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This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi)$, where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$, can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho)$ for all virtual representations ρ of the corresponding Weil group WE$W_E$. Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. [ABSTRACT FROM AUTHOR]

Developing Arithmetic Word Problem (AWP) -solving algorithms has recently become one of the hottest research areas because it can simultaneously advance general artificial intelligence and the application of AI technology in education. This paper presents a novel algorithm for solving AWPs by synergizing Large Language Models (LLMs) with the Scene-Aware Syntax–Semantics ( S 2 ) method. The innovation of this algorithm lies in leveraging the LLM to divide problems into multiple scenes, thereby enhancing the relation-flow approach in the processes of relation extraction and reasoning. Our algorithm consists of three components: scene decomposer, relation extractor, and symbolic solver. In the scene decomposer, we propose the Chain-Of-Scene (COS) method. It dynamically constructs prompts for the LLM using a retrieval-augmented strategy, thus enabling the chain-formed generation of scenes from the input problem. In the relation extractor, we introduce the Scene-Aware S 2 method, which utilizes syntax–semantics models to match the text within each scene and convert it into relations. This allows for the efficient and accurate extraction of explicit and implicit relations. Finally, a symbolic solver is employed to reason through the set of relations to derive the solution. Experimental results on six authoritative datasets demonstrate that the proposed algorithm achieves an average solving accuracy of 90.4%, outperforming the State-Of-The-Art (SOTA) algorithm by 1.1%. The case study further illustrates that it outputs more reliable solutions than baseline algorithms. These findings have significant implications for promoting smart education and developing personalized intelligent tutoring systems. [ABSTRACT FROM AUTHOR]

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]

An image encryption algorithm is proposed based on the combination of genetic (DNA) random coding and chaotic mapping. An image encryption algorithm based on an improved new four-dimensional chaotic system and DNA coding is proposed to address the problem that a single coding method is prone to selecting plaintext attacks. Based on a four-dimensional chaotic system existing in the literature, a new four-dimensional hyperchaotic system is obtained through improvement. The initial value of the system is generated based on SHA-256, zigzag transform. The input key and four pseudo-random chaotic sequences are generated iteratively. DNA chunking encoding, arithmetic operation, and decoding are implemented for the image disrupted based on the zigzag transform. The two-dimensional matrix constituted based on the Chaotic Sequence of Chebyshev to obtain the scrambled and diffused ciphertext image. Simulation experiments and security performance analysis show that the algorithm enhances the correlation between the key and the plaintext, the randomness of the encryption process, and effectively improves the anti-attack capability [H. Chen, J. Zhao et al., Appl. Res. Comput. 10, 0434 (2021)]. In this paper, 512 × 512 × 3 peppers color images are used for testing, and the correlation coefficients of adjacent pixels of the encrypted images are all close to 0, and the information entropy is all more significant than 7.997, which is relative to the theoretical value of 8. The experimental results show that the proposed algorithm improves the security of the ciphertext, increases the critical space, and, at the same time, resists various attack methods. [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]

Assume that we are given a sequence F of k homogeneous polynomials in n variables of degree at most d and the ideal ℐ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity d O (n) , whether F is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if F is regular then we can transform ℐ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity d O (n 2) . Finally, under the same assumption, we establish the new upper bound 2 (d k / 2) 2 n − k − 1 for the maximum degree of the elements of any reduced Gröbner basis of ℐ in the case that n > k. [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]

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions). [ABSTRACT FROM AUTHOR]

Large language models such as ChatGPT have been shown to excel in solving complex math problems. However, they cannot solve basic arithmetic problems such as 758*639 = 484,362. This makes us ponder if LLMs have been trained to solve math and science problems in the right way. When a student learns math at school, she or he starts with arithmetic, then moves to word problems, polynomials, and calculus. Each skill she or he acquires will be used in the next stage to solve more advanced problems. In this paper we propose Scaffolding Learning for LLMs, which imitates how a student learns a subject in a step-by-step manner. For example, we first train an LLM to perform highly specific operations such as multiplication and division, and then apply such "skills" in a more generic task such as solving word problems. This is related to Curriculum Training, which trains a model on tasks following a specific order, such as training on easy tasks first and then gradually increases the difficulty. Our proposed approach goes from specific tasks to generic ones, which can be considered as a special case of Curriculum Training. Our empirical studies show that when an LLM has "mastered" a specific skill, only a small amount of training is required to teach it to apply the skill to a more generic application. [ABSTRACT FROM AUTHOR]

In this paper, the multidigit arithmetic-related strategy adaptivity, strategy flexibility and solution accuracy of Danish compulsory school students is examined. Participants, 749 grade three, 731 grade six and 818 grade eight, were drawn from twenty demographically different schools. Drawing on a tri-phase assessment tool, each student completed a series of tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy; those using shortcut strategies were construed as adaptive for that task. Second, students solved the same tasks by means of whatever alternative strategies they had available; those offering at least two strategies were construed as flexible for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Across all grades, students were more flexible than adaptive. Overall, sixth graders exhibited higher levels of flexibility than third graders and marginally lower levels than eighth graders. Sixth graders exhibited higher levels of adaptivity than those in either grade three or grade eight. Students' accuracy, which improved with maturation, was influenced positively by both adaptivity and flexibility, with flexibility having the greatest influence in grade three and adaptivity in grade six. The findings raise further questions concerning, inter alia, culture's influence on students' strategy choices and the interaction of adaptivity, flexibility and maturity on accuracy. [ABSTRACT FROM AUTHOR]

Let T be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object X with respect to a homogeneous ideal a , width R (a , X) and hd (a , X) , respectively, are introduced. The local nature and some basic properties of width R (a , X) and hd (a , X) are provided. In addition, we give an upper bound and lower bound of width R (a , X) . What is more, we give the relationship between the local homology dimension hd (a , X) and the arithmetic rank of a and dim R . [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]

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]

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]

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]

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]

In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]

Solving arithmetic word problems that entail deep implicit relations is still a challenging problem. However, significant progress has been made in solving Arithmetic Word Problems (AWP) over the past six decades. This paper proposes to discover deep implicit relations by qualia inference to solve Arithmetic Word Problems entailing Deep Implicit Relations (DIR-AWP), such as entailing commonsense or subject-domain knowledge involved in the problem-solving process. This paper proposes to take three steps to solve DIR-AWPs, in which the first three steps are used to conduct the qualia inference process. The first step uses the prepared set of qualia-quantity models to identify qualia scenes from the explicit relations extracted by the Syntax-Semantic (S²) method from the given problem. The second step adds missing entities and deep implicit relations in order using the identified qualia scenes and the qualia-quantity models, respectively. The third step distills the relations for solving the given problem by pruning the spare branches of the qualia dependency graph of all the acquired relations. The research contributes to the field by presenting a comprehensive approach combining explicit and implicit knowledge to enhance reasoning abilities. The experimental results on Math23K demonstrate hat the proposed algorithm is superior to the baseline algorithms in solving AWPs requiring deep implicit relations. [ABSTRACT FROM AUTHOR]

GPUs are widely used for diverse applications, particularly data-parallel tasks like machine learning and scientific computing. However, their efficiency is hindered by architectural limitations, inherited from historical RISC processors, in handling memory loads causing high register file contention. We observe that a significant number (around 26%) of values present in the register file are typically used only once, contributing to more than 25% of the total register file bank conflicts, on average. This paper addresses the challenge of single-use memory values in the GPU register file (i.e. data values used only once) which wastes space and increases latency. To this end, we introduce a novel mechanism inspired by CISC architectures. It replaces single-use loads with direct memory operands in arithmetic operations. Our approach improves performance by 20% and reduces energy consumption by 18%, on average, with negligible (<1%) hardware overhead. [ABSTRACT FROM AUTHOR]

Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higherdimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions. [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]

This paper provides an analysis of existing single-digit binary adders from the point of view of their implementation on fans built based on metal-oxide-semiconductor field-effect transistor—MOSFETs. The synthesis of a single-digit adder with a conditional sum is carried out. The considered adders are compared in terms of speed and hardware complexity (by the number of MOSFETs). Adders perform arithmetic operations on numbers. In combination with other logical operations, adders are the core of the circuits of arithmetic logic devices that implement several different operations; they are an integral part of different processors. The most important parameters of adders are their hardware complexity and performance; therefore, many options for single-bit and multi-bit connectors with serial, parallel and combined transmissions have been developed. In the final part, a scheme of a multi-bit adder with consecutive transfers on adders with a conditional sum is given. An example of performing addition operations is given. [ABSTRACT FROM AUTHOR]

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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]

In this paper traveling wave solutions of the modified regularized and regularized long wave equations (MRLW & RLW equations) are simulated using modified trigonometric B-spline functions in collocation method. We validate the adaptability of the proposed method successfully by simulating single solitary wave motion, two and three solitary wave interactions. We also compute the three motion invariants numerically and these are found to be very very close with their analytical values. We also calculate 4 and Loo error norms for single solitary wave motion. Stability of the method is also discussed. Convergence rate for the proposed scheme is obtained numerically with respect to space step size. Computational complexity is analysed and it is found that the developed approach gives very good results with very less computational cost. in terms of the number of arithmetic operations, the computational complexity of the proposed method is found to be linear. [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]

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 this paper, we consider the forward rounding errors of the partial pivoting quaternion LU decomposition. To carry out error analysis of algorithms in quaternion arithmetic, we first propose the accuracy model for the basic quaternion arithmetic operations by the real structure-preserving method. And then, we study the forward rounding errors of the partial pivoting quaternion LU decomposition. Finally, by using the results of the forward rounding errors of the partial pivoting quaternion LU decomposition, we analyze the forward rounding errors of the nonsingular quaternion linear system. [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]

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]

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula ϕ (F) , with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA 2 . [ABSTRACT FROM AUTHOR]

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of 'ordinary' mathematics? (2) How should Frege's theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won't give rise to different numbers at different worlds? (3) Can one reconstruct Frege's theory of numbers in a non-modal setting without mathematical primitives such as "the number of Fs" ( # F ) or mathematical axioms such as Hume's Principle? Our answer to question (1) is 'None'. Our answer to question (2) begins by defining 'x numbers G' as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege's derivation of the Dedekind-Peano axioms in a non-modal setting. [ABSTRACT FROM AUTHOR]

The mature Wittgenstein's groundbreaking analyses of sense and the logical must—and the powerful new method that made them possible—were the result of a multi‐year process of writing, re‐arranging, re‐writing and one large‐scale revision that eventually produced the Philosophical Investigations and RFM I. In contrast, his struggles during the same period with questions of arithmetic and higher mathematics remained largely in first‐draft form, and he drops the topic entirely after 1945. In this paper, I argue that Wittgenstein's new method can be applied to the cases of arithmetic and set theory and that the result is innovative, recognizably Wittgensteinian, and independently appealing. I conclude by acknowledging the reasons Wittgenstein himself might have had to resist applying his own proven method to the case of mathematics—particularly to set theory—and by indicating why I think those reasons are ultimately unsound. [ABSTRACT FROM AUTHOR]

Let F be a field of formal series over a finite field and ℬ d be the affine building associated to PGL d (F). Given a lattice Γ in PGL d (F) , the complex arising as a quotient Γ \ ℬ d is called weakly Ramanujan if every nontrivial discrete simultaneous spectrum of the colored adjacency operators A 1 , A 2 , ... , A d − 1 acting on L 2 (Γ \ ℬ d) is contained in the simultaneous spectrum of those operators acting on L 2 (ℬ d). In this paper, we prove that the standard non-uniform arithmetic quotient PGL 4 ( q [ t ]) \ ℬ 4 of PGL 4 (F) is weakly Ramanujan. [ABSTRACT FROM AUTHOR]

For a prime p, positive integers r , n , and a polynomial f with coefficients in p r , let W p , r , n (f) = f n p r \ f n + 1 p r . As n varies, the W p , r , n (f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on p r . In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of p r lying in a given W p , r , n (f). Moreover, when we generalize our definition of W p , r , n (f) , we obtain both upper and lower bounds for the resulting averages. [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]

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. [ABSTRACT FROM AUTHOR]

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In [a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641], Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin–Schreier covers. [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 aims to explore the arithmetic properties of Fu's k dots bracelet partition where k = pα, p is a prime number with p ≥ 5 and α is an integer with α ≥ 0. For pα dots bracelet partitions with p = 5, 7 and 11, we found several exciting Ramanujan-like congruences modulo p. We also used Newman's theorems to demonstrate certain congruence modulo p. [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]

Foundation mathematics courses play a crucial role in allowing students who have not achieved the pre-requisite mathematics credits for tertiary studies to re-engage with STEM studies. This paper describes how targeting self-efficacy, skill development and social connectedness have influenced the design of a foundation mathematics course at a New Zealand university. These design goals are grounded in education literature, and the paper outlines how focusing on these goals has resulted in the design of a course that incorporates novel approaches to learning and assessment, including online and face-to-face components. These approaches include the use of large, automatically graded weekly quizzes; adaptive learning quizzes for essential skills such as fraction arithmetic; and an emphasis on face-to-face learning and connection through tutorials and collaborative problem-solving workshops. The paper concludes with reflections on the success of the course so far and raising questions for future investigation. [ABSTRACT FROM AUTHOR]

The author's last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers. [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]