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All-pass transformation (APT)-based variable digital filters (VDFs) can be used in different biomedical signal-processing applications, particularly in electrocardiograph denoising. In this paper, arithmetic optimised APT-VDF is proposed by modifying the hardware structure of APT-VDF based on arithmetic perception to enhance the performance in terms of area, speed and power consumption. The core blocks of the suggested Arithmetic Optimised APT-VDF (AOAPT-VDF)are adders and multipliers. This paper introduces a new reusable Vedic multiplier with redundant detection unit to identify redundant computations while generating the partial products. Also, a new Carry select adder (CSLA) with simplified combinational logics is proposed to improve the performance of the CSLA by simplifying the partial sum and carry generation logics without requiring Ripple Carry Adders. The proposed Reusable Vedic Multiplier with Redundant Detection Logic (RVM-RDL) also reuses the same vertical and cross-product generation units repetitively to reduce the area while increasing the bit length. The effectiveness of the suggested AOAPT is analysed utilising the MIT-BIH Arrhythmia Database. The proposed AOAPT-VDF consumes only 0.21 $$W$$ W power and achieves a 235.50 MHz operating frequency. [ABSTRACT FROM AUTHOR]

AbstractIn a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [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 positive real α, we can consider the additive submonoid M of the real line that is generated by the nonnegative powers of α. When α is transcendental, M is a unique factorization monoid. However, when α is algebraic, M may not be atomic, and even when M is atomic, it may contain elements having more than one factorization (i.e., decomposition as a sum of irreducibles). The main purpose of this paper is to study the phenomenon of multiple factorizations inside M. When α is algebraic but not rational, the arithmetic of factorizations in M is highly interesting and complex. In order to arrive to that conclusion, we investigate various factorization invariants of M, including the sets of lengths, sets of Betti elements, and catenary degrees. Our investigation gives continuity to recent studies carried out by Chapman et al. in 2020 and by Correa-Morris and Gotti in 2022. [ABSTRACT FROM AUTHOR]

This paper studies conjugation-invariant norms on SL 2 (R) for R a ring of S-algebraic integers with infinitely many units in a number field. We will show that barring some differences in the proofs the properties of these norms strongly resemble related properties of conjugation-invariant norms on arithmetic Chevalley groups of higher rank. Specifically, we prove two properties: First, we show that the diameter of each word norm on SL 2 (R) defined by finitely many conjugacy classes has an upper bound linear in the number of conjugacy-classes. Second, we show that all non-discrete conjugation-invariant norms on SL 2 (R) have a profinite norm-completion. Both of these properties of conjugation-invariant norms are already known for higher rank arithmetic Chevalley groups. [ABSTRACT FROM AUTHOR]

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pricing of basket-spread options with default risk under Hawkes jump-diffusion processes. A self-exciting Hawkes process is employed to describe jump clustering, and jump amplitudes of different assets in baskets are all correlated. In addition, the diffusive components of assets are also assumed to be correlated with each other. We obtain option prices by approximating the arithmetic average of the underlying assets in the basket with their second moment-matched geometric average values, and numerical experiments show that our approximated prices are quite accurate, spanning different underlying asset numbers and alternative strike prices. Finally, we illustrate the effects of default risk and clustered jump risk on the prices of basket-spread options. [ABSTRACT FROM AUTHOR]

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R⟨X⟩. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38] related to the transfer of Prüfer conditions to these two constructions. [ABSTRACT FROM AUTHOR]

Number sense and working memory contribute to mathematical development throughout primary school. However, it is still unclear how the contributions of each of these predictors may change across development and whether the cognitive contribution is the same for children with and without mathematical difficulties. The aim of the two studies in this paper was to shed light on these topics. In a cross-sectional design, a typically developing group of children (study 1; N = 459, Grades 1–4) and a group with mathematical difficulties (study 2; N = 61, Grades 4–6) completed a battery of number sense and working memory tests, as well as a measure of arithmetic competence. Results of study 1 indicated that number sense was important in first grade, while working memory gained importance in second grade, before predictive value of both predictors waned. Number sense and working memory supported mathematics development independently from one another from Grade 1. Analysis of task demands showed that typically developing children rely on comprehension and visualization of quantity-to-number associations in early development. Later in development, pupils rely on comparing larger numerals and working memory until automatization. Children with mathematical difficulties were less able to employ number sense during mathematical operations, and thus might remain dependent on their working memory resources during arithmetic tasks. This suggests that children with mathematical difficulties need aid to employ working memory for mathematics from an early age to be able to automatize mathematical abilities later in development. [ABSTRACT FROM AUTHOR]

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p ′ -order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors. [ABSTRACT FROM AUTHOR]

AbstractThe Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. García-Sánchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings k[Γ] can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup Γ. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where Γ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences. [ABSTRACT FROM AUTHOR]

AbstractArithmetic achievement holds particular significance in contemporary times; however, students continue to encounter difficulties in this domain. Researchers and organizations are actively seeking factors that can support students in navigating mathematical content. To address this gap, we propose an examination of the role played by six soft skills (adaptability, curiosity, initiative, leadership, perseverance, and social awareness), alongside self-regulated learning strategies, academic motivation, achievement emotions, and fluid reasoning abilities, in students’ performance on standardized arithmetic tasks. Data were gathered from 551 Italian students aged 11–18 through classroom-based paper-and-pencil tasks and questionnaires. Path analysis results reveal that only fluid reasoning abilities, academic motivation, and negative achievement emotions significantly correlate with arithmetic performance. Soft skills demonstrate an indirect relationship with arithmetic performance, mediated by negative achievement emotions and academic motivation (while not mediated by positive achievement emotions and self-regulated learning strategies). These findings underscore the increasingly recognized significance of soft skills and expand their relevance to arithmetic achievement, potentially suggesting novel psychoeducational interventions to bolster students’ performance in this realm. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a new 4-2 compressor which can be used in high-speed multipliers. The proposed compressor features eliminated glitch at the output waveform. By optimum tuning of the width of the transistors, Cout is produced more quickly, and therefore higher operating speeds can be achieved. The effect of process variation on the circuit has been studied and a new structure is proposed to overcome the effect of process variation by utilizing a reference voltage generator circuit. A 32 × 32-bit multiplier has been developed utilizing these compressors in order to evaluate the new compressor in a practical environment. The total multiplier circuit was simulated with HSPICE simulator (BSIM3v3 parameters) using CMOS standard cell library of 0.18 µm and the Layouts were extracted with Cadence Virtuoso v 5.1 Layout Plus tool. The simulation results represent 185 × 10−3 pj Power-Delay-Product (PDP) and 40 pj × ps Energy Delay Product (EDP) with 221 ps delay from input to output. [ABSTRACT FROM AUTHOR]

A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains. [ABSTRACT FROM AUTHOR]

This paper uses a post-qualitative philosophical perspective to find new ways of understanding teaching and learning. The paper presents a series of examples that were used in a longitudinal study, with the aim of creating variation patterns that would make it possible for students to discern the use of the four basic arithmetic operations in different situations. The focus of this article is the potential of the examples to systematically create variation patterns that students need to perceive in order to make generalizations. The result demonstrates that well-thought-out examples help identify the correct arithmetic operation in different situations, and provide a basis from which students can discern the connection between text and the use of operation in mathematical example. The result also demonstrates that students develop rhizomatic thinking through the creation of new links between aspects of the object of learning, association and linking of different aspects to each other and the creation of a whole with unique and specific characteristics that cannot be explained by simply adding the characteristics of the individual parts. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of affine semigroups generated by integral vectors, whose components are in generalized arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that it preserves some nice properties, including Cohen-Macaulayness, when the constituent semigroups have those properties. [ABSTRACT FROM AUTHOR]

In this paper, we give a new characterization for the perspective of a continuous function under certain assumptions. This result generalizes a non-commutative analogue of the arithmetic–geometric mean inequality. Our main result extends and strengthens the existing result concerning the characterization of the geometric operator mean. We provide the characterization of other operator means, in particular, the harmonic operator mean and the arithmetic operator mean. [ABSTRACT FROM AUTHOR]

A couple of Sylvester matrix equations (CSME) are required to be solved simultaneously in many applications, especially in analysing the stability of control systems. However, there are some situations in which the crisp CSME are not well equipped to deal with the uncertainty problem during the stability analysis of control systems. Thus, this paper proposes a new method for solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation (ACTrFFSME). New arithmetic fuzzy multiplication operations are developed and applied to convert the ACTrFFSME to a reduced system of non-linear equations. Then it is converted to a system of absolute equations where the arbitrary fuzzy solution is obtained by solving that system. The proposed method can solve many arbitrary fuzzy equations, such as fully fuzzy matrix equations, Sylvester and Lyapunov fully fuzzy matrix equations with triangular and trapezoidal fuzzy numbers without any restrictions. We illustrate the proposed method by solving two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce provability multilattice logic P M L n and multilattice arithmetic M P A n which extends first-order multilattice logic with equality by multilattice versions of Peano axioms. We show that P M L n has the provability interpretation with respect to M P A n and prove the arithmetic completeness theorem for it. We formulate P M L n in the form of a nested sequent calculus and show that cut is admissible in it. We introduce the notion of a provability multilattice and develop algebraic semantics for P M L n on its basis, by the method of Lindenbaum-Tarski algebras we prove the algebraic completeness theorem. We present Kripke semantics for P M L n and establish the Kripke completeness theorem via syntactical and semantic embeddings from P M L n into G L and vice versa. Last but not least, the decidability of P M L n is shown. [ABSTRACT FROM AUTHOR]

In this paper, a low-complexity quantization table is proposed for the baseline JPEG encoder. The proposed scheme does not require any multiplications or additions; only bit-shift operations are involved. The computational complexity should be drastically reduced without almost incurring any significant performance loss. The resulting approximate quantization table has extremely low arithmetic cost and is compared to state-of-the-art schemes in terms of computational complexity. Comparisons showed that our proposal exhibits better or similar results. The proposed low-complexity quantization table is also investigated in image compression context. It could outperforms other related schemes in low and high-compression scenarios. Comparisons with respect to average PSNR and SSIM are conducted. Experiment outcomes show up to 4.03 dB improvement in average PSNR over the state-of-the-art competitor. The proposed quantization table has a considerably lower computational cost while maintaining much higher image quality. [ABSTRACT FROM AUTHOR]

Spectral data and spatial information such as shape and texture features can be fused to improve the classification of the hyperspectral images. In this paper, a novel approach of the spectral and spatial features (texture features and shape features) fusion in the probabilistic framework is proposed. The Gabor filters are applied to obtain the texture features and the morphological profiles (MPs) are used to obtain the shape features. These features are classified separately by the support vector machine (SVM); therefore, the per-pixel probabilities can be estimated. A novel meta-heuristic optimization method called Arithmetic Optimization Algorithm (AOA) is used to weighted combinations of these probabilities. Three parameters, α, β and γ determine the weight of each feature in the combination. The optimal value of these parameters is calculated by AOA. The proposed method is evaluated on three useful hyperspectral data sets: Indian Pines, Pavia University and Salinas. The experimental results demonstrate the effectiveness of the proposed combination in hyperspectral image classification, particularly with few labelled samples. As well as, this method is more accurate than a number of new spectral-spatial classification methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyse the reasoning and symbolisations used by sixth grade Moroccan students in solving a task based on figurative patterns. Our analysis aims at identifying the systems of actions elaborated by the students to give the general expression of the sequence, according to their perceptions of the sequence of its patterns. The results of this study show that the students can produce a wide variety of generalisations, without using the letter, by resorting to verbal expressions, the majority of which remain in the class of a factual or contextual generalisation. However, several students were not able to determine the general term of the sequence in the question, through an expression that is detached from the context of the situation. This study also contributes to raising the consciousness of Moroccan teachers and researchers of mathematics didactics about the important role played by algebraic thinking, in particular the generalisation coming from an activity based on the figurative patterns, which generates several equivalent and meaningful numerical or algebraic expressions. This article also sheds light on ways for programme designers to enrich primary school curricula, by proposing arithmetic activities with algebraic potential, like the one proposed in our study. [ABSTRACT FROM AUTHOR]

The photovoltaic power system is highly dependent on the quantity of solar irradiance incident on it and its temperature. These parameters are dynamically changing with time due to climatic changes, shadows formed by clouds, trees, buildings, and so on, resulting in multiple maximum power points (MPP). Out of which one is global MPP (GMPP) and the rest are local MPP (LMPP). The GMPP varies dynamically along with the weather conditions and connected load. Many GMPP tracking (GMPPT) algorithms were developed which are inefficient and ineffective under dynamic irradiance conditions. This paper proposes a new enhanced arithmetic optimization algorithm based on the levy flight (AOA-LF) as a GMPPT method, which improves the tracking efficiency and tracking speed because of its good exploration and exploitation due to the long jump with a variable step size. The suggested AOA-LF GMPPT method is implemented in MATLAB/SIMULINK and tested with eight different dynamic irradiance patterns whose tracking curves are compared with the existing GMPPT algorithms like JAYA-LF, DFO, AOA, PSO, and P&O, which reveals that it is excellent in the tracking of GMPP with increment in efficiency as high as 19% when compared with P&O method and the least time to reach the GMPP with percentage decrement as high as 69% as compared to the PSO method. Also, the proposed AOA-LF GMPPT method is validated with the help of OPAL-RT OP4510 Real-Time Digital Simulator. Hence, the proposed algorithm shows superior performance with zero steady-state oscillations and also enhanced exploration and exploitation. [ABSTRACT FROM AUTHOR]

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice. [ABSTRACT FROM AUTHOR]

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field F and exponents in an additive submonoid M of Q ≥ 0 is called a Puiseux algebra and denoted by F [ M ]. Here we study the atomic structure of Puiseux algebras. To begin with, we answer the isomorphism problem for the class of Puiseux algebras, that is, we show that for a field F if two Puiseux algebras F [ M 1 ] and F [ M 2 ] are isomorphic, then the monoids M1 and M2 must be isomorphic. Then we construct three classes of Puiseux algebras satisfying the following well-known atomic properties: the ACCP property, the bounded factorization property, and the finite factorization property. We show that there are bounded factorization Puiseux algebras with extremal systems of sets of lengths, which allows us to prove that Puiseux algebras cannot be determined (up to isomorphism) by their arithmetic of lengths. Finally, we give a full description of the seminormal closure, root closure, and complete integral closure of a Puiseux algebra, and we use this description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles). [ABSTRACT FROM AUTHOR]

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my bookAfter Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011).The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. These issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics. [ABSTRACT FROM PUBLISHER]

One of the main changes in European Renaissance mathematics was the progressive development of algebra from practical arithmetic, in which equations and operations began to be written with abbreviations and symbols, rather than in the rhetorical way found in earlier arithmetical texts. In Spain, the introduction of algebraic procedures was mainly achieved through certain commercial or arithmetical texts, in which a section was devoted to algebra or the ‘Arte Mayor’. This paper deals with the contents of the first arithmetical texts containing sections on algebra. These allow us to determine how algebraic ideas were introduced into Spain and what their main sources were. The first printed arithmetical Spanish text containing algebra was the Libro primero de Arithmetica Algebratica (1552) by Marco Aurel. Therefore, the aim of this paper is to analyse the possible sources of this book and show the major influence of the German text Coss (1525) by Christoff Rudolff, on Aurel's work. [ABSTRACT FROM AUTHOR]

This work presents an application of upgraded arithmetic operations on trapezoidal fuzzy number to extend the reliability from point estimation to interval estimation. A realistic example of an accidental case has been taken to show the desired results. As we know that uncertainty is a disregardable phenomenon and in this situation crisp reliability may not be exact. To get a better hold on this situation, we introduce a method of trapezoidal fuzzy number and upgraded arithmetic operations. In this paper, the reliability of each factor is represented by the trapezoidal fuzzy number and the final reliability is obtained in the form of trapezoidal fuzzy number. Hence, the dejection of fuzzy number is preserved and the crisp region is increased to illustrate the highest membership grade as an interval. So it can help the decision maker to analyse the behaviour of the system and take effective decision accordingly to reduce the chance of mishappening. [ABSTRACT FROM AUTHOR]

When using educational video games, particularly drill-and-practice video games, there are several ways of providing an answer to a quiz. The majority of paper-based options can be classified as being either multiple-choice or constructed-response. Therefore, in the process of creating an educational drill-and-practice video game, one fundamental question to be answered is which is the most appropriate response format? This paper answers this question, proposing a hybrid format that we call fine-grained multiple-choice, where the player constructs their response one element at a time, with each element chosen from a pre-defined set. A comparative study was conducted with second grade students. When comparing the impact of an educational drill-and-practice video game that uses multiple-choice with one that uses fine-grained multiple-choice, we observe that the fine-grained multiple-choice game achieved better results. However, the multiple-choice game was substantially more successful at capturing the students’ interest and motivation. Considering the widespread use of the multiple-choice format in game-based learning activities, as well as in interactive activities, this research suggests rethinking what the most suitable answer format might be for each area of the curriculum and for the different types of games. [ABSTRACT FROM AUTHOR]

In this paper a practical power system planning and control strategy based on a new adaptive sine cosine arithmetic optimization algorithm (ASC_AOA) is proposed to enhance the technical performances of radial distribution (RD) systems. The main objective considered in this study is to optimize the location and the size of shunt compensators based STATCOM devices and multi distributed generation to reduce the total power losses, and to maximize the loading margin stability of practical RD systems. As well confirmed in many recent researches, the success of metaheuristic algorithms is related to the interactivity between intensification and diversification stages. In this study, an adaptive process based on sine and cosine functions is incorporated within the standard AOA to guide the search process toward the best solution. The particularity of the proposed variant (ASC_AOA) has been validated on many benchmark functions, and also applied for optimal automation control of multi DGs and shunt compensators based STATCOM Controllers to enhance the performances of various types of RD systems, such as the 33-bus, the 69-bus, and 85-bus. Obtained results are compared to many recent optimization methods. It is confirmed that the proposed variant achieves the best solution in all of the cases studies elaborated. The proposed variant seems to be a competitive technique and an alternative tool for solving various combinatorial planning and control problems of modern RD systems. [ABSTRACT FROM AUTHOR]

Shortly after Kurt Gödel had announced an early version of the 1st incompleteness theorem, John von Neumann wrote a letter to inform him of a remarkable discovery, i.e. that the consistency of a formal system containing arithmetic is unprovable, now known as the 2nd incompleteness theorem. Although today von Neumann's proof of the theorem is considered lost, recent literature has explored many of the issues surrounding his discovery. Yet, one question still awaits a satisfactory answer: how did von Neumann achieve his result, knowing as little as he seemingly did about the 1st incompleteness theorem? In this article, I shall advance a conjectural argument to answer this question, after having rejected the argument widely shared in the literature and having analyzed the relevant documents surrounding his discovery. The argument I shall advance strictly links two of the three letters written by von Neumann to Gödel in the late 1930 and early 1931 (i.e. respectively that of November 20, 1930 and that of January 12, 1931) and finds the key for von Neumann's discovery in his prompt understanding of the Gödel sentence A – as the documents refer to it – as expressing consistency for a formal system that contains arithmetic. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained. [ABSTRACT FROM AUTHOR]

In mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper, we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases. [ABSTRACT FROM AUTHOR]

The paper presents an extension of the differentiation method for model order reduction of large-scale interval systems. This is an alternative approach to the existing differentiation method of interval systems. The proposed method has been applied for both continuous and discrete-time interval systems. The reduction of discrete-time interval systems is achieved by using simple linear transformation z = w + 1 and bilinear transformation z = (1 + w) / (1 − w) , where w ≠ 1. The proposed method always generates stable reduced-order models, and also it retains the zeroth-order interval time moment. Four numerical examples exemplify the accuracy of the method and computational simplicity. Furthermore, the difficulties associated with the extension of Routh-based approximations to interval systems for obtaining stable reduced-order models are discussed. The stability of interval systems is verified by using Kharitonov's theorem. [ABSTRACT FROM AUTHOR]

Since Saul Kripke's influential work in the 1970s, the revisionary approach to semantic paradox—the idea that semantic paradoxes must be solved by weakening classical logic—has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes that they are unable to block. [ABSTRACT FROM AUTHOR]

Let pk,3(n) count the number of 2-color partition triples of n where one of the colors appears only in parts that are multiples of k and Bk,ℓ(n) denote the number of (k, ℓ)-regular bipartitions of n. In this paper, we prove two infinite families of congruences modulo 5 for p5,3(n), three infinite families of congruences modulo powers of 5 for p25,3(n), and six infinite families of congruences modulo powers of 5 for B5,25(n). For instance, for any integers n ≥ 0 and α ≥ 1, we have and [ABSTRACT FROM AUTHOR]

The method of political arithmetic elaborated by William Petty in the 17th century is usually taken to be an instrument of better governance aimed at practical utility, and lacking any theoretical support in a natural philosophical sense. This paper argues otherwise by showing how Petty's scarce discussions of First Principles and mechanical philosophy could be viewed as constituting a theoretical background against which his insistence on using arithmetic to account for social phenomena could be interpreted more charitably. Reconstructing Petty's philosophy also helps answering why the accuracy of his numbers was not of primary importance for his scientific projects. [ABSTRACT FROM AUTHOR]

With the increasing accessibility of high or very high spatial resolution remote sensing images, more and more change detection works are conducted on the unit of image object, i.e. the object-based change detection (OBCD). Change vector analysis (CVA) is a promising tool for unsupervised OBCD because it can provide reasonable interpretation of the change and an insight into the type of change. However, various features and attributes of image object produce complex high dimensional feature space that poses new challenges to measure the overall change magnitude and direction in CVA. This paper presents a new approach to measure the overall change magnitude of an image object by integrating its spectral and textural attributes, self-adaptively. The new approach was compared with the standard CVA magnitude, Mahalanobis distance, and a recent Self-Adaptive Weight CVA (SAW-CVA) magnitude for unsupervised OBCD with the same OTSU auto-thresholding method. Two cases were investigated where Case I used bi-temporal WorldView multispectral images (4-band) and Case II employed bi-temporal three-band images obtained from Google Earth. Results in the two cases both demonstrated that the new approach outperforms the others for unsupervised OBCD. The percentage correct classification of the new approach, SAW-CVA, standard CVA, and Mahalanobis are 80.28%, 77.93%, 72.77%, and 61.50% in Case I, and 91.80%, 90.16%, 88.52%, and 90.16% in Case II. Further analysis indicated that the new approach gives more reasonable self-adaptive weights and it does not require the empirical self-adaptive index as compared with the recent SAW-CVA method. [ABSTRACT FROM AUTHOR]

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege's treatment of definitions. Specifically, I pursue this issue via an examination of Frege's views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege's logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen's fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature. [ABSTRACT FROM AUTHOR]

Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197-208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results. [ABSTRACT FROM AUTHOR]

Ante rem structuralists claim that mathematical objects are places in ante rem structural universals. They also hold that the places in these structural universals instantiate themselves. This paper is an investigation of this self-instantiation thesis. I begin by pointing out that this thesis is of central importance: unless the places of a mathematical structure, such as the places of the natural number structure, themselves instantiate the structure, they cannot have any arithmetical properties. But if places do not have arithmetical properties, then they cannot be the natural numbers. The self-instantiation thesis turns out to be crucial for the identification of mathematical objects with places in structures. Unfortunately, we have no reason to believe that the self-instantiation thesis is true. [ABSTRACT FROM AUTHOR]

A new approach for designing a quaternary arithmetic logic unit (ALU) based on carbon nanotube field-effect transistors (CNTFETs) is presented in this paper. The proposed design performs basic arithmetic and logic functions and is extendable to perform additional functions (both arithmetic and logic functions). Efficient circuit-level optimisation techniques such as data gating are used to reduce the power consumption and enhance the performance of the proposed design. In addition, the unique properties of CNTFETs such as adjusting the threshold voltages of CNTFETs by tuning the CNTs diameters are utilised for designing a quaternary ALU. The proposed design is simulated and examined in different conditions and in the presence of major process variations using the Synopsys HSPICE simulator with Stanford 32nm CNTFET technology. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix – a special matrix which has only the natural numbers as entries and is closely related to the well-known Pascal matrix. By this means, we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations. [ABSTRACT FROM PUBLISHER]

In this paper I adopt the approach advocated by Sibum (1995, 73–106) in my reworking of the use of the medieval counting table during the early modern period. Whilst textbooks written in the vernacular from 1532 onwards include sections on using the counting table to perform calculations, the approaches given may not have been those adopted by the general populace. Using questions from these texts I will perform the calculations, manipulating the counters in a variety of ways, attempting to establish which feels most comfortable for the counter-caster. In the first part of the paper I focus on the ubiquity of the medieval counting table, its place in literature and in the early modern household, evidence for the latter being provided by a range of testamentary evidence. This will then be followed by a section in which I will give a detailed report of my experiences using a counting table and comparing the approaches I used with those outlined in the earliest books written in English,An introduction for to lerne to recken with pen or with counters(1539) and Robert Recorde’sThe grounde of artes(1543). In doing so, I will endeavour to establish the range and extent of tacit, practical knowledge and that the skilled practitioner acquired learning and skills beyond those in the written text. [ABSTRACT FROM PUBLISHER]

In Peru, national assessments repeatedly rank Indigenous mathematics students as the lowest performing across the entirety of Latin America and South America. Whereas lack of financial resources often predicts low measures, the history of educational practice teaches us that students' poverty need not predict their educational outcomes – creative instructional approaches may turn the tables. Here we report on an innovative, body-based arithmetic technique, the Botetano Method, that has been enabling poverty rural children from remote mountainous regions of Peru to match and even greatly surpass their urban peers on comparable test items. The article explains the method's guiding humanistic and cognitive principles and then reports on findings from explorative action research that implemented and evaluated the method. Using observational methodologies, we argue that the students developed in their conceptual understanding of the content as well as in their attraction to the discipline, their professional identity, their personal pride in their achievement, and their general epistemic capacity for concentration and self-regulation. Throughout, we emphasize the methodological limitations of this grassroots proof-of-concept action research, which threaten the validity of the assertions. We speculate on early extensions of the method to literacy studies. [ABSTRACT FROM AUTHOR]

A Z-number (ZN) presented by Zadeh is a new framework of fuzzy sets that combines a restriction value with a reliability measure. Based on this motivation of the concept of ZN, this study first proposes a cubic Z-number (CZN) set and operational relations of CZNs, and then defines two expected functions of CZNs to rank CZNs. Next, we present a CZN hybrid weighted arithmetic and geometric averaging (CZNHWAGA) operator and its properties. Then, a multicriteria DM model with unknown criterion weights is established based on the proposed CZNHWAGA operator, the expected functions, and the standard deviation of the expected values in the setting of CZNs. Lastly, the established DM model is applied to the selection of solar cells to illustrate its practicability and effectiveness in the setting of CZNs. However, the established DM model can solve the problem of unknown criterion weights and strengthen the DM reliability and flexibility. [ABSTRACT FROM AUTHOR]

We consider generalized Paley graphs $ \Gamma (k,q) $ Γ (k , q) , generalized Paley sum graphs $ \Gamma ^+(k,q) $ Γ + (k , q) , and their corresponding complements $ \bar \Gamma (k,q) $ Γ ¯ (k , q) and $ \bar \Gamma ^+(k,q) $ Γ ¯ + (k , q) , for k = 3, 4. Denote by $ \Gamma =\Gamma ^*(k,q) $ Γ = Γ ∗ (k , q) either $ \Gamma (k,q) $ Γ (k , q) or $ \Gamma ^+(k,q) $ Γ + (k , q). We compute the spectra of $ \Gamma (3,q) $ Γ (3 , q) and $ \Gamma (4,q) $ Γ (4 , q) and from them we obtain the spectra of $ \Gamma ^+(3,q) $ Γ + (3 , q) and $ \Gamma ^+(4,q) $ Γ + (4 , q) also. Then we show that, in the non-semiprimitive case, the spectrum of $ \Gamma (3,p^{3\ell }) $ Γ (3 , p 3 ℓ) and $ \Gamma (4,p^{4\ell }) $ Γ (4 , p 4 ℓ) with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs $ \Gamma (3,p) $ Γ (3 , p) and $ \Gamma (4,p) $ Γ (4 , p) for any $ \ell \in \mathbb{N} $ ℓ ∈ N , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of $ \Gamma ^*(k,q) $ Γ ∗ (k , q) such that $ \Gamma ^*(k,q) $ Γ ∗ (k , q) and $ \bar \Gamma ^*(k,q) $ Γ ¯ ∗ (k , q) are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs $ \Gamma _{\rm bip} $ Γ bip and all strongly regular graphs $ \Gamma _{\rm srg} $ Γ srg which are complementary equienergetic, i.e. $ \{\Gamma _{\rm bip}, \bar {\Gamma }_{\rm bip}\} $ { Γ bip , Γ ¯ bip } and $ \{\Gamma _{\rm srg}, \bar {\Gamma }_{\rm srg}\} $ { Γ srg , Γ ¯ srg } are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $ \{\Gamma, \bar \Gamma \} $ { Γ , Γ ¯ } which are neither bipartite nor strongly regular. [ABSTRACT FROM AUTHOR]

Many colleges are finding that the use of acceleration in developmental education is a promising direction for improved student progress toward a degree or certificate. Acceleration has been defined in the literature as the reorganization of curricula and instruction in ways that facilitate the completion of educational requirements in an expedited manner. A commonly cited technique for acceleration in developmental mathematics is to contextualize material from a prerequisite course into a later course, thereby eliminating the need for the prerequisite course. In this paper, we present a case study of how an arithmetic course was contextualized into an elementary algebra course. We address policy and structural prerequisites, design and implementation, and discuss some consequences of this approach. We conclude with evidence of the success of our approach using quasi-experimental methods. Our intention in this manuscript is to offer concrete examples of how we implemented this work, while maintaining sufficient generality so that abstraction of the basic principles will allow others to replicate this work within different courses. [ABSTRACT FROM PUBLISHER]

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamouspenchantfor mathematical ‘myth-making’. [ABSTRACT FROM PUBLISHER]