Arithmetic Operations Research Articles

Simple arithmetic operation in latent space can generate a novel three-dimensional graph metamaterials

Recent advancements in artificial intelligence (AI)-based design strategies for metamaterials have revolutionized the creation of customizable architectures spanning nano- to macro-scale dimensions. However, their increasing complexity poses challenges in generating diverse metamaterials, hindering widespread adoption. Here, we introduce an innovative design strategy for three-dimensional graph metamaterials through simple arithmetic operations within the latent space. By leveraging carefully designed hidden representations of disentangled latent space and diffusion processes, our method unravels design space complexity, generating diverse graph metamaterials with comprehensive understanding. This versatile methodology facilitates the creation of graph metamaterials ranging from repetitive lattices to functionally graded materials. We believe that this methodology represents a foundational step in advancing our comprehension of the intricate latent design space, offering the potential to establish a unified model for various traditional generative models in the realm of graph metamaterials.

Non-interactive Private Multivariate Function Evaluation using Homomorphic Table Lookup

To address security issues in cloud computing, fully homomorphic encryption (FHE) enables a third party to evaluate functions using ciphertexts that do not leak information to the cloud server. The remaining problems of FHE include high computational costs and limited arithmetic operations, only evaluating additions and multiplications. Arbitrary functions can be evaluated using a precomputed lookup table (LUT), which is one of the solutions for those problems. Previous studies proposed LUT-enabled computation methods 1) with bit-wise FHE and 2) with word-wise FHE. The performance of LUT-enabled computation with bit-wise FHE drops quickly when evaluating BigNum functions because of the complexity being O(s·2^d·m), where m represents the number of inputs, d and s represent the bit lengths of the inputs and outputs, respectively. Thus, LUT-enabled computation with word-wise FHE, which handles a set of bits with one operation, has also been proposed; however, previous studies are limited in evaluating multivariate functions within two inputs and cannot speed up the evaluation when the domain size of the integer exceeds 2N, where N is the number of elements packed into a single ciphertext. In this study, we propose a non-interactive model, in which no decryption is required, to evaluate arbitrary multivariate functions using homomorphic table lookup with word-wise FHE. The proposed LUT-enabled computation method 1) decreases the complexity to O(2^d·m/l), where l is the element size of FHE packing; 2) extends the input and output domain sizes to evaluate multivariate functions over two inputs; and 3) adopts a multidimensional table for enabling multithreading to reduce latency. The experimental results demonstrate that evaluating a 10-bit two-input function and a 5-bit three-input function takes approximately 90.5 and 105.5 s with 16-thread, respectively. Our proposed method achieves 3.2x and 23.1x speedup to evaluate two-bit and three-bit 3-input functions compared with naive LUT-enabled computation with bit-wise FHE.

Synchronization of Gold sequences based on fast transform in a truncated basis of Walsh–Hadamard functions

Based on the analysis of the structures of isomorphic multiplicative groups of extended Galois fields, it is established that any cyclic shift of a pseudo-random Gold sequence can be transformed into a function belonging to the complete set of analogues of Rademacher functions of the corresponding dimension. This made it possible to develop a new algorithm for fast synchronization of Gold sequences based on the calculation of their discrete convolution using fast spectral transformation in a truncated basis of Walsh–Hadamard functions. The gain of the developed algorithm in terms of the number of arithmetic operations compared to the traditional method of calculating discrete convolution increases with increasing sequence length N and for N=511.1023 is approximately 3.4 times.

The "effects of wind turbine sound on working memory" (a pre-registration plan)

Although research on the effects of wind turbine sound (WTS) over mental health factors is increasing, very little attention has been given to the potential effects of WTS over subtle cognitive processes, such as concentration or memory. The "brainwave entrainment hypothesis" is rooted in neuropsychological research demonstrating that brain oscillations are modulated by external rhythms. In addition, perceived annoyance to wind turbines seems to be a phenomenon in which both visual and psychological variables interact with acoustic stimulation. However, annoyance is a concept rarely operationalised in the literature. Our study aims to experimentally assess the effects of WTS on working memory. Working memory is recognised as a basic process, playing a crucial role in unconscious everyday processes (e.g., perceiving an object's trajectory despite input interruptions when blinking), as well as deliberate processes (e.g., carrying out arithmetic operations). By exploring variations in performance during a memory task, while being exposed to WTS samples, we could highlight acoustic characteristics of the WTS stimuli that interfere-or not-with cognitive processes. These data, combined with within-block ratings of annoyance, can provide us with index of annoyance directly related to the acoustic properties present in WTS recordings.

No evidence for unconscious initiation and following of arithmetic rules: A replication study.

The field of consciousness studies has yielded various-sometimes contradicting-accounts regarding the function of consciousness, ranging from denying it has such function to claiming that any high-level cognitive function requires consciousness. Empirical findings supporting both accounts were reported, yet some of them have been recently revisited based on failures to replicate. Here, we aimed at replicating a remarkable finding reported by Ric and Muller (2012); participants were able to follow an unseen instruction, integrate it with a subsequently presented pair of unseen digits, and accordingly either add the digits (resulting in a priming effect), or simply represent them. This finding thus demonstrates unconscious task-switching, temporal integration (involving mental chaining), and arithmetic operation. Finding such high-level processes in the absence of awareness is of pivotal importance to our understanding of consciousness, as it challenges prominent theories in the field (e.g., the global neuronal workspace). Accordingly, in light of the self-correction wave in psychological science in general and in the field of consciousness studies in particular, we report here a preregistered replication aimed at testing the reproducibility of this finding, while also better controlling for subjects' awareness of both the instruction and the digits. Across two highly powered experiments, our results failed to replicate the original effect. We, therefore, conclude that the current evidence does not support the claim that arithmetic operations (specifically, addition) can be flexibly initiated without awareness, in line with the current arguments for a more limited scope of unconscious processing. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Low-Cost Constant Time Signed Digit Selection for Most Significant Bit First Multiplication

Serial binary multiplication is frequently used in many digital applications. In particular, left-to-right (aka online) manipulation of operands promotes the real-time generation of product digits for immediate utilization in subsequent online computations (e.g., successive layers of a neural network). In the left-to-right arithmetic operations, where a residual is maintained for digit selection, utilization of a redundant number system for the representation of outputs is mandatory, while the input operands and the residual may be redundant or non-redundant. However, when the input data paths are narrow (e.g., eight bits as in BFloat16), conventional non-redundant representations of inputs and residual provide some advantages. For example, the immediate and costless sign detection of the residual that is necessary for the next digit selection; a property not shared by redundant numbers. Nevertheless, digit selection, as practiced in the previous realizations, with both redundant and non-redundant inputs and/or residual, is slow and rather complex. Therefore, in this paper, we offer an imprecise, but faster digit selection scheme, with the required correction in the next cycle. Analytical evaluations and synthesis of the proposed circuits on FPGA platform, shows 30% speedup and less cost with respect to both cases with redundant and non-redundant inputs and residual.

ENHANCED ALGORITHM FOR COMPUTING CAYLEY’S FIRST HYPERDETERMINANT

Combinatorial hyperdeterminant DET – is the homogeneous polynomial in the entries of a hypermatrix of even number of indices, which is also a unique SL-invariant of minimal degree. It was first studied by Cayley in the middle of 19-th century. Given its fundamental nature, the computation of this polynomial is an important task. For fixed d and a cubical hypermatrix X of length n Barvinok introduced an algorithm of computing hyperdeterminant in 0(2nd nd-1) . Since the problem of deciding whether for the given hypermatrix X the hyperdeterminant DET(X) is equal to zero is NP-hard, it is essential to develop efficient algorithm for computing hyperdeterminant, as the size of problem grows exponentially. We provide enhanced algorithm of computing hyperdeterminant that requires 0(2 n(d-1) nd-1) arithmetic operations.

S-слівна арифметика та високоточні обчислення

The intricacies of using S-word arithmetic, the influence of the value of the parameter S on the estimation of the rounding error are analyzed; what are high-precision calculations and where they are used. The problems of two-key cryptography, computer steganography and the problem of transcomputational complexity are considered as areas of application of S-word arithmetic. For the development of S-word arithmetic algorithms, sequential, parallel, quantum computing models are used, and systems of residual classes are used. The architectural features of the computer system for the implementation of an effective algorithm in various models of calculations are considered. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Peculiarities of transferring algorithms to another computing model are considered. An analysis of the complexity of implementing S-word arithmetic operations in sequential, parallel, and quantum computing models is carried out. For the parallel computing model, the importance of reducing the connected steps is indicated, which can increase the amount of processed data, but allows to involve a larger number of parallel processors. This approach is in conflict with a method that reduces the amount of processed data, and there is a need to maintain a balance between these two methods in a parallel computing model. For the quantum computing model, the connection of qubits is a key factor in determining the quantum volume. The physical scheme determines which pairs of qubits can be entangled in a quantum computer. Information is provided about the ongoing scientific forum «Calculation optimization issues», the subject of which is closely related to the topic (1969-2023)

Kashiwara’s theorem for twisted arithmetic differential operators

Kashiwara’s theorem for twisted arithmetic differential operators

Accelerating large-scale DEA computation using sequential categorization and dynamic reference set selection

Data envelopment analysis (DEA) is a well-known data-enabled analytic tool for evaluating relative efficiency of units with multiple inputs and multiple outputs. The DEA computation increases substantially in the presence of large samples. In this study, we first recall two lemmas to distinguish efficient units using arithmetic operations without solving linear programming (LP). Using the used lemmas, the total sample of units is partitioned into several sequential blocks, where units in the preceding blocks are relatively efficient to those in the subsequent blocks. A novel reference set selection procedure is then formulated. We implement the proposed approach into one of the fastest existing methods and demonstrate a significant improvement in elapsed time. We conduct simulation experiments and illustrate the outcomes across varying dimensions, cardinality, and density.

Matemáticas en el mundo árabe: Un recorrido por la edad de oro del saber

The history of Arabic mathematics is crucial to the development of mathematics. During the Islamic Golden Age (8th-14th centuries), Arab mathematicians made important contributions in algebra, geometry, trigonometry, and arithmetic, applying their knowledge in astronomy, navigation, and engineering. A significant advance was the introduction of the Hindu-Arabic numeral system, which facilitated arithmetic operations. Prominent mathematicians such as Al-Khwarizmi, who developed methods for solving equations and wrote about algebra, and Omar Khayyam, known for his work in algebraic calculus, made fundamental contributions. Thabit ibn Qurra also contributed to geometry and number theory. In addition, Arabic mathematics preserved and translated works of ancient civilizations, which allowed the transmission of this knowledge to Europe and its influence on the Renaissance.

Selection of an Appropriate Global Partner for Companies Using the Innovative Extension of the TOPSIS Method with Intuitionistic Hesitant Fuzzy Rough Information

In this research, we introduce the intuitionistic hesitant fuzzy rough set by integrating the notions of an intuitionistic hesitant fuzzy set and rough set and present some intuitionistic hesitant fuzzy rough set theoretical operations. We compile a list of aggregation operators based on the intuitionistic hesitant fuzzy rough set, including the intuitionistic hesitant fuzzy rough Dombi weighted arithmetic averaging aggregation operator, the intuitionistic hesitant fuzzy rough Dombi ordered weighted arithmetic averaging aggregation operator, and the intuitionistic hesitant fuzzy rough Dombi hybrid weighted arithmetic averaging aggregation operator, and demonstrate several essential characteristics of the aforementioned aggregation operators. Furthermore, we provide a multi attribute decision-making approach and the technique of the suggested approach in the context of the intuitionistic hesitant fuzzy rough set. A real-world problem for selecting a suitable worldwide partner for companies is employed to demonstrate the effectiveness of the suggested approach. The sensitivity analysis of the decision-making results of the suggested aggregation operators are evaluated. The demonstrative analysis reveals that the outlined strategy has applicability and flexibility in aggregating intuitionistic hesitant fuzzy rough information and is feasible and insightful for dealing with multi attribute decision making issues based on the intuitionistic hesitant fuzzy rough set. In addition, we present a comparison study with the TOPSIS approach to illustrate the advantages and authenticity of the novel procedure. Furthermore, the characteristics and analytic comparison of the current technique to those outlined in the literature are addressed.

Encryption algorithm based on improved four-dimensional chaotic system and dynamic DNA encoding

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.

Expected value of generalized trapezoidal bipolar fuzzy number to solve a multi-item marketing planning inventory model with allowable shortages

Decision makers often grapple with the complexity of multi-faceted real-life problems, where uncertainty arises from both positive and negative states of mind. This duality reflects the decision maker’s optimistic and pessimistic perspectives during the decision-making process. To address this, our paper introduces innovative concepts and frameworks for generalized bipolar trapezoidal fuzzy numbers.We propose novel arithmetic operations on these fuzzy numbers, computing the negative α-cut and positive β-cut methods. Furthermore, we uniquely compute the convex combination of expected values from both the positive and negative membership parts.These theoretical advancements are applied to a practical case study: a multi-item marketing planning inventory model with allowable shortages. Our proposed method’s efficacy is highlighted through detailed numerical illustrations, sensitivity analyses and comparative studies complemented by compelling graphical presentations

A linear programming aggregation method based on generalized Zhenyuan integral in q-ROFN environment and the application of talent recruitment in universities

The reasonable ranking of binary pairs that characterize fuzzy information in many fuzzy decision problems is very important. To overcome some defects of the existing score functions for the q-rung orthopair fuzzy numbers (q-ROFNs), a novel score function and ranking criterion are proposed by the q-compression transformation and hesitation factor. The main motivation is to introduce the generalized Zhenyuan (GZ)-integral into the q-ROFN environment, and cleverly transform the aggregation operations into a linear programming problem through the arithmetic operations of q-ROFNs. The main contribution is to solve the aggregation problem of q-rung orthopair fuzzy generalized Zhenyuan integral ordered weighted average (q-ROFGZIOWA) operator through the optimization technique of linear programming, and a new decision making method is established by using the q-ROFGZIOWA operator and ranking criterion. The main innovation is to map all q-ROFNs to the unit triangle in the first quadrant (converted into intuitionistic fuzzy numbers, IFNs) according to the q-compression transformation in geometric significance, and the novel score function and its ranking criterion are proposed by combining hesitation factor, and then the aggregation operation based on generalized Z-integral is converted to an optimization problem in linear programming. Finally, the superiority of the proposed method are verified by comparing the aggregation results of two integral operators through an example, and apply the proposed method to the optimal decision-making of talent recruitment in universities. The proposed method can not only correct some flaws in the ranking of existing q-ROFNs, but also overcomes some defects of existing Choquet integral average (geometric) operators in a q-ROFN environment. These results are of great significance for further research on the widespread application of q-ROFNs.

Design of ternary reversible Feynman and Toffoli gates in ternary quantum-dot cellular automata

The use of reversible logic gates leads to a reduction in energy loss in logic circuits by preventing information loss. New computing methods, such as quantum-dot cellular automata (QCA), have been offered by nanotechnology emerging with nanoelectronics to make more comprehensive logic circuits. In nanotechnology-based systems, some bits are erased when the system performs any computation, and this causes heat dissipation and energy loss in systems. Adder circuits are the basis of any arithmetic operation and one of the main parts of many circuits for creating complex hardware; therefore, the use of enhanced adder circuits leads to high performance in logic circuits. In irreversible logic, the energy that is transferred from the power supply to the circuit is converted into heat, and energy loss occurs. Power management plays a vital role in modern computational systems, and using ternary logic instead of previous technologies leads to better performance. The main purpose of our study is to design ternary quantum-dot cellular automata (TQCA) reversible logic gates based on ternary quantum-dot cellular technology. Reversible gates are the basis of creating a reversible circuit. In this paper, the Muthukrishnan-Stroud (M-S) gate, which is the basis of all other reversible ternary gates, is implemented in ternary QCA technology, and then, reversible ternary Feynman and Toffoli (C2NOT) gates are designed. More optimal adder circuits can be realized in three-valued technology using Feynman and Toffoli gates. The area, delay, and cell count of the proposed TQCA designs are compared with those of other related works, and the effect of fault on the designs in the presence of cell omission defect is determined. The occupied areas of the proposed Feynman and Toffoli gate designs are 0.069 μm2 and 0.073 μm2, respectively. Moreover, the fault tolerance levels of these TQCA gates are 77% and 92%, respectively.

Zoom in: Exploring perceptions of the multiplication symbol (×) up close

This study explored the perceptions of 51 prospective elementary school teachers selected using purposive sampling technique regarding the multiplication symbol (×) in arithmetic operations. Using a qualitative hermeneutic phenomenological approach, data were collected through task-based interviews administered to education students specializing in elementary education. The tasks were designed to explore candidates’ understanding and interpretation of the ‘×’ symbol, including uncovering conceptual images, understanding the ‘×’ symbol in different contexts, ability to represent the ‘×’ symbol, flexibility in relating the ‘×’ symbol among concepts, and problem-solving skills. Data analysis involved thematic coding and interpretive analysis to uncover patterns and insights into candidates’ cognitive frameworks. The findings revealed significant variation in candidates’ understanding of the ‘×’ symbol, influenced by their educational background and personal experiences with mathematics. This study highlights the need for improved mathematics instruction and curriculum design to equip future teachers with a deep and accurate understanding of arithmetic symbols, which is critical for effective mathematics teaching at the elementary level.

A low-time-consumption image encryption combining 2D parametric Pascal matrix chaotic system and elementary operation

The rapid development of the big data era has resulted in traditional image encryption algorithms consuming more time in handling the huge amount of data. The consumption of time cost needs to be reduced while ensuring the security of encryption algorithms. With this in mind, the paper proposes a low-time-consumption image encryption (LTC-IE) combining 2D parametric Pascal matrix chaotic system (2D-PPMCS) and elementary operation. First, the 2D-PPMCS with robustness and complex chaotic behavior is adopted. Second, the SHA-256 hash values are applied to the chaotic sequences generated by 2D-PPMCS, which are processed and applied to image permutation and diffusion encryption. In the permutation stage, the pixel matrix is permutation encrypted based on the permutation matrix generated from the chaotic sequences. For diffusion encryption, elementary operations are utilized to construct the model, such as exclusive or, modulo, and arithmetic operations (addition, subtraction, multiplication, and division). After analyzing the security experiments, the LTC-IE algorithm ensures security and robustness while reducing the time cost consumption.

Reliability analysis of landing architecture of aircraft using Fermatean fuzzy arithmetic operation

Reliability analysis of landing architecture of aircraft using Fermatean fuzzy arithmetic operation

Reversible arithmetic and logic unit using a novel reversible NRRG gate in quantum dot technology

Quantum-dot Cellular Automata (QCA) has become one of the promising studies for nano-scale computing. QCA is one of the candidate technologies to be replaced with CMOS technology. QCA technology not only reduces power consumption and delay but also increases operating frequency and speed. The arithmetic logic unit is the essential component in a processor that performs arithmetic and logical operations. This paper presents a novel 5 × 5 reversible logic gate called the NRRG (Norouzi_Rasouli Reversible Gate) which can be used as the basic building block of 4:1 and 8:1 reversible multiplexers. Then, we have designed a RALU (reversible arithmetic and logic unit) using this gate. Our design can perform 20 operations such as AND, NAND, OR, XOR, XNOR, COPY, addition, and increment. The proposed QCA RALU requires 0.44 μm2 area, 480 QCA cells, and 10 clock phases. The proposed design needs less cell count, delay, and cost of QCA compared to previous works. The structure is implemented without any rotated cells and only uses one layer which improves the manufacturability of the design. The architectures are designed and simulated using QCA Designer 2.0.3.