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A lack of access to a sequence of introductory to top-level curricula in rural schools has widened the achievement disparity between rural and urban students and influenced the number of college degree holders in rural areas. Recently, surging diversity and differentiation in income levels across nonurbanized areas can serve as determinants of the development of academically rigorous course pipelines in rural schools. This study examines differentiation in spatial access to the curriculum ladder for rigorous mathematics in three southern U.S. states during the 2017–18 school year: Alabama, Arkansas, and Mississippi. Policy variations in these states yield different effects on the sequence of rigorous mathematics courses in rural high schools. Given that the mathematics course pipeline reinforces college access, access to academically challenging courses likely outweighs the rural disadvantages of college readiness in non-urban students and the persistence of inequality in public education. [ABSTRACT FROM AUTHOR]

To any associative algebra A is associated a numerical sequence c_n^{\delta }(A), n\ge 1, called the sequence of proper central codimensions of A. It gives information on the growth of the proper central polynomials of the algebra. If A is a PI-algebra over a field of characteristic zero it has been recently shown that such a sequence either grows exponentially or is polynomially bounded. Here we classify, up to PI-equivalence, the algebras A for which the sequence c_n^{\delta }(A), n\ge 1, has almost polynomial growth. Then we face a similar problem in the setting of group-graded algebras and we obtain a classification also in this case when the corresponding sequence of proper central codimensions has almost polynomial growth. [ABSTRACT FROM AUTHOR]

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as "ten-roots" (mūla), and we discover that these arrangements form a regular numerical sequence, called "oblong numbers". Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics. [ABSTRACT FROM AUTHOR]

Background: Several computational and mathematical models of protein synthesis have been explored to accomplish the quantitative analysis of protein synthesis components and polysome structure. The effect of gene sequence (coding and non-coding region) in protein synthesis, mutation in gene sequence, and functional model of ribosome needs to be explored to investigate the relationship among protein synthesis components further. Ribosomal computing is implemented by imitating the functional property of protein synthesis. Result: In the proposed work, a general framework of ribosomal computing is demonstrated by developing a computational model to present the relationship between biological details of protein synthesis and computing principles. Here, mathematical abstractions are chosen carefully without probing into intricate chemical details of the micro-operations of protein synthesis for ease of understanding. This model demonstrates the cause and effect of ribosome stalling during protein synthesis and the relationship between functional protein and gene sequence. Moreover, it also reveals the computing nature of ribosome molecules and other protein synthesis components. The effect of gene mutation on protein synthesis is also explored in this model. Conclusion: The computational model for ribosomal computing is implemented in this work. The proposed model demonstrates the relationship among gene sequences and protein synthesis components. This model also helps to implement a simulation environment (a simulator) for generating protein chains from gene sequences and can spot the problem during protein synthesis. Thus, this simulator can identify a disease that can happen due to a protein synthesis problem and suggest precautions for it. [ABSTRACT FROM AUTHOR]

Following the definition of interval-valued fuzzy n-normed linear space given by S. Vijayabalaji et al., in this paper, notion of interval-valued generalized fuzzy n-normed linear space is introduced. The notion of convergent sequence, Cauchy sequence and their relation are studied. Some basic results are established on finite dimensional interval-valued generalized fuzzy n-normed linear space. [ABSTRACT FROM AUTHOR]

The question of precollege physics access and performance has been a persistent concern when considering the goal of diversifying participation in post-secondary STEM study and careers. This observational study examined school-level academic and demographic predictors of high school physics enrollment and performance in the USA. Due to the sequential and hierarchical nature of high school science and mathematics coursework, related predictors were included in multivariable regression and structural equation models to understand whether selectivity constrains the STEM pipeline. Descriptive and inferential analyses of state-level data were conducted (N = 663 high schools enrolling 559,044 students) to define the scope of physics enrollment and performance, and the predictive value of ethnicity, socioeconomic status, and science and mathematics course enrollments and performance. Physics was taken by fewer students when compared to biology, earth science, and chemistry. Correlations, multiple regression, and structural equation modeling indicated that although socioeconomic status was the main predictor of student enrollment in physics, this was mediated by schoolwide enrollment in chemistry and algebra II as well as schoolwide performance in chemistry and geometry. School-level performance in physics was negatively predicted by the percentage of students traditionally underrepresented in STEM, yet the predictive value was mediated by chemistry and algebra II performance. Results suggest that the prevalent science and mathematics sequence correlates to diminished physics enrollment, particularly with regard to students traditionally marginalized in STEM. US school leaders and policy makers should implement more proactive interventions earlier in the STEM pipeline to promote diverse physics participation and more equitable performance outcomes. [ABSTRACT FROM AUTHOR]

AbstractQuantum computing, grounded in the principles of quantum mechanics, has ushered in a new era of intriguing possibilities and fresh scientific insights. In this document, we present an innovative algorithm tailored for the encryption and decryption of images. This algorithm harnesses the quasi-probability values extracted from each state within the quantum system, represented as |ψ〉. These numerical values are seamlessly integrated with conventional data to facilitate image encryption and decryption processes. Our methodology undergoes comprehensive validation and evaluation through extensive simulations performed on the IBM Qiskit Aer simulator. We subject the generated numerical sequence to rigorous statistical scrutiny using the NIST SP 800-22 suite, ensuring its authenticity and reliability. Employing a quantum system comprising 3 qubits and uniform rotation gates, our algorithm produces a dataset consisting of 41,943,040 binary digits. Additionally, we conduct a thorough assessment encompassing both visual and statistical analyses to ascertain the effectiveness and robustness of our proposed algorithm. Furthermore, we emphasize the critical importance of achieving a quasi-probability error rate of 10−6 for real-world implementation, highlighting the practical viability and scalability of our approach. Through comparative analysis and meticulous examination of correlation patterns, our study provides invaluable insights into the distinctive characteristics and efficacy of our encryption scheme. [ABSTRACT FROM AUTHOR]

The given text contains a collection of mathematical formulas, equations, and graphs related to various topics such as the calculation of pi, the Fibonacci sequence, and the arcsine function. The text provides specific formulas and calculations, as well as proofs for some of the formulas. [Extracted from the article]

How do task experiences support proximal and longer‐term desires to reengage? Central to this question is the interplay between perceptions of task difficulty and the situational interest. This interplay is increasingly researched but not yet well understood. Micro‐analytic, latent studies are important tools for replicating past findings and extending our understanding in this area. This study aimed to extend our understanding of the interplay between readiness (prior knowledge and individual interest), a sequence of mathematics task experiences (situational interest and perceived difficulty) and reengagement beyond the set of tasks (i.e. optional follow‐up engagement). Participants (ages 20–49; n = 301) completed a series of maths tasks online, self‐reporting their task experiences after each task. Participants were offered an optional opportunity to follow‐up with the task content as a measure of their continued interest. Both readiness components assessed were important predictors of future situational interest (positive) and perceptions of task difficulty (negative). Persistent interconnections between the task experience components across time supported past evidence regarding the important role of situational interest and the challenges of perceived difficulty for sustained engagement. Modelling indicated that task success (β = 0.36), situational interest (β = 0.16) and perceptions of task difficulty (β = 0.21) can converge on choices to reengage with an object going forward. [ABSTRACT FROM AUTHOR]

Ethereum is a blockchain network that allows developers to create smart contracts and programs that run on the blockchain. Smart contracts contain logic to transfer assets based on pre-defined conditions. With over 100,000 new smart contracts being deployed every day, the potential for coding errors is high, making the contracts vulnerable to exploits. A key limitation is that once deployed, smart contracts are immutable and cannot be updated, even if flaws are found. This inflexibility puts funds at risk of theft and loss. The rapid pace of deployment outpaces security audits, increasing vulnerabilities that put users' cryptocurrency at risk. To reduce the risk caused by smart contract vulnerabilities, we applied deep learning techniques. To develop a deep learning model capable of detecting vulnerabilities, we first created a dataset by replaying real transactions on the Ethereum Mainnet, collecting opcode sequences from real Ethereum contracts, and labeling them using the SODA plugin. We pre-processed this opcode data by removing duplicates, normalizing sequence lengths, simplifying opcodes into representative groups, and converting sequences into numerical vectors to ultimately obtain an optimal representation of the data. We then trained and evaluated three different neural network architectures on this dataset. Our best-performing model achieved an average accuracy of 88% in detecting seven types of vulnerabilities. Further analysis showed that the model was effective at identifying potential problems in smart contracts, which was an important capability for securing funds and executing logic in live contracts. [ABSTRACT FROM AUTHOR]

Conceptualized in France during the sixties, the Theory of Didactical Situations [TDS] is a pivotal framework for developing mathematics teaching and learning processes. Despite the increasing qualitative studies over the last ten years, there remains a dearth of analysis of its effects on mathematics education. Through a Grounded Formal Theory approach, this paper presents the metasynthesis of 28 studies screened from four research databases--ProQuest, EBSCO, SCOPUS, and JSTOR. Data extraction was performed to enumerate the author/s, country, publication year, and connected theoretical approaches with TDS. Subsequently, the studies were categorized into three axial codes based on the studies' goals of applying TDS: lesson sequence, teacher development, and learning innovations. The synthesis underscores TDS' capacity to improve the sequence of mathematics lessons, particularly geometry and number patterns; develop teachers' pedagogical practices in teaching elementary, high school, and college students; and be a practical tool in innovating didactical resources for learning mathematics. TDS was also associated with other theoretical approaches, such as constructivism, realistic mathematics education, gamification, and technology-based instruction. Most of these effects were rooted in the challenges experienced during the COVID-19 pandemic and the adaptation of TDS to the digital era, including distance and online learning. While the European and Western Asian countries have been at the forefront of TDS integration, the paper advocates for broader global adoption of this theory to enrich mathematics education worldwide. [ABSTRACT FROM AUTHOR]

The goal of this research article is to study some fixed point results in dislocated quasi metric spaces. These results are proved for certain generalized contraction conditions involving linear and rational expressions. Our results extend and generalize some of the well-known fixed point results of the literature available in fixed point theory. Suitable examples of the established results are also given. [ABSTRACT FROM AUTHOR]

In this paper we consider rectangular billiard tables having vertices with integer coordinates, and side lengths equal to integer multiples of the norms of the side directions. We also assume that all the bouncing points of a billiard ball are constrained to belong to the integer lattice Z 2. We address several questions concerning combinatorial and geometric properties of the allowed orbits, that, due to the integer constraint, are called integer orbits. We give a complete classification of integer orbits, and the parameters contributing to their structure are precisely determined. This leads to understand how the orbit fills the lattice billiard before it really propagates. In particular, one can characterize the trajectories that reach a billiard pocket, as well as all the closed orbits, by the simple knowledge of the size of the billiard table, and of the starting moving direction. The characterization bases on the explicit determination of the numerical sequences corresponding to clockwise, and counterclockwise, bouncing. We also investigate the geometrical structure of an allowed orbit in terms of special sub-patterns, called Z -paths, pointing out the allowed lengths of different Z -paths in a same orbit. This is of independent interest, and is related to the configurations known as switching components, that play a crucial role in discrete tomography, and in problems concerning image reconstruction. [ABSTRACT FROM AUTHOR]

The 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]

Given the challenges associated with the low accuracy, complexity of the equipment, and poor interference resistance observed in current wireless multipath channel measurements, this study introduces a novel algorithm called KFSC-WRELAX. This algorithm integrates techniques involving pseudorandom noise (PN) sequences, Kalman filtering (KF), sliding correlation, and weighted Fourier transform combined with the RELAXation (WRELAX) algorithm. An m-sequence is employed as the probing sequence for channel detection. The effectiveness of the KFSC-WRELAX algorithm is demonstrated through both simulation experiments and corridor testing, showing that it can accurately determine the delays in various paths with robust performance at a signal-to-noise ratio (SNR) of −5 dB or higher. [ABSTRACT FROM AUTHOR]

Sequences formed by symbols are found in diverse fields, including genome sequences, written texts and computer codes. An interesting question is whether a sequence of symbols contains correlated structures. Existing methods to characterize correlations require a numerical representation of the sequence. In this regard, mapping a sequence of text into a sequence of numerical values is a key step for assessing correlation analysis. This work proposes a methodology to study correlations in a sequence of symbols. In the first step, the sequence of symbols is mapped in a multivariate numerical sequence formed by unit vectors in a vectorial space. The main feature of such mapping is that symbols are equally weighted, thus avoiding the numerical overrepresentation of symbols. In the second step, a multivariate version of the detrended fluctuation analysis is used to quantify correlations in the numerical sequence. Genome sequences (first COVID-19), written English texts and comovements between Bitcoin and gold markets were used to illustrate the proposed methodology's performance. The results showed that the balanced numerical mapping of symbolic sequences and the multivariate DFA provides valuable insights into the correlations in a sequence of symbols. [ABSTRACT FROM AUTHOR]

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series is nowadays beyond dispute. The Weniger transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better-known resummation techniques, like, for instance, Padé approximants. However, our theoretical understanding of Levin-type transformations is still far from being satisfactory and is particularly bad as far as the decoding of factorially divergent series is concerned. The Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the converging factor of any order of typical Stieltjes series can be expressed as an inverse factorial series, whose terms are analytically retrieved through a simple recursive algorithm. A few examples of applications are presented, in order to show the effectiveness and implementation ease of the algorithm itself. We believe that further investigations of the asymptotic forms of the remainder terms, encoded within the converging factors, could eventually lead toward a more general theory of the asymptotic behavior of the Weniger transformation when it is applied to Stieltjes series in high transformation orders. It is a rather ambitious project, which should be worthy of being pursued in the future. [ABSTRACT FROM AUTHOR]

The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces. [ABSTRACT FROM AUTHOR]

Quantum mechanical phenomena are revolutionizing classical engineering fields such as signal processing or cryptography. When randomness plays an important role, like in cryptography where random bit sequences guarantee certain levels of security, quantum mechanical phenomena allow new ways of generating random bit sequences. Such sequences have a lot of applications in the communication sector, e.g., regarding data transmission, simulation, sensors or radars, and beyond. They can be generated deterministically (e.g., by using polynomials, resulting in pseudo-random sequences) or in a non-deterministic way (e.g., by using physical noise sources like external devices or sensors, resulting in random sequences). Important characteristics of such binary sequences can be modelled by gap processes in conjunction with the probability theory. Recently, all-optical approaches have attracted a lot of research interest. In this work, an adaptation of the quantum key distribution setup is utilized for generating randomised bit sequences. The simulation results show that all-optically generated sequences very well resemble the theoretically ideal probability density characteristic. Additionally, an experimental optical setup is developed that confirms the simulation results. Furthermore, m-sequences show very promising results as well as Gold sequences. Additionally, the level of burstiness, i.e., the distribution of ones and zeros throughout the sequence, is studied for the different sequences. The results enable the finding that generator polynomials with concentrated non-zero coefficients lead to more bursty bit sequences. [ABSTRACT FROM AUTHOR]

Establishing a national comprehensive PNT (Positioning, Navigation, and Timing) system has become a consensus among major countries worldwide. As a crucial component in completing the entire PNT system, the 5G NR (new radio) time service signal plays a vital role. This paper proposes a 5G NR time service signal that uses a spread spectrum system, shares the 5G signal frequency band, but does not occupy the bandwidth of the 5G communication signal. This timing service signal has relatively low power, making it appear "submerged" within the power of the 5G communication signal. The spread spectrum code for this timing signal employs the chaotic orthogonal composite sequence proposed in this paper. Compared to traditional spread spectrum sequences, this sequence offers better security than m-sequences, improved autocorrelation than Walsh sequences, and an effective suppression of the short-period characteristics exhibited when the Skew Tent-Map chaotic sequence takes special values. This paper simulates the capture of the 5G NR time service signal in an environment with a signal-to-noise ratio of 10 dB using an FFT-based parallel code phase search algorithm, successfully capturing the 5G NR time service signal and verifying the feasibility of the proposed chaotic orthogonal composite sequence as a spread spectrum code. [ABSTRACT FROM AUTHOR]

This paper presents the outcomes of a sequence of three-dimensional numerical simulation models conducted in linear cutting mode, employing the LS-DYNA software. The primary objective was to scrutinize the side force values exerted on the gauge cutters. To accomplish this, an assortment of simulation models was meticulously constructed and executed, employing the Lagrange and SPG computational algorithms, along with the JHC and RHT constitutive material models. By comparing the outcomes of these models with the linear cutting experiments documented in the literature, specifically concerning the Colorado red Granite, it was determined that the SPG computational algorithm and the RHT constitutive material model proved to be the most suitable for simulating rock-cutting scenarios. Building upon the chosen computational algorithm and constitutive material model, a series of numerical models was then crafted to simulate the penetration of gauge cutters in both longitudinal and transverse orientations relative to their tips. Subsequently, the applied side forces were extracted, and a statistical model was developed to accurately predict the side forces experienced by the gauge cutters, yielding a coefficient of determination of 93%. [ABSTRACT FROM AUTHOR]

This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors "Leonardo sequence", "Leonardo numbers", "complexification", "generalization", "hybrids", and "combinatorial model" were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil. [ABSTRACT FROM AUTHOR]

This article is a revised version of the lectures given by the author at KROMSH-2019. These lectures are devoted to describing a few different ways of constructing a model representation for self-adjoint and unitary operators acting in Pontryagin spaces, and a comparison between them. Two of these models are based on the regularized integral Krein–Langer representation of a numerical sequence generated by the powers of a self-adjoint (in the sense of Pontryagin spaces) operator. The steps to deduce both this representation and the spectral function of the corresponding operator are given. In both models (first of which belongs to the author of this paper), the operator is realized as an operator of multiplication by an independent variable, but the space of functions in which it acts is different for each of the models. The third model, introduced by Shulman, is based on his own concept of a quasivector. [ABSTRACT FROM AUTHOR]

Tumour growth is a complex process influenced by various factors, including cell proliferation, migration, and chemotaxis. In this study, a biphasic chemotaxis model for tumour growth is considered, and the effect of chemotherapy on the growth process is investigated. We use optimal control theory to derive the optimized treatment strategy that minimises the tumour size while minimising the toxicity associated with chemotherapy. Moreover, the existence, uniqueness, and strong solution estimates for the biphasic chemotaxis model subsystem in one dimension are derived. These results are achieved through semigroup theory and the truncation method. In addition, the research provides evidence of the existence of an optimal pair through the utilization of the minimising sequence technique. It also demonstrates the differentiability of the mapping from control variable to state variable and establishes the first-order necessary optimality condition. Lastly, a sequence of numerical simulations are presented to showcase the impact of chemotherapy and the influence of parameters in restraining tumour growth when applied in an optimized manner. Our results show that optimal control can provide a more effective and personalised treatment for cancer patients, and the approach can be extended to other tumour growth models. [ABSTRACT FROM AUTHOR]

The problem of diffraction of ultrasonic waves by a sharp-angled rigid cone is studied. In the framework of the parabolic equation method, an analytical solution of the problem with an arbitrarily located point source is constructed. Namely, the problem is reduced to the Volterra boundary integral equation, which can be solved using the Fourier transform. An experimental measurement of the diffracted field is performed. The experiment is based on the M-sequence method adapted for narrowband sound sources. The experimental and theoretical results are compared. [ABSTRACT FROM AUTHOR]

This article provides an overview of Ibn Arabi's Fihrist, a catalog of his works, and delves into its structure and thematic considerations. The authors emphasize the importance of understanding this text as it forms the foundation for comprehending Ibn Arabi's work. They highlight the thematic connections among the book titles and their association with Quranic verses and Hadith. The article also explores the use of abjad methodology in the Fihrist and its significance in understanding Ibn Arabi's contemplation of the mysteries of Reality. Overall, the text offers insights into the structure and meaning of Ibn Arabi's work. [Extracted from the article]

Many models of choice assume that people retrieve memories of past experiences and use them to guide evaluation and choice. In this paper, we examine whether samples of recalled past experiences do indeed underpin our evaluations of options. We showed participants sequences of numerical values and asked them to recall as many of those values as possible and also to state how much they would be willing to pay for another draw from the sequence. Using Bayesian mixed effects modeling, we predicted participants' evaluation of the sequences at the group level from either the average of the values they recalled or the average of the values they saw. Contrary to the predictions of recall-based models, people's evaluations appear to be sensitive to information beyond what was actually recalled. Moreover, we did not find consistent evidence that memory for specific items is sufficient to predict evaluation of sequences. We discuss the implications for sampling models of memory and decision-making and alternative explanations. [ABSTRACT FROM AUTHOR]

The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for the developmental sequence of mathematical drawings and the progression of children's strategies for multiplicative situations were an interpretive lens through which to analyse the drawings. Children used pictographic and iconic drawing types to represent the "story" in the problem and the multiplicative strategies employed to solve the tasks. Exploration of the children's drawings suggested that as children's drawings become more structural, schematic in nature, it may be easier for children to show their understanding of the structural elements of multiplicative relationships. Results revealed that structural elements of multiplicative relationships were more easily seen in iconic representations; however, both pictographic and iconic drawings were useful to observe counting, additive, and multiplicative strategies when mathematical elements of the problem were visible. Additional representations attached to the drawings (e.g. numerical) were needed to confirm children's strategies when their drawings lacked structure. These findings have implications for how young children's drawings are interpreted by classroom teachers. The interpretation of these drawings suggested that some children may not yet realise how their drawings in mathematics need to shift from illustrations of the problem's story context to representing mathematical ideas and processes — which requires intentional teaching of the purpose of drawings for mathematical contexts. [ABSTRACT FROM AUTHOR]

This article studies the bidimensional versions of four number sequences: balancing, Lucasbalancing, Lucas-cobalancing and cobalancing. Some of the identities and generating functions were presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce dual numbers with components including Leonardo number sequences. This novel approach facilitates our understanding of dual numbers and properties of Leonardo sequences. We also investigate fundamental properties and identities associated with Leonardo number sequences, such as Binet's formula and Catalan's, Cassini's and D'ocagne's identities. Furthermore, we also introduce a dual vector with components including Leonardo number sequences and dual angles. This extension not only deepens our understanding of dual numbers, it also highlights the interconnectedness between numerical sequences and geometric concepts. In the future it would be valuable to replicate a similar exploration and development of our findings on dual numbers with Leonardo number sequences. [ABSTRACT FROM AUTHOR]

The article presents the analysis, design, and low-cost implementation of application-specific AD converters for M-sequence-based UWB applications to minimize and integrate the whole UWB sensor system. Therefore, the main goal of this article is to integrate the AD converter's own design with the UWB analog part into the system-in-package (SiP) or directly into the system-on-a-chip (SoC), which cannot be implemented with commercial AD converters, or which would be disproportionately expensive. Based on the current and used UWB sensor system requirements, to achieve the maximum possible bandwidth in the proposed semiconductor technology, a parallel converter structure is designed and presented in this article. Moreover, 5-bit and 4-bit parallel flash AD converters were initially designed as part of the research and design of UWB M-sequence radar systems for specific applications, and are briefly introduced in this article. The requirements of the newly proposed specific UWB M-sequence systems were established based on the knowledge gained from these initial designs. After thorough testing and evaluation of the concept of the early proposed AD converters for these specific UWB M-sequence systems, the design of a new AD converter was initiated. After confirming sufficient characteristics based on the requirements of UWB M-sequence systems for specific applications, a 7-bit AD converter in low-cost 0.35 µm SiGe BiCMOS technology from AMS was designed, fabricated, and presented in this article. The proposed 7-bit AD converter achieves the following parameters: ENOB = 6.4 bits, SINAD = 38 dB, SFDR = 42 dBc, INL = ±2-bit LSB, and DNL = ±1.5 LSB. The maximum sampling rate reaches 1.4 Gs/s, the power consumption at 20 Ms/s is 1050 mW, and at 1.4 Gs/s is 1290 mW, with a power supply of −3.3 V. [ABSTRACT FROM AUTHOR]

This paper proposes a spatiotemporal attention convolutional network (STAC-Pred) that leverages deep learning techniques to model the spatiotemporal features of tropical cyclones (TCs) and enable real-time prediction of their intensity. The proposed model employs dual branches to concurrently extract and integrate features from intensity heatmaps and satellite cloud imagery. Additionally, a residual attention (RA) module is integrated into the three-channel cloud imagery convolution process to automatically respond to high wind speed regions. TC's longitude, latitude, and radius of winds are injected into the multi-timepoint prediction model to assist in the prediction task. Furthermore, a rolling mechanism (RM) is employed to smooth the fluctuation of losses, achieving accurate prediction of TC intensity. We use several TC records to evaluate and validate the universality and effectiveness of the model. The results indicate that STAC-Pred achieves satisfactory performance. Specifically, the STAC-Pred model improves prediction performance by 47.69% and 28.26% compared to the baseline (official institutions) at 3- and 6-h intervals, respectively. Significance Statement: Tropical cyclones are one of the most deadly and damaging natural disasters in coastal areas worldwide. Early prediction can significantly reduce casualties and property losses. This study innovatively conducts dimensionality augmentation on one-dimensional intensity numerical sequences and proposes a new network model for rolling forecast of their future intensity. The proposed prototype model (not yet incorporating any atmospheric conditions) shows promising results for 3- and 6-h advance forecasts, providing valuable guidance for forecasters regarding real-time operational predictions of short-term tropical cyclone intensity. [ABSTRACT FROM AUTHOR]

This paper presents a model of recurrent multinomial sequences. Though there exists a quite considerable literature on modeling autocorrelation in numerical data and sequences of categorical outcomes, there is currently no systematic method of modeling patterns of recurrence in categorical sequences. This paper develops a means of discovering recurrent patterns by employing a more restrictive Markov assumption. The resulting model, which I call the recurrent multinomial model, provides a parsimonious representation of recurrent sequences, enabling the investigation of recurrences on longer time scales than existing models. The utility of recurrent multinomial models is demonstrated by applying them to the case of conversational turn-taking in meetings of the Federal Open Market Committee (FOMC). Analyses are effectively able to discover norms around turn-reclaiming, participation, and suppression and to evaluate how these norms vary throughout the course of the meeting. [ABSTRACT FROM AUTHOR]

According to contemporary research, there exists an imbalance within the disciplines of Science, Technology, Engineering, and Mathematics (STEM), wherein certain subjects are lacking representation due to the neglect or omission of mathematical elements. The purpose of this study is to address this issue through the analysis of an established learning sequence that has been well-tested to promote mathematical skills. We adapted the selected biology-based learning sequence to foster mathematical problem solving and conducted it with a school class. The qualitative analysis of the recorded video footage of this adapted learning sequence revealed that the modified task effectively stimulated mathematical problem-solving skills. This successful adaptation demonstrates one approach through which mathematics can be strengthened and effectively utilized in STEM subjects. [ABSTRACT FROM AUTHOR]

The study of numerical sequences usually focuses on the use of symbols or finding generalizations, that is, a more formal notation. However, there is several research on the different representations, but little research in representations that we consider as drawings and illustrations and that, in general, are commonly used in textbooks when both a numerical sequence and a figurative sequence are shown. The present research with a qualitative approach sought to identify the representations of sequences contained in Middle school textbooks. Nine textbooks were analyzed using the theoretical-methodological proposal known as Mathematical Content Analysis, where 28 different figures were found to represent a numerical sequence. The findings showed that there are three types of representations in the textbooks analyzed: abstract figures, real figures, and pictorial figures. The results show that the representations that appear most frequently in textbooks are pictorial figures. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]

We consider in this paper numerical approximation and simulation of a two-species Keller-Segel model. The model enjoys an energy dissipation law, mass conservation and bound or positivity preserving for the population density of two species. We construct a class of very efficient numerical schemes based on the generalized scalar auxiliary variable with relaxation which preserve unconditionally the essential properties of the model at the discrete level. We conduct a sequence of numerical tests to validate the properties of these schemes, and to study the blow-up phenomena of the model in a three-dimensional domain in parabolic-elliptic form and parabolic-parabolic form. [ABSTRACT FROM AUTHOR]

Amplification of wideband high-frequency and microwave signals is a fundamental element within every high-frequency circuit and device. Ultra-wideband (UWB) sensor applications use circuits designed for their specific application. The article presents the analysis, design, and implementation of ultra-wideband differential amplifiers for M-sequence-based UWB applications. The designed differential amplifiers are based on the Cherry–Hooper structure and are implemented in a low-cost 0.35 µm SiGe BiCMOS semiconductor process. The article presents an analysis and realization of several designs focused on different modifications of the Cherry–Hooper amplifier structure. The proposed amplifier modifications are focused on achieving the best result in one main parameter's performance. Amplifier designs modified by capacitive peaking to achieve the largest bandwidth, amplifiers with the lowest possible noise figure, and designs focused on achieving the highest common mode rejection ratio (CMRR) are described. The layout of the differential amplifiers was created and the chip was manufactured and wire-bonded to the QFN package. For evaluation purposes, a high-frequency PCB board was designed. Schematic simulations, post-layout simulations, and measurements of the individual parameters of the designed amplifiers were performed. The designed and fabricated ultra-wideband differential amplifiers have the following parameters: a supply current of 100–160 mA at −3.3 V or 3.3 V, bandwidth from 6 to 12 GHz, gain (at 1 GHz) from 12 to 16 dB, noise figure from 7 to 13 dB, and a common mode rejection ratio of up to 70 dB. [ABSTRACT FROM AUTHOR]

In a pre-service mathematics methods class I taught, in which we mathematized political districting (first horizontally, then vertically), student questions led to engaging mathematics, in particular, the development of a new number sequence. [ABSTRACT FROM AUTHOR]

RNA modification, a posttranscriptional regulatory mechanism, significantly influences RNA biogenesis and function. The accurate identification of modification sites is paramount for investigating their biological implications. Methods for encoding RNA sequence into numerical data play a crucial role in developing robust models for predicting modification sites. However, existing techniques suffer from limitations, including inadequate information representation, challenges in effectively integrating positional and sequential information, and the generation of irrelevant or redundant features when combining multiple approaches. These deficiencies hinder the effectiveness of machine learning models in addressing the performance challenges associated with predicting RNA modification sites. Here, we introduce a novel RNA sequence feature representation method, named BiPSTP, which utilizes bidirectional trinucleotide position-specific propensities. We employ the parameter ξ to denote the interval between the current nucleotide and its adjacent forward or backward dinucleotide, enabling the extraction of positional and sequential information from RNA sequences. Leveraging the BiPSTP method, we have developed the prediction model mRNAPred using support vector machine classifier to identify multiple types of RNA modification sites. We evaluate the performance of our BiPSTP method and mRNAPred model across 12 distinct RNA modification types. Our experimental results demonstrate the superiority of the mRNAPred model compared to state-of-art models in the domain of RNA modification sites identification. Importantly, our BiPSTP method enhances the robustness and generalization performance of prediction models. Notably, it can be applied to feature extraction from DNA sequences to predict other biological modification sites. [ABSTRACT FROM AUTHOR]

Let λ = (λn)n∈N0 be a non-negative sequence increasing to +∞, τ (λ) = limn→∞(ln n/λn), and D0(λ) be the class of all Dirichlet series of the form F(s) = P∞ n=0 an(F)esλn absolutely convergent in the half-plane Re s < 0 with an(F) ̸= 0 for at least one integer n ≥ 0. Also, let α be a continuous function on [x0,+∞) increasing to +∞, β be a continuous function on [a, 0) such that β(σ) → +∞ as σ ↑ 0, and γ be a continuous positive function on [b, 0). In the article, we investigate the growth of a Dirichlet series F ∈ D0(λ) depending on the behavior of the sequence (|an(F)|) in terms of its α, β, γ-orders determined by the equalities... where μ(σ) = max{|an(F)|eσλn: n ≥ 0} and M(σ) = sup{|F(s)|: Re s = σ} are the maximal term and the supremum modulus of the series F, respectively. In particular, if for every fixed t > 0 we have α(tx) ∼ α(x) as x → +∞, β(tσ) ∼ t-ρβ(σ) as σ ↑ 0 for some fixed ρ > 0, 0 < limσ↑0 γ(tσ)/γ(σ) ≤ limσ↑0 γ(tσ)/γ(σ) < +∞, Φ(σ) = α-1(β(σ))/γ(σ) for all σ ∈ [σ0, 0), eΦ(x) = max{xσ -Φ(σ): σ ∈ [σ0, 0)} for all x ∈ R, and ΔΦ(λ) = limn→∞(-ln n/eΦ(λn)), then: (a) for each Dirichlet series F ∈ D0(λ) we have... (b) if τ (λ) > 0, then for each p0 ∈ [0,+∞] and any positive function Ψ on [c, 0) there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and M(σ, F) ≥ Ψ(σ) for all σ ∈ [σ0, 0); (c) if τ (λ) = 0, then (Rα,β,γ(F))1/ρ ≤ (R∗ α,β,γ(F))1/ρ + ΔΦ(λ) for every Dirichlet series F ∈ D0(λ); (d) if τ (λ) = 0, then for each p0 ∈ [0,+∞] there exists a Dirichlet series F ∈ D0(λ) such that R∗ α,β,γ(F) = p0 and (Rα,β,γ(F))1/ρ = (R∗ α,β,γ(F))1/ρ + ΔΦ(λ). [ABSTRACT FROM AUTHOR]

This paper explores the specialized knowledge mobilized by a mathematics teacher in the selection and use of examples for teaching sequences. Taking an experimental case study approach, we analyse the examples deployed in a series of third-year secondary level lessons on sequences and identify the different knowledge subdomains activated according to the mathematics teachers' specialized knowledge analytical model. We will analyse active and passive examples, pointing out the mathematical entity that is being exemplified and the aspect of this entity which is being emphasized by the example. The results identify the different subdomains and categories which are drawn on in the selection and use of examples, along with the various interconnections across knowledge subdomains which interact in the process. [ABSTRACT FROM AUTHOR]

This paper shows our discovery of the fiboquadratic numbers, quadratic Fibonacci numbers in rithmomachia. This board game consists in performing mathematical operations to attack an opponent through geometric numbered pieces. We will show that the numbers on the board pieces are the first terms of six sequences that we will define through a recursive pattern of superpartients. We then prove that this extension is the solution of a recurrence relation with initial conditions, and therefore, is unique. The fiboquadratic numbers we introduce in this work are not limited to the context of this board. In general, we can generate any fiboquadratic sequence with arbitrary initial conditions. [ABSTRACT FROM AUTHOR]

DNA steganography is a technique for securely transmitting important data using DNA sequences. It involves encrypting and hiding messages within DNA sequences to prevent unauthorized access and decoding of sensitive information. Biometric systems, such as fingerprinting and iris scanning, are used for individual recognition. Since biometric information cannot be changed if compromised, it is essential to ensure its security. This research aims to develop a secure technique that combines steganography and cryptography to protect fingerprint images during communication while maintaining confidentiality. The technique converts fingerprint images into binary data, encrypts them, and embeds them into the DNA sequence. It utilizes the Feistel network encryption process, along with a mathematical function and an insertion technique for hiding the data. The proposed method offers a low probability of being cracked, a high number of hiding positions, and efficient execution times. Four randomly chosen keys are used for hiding and decoding, providing a large key space and enhanced key sensitivity. The technique undergoes evaluation using the NIST statistical test suite and is compared with other research papers. It demonstrates resilience against various attacks, including known-plaintext and chosen-plaintext attacks. To enhance security, random ambiguous bits are introduced at random locations in the fingerprint image, increasing noise. However, it is important to note that this technique is limited to hiding small images within DNA sequences and cannot handle video, audio, or large images. [ABSTRACT FROM AUTHOR]

Figured numbers are sequences of natural numbers that, when represented by dots, form geometric configurations. The immediate sequence of figured numbers presents in its configuration the equilateral triangles, being the basis for finding several others, as long as a relationship between them is discovered. This article shows the construction of polygonal, stellar and three-dimensional figured numbers that can be constructed using the GeoGebra software as a resource to visualize the properties that relate these numbers to the triangular numbers. It is a way of observing the relationship that the figures shown have to each other, how we can understand the structure of these sequences, when presented in the form of figures, and the possibility of building figures that represent some of the most known sequences within the number's figures such as 2D figured numbers such as square, pentagonal, centered triangular and star numbers within 3D figured numbers. Thus, this work presents such construction through the properties of pyramidal numbers, cubic numbers, octahedral numbers and two forms of construction of the stella octangula numbers, with emphasis on the use of triangular numbers or using the other sequences that depart from this one. Furthermore, the work shows the possibility of constructing other sequences as well as the discovery of possible properties that relate them. [ABSTRACT FROM AUTHOR]

Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

The underwater acoustic communication (UAC) operating in very shallow-water should ensure reliable transmission in conditions of strong multipath propagation, significantly disturbing the received signal. One of the techniques to achieve this goal is the direct sequence spread spectrum (DSSS) technique, which consists in binary phase shift keying (BPSK) according to a pseudo-random spreading sequence. This paper describes the DSSS data transmission tests in the simulation and experimental environment, using different types of pseudo-noise sequences: m-sequences and Kasami codes of the order 6 and 8. The transmitted signals are of different bandwidth and the detection at the receiver side was performed using two detection methods: non-differential and differential. The performed experiments allowed to draw important conclusions for the designing of a physical layer of the shallow-water UAC system. Both, m-sequences and Kasami codes allow to achieve a similar bit error rate, which at best was less than 10-3. At the same time, the 6th order sequences are not long enough to achieve an acceptable BER under strong multipath conditions. In the case of transmission of wideband signals the differential detection algorithm allows to achieve a significantly better BER (less than 10-2) than nondifferential one (BER not less than 10-1). In the case of narrowband signals the simulation tests have shown that the non-differential algorithm gives a better BER, but experimental tests under conditions of strong multipath propagation did not confirm it. The differential algorithm allowed to achieve a BER less than 10-2 in experimental tests, while the second algorithm allowed to obtain, at best, a BER less than 10-1. In addition, two indicators have been proposed for a rough assessment which of the detection algorithms under current propagation conditions in the channel will allow to obtain a better BER. [ABSTRACT FROM AUTHOR]