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In 1848 Ch. Hermite formulated a question on periodic representation of cubic numbers. In this paper we introduce a new Jacobi–Perron type algorithm that provides periodicity for all cubic irrationalities in the input and prove the periodicity. This is the first subtractive algorithm for which the algebraic periodicity is proven. The algorithm is designed for the totally real case, providing the answer to Hermite's original question in this case. [ABSTRACT FROM AUTHOR]

For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the two components of a directed chord on which the algebra operates, so that chord transformations within one chord cell or towards other cells, and paths or simple circuits over the chord network can be determined without resorting to computational algorithms or geometrical representations. The one-step transformations over the edges of the chord network associated with the $ k{-}1 $ k − 1 drift operators generalize the basic operators P, R and L of the Neo-Riemmanian triadic progressions and the maximally smooth cycles of the 12-TET system to any higher dimensional space. [ABSTRACT FROM AUTHOR]

In 1972, John Brillhart described an algorithm for expressing a prime p ≡ 1 (mod 4) as the sum of two squares. Brillhart's algorithm, which is based on the Euclidean algorithm, is simplicity itself. However, Brillhart's proof of his algorithm's correctness uses several previous results, and subsequent simplifications of his argument still retain something of an air of mystery. We provide a geometric interpretation of Brillhart's algorithm, which not only proves that it works, but also sheds light on a surprising palindromic property proved by Perron, and used by Brillhart in his proof. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {O} $ O denote the ring of integers of a quadratic field $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7). In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form $ a_1X_1^2+a_2X_2^2+a_3X_3^2 $ a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where $ a_i\in \mathbb {\mathcal {O}} $ a i ∈ O for $ 1\leq i \leq ~3 $ 1 ≤ i ≤ 3 for representing all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O} $ O . Let K denote a quadratic field such that its ring of integers $ \mathcal {O}_K $ O K is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7) is the only imaginary quadratic field with the above properties. Let $ D_K $ D K denote the discriminant of K. We have $ D_K\equiv 1(\text{mod }8) $ D K ≡ 1 (mod 8) if and only if 2 is a product of two distinct primes in $ \mathcal {O}_K $ O K . With $ \mathcal {O}_K $ O K as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form $ {\sum _{i=1}^{m}a_iX_i^2} $ ∑ i = 1 m a i X i 2 where $ a_i\in \mathcal {O}_K $ a i ∈ O K , $ 1\leq i \leq m $ 1 ≤ i ≤ m to represent all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O}_K $ O K . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022]. [ABSTRACT FROM AUTHOR]

The graph a K m , m ∇ C n is named the generalized double-wheel graph. A graph G is said to be M-integral (resp. A-integral, D-integral, D L -integral or D Q -integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G) , distance matrix D(G) , distance Laplacian matrix D L (G) or distance signless Laplacian matrix D Q (G) ) are integers. In this paper, we completely determine all D-integral, D L -integral and D Q -integral generalized double-wheel graphs respectively. [ABSTRACT FROM AUTHOR]

Multiplicative calculus is a mathematical system that offers an alternative to traditional calculus. Instead of using addition and subtraction to measure change, as in traditional calculus, it uses multiplication and division. The framework of nonlinear equations is an incredibly powerful tool that has proven invaluable in advancing our understanding of various phenomena across a wide range of applied sciences. This framework has enabled researchers to gain deeper insights into a vast array of scientific problems. The physical interpretation of iterative methods for nonlinear equations using multiplicative calculus offers a unique perspective on solving such equations and opens up potential applications across various scientific disciplines. Multiplicative calculus naturally aligns with processes characterized by exponential growth or decay. In many physical, biological, and economic systems, quantities change in a manner proportional to their current state. Multiplicative calculus models these processes more accurately than traditional additive approaches. For example, population growth, radioactive decay, and compound interest are all better described multiplicatively. The primary objective of this work is to modify and implement the Adomian decomposition method within the multiplicative calculus framework and to develop an effective class of multiplicative numerical algorithms for obtaining the best approximation of the solution of nonlinear equations. We build up the convergence criteria of the multiplicative iterative methods. To demonstrate the application and effectiveness of these new recurrence relations, we consider some numerical examples. Comparison of the multiplicative iterative methods with the similar ordinary existing methods is presented. Graphical comparison is also provided by plotting log of residuals. The purpose in constructing new algorithms is to show the implementation and effectiveness of multiplicative calculus.

We give an elementary approach to studying whether rings of S-integers in complex quadratic fields are Euclidean with respect to the S-norm. [ABSTRACT FROM AUTHOR]

For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iterates and cardinals, together with the elementary intervals between them. In the keyboard, generalized diatonic and chromatic intervals are easily identified. Two factor decompositions of the scale tones, which are particular cases of duality, make evident several properties of the sequence of intervals composing the octave, such as the number of repeated adjacent intervals and the composition of the generic step-intervals. The keyboard is associated with two matrix forms. When they are mutually transposed, the keyboard is reversible, as in the 12-tone Pythagorean scale. In this case, the relationship between the two main factor decompositions is given by an involutory matrix. [ABSTRACT FROM AUTHOR]

In 2018, Jungin Lee [5] gave a necessary and sufficient condition for a diagonal quadratic form ∑ i = 1 m a i X i 2 where a i ∈ Z for all i, 1 ≤ i ≤ m for representing all 2 × 2 matrices over Z . In this paper, we will consider the imaginary quadratic field Q (- 7) . Its ring of integers O is a principal ideal domain. Q (- 7) is the only imaginary quadratic field such that O is a principal ideal domain and 2 is a product of two distinct primes in O (upto units). With O as above, in this paper we give a necessary and sufficient condition for a diagonal quadratic form a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where a 1 , a 2 , a 3 ∈ O to represent all 2 × 2 matrices over O . [ABSTRACT FROM AUTHOR]

Let γ 0 = 5 - 1 2 = 0.618 ... . We prove that, for any ε > 0 and any trigonometric polynomial f with frequencies in the set { n 2 : N ⩽ n ⩽ N + N γ 0 - ε } , the inequality ‖ f ‖ 4 ≪ ε - 1 / 4 ‖ f ‖ 2 holds, which makes a progress on a conjecture of Cilleruelo and Córdoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any ε > 0 , there is C (ε) > 0 such that each positive integer N has at most C (ε) divisors in the interval [ N 1 / 2 , N 1 / 2 + N 1 / 2 - ε ] . [ABSTRACT FROM AUTHOR]

Geometric calculus is a special case of the Non-Newtonian Calculus introduced by Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Also, it is a more convenient calculation method for situations where the geometric increment is more meaningful than the arithmetic increment. In this study, geometric curves are defined and geometric Frenet–Serret formulas for these curves are presented. Furthermore, we give applications of these concepts to geometric magnetic curves. [ABSTRACT FROM AUTHOR]

The complex and changing environment in the field brings a more significant detection challenge to camera surveillance. This paper proposes target detection in camera surveillance images using the improved YOLOv5s algorithm for target tracking in surveillance images. Firstly, to address the weak feature extraction ability of YOLOv5s for small-scale and overlapping targets, the feature extraction performance of YOLOv5s is improved by combining the attention mechanism and substitution loss function in the deep learning network to improve the feature extraction performance of YOLOv5s for target detection in field camera surveillance images. Then, to test the improved algorithm’s performance, the improved algorithm in this paper is compared with SSD, Faster R-CNN, and YOLOv5s detection methods, and performance comparison experiments are done on the basis of the dataset. The results show that the average detection accuracy (MAP) of the algorithm in this paper is improved by 19%, 14.5% and 6.3% than SSD, Faster R-CNN, and YOLOv5s, respectively, and the average speed of detection is 324 FPS, the accuracy is improved, the detection is faster, and for the YOLOv5 m with DA and PT has a higher AP than the other models in this paper. This study enhances the scalability of YOLOv5s algorithm in complex environments, which is crucial for advancing image target detection.

The traditional intellectual property law course cannot adapt to the needs of cultivating students’ active learning ability, practical ability and innovation ability in the digital era due to the teaching concept, teaching means, teaching method and assessment form. This study proposes an information-based teaching model, which builds a course knowledge organization model using a formal expression language, through atomized knowledge points and knowledge granularity. The k-means algorithm has been improved from the perspective of center selection and constraint rules to adapt to the characteristics of the intellectual property law course and complete the clustering and extraction of knowledge points. The frequent item-based method analyzes the students’ answers and generates the corresponding association rules to further develop the analysis of learning situations. For a college law school study, it was found that 925 simultaneous wrong answers and 8152 simultaneous correct answers were successfully mined using clustering. In applying this college students’ evaluation scores of this information technology teaching in terms of content design, presentation design, structural design and relationship design of knowledge clustering are 0.796, 0.818, 0.788, 0.794, and overall good results are achieved.

The escalating spread of industrial wastelands has spotlighted the need for effective ecological restoration. This paper introduces an evaluation system for assessing the environmental restoration benefits of such wastelands, integrating the entropy weight and optimal assignment method for index quantification. Enhancements to decision-making elements and the incorporation of the fuzzy comprehensive evaluation method have culminated in developing a D-FCE model-based ecological restoration benefit assessment model. An empirical analysis on selected sites revealed notable increases in soil organic carbon content post-restoration—ranging from 1.73% to 2.98% in ERL areas, 1.69% to 2.45% in GL areas, and 1.25% to 2.08% in AL areas—demonstrating significant carbon sequestration. Additionally, a 38% rise in ecological benefits was observed, translating to an economic boon of 8563.2 yuan/year. This study furnishes a scientific method for evaluating industrial wasteland restoration and underpins the formulation of pertinent policies.

In the era of e-commerce economy, how to mine and study the different values of customers is an essential factor affecting the development of e-commerce enterprises. Based on this, this paper uses a clustering algorithm to mine and study the value of e-commerce customers. The research firstly constructs RFM e-commerce customer value identification framework, designs e-commerce customer value evaluation indexes and calculates the weights of the value indexes, and then uses K-means clustering algorithm and classical artificial bee colony algorithm to improve the Calculation, and completes e-commerce customer refinement based on the value of the user. The results show that the type of e-commerce customer value based on the RFMC model can be divided into eight types, based on which the e-commerce customer value is further subdivided into four types by using the clustering algorithm, namely, core customers (13.86%), supportive customers (45.38%), habitual customers (21.08%) and risky customers (19.68%). Based on the clustering algorithm, the customer value was classified into type 0 customers (low value customers), type 1 customers (high value customers), and type 2 customers (medium value customers).

Addressing the complex equipment and network challenges in new energy power plant industrial control systems, this study introduces a Markov time-varying machine learning algorithm, leveraging classification-constrained Boltzmann machines for real-time network security risk prediction. By employing a hybrid training mode for innovative feature extraction and classification, the algorithm forecasts future security risk states through an up-to-date state transition probability matrix. Integrating with the Markov time-varying model enhances the efficiency over traditional Boltzmann machines, facilitating nuanced network state analyses. The proposed model demonstrates high effectiveness against various network attacks, with average precision, recall, and F1 scores of 0.96, 0.93, and 0.94, respectively, and maintains over 80% accuracy under noise levels up to 40 dB. This research provides a solid foundation for proactive security defense mechanisms in industrial control systems.

We give necessary and sufficient conditions for the divisibility of two finite geometric series G n (x) = 1 + x + x 2 + ⋯ + x n - 1 over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. We consider sums of type ∑ n 1 ⋯ n k ≤ x F (n 1 , ... , n k) , taken over the hyperbolic region { (n 1 , ... , n k) ∈ N k : n 1 ⋯ n k ≤ x } , where F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ∑ n 1 ⋯ n k ≤ x f ((n 1 , ... , n k)) and ∑ n 1 ⋯ n k ≤ x f ([ n 1 , ... , n k ]) , involving the GCD and LCM of the integers n 1 , ... , n k , where f : N → C belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k = 2 . [ABSTRACT FROM AUTHOR]

We obtain bounds for the error term in the prime number theorem of the form π (x) - Li (x) ≤ 9.2211 x log (x) exp - 0.8476 log (x) for all x ≥ 2 , as well as other classical forms, improving upon the various constants and ranges compared to those in the literature. The strength and originality of our methods come from leveraging numerical results for small x in order to improve both the asymptotic and numerical bounds one obtains. We develop algorithms and formulas optimizing the conversion of both asymptotic and explicit numerical bounds from the prime counting function ψ (x) to both θ (x) and π (x) . [ABSTRACT FROM AUTHOR]

In this note, for any integers m, n ≥ 2, we find a condition on a positive integer c under which there exists a monic polynomial f ∈ ℤ [ x ] of degree n for which f(x)m – c has mn integral roots counting with multiplicities. This is the case if and only if m = 2 and c is a constant that comes from a solution of the Prouhet-Tarry-Escott problem of size n. For example, the smallest positive integer c for which there exists a monic degree 7 polynomial f ∈ ℤ [ x ] such that f(x)2–c has 14 integral roots is c = 6620176679276160000. [ABSTRACT FROM AUTHOR]

Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the Z -gradings on E having the form E = E (r 1 , r 2 , r 3) (v 1 , v 2 , v 3) , in which each element of a basis of L has Z -degree r 1 , r 2 , or r 3 . We provide a criterion for the support of this structure to coincide with a subgroup of the group Z , and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A. Z -gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332–353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020]. [ABSTRACT FROM AUTHOR]

We use the estimation of the number of integers n such that ⌊nc⌋ belongs to an arithmetic progression to study the coprimality of integers in ℕc = {⌊nc⌋}n∈ℕ, c > 1, c ∉ ℕ. [ABSTRACT FROM AUTHOR]

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/(n log n) for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that ∑n−λ over a primitive set is maximized by the primes if and only if λ is at least the critical exponent τ1 ≈ 1.14. A set is k-primitive if no member divides any product of up to k other distinct members. One may similarly consider the critical exponent τk for which the primes are maximal among k-primitive sets. In recent work the authors showed that τ2 < 0.8, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that τk tends to zero as k → ∞. [ABSTRACT FROM AUTHOR]

We prove Graham conjecture for some cases only by elementary methods in basic number theory. Extensive notes and remarks on new results on searching for the elementary proof of Graham conjecture are presented and discussed, e.g., by induction or directed graph. [ABSTRACT FROM AUTHOR]

Weinberger [16] in 1972, proved that the ring of integers of a number field with unit rank at least 1 is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra [13], extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least 1 is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that K 1 and K 2 are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over Q . If K 1 K 2 is ramified over K i , then at least one K i ( i ∈ { 1 , 2 } ) must have a Euclidean ideal class. [ABSTRACT FROM AUTHOR]

The fractional chromatic number of a graph is defined as the optimum of a rather unwieldy linear program. (Setting up the program requires generating all independent sets of the given graph.) Using combinatorial arguments we construct a more manageable linear program whose optimum value provides an upper estimate for the fractional chromatic number. In order to assess the feasibility of the proposal and in order to check the accuracy of the estimates we carry out numerical experiments.

Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L (m , a) = ⟨ (1 , a) , (0 , m) ⟩ where 1 ≤ a < m are integers with gcd (m , a) = 1 . We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a. [ABSTRACT FROM AUTHOR]

This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field ℚ(ζ11),$ \mathbb{Q}\left( {{\zeta }_{11}} \right), $where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.

We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp −cM(loglog⁡x)3(logloglog⁡x)2$\begin{array}{} \displaystyle \left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right) \end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored.

We prove that if $A \subseteq [X,\,2X]$ and $B \subseteq [Y,\,2Y]$ are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then $|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$. This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments. [ABSTRACT FROM AUTHOR]