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

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]

Abstract Basic numerical skills provide an important foundation for the learning of mathematics. Thus, it is critical that researchers and educators have access to valid and reliable ways of assessing young children's numerical skills. The purpose of this study was to evaluate the concurrent, predictive, and incremental validity of a two-minute paper-and-pencil measure of children's symbolic (Arabic numerals) and non-symbolic (dot arrays) comparison skills. A sample of kindergarten children (M age = 5.86, N = 439) were assessed on the measure along with a number line estimation task, a measure of arithmetic, and several control measures. Results indicated that performance on the symbolic comparison task explained unique variance in children's arithmetic performance in kindergarten. Longitudinal analyses demonstrated that both symbolic comparison and number line estimation in kindergarten were independent predictors of 1st grade mathematics achievement. However, only symbolic comparison remained a unique predictor once language skills and processing speed were taken into account. These results suggest that a two-minute paper-and-pencil measure of children's symbolic number comparison is a reliable predictor of children's early mathematics performance. Highlights • We tested the validity and reliability of a recently developed Numeracy Screener. • Performance on the symbolic comparison task predicted kindergarten children's arithmetic performance. • Performance on the symbolic comparison task in kindergarten predicted school mathematics achievement one year later. • The Numeracy Screener may be a useful tool for early mathematics researchers and educators alike. [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]

The need to complete multiple tasks concurrently is a common occurrence both daily life and in occupational activities, which can often include simultaneous cognitive and physical demands. As one example, there is increasing availability of head-worn display technologies that can be employed when a user is mobile (e.g., while walking). This new method of information presentation may, however, introduce risks of adverse outcomes such as a decrement to gait performance. The goal of this study was thus to quantify the effects of a head-worn display (i.e., smart glasses) on motor variability during gait and to compare these effects with those of other common information displays (i.e., smartphone and paper-based system). Twenty participants completed four walking conditions, as a single task and in three dual-task conditions (three information displays). In the dual-task conditions, the information display was used to present several cognitive tasks. Three different measures were used to quantify variability in gait parameters for each walking condition (using the cycle-to-cycle standard deviation, sample entropy, and the “goal-equivalent manifold” approach). Our results indicated that participants used less adaptable gait strategies in dual-task walking using the paper-based system and smartphone conditions compared with single-task walking. Gait performance, however, was less affected during dual-task walking with the smart glasses. We conclude that the risk of an adverse gait event (e.g., a fall) in head-down walking conditions (i.e., the paper-based system and smartphone conditions) were higher than in single-task walking, and that head-worn displays might help reduce the risk of such events during dual-task gait conditions. [ABSTRACT FROM AUTHOR]

"Which graphs are determined by their spectra (DS for short)?" is an important and challenging topic in spectral graph theory. Let G be an n -vertex graph with adjacency matrix A (G) and adjacency walk matrix W A (G) = [ e , A (G) e , ... , A n − 1 (G) e ]. We call G controllable if the rank of W A (G) is n over R and noncontrollable otherwise. In Wang (2017), the author gave a simple condition for a controllable graph to be determined by their generalized adjacency spectrum (DGAS for short). However, the result fails for noncontrollable graphs. In this paper, we consider the problem in the context of the generalized distance spectrum. Let G be a connected graph with distance matrix D. A graph G is determined by its generalized distance spectrum (DG D S for short) if any graph H sharing the same generalized distance spectrum with G must be isomorphic to G. The paper aims to extend the result in Wang (2017) from generalized adjacency spectrum to generalized distance spectrum. Besides, a simple arithmetic criterion is given to determine whether a connected graph is DG D S or not, which is applicable to some kind of noncontrollable graphs, e.g., regular graphs. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

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

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

This paper discusses a longitudinal study with children with Apert syndrome aged between 4 and 11 years. There has long been an interest in the role of fingers in the development of early number skills and arithmetic. As children with Apert syndrome are born with complex fusions of their fingers, they have to undergo several surgical procedures in order to obtain individuated fingers. This has implications for their finger mobility and finger awareness. It has been suggested that children with Apert syndrome have specific difficulties with early number and arithmetic activities. The findings from this study suggest that engaging children with Apert syndrome in activities that develop finger awareness (finger gnosis) and finger mobility (fine motor skills) may have a positive impact on their ability to engage with appropriate mathematics curricula at school. This is relevant to all those involved in the care of children with Apert syndrome and will be of particular relevance to those involved in early childhood and primary education. This study also provides new insights into the role of finger use in the development of skills and understanding in early number and arithmetic. [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]

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

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In [a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641], Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach to arbitrary powers of the Cartier operator, yielding bounds for the dimension of the kernel. These bounds give new restrictions on the Ekedahl-Oort type of Artin–Schreier covers. [ABSTRACT FROM AUTHOR]

A new algorithm for computing Hecke operators for SL n was introduced in [14]. The algorithm uses tempered perfect lattices , which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL 2 (Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields. [ABSTRACT FROM AUTHOR]

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

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]

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]

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]

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]

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

The equation |$x^2 + 1 = 0\mod p$| has solutions whenever p  = 2 or |$4n + 1$|⁠. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of |$SL_2(\mathbb{R})$|⁠. Since our Poincare series have a nontrivial dependence on their Iwasawa θ -coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work. [ABSTRACT FROM AUTHOR]

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve , or a formal germ of a p-adic curve , and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of $\pi _1(X)$ splits , then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite. [ABSTRACT FROM AUTHOR]

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

Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form y 2 = x 3 − m x , and quartic curves of the form X 2 − d Y 4 = k. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients d , k. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of k. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of k. [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]

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]

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]

A group G is called residually finite if for every non-trivial element g ∈ G , there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g , for example by using the word norm ‖ g ‖ G if the group G is assumed to be finitely generated. The residual finiteness growth RF G : N → N is then defined as the smallest function such that if ‖ g ‖ G ≤ r , there exists a morphism φ : G → Q to a finite group Q with | Q | ≤ RF G (r) and φ (g) ≠ e Q. Although upper bounds have been established for several classes of groups, exact asymptotics for the function RF G are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals log k for some k ∈ N , where the value k is given by an explicit expression. As an application, we show that for every m ≥ 1 and every 1 ≤ k ≤ m , there exists a group G containing a normal abelian subgroup of rank m and with RF G ≈ log k. [ABSTRACT FROM AUTHOR]

The article discusses the use of paper folding to build understanding of multiplicative relationships, which is a key goal of mathematics instruction in the upper elementary and middle grades. Multiplicative thinking includes comparing numbers through many processes, multiplication and division, ratio, proportions, stretching and shrinking, magnification, scaling, and splitting. The author explains that students tend to reason additively about multiplicative situations first, and are resistant to changing this approach, so teachers need to provide tasks that help students recognize and reason about multiplicative relationships. Paper folding is presented as a means of addressing both the development of multiplicative thinking and facilitating discussion of mathematical problems.

In this paper, the general case of normal discrete Hausdorff operators in L2(Rd) is studied. The main result states that under some natural arithmetic condition, the spectrum of such an operator is rotationally invariant. Several special cases and applications are considered. [ABSTRACT FROM AUTHOR]

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

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]

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]

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli. [ABSTRACT FROM AUTHOR]

It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by "(semi-)classical" invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant. In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the m reduction at large m limit of the ℓ -loop part of the refinement of the LMO invariant for ℓ ≤ 5 , we formulate an ℓ -variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant. [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]

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

The patient admission scheduling problem (PASP) has been studied for many years as one of the most important scheduling problems in the health sector. The primary goal of PASP is to assign patients to appropriate hospital beds while considering some hard and soft constraints. PASP is an NP-hard problem, which implies that the optimization approach is one of the best approaches that can be used to solve it. The arithmetic optimization algorithm (AOA) is a new optimization algorithm that can effectively solve both continuous and discrete optimization problems. However, it may suffer from poor exploration and premature convergence to sub-optimal solutions due to some problems with its exploration operators. In this paper, we propose a new method for the PASP using an improved AOA algorithm called the island-based AOA (iAOA). The new algorithm is based on a structured population model called the island model. This model distributes the population of candidate solutions among islands that periodically exchange some of the candidate solutions among each other based on a migration protocol. In iAOA, some modifications were applied to AOA's parameters to make it capable of handling discrete space. We evaluated iAOA using a public benchmark for the PASP and compared our results with those of other baseline algorithms (DFPA, SA, HS, GD-NLGD, HTS, TS, I-BBO, and MBBO). The simulation results revealed that iAOA achieved the minimum average execution time on 5 out of 6 instances of the public benchmark. It also showed that iAOA achieved the second minimum average violation of the objective function over 10 independent runs. Finally, the statistical examination of the experimental results using Friedman and Wilcoxon tests confirms the significance of the results of iAOA compared with the other baseline algorithms. [ABSTRACT FROM AUTHOR]

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

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]

Consider the set M a , b = { n ∈ Z ≥ 1 : n ≡ a mod b } ∪ { 1 } for a , b ∈ Z ≥ 1 . If a 2 ≡ a mod b , then M a , b is closed under multiplication and known as an arithmetic congruence monoid (ACM). A non-unit n ∈ M a , b is an atom if it cannot be expressed as a product of non-units, and the atomic density of M a , b is the limiting proportion of elements that are atoms. In this paper, we characterize the atomic density of M a , b in terms of a and b. [ABSTRACT FROM AUTHOR]

This paper aims to present Bipolar valued probabilistic hesitant fuzzy sets (BVPHFSs) by combining bipolar fuzzy sets and probabilistic hesitant fuzzy sets (PHFSs). PHFSs are a strong version of hesitant fuzzy sets (HFSs) in terms of evaluated as probabilistic of each element. Probabilistic hesitant fuzzy sets (PHFSs) are a set structure that argues that each alternative should be evaluated probabilistically. In this framework, the proposed cluster allows probabilistic evaluation of decision- makers' opinions as negative. Thus, this case proposes flexibility about selection of an element and aids to overcome with noise channels. Furthermore, some new aggregation operators are discussed called bipolar valued probabilistic hesitant fuzzy weighted average operator (BVPHFWA), Generalized bipolar valued probabilistic hesitant fuzzy weighted average operator (GBVPHFWA), bipolar valued probabilistic hesitant fuzzy weighted geometric operator (BVPHFWG), Generalized bipolar valued probabilistic hesitant fuzzy weighted geometric operator (GBVPHFWG), bipolar valued probabilistic hesitant fuzzy hybrid weighted arithmetic and geometric operator (BVPHFHWAG) and Generalized bipolar valued probabilistic hesitant fuzzy hybrid weighted arithmetic and geometric (GBVPHFHWAG) and some basic properties are presented. A score function is defined ranking alternatives. Moreover, two different algorithms are put forward with helping to TOPSIS method and by using aggregation operators over BVPHFSs. The validity of proposed operators are analyzed with an example and results are compared in their own. [ABSTRACT FROM AUTHOR]

Panchromatic and multi-spectral image fusion, called panchromatic sharpening, is the process of combining the spatial and spectral information of the source image into the fused image to give the image a higher spatial and spectral resolution. In order to improve the spatial resolution and spectral information quality of the image, an adaptive multi-spectral image fusion method based on an improved arithmetic optimization algorithm is proposed. This paper proposed improved arithmetic optimization algorithm, which uses dynamic stochastic search technique and oppositional learning operator, to perform local search and behavioral complementation of population individuals, and to improve the ability of population individuals to jump out of the local optimum. The method combines adaptive methods to calculate the weights of linear combinations of panchromatic and multi-spectral gradients to improve the quality of fused images. This study not only improves the quality and effect of image fusion, but also focuses on optimizing the operation efficiency of the algorithm to have real-time and high efficiency. Experimental results show that the proposed method exhibits strong performance on different datasets, improves the spatial resolution and spectral information quality of the fused images, and has good adaptability and robustness. The source code is available at: . [ABSTRACT FROM AUTHOR]