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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 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]

As an emerging computing technology, approximate computing enables computing systems to utilize hardware resources efficiently. Recently, approximate arithmetic units have received extensive attention and have been employed as hardware modules to build approximate circuit systems, such as approximate accelerators. In order to make the approximate circuit system meet the application requirements, it is imperative to quickly estimate the error quality caused by the approximate unit, especially in the high-level synthesis of the approximate circuit. However, there are few studies in the literature on how to efficiently evaluate the errors in the approximate circuit system. Hence, this paper focuses on error evaluation techniques for circuit systems consisting of approximate adders and approximate multipliers, which are the key hardware components in fault-tolerant applications. In this paper, the characteristics of probability mass function (PMF) based estimation are analyzed initially, and then an optimization technique for PMF-based estimation is proposed with consideration of these features. Finally, experiments prove that the optimization technology can reduce the time required for PMF-based estimation and improve the estimation quality. [ABSTRACT FROM AUTHOR]

This paper presents a relation-centric algorithm for solving arithmetic word problems (AWPs) by synergizing a syntax-semantics extractor for extracting explicit relations, and a neural network miner for mining implicit relations. This is the first algorithm that has a specific component to acquire implicit knowledge items for solving AWPs. This paper proposes a three-phase scheme to decompose the challenging task of designing an algorithm for solving AWPs into three smaller tasks. The first phase proposes a state-action paradigm; the second phase instantiates the paradigm into a relation-centric approach; and the third phase implements a relation-centric algorithm for solving AWPs. There are two main steps in the proposed algorithm: problem understanding and symbolic solver. By adopting the relation-centric approach, problem understanding becomes a task of relation acquisition. For conducting the task of relation acquisition, a relaxed syntax-semantics method first extracts a group of explicit relation candidates. In parallel, a neural network miner acquires implicit relation candidates. The miner computes the vectors encoded by BERT to determine which implicit relations should be added. Thus, problem understanding can acquire both explicit relations and implicit relations, which addresses the challenge of building a problem understanding method that can acquire all the knowledge items to find the solution. In the subsequent step of symbolic solver, a fusion procedure forms a distilled set of relations from all the candidates by discarding unnecessary relations. Experimentation on nine benchmark datasets validates the superiority of the proposed algorithm that outperforms the state-of-the-art algorithms. [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]

Runge–Kutta embedded pairs of high algebraic order are frequently utilized when strict tolerances are required. When creating such pairings of orders nine and eight for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients' magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders 9 and 8 with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems. [ABSTRACT FROM AUTHOR]

The emergence of Mobile Edge Computing (MEC) not only provides low-latency computing services for the User Equipment (UE), but also extends the battery life of the UE. However, the computational resources of MEC servers are usually limited, and how to efficiently offload UE's task and allocate the resources of MEC servers has become a research hotspot in MEC. In this paper, we develop an improved arithmetic optimization algorithm (IAOA) to optimize the convergence speed and convergence accuracy of the arithmetic optimization algorithm. Then a task offloading algorithm based on IAOA is designed to reduce the cost of offloading tasks in the framework including a single MEC server and multi-UE. The proposed algorithm jointly optimizes the task offloading strategy of the UEs and the resource allocation of the MEC server, meanwhile, models the weighted sum of delay and energy consumption as the system cost, with the goal of minimizing the system cost while satisfying the delay and energy consumption constraints of the tasks. Simulation results show that the proposed algorithm can effectively reduce the system cost and achieve a performance improvement of up to 20% compared with the benchmark algorithm. [ABSTRACT FROM AUTHOR]

This paper investigates the analysis of critical path delay in various infinite impulse response (IIR) digital filter structure implementations. The critical path delay increases with the order of filter increasing in all conventional structures. So, it is needed to reduce the critical path delay for the faster realization of a digital filter in real-time processing. This paper proposed the rules for taking efficient cutset to retime the higher-order IIR filter structure. By applying this proposed retiming/pipelining techniques to EEG preprocessing filter, the critical path delay is 28 % , 23 % and 18 % more reduced as compared to the traditional retiming in direct form, parallel and lattice-ladder structure, respectively. These newly retimed structures are simulated in MATLAB Simulink and converted to fixed-point arithmetic for the synthesis of structures on Virtex-5 implementation. The results in different structures show that the allpass-based IIR structures has the lowest critical path delay and lowest computational complexity as compared to the conventional structures. Also, the proposed retimed structure has a smaller slice-delay product and support the largest maximum sampling frequency but requires more slice lookup tables (LUTs) than a conventional structure. [ABSTRACT FROM AUTHOR]

This paper presents an improved arithmetic optimization algorithm that incorporates hybrid elite pool strategies to address the limitations of the arithmetic optimization algorithm (AOA). In AOA, the linear mathematical optimization acceleration (MOA) function cannot balance global exploitation and local exploration well. Therefore, the accuracy and convergence speed of the algorithm cannot be guaranteed. To improve the performance of AOA, this paper reconstructed a nonlinear MOA function, which is expected to balance the exploitation and the exploration of AOA. Furthermore, four hybrid elite pool strategies are integrated to enhance the ability to escape local optima. The proposed algorithm inherits the fast convergence of AOA and develops the performance of escaping local optima. Numerical experiment results on benchmark functions and engineering problems show that the proposed algorithm outperforms other compared meta-heuristic algorithms in terms of convergence speed and accuracy. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses various topics within the issue on the topics including random arithmetic formulas, approximate degree and quantum query complexity for total Boolean functions and uniformity and circuit lower bounds.

To satisfy requirements of high security and transmission efficiency in application scenarios with high image quality, an image lossless encryption and compression algorithm based on four-dimensional hyperchaos and embedded block coding with optimal truncation (EBCOT) is proposed in this paper. First, according to a character that the amplitude of the high-frequency part of the wavelet coefficient is less than the amplitude of the low-frequency part, an encryption algorithm for the wavelet coefficients is proposed to improve the security while reducing the impact on the compression performance. Second, the bit-plane coding and arithmetic coding in EBCOT Tier1 are embedded with encryption points, and the encryption process and compression process are combined to propose a secure EBCOT Tier1 code, which could further improve the security of the algorithm. Furthermore, this paper proposes a new four-dimensional hyperchaotic system, using Secure Hash Algorithm-256 (SHA-256) to generate initial values of the chaotic system, so that the algorithm could resist known plaintext and selected plaintext attacks. The results of the mean square error of all restored images are 0 and the information entropy of this algorithm is close to the theoretical value. The experimental results show that the algorithm has high security and lossless compression performance. [ABSTRACT FROM AUTHOR]

Let Γ ⊆ PSL (2 , R) correspond to the group of units of norm 1 in an Eichler order O of an indefinite quaternion algebra over Q . Closed geodesics on Γ \ H correspond to optimal embeddings of real quadratic orders into O . The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper. [ABSTRACT FROM AUTHOR]

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a Π 2 -property formalized in an appropriate language for second order number theory is forcible from some T ⊇ ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of H ω 1 is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for Δ 0 -properties and for all universally Baire sets of reals. We will extend these results also to the theory of H ℵ 2 in a follow up of this paper. [ABSTRACT FROM AUTHOR]

A common practice in statistics is to take the log transformation of highly skewed data and construct confidence intervals for the population average on the basis of transformed data. However, when computed based on log-transformed data, the confidence interval is for the geometric instead of the arithmetic average and neglecting this can lead to misleading conclusions. In this paper, we consider an approach based on a regression of the two sample averages to convert the confidence interval for the geometric average in a confidence interval for the arithmetic average of the original untransformed data. The proposed approach is substantially simpler to implement when compared to the existing methods and the extensive Monte Carlo and bootstrapping simulation study suggests outperforming in terms of coverage probabilities even at very small sample sizes. Some real data examples have been analyzed, which support the simulation findings of the paper. [ABSTRACT FROM AUTHOR]

Nowadays, computing on encrypted data seems to be more practical than a few years ago, thanks to the emergence of new Homomorphic Encryption schemes. In this paper, an algorithm based on Homomorphic Encryption for Arithmetic of Approximate Numbers (Cheon et al., in: Takagi, Peyrin (eds) Advances in cryptology—ASIACRYPT 2017, Springer, Cham, pp 409–437, 2017) (HEAAN, or also CKKS) scheme, that is able to perform a secure k-means algorithm which processes encrypted data, has been studied and presented. The performance of the classifier running on encrypted data has been evaluated using a standard k-means algorithm that works on plain data as a supervised structure, since the results are obtained by approximated computations. The main point of this paper is to take existent theoretical techniques (for example approximations of sgn (x) ), to use them and to observe if they are valid in practical applications. The output of the algorithm is a set of k encrypted masks that can be applied to the original dataset in order to obtain different clusters. The setting is a standard client–server one. The workload is heavily server-centric, as the client only has to execute a light masking algorithm at the end of each iteration, which, excluding the decryption, is faster than a plain k-means iteration; the main disadvantage concerns the accuracy of the results. Experiments show that the algorithm can be executed fairly quickly: the execution time of the training phase is on the order of seconds, while classification is on the order of tenths of a second. [ABSTRACT FROM AUTHOR]

In recent years, long short-term memory (LSTM) networks have been extensively used in various domains such as machine interpretation, language displaying, and so forth. The paper proposes two architectures for accelerating matrix vector multiplication (MVM) in the LSTM network. The first proposal introduces an architecture for the MVM unit by incorporating the concept of Activity Span Reduction (ASR). Here, the partial products for inner product computations (IPC) are generated using ASR logic. This logic ensures that the computational units are active for an optimal duration, effectively utilizing time without compromising the functionality. This approach reduces power consumption as well as area requirements. The ASR technique is able to achieve a 65.9% reduction in power for MVM and occupies 43.77% fewer logic cell units when compared to the state of the art. The second proposal emphasizes the development of a two-stage recursive MVM architecture for the LSTM network. A partial product generator unit using recursive arithmetic architecture is proposed for the IPC in MVM, intended to reduce the power and area consumption. This MVM architecture provides a 6.8% improvement in the maximum clock frequency without an increase in power consumption. The utilization of the proposed designs in the development of the LSTM architecture provides additional evidence supporting the effectiveness of the proposed MVMs. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for encryption of RGB color images by combining two encryption approaches: the spatial approach and the transformation approach. The proposed method uses the 3D fractional modified Henon map (3D FrMHM) and the discrete fractional Krawtchouk moments (FrDKM). We have also proposed a new hybrid optimization algorithm (H-SSAOA) to optimize the parameters of the proposed Henon map and the parameters of the Krawtchouk fractional moments. This algorithm is based on the hybridization of two metaheuristic algorithms: the "Salp Swarm Algorithm" (SSA) and the "Arithmetic Optimization Algorithm" (AOA). The simulation results reveal the optimization efficiency of the proposed hybrid algorithm H-SSAOA compared to other meta-heuristic algorithms and the efficiency of the suggested encryption method for encrypting RGB color images in terms of sensitivity to the security key and resistance to different attacks. [ABSTRACT FROM AUTHOR]

Recently, with the rapid development of cloud storage, secure cloud computing and privacy protection have attracted widespread attention. Reversible data hiding in encrypted images (RDHEI) plays an important role in it and has been paid more and more attention. In this paper, an improved RHDEI is proposed. By combining adaptive Arithmetic coding and static Huffman coding, the image bit-plane is effectively compressed and a lot of space is made for data embedding. The security of image and embedded data is guaranteed by XOR-encryption and scrambling encryption. The data extractor can embed and extract data without decrypting the carrier image to protect the privacy of the image owner. Experimental results show that the scheme can achieve an average embedding capacity (EC) of up to 3 bpp while ensuring lossless recovery of carrier images and correct extraction of embedded data. Compared with state-of-the-art RDHEI methods, our scheme achieves higher EC and better security. [ABSTRACT FROM AUTHOR]

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as 'arithmetic chaos'. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a 'linear ramp'). We 'bootstrap' the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]×[interval] wormhole amplitude in AdS3 gravity. [ABSTRACT FROM AUTHOR]

Programming is an interdisciplinary practice with applications in both mathematics and computer science. Mathematics concerns rigor, abstraction, and generalization. Computer science predominantly concerns efficiency, concreteness, and physicality. This makes programming a medium for problem solving that mediates between mathematics and computer science in intriguing ways. Behind programming practices is computational thinking (CT), a mode of thinking involved in formulating and solving problems so that the solutions could be represented and carried out by computing means. In this paper, CT is seen as a boundary object connecting mathematics and computer science in a school problem-solving context. In particular, we examine and analyse middle school students' work upon engaging in mathematical problem solving-in a programming environment, taking CT as a boundary object embedded in the block-based programming environment, Scratch. The analysis is guided by observing boundary crossing features of CT in the students' artefacts produced in Scratch while solving mathematical problems related to symmetry and arithmetic sequence. The findings of this study open up new dimensions to explore CT as a boundary object in integrated STEM pedagogy. [ABSTRACT FROM AUTHOR]

A multiplier is a key component in arithmetic and logical units. So low power consuming, area efficient speedy multiplier architecture is need of today's Arithmetic and Logic Unit. In the ancient Indian Vedic mathematics, the Urdhvatiryagbhyam sutra gives a multiplication methodology. Vedic multiplier improves the performance parameters of the digital circuit logically as well as physically. The performance of the multiplier depends upon a reduction in partial product generation. This is achieved with the help of a 4:2 compressor. In this paper, a logically improved, area-optimized 4:2 compressor is proposed. The proposed design is implemented in standard CMOS 45 nm technology and analyzed the results post layout design. For performance analysis, the novel proposed 4:2 compressor is used in a Vedic multiplier. After simulation, we get results which shows that the proposed compressor has 62.95% of power reduction and 53.31% delay reduction compared to the conventional compressor. The compressor shows a 44.34–10.57% reduction in ADP and 149.84–12.59% reduction in PDP against present variants. The proposed 4 Bit Vedic multiplier shows a 9.633% power reduction compared to the Vedic multiplier with a conventional compressor. The 4 Bit Vedic multiplier shows a 44.877–7.14% reduction in Area-Delay product and 27.99–0.22% reduction in Power-Delay product against present variants. [ABSTRACT FROM AUTHOR]

Let α ≥ 2 , m ≥ 2 be integers, p be an odd prime with p ∤ m (m + 1) , 0 < λ 1 , λ 2 ≤ 1 be real numbers, q = p α > max { [ 1 λ 1 ] , [ 1 λ 2 ] } . For any integer n with (n , q) = 1 and a nonnegative integer k, we define M λ 1 , λ 2 (m , n , k ; q) = ∑ ′ a = 1 q ∑ ′ b = 1 [ λ 1 q ] ∑ ′ c = 1 [ λ 2 q ] a b ≡ 1 (mod q) c ≡ a m (mod q) n ∤ b + c (b − c) 2 k. In this paper, we study the arithmetic properties of these generalized Kloosterman sums and give an upper bound estimation for it. By using the upper bound estimation, we discuss the properties of M λ 1 , λ 2 (m , n , k ; q) and obtain an asymptotic formula. [ABSTRACT FROM AUTHOR]

This paper focuses on the categoricity of arithmetic and determinacy of arithmetical truth. Several 'internal' categoricity results have been discussed in the recent literature. Against the background of the philosophical position called internalism, we propose and investigate truth-theoretic versions of internal categoricity based on a primitive truth predicate. We argue for the compatibility of a primitive truth predicate with internalism and provide a novel argument for (and proof of) a truth-theoretic version of internal categoricity and internal determinacy with some positive properties. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic nature of the value of a higher Green function on an orthogonal Shimura variety at a single CM point. This is motivated by a conjecture of Gross and Zagier in the setting of higher Green functions on the product of two modular curves. In the process, we will study analogue of harmonic Maass forms in the setting of Hilbert modular forms, and obtain results concerning the arithmetic of their holomorphic part Fourier coefficients. As a consequence, we answer a question of Zagier in his 1986 ICM proceeding. [ABSTRACT FROM AUTHOR]

We study the infinite linear independence of polyadic numbers , f1(λ) = where λ is a polyadic Liouville number. Here, denotes the Pochhammer symbol, i.e., and for . The considered series converge in any field . The results presented extends the author's previous results concerning the arithmetic properties of the polyadic numbers . The values of generalized hypergeometric series have been extensively studied. If the series parameters are rational numbers, then the series belong to the class of E-functions (if these series are entire functions), to the class of G-functions (if they have a finite nonzero radius of convergence), or to the class of F-series (in the case of a zero radius of convergence in the field of complex numbers, but they converge in the fields of p-adic numbers). In all these cases, the Siegel–Shidlovskii method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on Hermite–Padé approximations. In the considered case, the parameter is a transcendental number. Note that earlier A.I. Galochkin proved the algebraic independence of the values of E-functions at points that are real Liouville numbers. We also mention E.Yu. Yudenkova's papers (submitted for publication) on the values of F-series at polyadic Liouville points. It should be emphasized that this paper deals with the values, at polyadic transcendental points, of hypergeometric series with a parameter that is a polyadic transcendental (Liouville) number. [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]

In this paper we present a complete asymptotic expansion of the Archimedean compound of two symmetric homogeneous means and derive recursive algorithms for coefficients in this expansion. We also show some examples and obtain explicit expansions for the Archimedean compounds of the arithmetic, geometric and harmonic means. [ABSTRACT FROM AUTHOR]

This paper presents a new nonlinear feedback shift register (NLFSR) in the Galois configuration to generate the pseudorandom number sequences (PRNS) for a lightweight encryption scheme. In the NLFSR, the feedback function is applied to each state, not only the last bit of NLFSR. In the proposed design the size of the feedback is reduced using Galois configuration. The NLFSR is tested with NIST Statistical Test Suite to evaluate its quality randomness. The new NLFSR is used as an pseudorandom number generator to design a new lightweight image encryption algorithm. The encryption process can be represent in two phases permutation and diffusion. In the first phase, two different PRNSs generated by the new NLFSR for permute the row and column pixels of the plain image (PI). Then convert the permuted image into a 1D binary vector. In the diffusion phase, DNA arithmetic is applied between the 1D and another PRNS generated by the same NLFSR with different key. Several security analysis tests are performed on the proposed scheme (like histogram analysis, entropy, correlation coefficient, NPCR, UACI, MSE, and PSNR) to test the security strength of the encryption method. It was found that the new NLFSR had passes all the tests in NIST Statistical Test Suite, and the test results of the encryption scheme are also acceptable and shows potential security strength. [ABSTRACT FROM AUTHOR]

Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene's initial realizability laid the ground for more sophisticated notions such as Kreisel's modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely herbrandized modified realizability. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic. [ABSTRACT FROM AUTHOR]

The accurate model of the solar PV system is the principal organ that describes the performance of this resource. Several approaches based on optimizing algorithms were considered valuable tools to illustrate the I–V curve for improving the photovoltaic models. Their electrical parameters are estimated using optimization algorithms referring to the experimental database or manufacturer's datasheet. This paper proposes a novel developed a photovoltaic model based on improved arithmetic optimization algorithm (IAOA) to extract the solar cell parameters. Also, an experimental test bench is presented for obtaining the measured illustration of the I–V characteristics. Thus, the root mean square error value that describes the difference between measured and estimated results is considered the objective function for two different models, the simple-diode model and the one-diode model. The proposed IAOA results are compared with other research papers and optimization algorithms. Furthermore, the evaluation of the proposed IAOA has been discussed considering several statistical analysis tests. The presented results show that the effectiveness and accuracy of IAOA results are excellent, and their I–V characteristics coincide with experimental data. Moreover, the results obtained by the proposed algorithm show its high superiority in optimizing the solar cell parameters under a variety of operating conditions. [ABSTRACT FROM AUTHOR]

In evidence theory, conflicting evidence and fuzzy evidence have a significant impact on the results of evidence combination. Nevertheless, the existing weight assignment methods can hardly reflect the significant influence of fuzzy evidence on the combination results. In this paper, we address this issue by proposing a new method for assigning evidence weights and the corresponding combination rule. The proposed weight assignment method strengthens the consideration of fuzzy evidence. We further incorporate the Wasserstein distance to compute the clarity degree of the evidence. This is an important reference index for the weight assignment in the proposed combination rule and effectively reduces the impact of ambiguous evidence. Using experiments, we illustrate the significant impact of fuzzy evidence on the results of combination. This justifies its integration in the weight assignment process. The proposed combination rule with the new weight assignment method is also examined on a set of numerical arithmetic and Iris datasets. Our results confirm that compared with the four existing methods, the proposed method improves decision accuracy, F1 score, computational convergence and achieves more reliable fusion results. [ABSTRACT FROM AUTHOR]

In this paper, we explore efficient hardware implementation of feedforward artificial neural networks (ANNs) using approximate adders and multipliers. Due to a large area requirement in a parallel architecture, the ANNs are implemented under the time-multiplexed architecture where computing resources are re-used in the multiply accumulate (MAC) blocks. The efficient hardware implementation of ANNs is realized by replacing the exact adders and multipliers in the MAC blocks by the approximate ones taking into account the hardware accuracy. Additionally, an algorithm to determine the approximate level of multipliers and adders due to the expected accuracy is proposed. As an application, the MNIST and SVHN databases are considered. To examine the efficiency of the proposed method, various architectures and structures of ANNs are realized. Experimental results show that the ANNs designed using the proposed approximate multiplier have a smaller area and consume less energy than those designed using previously proposed prominent approximate multipliers. It is also observed that the use of both approximate adders and multipliers yields, respectively, up to 50% and 10% reduction in energy consumption and area of the ANN design with a small deviation or better hardware accuracy when compared to the exact adders and multipliers. [ABSTRACT FROM AUTHOR]

This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel's diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel. [ABSTRACT FROM AUTHOR]

Various intervention programs for fostering at-risk students' understanding of basic concepts (such as place value understanding or meanings of multiplication and division) have been developed and evaluated. However, little is known about the teacher expertise required to enact these intervention programs, and how this teacher expertise can be promoted. The article suggests a conceptual model for teacher expertise for fostering at-risk students' understanding based on three recurrent jobs: (a) specify learning content (in basic concepts), (b) monitor students' learning progress (in basic concepts) and (3) enhance students' understanding (of basic concepts). Mastering these jobs with productive teaching practices involves four orientations in particular (conceptual rather than procedural orientation, diagnostic rather than syllabus-led orientation, communicative rather than individualistic orientation, and long-term rather than short-term orientation) as well as detailed pedagogical content categories for unpacking relevant knowledge elements. The paper reports on the professional development program Mastering Math which aims at promoting this expertise and its evaluation using a pre–post-design. For 95 participating teachers, the practices for specifying goals and monitoring and enhancing at-risk students' understanding were captured in self-reports and situated in vignette-based activities for eliciting diagnostic judgments. Teachers' development across different aspects of their expertise from the beginning and the end of the 1-year PD reveals the first quantitative evidence that the PD was effective in promoting growth of expertise. Whereas specifying and monitoring practices had substantially developed, the enhancement practices were hindered by a persistent short-term orientation. [ABSTRACT FROM AUTHOR]

Protocols for secure multiparty computation enable a set of parties to compute a function of their inputs without revealing anything but the output. The security properties of the protocol must be preserved in the presence of adversarial behavior. The two classic adversary models considered are semi-honest (where the adversary follows the protocol specification but tries to learn more than allowed by examining the protocol transcript) and malicious (where the adversary may follow any arbitrary attack strategy). Protocols for semi-honest adversaries are often far more efficient, but in many cases the security guarantees are not strong enough. In this paper, we present new protocols for securely computing any functionality represented by an arithmetic circuit, assuming an honest majority exists. We utilize a new method for verifying that the adversary does not cheat, that yields a cost of just twice that of semi-honest protocols in some settings. Our protocols are information-theoretically secure in the presence of malicious adversaries. We present protocol variants for small and large fields, and show how to efficiently instantiate them based on replicated secret sharing and Shamir secret sharing. In particular, for large fields, our protocol requires each party to send just 2 field elements per multiplication gate in the three-party setting, and just 12 field elements per multiplication gate for any number of parties. As with previous works in this area aiming to achieve high efficiency, our protocol is secure with abort and does not achieve fairness, meaning that the adversary may receive output while the honest parties do not. We implemented our protocol and ran experiments for different numbers of parties, different network configurations and different circuit depths. Our protocol significantly outperforms the previous best for this setting (Lindell and Nof, CCS 2017); for a large number of parties (e.g., 100 parties), our implementation runs almost an order of magnitude faster than theirs. [ABSTRACT FROM AUTHOR]

The paper studies modules over a certain generalized ring. This ring is the noncommutative tensor square of the ring of integers. The modules in question are related to some interesting arithmetic problems. In particular, they are related to the solved Gauss class-number problem for imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with 1 and G = GL(n, R) the general linear group of degree n ≥ 3 over R. A goal of the paper is to calculate the relative centralizers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal A ⊴ R modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal B ⊴ R. Modulo congruence subgroups, the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups, they turned out to be quite tricky, and definitive answers are obtained only over commutative rings or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type. [ABSTRACT FROM AUTHOR]

Properties of arithmetic operations are important in mathematics and play a significant role in elementary school students' learning of mathematics. At the same time, it is quite challenging to help students learn arithmetic operational properties beyond memorization, so that they know when and how to use a specific arithmetic property. This paper reports a study containing two parts to explore students' learning of the Commutative Property of Addition (CPA) as related to the content presentation in textbooks. Part one is to assess selected students' performance on CPA at the end of Grade two, and part two is to explore possible contributions of textbooks to students' performance in Shanghai. Firstly, we evaluated the performance of 41 second-graders on CPA by collecting data through a paper-and-pencil test, which was followed up with individual interviews. Through analyzing how textbooks present and organize this topic, we sought to better understand from a curriculum standpoint how students' performance on CPA is impacted by textbooks. Results revealed that Shanghai textbooks make a positive contribution to students' learning performance. The findings from this study help to shed light on how those textbooks aid student learning, and provide an example for further reflection on student learning of arithmetic operational properties in different curricula. [ABSTRACT FROM AUTHOR]

In recent decades, near fault ground motions has been of great importance due to the difference in characteristics of earthquake records in the regions near to the active faults. Most of the developed control systems for structural vibration control have difficulties in dealing with these kinds of strong ground motions regarding the deficiencies of the human knowledge-based control systems such as fuzzy logic controllers for this purpose. Hence, the optimization of these controllers has been concerned in recent years. The main aim of this paper is to optimize the fuzzy controllers implemented in steel structures with nonlinear behavior in which the arithmetic optimization algorithm (AOA) is utilized as the main optimization algorithm while an improved version of this algorithm as IAOA is also proposed for performance enhancement of the standard algorithm. In the IAOA, a new parameter identification process is proposed in which the Levy flight as a well-known stochastic process with step length determined by levy distribution is implemented in the main loop of the AOA. The IAOA and AOA are utilized for optimization of the membership functions and the rule base of the fuzzy controllers implemented in a large-scale building structure. The overall performance of the IAOA is compared with the standard AOA and other metaheuristics. The obtained results of the improved method demonstrate the capability of this method in providing very competitive solutions which results in decreasing structural responses and damages of the considered building in dealing with the near-fault strong ground motions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new generalized p-dimensional multifarious radical reciprocal functional equation combining three classical means: arithmetic, geometric, and harmonic. Mainly, we find its general solution and stability related to the Ulam problem in modular spaces by using the fixed point method with suitable counterexamples. Importantly, in this paper, we illustrate the geometrical interpretation and applications of the introduced Pythagorean means multifarious functional equation in connection with the parallel circuit. Furthermore, we provide a formula for finding the equivalent resistance R eq of parallel electrical circuit using functional equations, which relates the electrical resistances and conductances with suitable examples. [ABSTRACT FROM AUTHOR]

In a seminal series of papers from the 80's, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for G L 2 (Deligne's theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with S L 2 (Q p) . As a result, they obtained explicit Ramanujan Cayley graphs from P S L 2 F p , as well as optimal topological generators ("Golden Gates") for the compact Lie group PU(2). In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for P U 3 by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with S L 3 (Q p) and S U 3 (Q p) , while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from P S L 3 (F p) and P S U 3 (F p) , as well as golden gates for PU(3). [ABSTRACT FROM AUTHOR]

The article discusses various reports published within the issue including "The Saturation Algorithm for Symbolic State Space Exploration," by Gianfranco Ciardo, Robert Marmorstein and Radu Siminiceanu, "Efficient BDD's for Bounded Arithmetic Constraints," by Constantinos Bartzis and Tevfik Bultan, "CAESAR SOLVE: A Generic Library for on-the-fly Resolution of Alternation-Free Boolean Equation Systems," by Radu Mateescu, "Pattern-Based Abstraction for Verifying Secrecy in Protocols," by Liana Bozga, Yassine Lakhnech and Michaël Périn and "Interactive Visualization of Large State Graphs," by Jan Frison Groote and Frank van Ham.

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author's paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles. [ABSTRACT FROM AUTHOR]

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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]

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7–8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking. [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]

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes E k and U k introduced in [1] are exactly the classes induced by Σ k and Π k respectively via the transformation procedure in any first-order theory. [ABSTRACT FROM AUTHOR]