and Unum-III (commonly known as Posits). In this paper, we provide a higher-order-logic formalization of Unum-III (posits). In particular, we formally model a posit format (binary encoding of a posit), which is comprised of the sign, exponent, regime and fraction bits, using the HOL Light theorem prover. In order to prove the correctness of a posit format, we formally verify various properties regarding conversions of a real number to a posit and a posit to a real number and the scaling factors of the regime, exponent and fraction bits of a posit using HOL Light. [ABSTRACT FROM AUTHOR]

Arithmetic optimization algorithm (AOA) is a meta-heuristic optimization method based on mathematical operators proposed in recent years. Although it has good performance, it can also lead to insufficient local search ability and falling into local optima when solving complex optimization problems. In order to make up for the above shortcomings, the optimization performance of AOA is further improved. This paper proposes a hybrid algorithm based on AOA and particle swarm optimization (PSO) called HAOAPSO. Firstly, a compound opposition-based learning (COBL) strategy is introduced to broaden the scope of finding optimal solutions to help the algorithm better jump out of local optima. Secondly, PSO is combined with AOA that integrates COBL to improve the algorithm's local search ability, so as to improve the overall search efficiency of the algorithm. In addition, experiments are performed on 23 classical benchmark functions with different characteristics and five engineering design optimization problems, and the experimental results of HAOAPSO are compared with those of other well-known optimization algorithms to comprehensively evaluate the performance of the proposed algorithm. The simulation results show that HAOAPSO can provide better solutions in most cases when solving global optimization problems such as engineering, with better convergence speed and accuracy. [ABSTRACT FROM AUTHOR]

Binary matrix factorization is an essential tool for identifying discrete patterns in binary data. In this paper, we consider the rank-k binary matrix factorization problem (k-BMF) under Boolean arithmetic: we are given an n × m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n × k and k × m, respectively, which minimize the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak linear programming (LP) relaxation, whereas the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank-k factorization. For one of the exponential size Ips, we describe a computational approach based on column generation. Experimental results on synthetic and real-world data sets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorizations. Funding: O. Günlük was partially supported by the Office of Naval Research [Grant N00014-21-1-2575]. R. Á. Kovács was supported by a doctoral scholarship from The Alan Turing Institute under the EPSRC [Grant EP/N510129/1] and the Office for National Statistics. R. A. Hauser was supported by The Alan Turing Institute under the EPSRC [Grant EP/N510129/1]. [ABSTRACT FROM AUTHOR]

In a Multiple-Input Multiple-Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) method, multiple antennas can be used on either the transmitter or receiver end to improve the system capacity, data throughput, and robustness. OFDM has been used as the modulation system that divides the data stream into multiple parallel low-rate subcarriers. MIMO enhances the system by utilizing spatial diversity and multiplexing abilities. Modulation classification in the MIMO-OFDM systems describes the process of recognizing the modulation scheme used by the communicated signals in a MIMO-OFDM communication system. This is a vital step in receiver design as it enables proper demodulation of the received signals. In this paper, an Enhanced Modulation Classification Approach using an Arithmetic Optimization Algorithm with Deep Learning (EMCA-AOADL) is developed for MIMO-OFDM systems. The goal of the presented EMCAAOADL technique is to detect and classify different types of modulation signals that exist in MIMO-OFDM systems. To accomplish this, the EMCA-AOADL technique performs a feature extraction process based on the Sevcik Fractal Dimension (SFD). For modulation classification, the EMCA-AOADL technique uses a Convolution Neural Network with Long Short-Term Memory (CNN-LSTM) approach. Finally, the hyperparameter values of the CNN-LSTM algorithm can be chosen by using AOA. To highlight the better recognition result of the EMCA-AOADL approach, a comprehensive range of simulations was performed. The simulation values illustrate the better results of the EMCA-AOADL algorithm. [ABSTRACT FROM AUTHOR]

We consider a random walk with zero mean and finite variance whose steps are arithmetic. The arcsine law for the time the walk reaches its maximum is well known. In this paper, we consider the distribution of the moment of reaching the maximum under the assumption that the maximum value itself is fixed. We show that, in the case of a moderate deviation of the maximum, the distribution of the moment of the maximum with appropriate normalization converges to the chi-square distribution with one degree of freedom. Similar results are obtained in the nonlattice case. [ABSTRACT FROM AUTHOR]

Hrotsvit of Gandersheim sees music and arithmetic as parts of philosophy. Presenting the motives for engaging in literary creativity and describing her writing process, she says that by glorifying God with her work, she incorporated threads and pieces of the ancient mantle of philosophy into it. The paper aims to provide insight into Hrotsvit’s knowledge of philosophy and the specificities and significance of her approach to the contents of the disciplines of arithmetic and music (in particular to musica humana and the harmony of a world) in her plays Paphnutius and Sapientia, to analyze different elements of her authorial self-perception and to emphasize the similarities between her and Augustine’s understanding of music, and the role of certain types of writing. The paper concludes that Hrotsvit incorporated her knowledge and understanding of ancient and medieval theories of music and numbers into her plays uniquely, showing originality in their interpretation and merging them with everyday moral life and education. In this way, the rarely treated philosophical elements of her oeuvre are presented and evaluated.

String constraint solving is the core of various testing and verification approaches for scripting languages. Among algorithms for solving string constraints, flattening is a well-known approach that is particularly useful in handling satisfiable instances. As string/integer conversion is an important function appearing in almost all scripting languages, Abdulla et al. extended the flattening approach to this function recently. However, their approach supports only a special flattening pattern and leaves the support of the general flat regular constraints as an open problem. In this paper, we fill the gap by proposing a complete flattening approach for the string/integer conversion. The approach is built upon a new quantifier elimination procedure for the linear-exponential arithmetic (namely, the extension of Presburger arithmetic with exponential functions, denoted by ExpPA) improved from the one proposed by Cherlin and Point in 1986. We analyze the complexity of our quantifier elimination procedure and show that the decision problem for existential ExpPA formulas is in 3-EXPTIME. Up to our knowledge, this is the first elementary complexity upper bound for this problem. While the quantifier elimination procedure is too expensive to be implemented efficiently, we propose various optimizations and provide a prototypical implementation. We evaluate the performance of our implementation on the benchmarks that are generated from the string hash functions as well as randomly. The experimental results show that our implementation outperforms the state-of-the-art solvers. [ABSTRACT FROM AUTHOR]

The Neutrosophic Bonferroni operator is a novel operator that we provide in this paper. Then the arithmetic operations for Neutrosophic Bonferroni operator is developed which tells the existence of Neutrosophic Bonferroni operator. Then its properties were discussed with special cases. To group decision-making issues with several attributes, arithmetic ranking operations and the Neutrosophic approach are used. The result is compared with the existing methodology. The suggested approach will more accurately give the decision maker the ideal attribute than the existing system does. Neutrophic multicriteria is a method of decision-making that makes use of ambiguity to integrate various criteria or factors--often with imprecise or ambiguous data--to reach a result. The neutrosophic multicriteria analysis enables the assessment of subjective and qualitative factors, which can assist in resolving conflicting goals and preferences. In Neutrosophic Multi-Attribute Group Decision Making (NMAGDM) problems, all the data supplied by the decision makers (DMs) is expressed in single-value Neutrosophic triangular and trapezoidal numbers, which are studied in this work and can improve the flexibility and precision of capturing uncertainty and aggregating preferences. Studying this operator is crucial because it can be utilised to resolve multi-attribute. [ABSTRACT FROM AUTHOR]

Algebraic operations on generalized hesitant fuzzy numbers are key tools to address the problems with decision uncertainty. In this paper, by studying the arithmetic operations on generalized trapezoidal hesitant fuzzy numbers, modified arithmetic operations are introduced for this class of numbers so that, using these arithmetic operations, the multiplication and division of two generalized trapezoidal hesitant fuzzy numbers are always generalized trapezoidal hesitant fuzzy numbers. Furthermore, a generalized trapezoidal hesitant fuzzy number raised to the power of a real number is a generalized trapezoidal hesitant fuzzy number, and in the defined division, the case where the denominator becomes zero is not considered. Numerical examples are used to show the shortcomings of the previous arithmetic operations as well as the efficiencies of the arithmetic operations proposed in this research for generalized trapezoidal hesitant fuzzy numbers. Finally, the application of the proposed new arithmetic operations to generalized trapezoidal hesitant fuzzy numbers in solving the generalized trapezoidal hesitant fully fuzzy equation is discussed. [ABSTRACT FROM AUTHOR]

After introducing the definition of hypergroups by Marty, the study of hyperstructures and its connections with other fields has been of great importance. In this paper, we continue the investigation between hyperstructure theory and number theory. More precisely, we define an Hv-module of complex valued functions over the Hv-ring of arithmetics, classify its complete parts, the strongly regular relations on it and identify its fundamental module. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform (FrFT) is a useful mathematical tool for signal and image processing. In some applications, the eigendecomposition-based discrete FrFT (DFrFT) is suitable due to its properties of orthogonality, additivity, reversibility and approximation of continuous FrFT. Although recent studies have introduced reduced arithmetic complexity algorithms for DFrFT computation, which are attractive for real-time and low-power consumption practical scenarios, reliable hardware architectures in this context are gaps in the literature. In this paper, we present two hardware architectures based on the referred algorithms to obtain N-point DFrFT ( N = 4 L , L is a positive integer). We validate and compare the performance of such architectures by employing field-programmable gate array implementations, co-designed with an embedded hard processor unit. In particular, we carry out computer experiments where synthesis, error and latency analyses are performed, and consider an application related to compact signal representation. [ABSTRACT FROM AUTHOR]

In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some R -specific operations in K like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Z n to K n , and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations). [ABSTRACT FROM AUTHOR]

In modern Digital System design, adders are widely used in many applications including Microprocessors, Digital Signal Processors, Arithmetic Logic Units and digital signal processing. Power efficiency is an important key factor for any integrated circuits. By considering the importance of Full Adder (FA) cells, their power consumption may play significant role in overall power consumption. So designing a low-power full adder is important to achieve the high-performance computing. Swing restored technique is commonly used method to reduce the signal swing at the internal nodes and it helps to reduce the power consumption and improves the circuit's speed. This paper presents a novel hybrid full adder cell design using the modified swing restored technique and complementary pass transistor logic. The Full Adder circuit was implemented using Cadence Virtuoso tool on gptk 180 nm CMOS process technology. The proposed full adder was compared with earlier reported circuits based on Delay, Power and PDP. The comparison shows that proposed circuit consumes less power with high performance. Also various process corners and noise immunity were analyzed to find the sensitivity of the circuit. Further an 8-bit Ripple Carry Adder was designed using the proposed full adder to verify the functionality in higher order bits. Simulations were performed and analyzed using the Cadence Spectre. • Designing low-power full adders is of paramount importance in modern digital system design. • The swing restored technique was employed to reduce the signal swing at the internal nodes of the full adder circuit, thereby reducing the power consumption and improving the circuit's speed. • This paper presents a novel design of a full adder circuit using the modified swing restored technique for reducing power consumption and enhancing performance. • The full adder cell introduced in this research was designed with 180 nm CMOS process technology. • A comparative study was performed between the proposed circuit and earlier reported circuits, revealing reduced power dissipation in the proposed design. [ABSTRACT FROM AUTHOR]

This research article introduces an innovative 1-bit full adder design, leveraging grounded keeper circuitry. To implement full adder, keeper based XOR-XNOR cell -based design approach is used. Achieving full swing output voltage is one of the critical challenges in the designing of full adder. In this paper 8-T XOR-XNOR cell is proposed and simulated using HSPICE software at 90 nm technology node. The introduction of keeper circuit, which decreases propagation delay and offer full output voltage swing, is the primary focus of this research. Furthermore, this work puts forth an original design for a voltage-resilient ultra high-speed low-power keeper-based 1-bit full adder (UHSLPFA). Our research delves into a comprehensive comparison of various full adder designs, focusing on power dissipation (PWR), propagation delay (tp), and power-delay product (PDP). Notably, our proposed 20-T full adder design boasts notably reduced propagation delay and power consumption when compared to the existing counterparts. The envisioned application scope for this voltage-resilient ultra-high-speed-low-power keeper-based 1-bit full adder extends to the development of arithmetic logic units, multipliers, calculators, and graphical processing units. To gauge its voltage resilience, our proposed UHSLPFA is subjected to simulation across a range of supply voltages, from 0.6 to 1.5 V. This evaluation uncovers variations in PWR, tp, and PDP, showcasing the superior resilience of our design compared to contemporary state-of-the-art alternatives. The performance of the proposed full adder is also evaluated in 4-bit ripple carry adder. [ABSTRACT FROM AUTHOR]

Quantitative Structure Activity Relation (QSAR) models are mathematical techniques used to link structural characteristics with biological activities, thus considered a useful tool in drug discovery, hazard evaluation, and identifying potentially lethal molecules. The QSAR regulations are determined by the Organization for Economic Cooperation and Development (OECD). QSAR models are helpful in discovering new drugs and chemicals to treat severe diseases. In order to improve the QSAR model's predictive power for biological activities of naturally occurring indoloquinoline derivatives against different cancer cell lines, a modified machine learning (ML) technique is presented in this paper. The Arithmetic Optimization Algorithm (AOA) operators are used in the suggested model to enhance the performance of the Sinh Cosh Optimizer (SCHO). Moreover, this improvement functions as a feature selection method that eliminates superfluous descriptors. An actual dataset gathered from previously published research is utilized to evaluate the performance of the suggested model. Moreover, a comparison is made between the outcomes of the suggested model and other established methodologies. In terms of pIC50 values for different indoloquinoline derivatives against human MV4-11 (leukemia), human HCT116 (colon cancer), and human A549 (lung cancer) cell lines, the suggested model achieves root mean square error (RMSE) of 0.6822, 0.6787, 0.4411, and 0.4477, respectively. The biological application of indoloquinoline derivatives as possible anticancer medicines is predicted with a high degree of accuracy by the suggested model, as evidenced by these findings. [Display omitted] • The primary contributions of this work can be summed up as follows: • Modified Sinh Cosh Optimizer (SCO)-QSAR model to predict pIC 50 values for various cells. • Improve the performance of SCO using Arithmetic Optimization Algorithm operators. • Assess the efficiency of the developed model using real-world QSAR datasets. [ABSTRACT FROM AUTHOR]

Background: This study aimed to investigate the cognitive and affective factors associated with cross-cultural differences in arithmetic tasks. Methods: A total of 404 third- and fourth- graders were recruited from China and Italy to complete exact arithmetic, arithmetic estimation and cognitive tasks (i.e., short-term memory, executive functions, and fluid reasoning). Their mathematical anxiety was also measured. Results: The results showed that Chinese children performed better than Italian children in both arithmetic tasks and in shifting task. Italian children performed better in visuospatial updating task and reported higher levels of mathematical anxiety than their Chinese peers. Multi-group path analyses showed that the patterns of relations among cognitive factors (i.e., short-term memory, inhibition and shifting), mathematical anxiety, and arithmetic performance were similar across groups. The only exception was that visuospatial updating uniquely predicted arithmetic estimation for Chinese but not for Italian children. Conclusions: Chinese children outperformed their Italian peers in the exact arithmetic task, likely due to the greater emphasis on arithmetic fluency in Chinese mathematics education, both in schools and at home. They also had a slight advantage than Italian peers in the arithmetic estimation task. The unique link between updating and arithmetic estimation found in Chinese children but not Italian children suggests that, although arithmetic estimation is not emphasized in the curricula of either country, instruction and practice in exact arithmetic may enhance Chinese children's efficiency in solving arithmetic estimation problems. [ABSTRACT FROM AUTHOR]

Current tunnel blasting hole layouts are mostly designed based on a two-dimensional plane at the workface, without considering the distribution of the minimum burden at the bottom of the blast holes. This results in a significant number of residual holes at the bottom, reducing excavation efficiency. To address this issue, this study proposes an easer hole design method based on the principle of minimum burden at the hole bottom. The method involved the arithmetic distribution for the minimum burden at the bottom of easer holes, using the difficulty of rock breaking as the design principle for hole positioning. Through theoretical analysis, numerical simulation, and field tests, it is proposed that the minimum burden at the bottom of the holes should increase progressively with the initiation sequence, and the relationship between burden distribution and blasting effect was investigated. This study indicates that using the new design principle achieves better blasting results than the model with an evenly distributed burden. When the control ratio of the minimum burden at the bottom of each row of easer holes is 1.3, an average residual hole depth of 36.7 cm and a maximum damage volume of 4.638 m3 can be achieved, yielding the best overall blasting effect. The application of this blasting scheme in the field significantly improved the residual hole problem, reducing the average residual hole depth to 39.5 cm, which is a 43.4% reduction compared to the previous scheme. Additionally, the utilization rate of blast holes in the new scheme increased to 91.3%, an improvement of 11.0% over the previous scheme. This study provides new insights and methods for tunnel blasting hole layout design, offering significant engineering application value. [ABSTRACT FROM AUTHOR]