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The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than $ ({q}/{2})+1 $ (q / 2) + 1. [ABSTRACT FROM AUTHOR]

In general, submarines are designed to optimize operation below the water surface because they spend most of their time underwater. On the other hand, the performance in the free surface condition is also important because submarines face a variety of scenarios to complete operational missions, and the free surface condition is unavoidable for port departure and arrival. In the case of a submarine, the numerical accuracy of the potential theory for seakeeping analysis is excellent in submerged conditions, but it is poor in free surface conditions because of nonlinear effects near the free surface area. In this study, STAR-CCM+ was used as a Reynolds-averaged Navier-Stokes (RANS) solver to estimate the seakeeping performance of a Canadian Victoria Class submarine in irregular waves. The results were compared with those of model tests from a published paper. In addition, the potential theory code was also used to assess the seakeeping performance and compare it with computational fluid dynamics (CFD) results. From the calculation results, the motion responses in irregular waves using CFD showed similar trends to the experimental results. In contrast, motion responses from potential code showed significantly larger values than the experimental results. In conclusion, CFD simulations with irregular waves can be a good solution to predict the seakeeping performance of submarines in free surface conditions. [ABSTRACT FROM AUTHOR]

Boolean functions used in symmetric-key encryption should have high higher-order nonlinearity to resist several known cryptographic attacks, such as algebraic attacks and low-degree approximation attacks. The higher-order nonlinearity also plays an important role in coding theory and theoretical computer science, since it relates to the covering radius of Reed–Muller codes and the Gowers norm, respectively. It is well-known that bent functions have the highest nonlinearity in an even number of variables and thus they possess the best ability to withstand fast correlation attacks and best affine approximation attacks. However, there is currently limited knowledge regarding the higher-order nonlinearity of bent functions because computing the higher-order nonlinearity, or even providing tight lower bounds, is an extremely hard task. In 1974, Dillon proposed two well-known classes of bent functions based on partial spread (in brief, PS), called PS − and PS + , respectively. He also exhibited a subclass of bent functions in PS − , known as partial spread affine plane(PS a p for short). In this paper, we provide a lower bound on the third-order nonlinearity of the simplest PS a p bent functions in n variables, where n ≥ 6 is even, by calculating the nonlinearities of all second-order derivatives of this kind of bent functions. Compared to the two known lower bounds on the third-order nonlinearity given by Carlet and Tang et al. respectively, our lower bound is much better than these two ones. [ABSTRACT FROM AUTHOR]

In the study of bodily awareness, the predictive coding theory has revealed that our brain continuously modulates sensory experiences to integrate them into a unitary body representation. Indeed, during multisensory illusions (e.g., the rubber hand illusion, RHI), the synchronous stroking of the participant’s concealed hand and a fake visible one creates a visuotactile conflict, generating a prediction error. Within the predictive coding framework, through sensory processing modulation, prediction errors are solved, inducing participants to feel as if touches originated from the fake hand, thus ascribing the fake hand to their own body. Here, we aimed to address sensory processing modulation under multisensory conflict, by disentangling somatosensory and visual stimuli processing that are intrinsically associated during the illusion induction. To this aim, we designed two EEG experiments, in which somatosensory- (SEPs; Experiment 1; N = 18; F = 10) and visual-evoked potentials (VEPs; Experiment 2; N = 18; F = 9) were recorded in human males and females following the RHI. Our results show that, in both experiments, ERP amplitude is significantly modulated in the illusion as compared with both control and baseline conditions, with a modality-dependent diametrical pattern showing decreased SEP amplitude and increased VEP amplitude. Importantly, both somatosensory and visual modulations occur in long-latency time windows previously associated with tactile and visual awareness, thus explaining the illusion of perceiving touch at the sight location. In conclusion, we describe a diametrical modulation of somatosensory and visual processing as the neural mechanism that allows maintaining a stable body representation, by restoring visuotactile congruency under the occurrence of multisensory conflicts. [ABSTRACT FROM AUTHOR]

The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann–Roch space associated with these degree-three places, aiming to determine essential characteristics such as the basis. The investigation then turns to Hermitian codes, where we analyze both functional and differential codes of degree-three places, focusing on their parameters and automorphisms. In addition, we explore the study of subfield subcodes and trace codes, determining their structure by giving lower bounds for their dimensions. This presents a complex problem in coding theory. Based on numerical experiments, we formulate a conjecture for the dimension of some subfield subcodes of Hermitian codes. Our comprehensive exploration seeks to deepen the understanding of Hermitian codes and their associated subfield subcodes related to degree-three places, thus contributing to the advancement of algebraic coding theory and code-based cryptography. [ABSTRACT FROM AUTHOR]

Determining the maximum number of unit vectors in Rr$\mathbb {R}^r$ with no pairwise inner product exceeding α$\alpha$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all α⩽0$\alpha \leqslant 0$, and in this paper, we show that the maximum is (2+o(1))r$(2+o(1))r$ for all 0⩽α≪r−2/3$0 \leqslant \alpha \ll r^{-2/3}$, answering a question of Bukh and Cox. Moreover, the exponent −2/3$-2/3$ is best possible. As a consequence, we obtain an upper bound on the size of a q$q$‐ary code with block length r$r$ and distance (1−1/q)r−o(r1/3)$(1-1/q)r - o(r^{1/3})$, which is tight up to the multiplicative factor 2(1−1/q)+o(1)$2(1 - 1/q) + o(1)$ for any prime power q$q$ and infinitely many r$r$. When q=2$q = 2$, this resolves a conjecture of Tietäväinen from 1980 in a strong form and the exponent 1/3$1/3$ is best possible. Finally, using a recently discovered connection to q$q$‐ary codes, we obtain analogous results for set‐coloring Ramsey numbers. [ABSTRACT FROM AUTHOR]

Qualitative secondary analysis has generated heated debate regarding the epistemology of qualitative research. We argue that shifting to an abductive approach provides a fruitful avenue for qualitative secondary analysts who are oriented towards theory-building. However, the concrete implementation of abduction remains underdeveloped—especially for coding. We address this key gap by outlining a set of tactics for abductive analysis that can be applied for qualitative analysis. Our approach applies Timmermans and Tavory's (Timmermans and Tavory 2012 ; Tavory and Timmermans 2014) three stages of abduction in three steps for qualitative (secondary) analysis: Generating an Abductive Codebook, Abductive Data Reduction through Code Equations, and In-Depth Abductive Qualitative Analysis. A key contribution of our article is the development of "code equations"—defined as the combination of codes to operationalize phenomena that span individual codes. Code equations are an important resource for abduction and other qualitative approaches that leverage qualitative data to build theory. [ABSTRACT FROM AUTHOR]

We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with 2 × 2 matrices with determinant 1. [ABSTRACT FROM AUTHOR]

The notion of weighted Renyi's entropy for truncated random variables has recently been proposed in the information-theoretic literature. In this paper, we introduce a generalized measure of it for double truncated distribution, namely weighted generalized interval entropy (WGIE), and study it in the context of reliability analysis. Several properties, including monotonicity, bounds and uniqueness of WGIE are investigated. Moreover, the proposed measure is estimated using parametric approach and a simulation study is carried out to demonstrate the performance of the estimates for a real data set. The role of WGIE in reliability modeling has been investigated. We also provide an application of the proposed concept in coding theory. [ABSTRACT FROM AUTHOR]

Background: Recovery from addiction is frequently equated with abstinence. However, some individuals who resolve an addiction continue to use substances, including via substitution (i.e., increased use of one substance after eliminating/ reducing another). Substitution may play a distinct role during early recovery (≤ 1 year), as this period is marked by dramatic change and adjustment. Cannabis is one of the most used substances and is legal for medical and recreational use in an increasing number of states. Consequently, cannabis an increasingly accessible substitute for substances, like fentanyl, heroin, cocaine and methamphetamine, with higher risk profiles (e.g., associated with risk for withdrawal, overdose, and incarceration). Methods: Fourteen participants reported that they had resolved a primary opioid or stimulant addiction and subsequently increased their cannabis use within the previous 12 months. Using grounded theory, the interviewer explored their experiences of cannabis use during early recovery. Data were analyzed in three stages: line by line coding for all text related to cannabis use and recovery, focused coding, and axial coding to generate a theory about recovery with cannabis substitution. The motivational model of substance use provided sensitizing concepts. Results & discussion: The final sample included eight men and six women ranging in age from 20 to 50 years old. Three participants resolved an addiction to methamphetamine and the remaining 11, an addiction to opioids. Participants explained that cannabis was appealing because of its less harmful profile (e.g., no overdose risk, safe supply, few side effects). Participants' primary motives for cannabis use included mitigation of psychiatric symptoms, withdrawal/ cravings, and boredom. While cannabis was effective toward these ends, participants also reported some negative side effects (e.g., decreased productivity, social anxiety). All participants described typical benefits of recovery (e.g., improved self-concept, better relationships) while continuing to use cannabis. Their experiences with and beliefs about substitution suggest it can be an effective strategy for some individuals during early recovery. Conclusions: Cannabis use may benefit some adults who are reducing their opioid or stimulant use, especially during early recovery. The addiction field's focus on abstinence has limited our knowledge about non-abstinent recovery. Longitudinal studies are needed to understand the nature of substitution and its impact on recovery over time. [ABSTRACT FROM AUTHOR]

Many papers have been devoted to applying fuzzy sets to algebraic structures. In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory, lattice theory, and coding theory points of view, and some applications of fuzzy sets in residuated lattices are presented. Since ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic, first, we investigate the lattice of fuzzy ideals in residuated lattices and study some connections between fuzzy sets associated to ideals and Hadamard codes. Finally, we present applications of fuzzy sets in coding theory. [ABSTRACT FROM AUTHOR]

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z 4 codes, which play a significant role in coding theory. [ABSTRACT FROM AUTHOR]

Despite the needs of assessing the time-varying queue length for incident traffic management, such imperative information has not yet been available in practice, due partly to the lack of reliable sensor data on many highways plagued by nonrecurrent congestion. Hence, this study first presents an incident queue impact assessment system that allows highway agencies to perform reliable estimate of the queue distance during the incident clearance period with available detector data, and then discusses two alternative models for supporting approximating such impacts on roadway networks without traffic sensors. The proposed system’s major model, grounded in the classical shockwave theory, has augmented the formulations for queue propagation dynamics with the impacts from incoming drivers’ perceptions and responses to the progressively constrained traffic conditions. To cope with the lack of real-time traffic flow information, the first alternative model integrates the increasing available speed data from probe vehicles or other sources with lane blockage patterns to approximate the discharge flow rate of the incident. The second alternative model centers on the methodology and calibration methods for constructing a set of speed– flow rate relationships for different times of day with archived traffic data to approximate the arriving flow rate. With such information, responsible agencies can then proceed the approximation of incident queue variation length and take necessary actions. Performance evaluation of the proposed system with both field incident data and simulated scenarios has confirmed its promising properties, especially for highway networks without reliable traffic surveillance systems. [ABSTRACT FROM AUTHOR]

Definition: Contemporary educators have increasingly recognised the diversity of their student population and, hence, have attempted to use multimodal teaching methods for additional student learning benefits. One popular example is repurposing film and TV content for higher education pedagogies. However, integrating these materials into teaching effectively often proves more complex than lecturers might anticipate. This entry investigates the merits and challenges of using FF/TV in teaching to determine the factors that impact development of an effective FF/TV pedagogy for student learning, through an interdisciplinary review of the existing literature, followed by a qualitative survey and semi-structured interviews with lecturers across disciplines at Australian universities. Using visual literacy theory, cognitive load theory, and dual coding theory, data analysis reveals that the pros and cons of integrating film and TV in teaching are in fact interconnected, and the main role of the teacher is to pedagogically balance them. Evidence-based and theory-grounded suggestions for application are detailed throughout the discussions. [ABSTRACT FROM AUTHOR]

ChatGPT was released to the public in November 30, 2022. This study examines how ChatGPT can be used by educators and students to promote learning and what are the challenges and limitations. This study is unique in providing one of the first systematic reviews using peer review studies to provide an early examination of the field. Using PRISMA principles, 44 articles were selected for review. Grounded coding was then used to reveal trends in the data. The findings show that educators can use ChatGPT for teaching support, task automation, and professional development. These were further delineated further by axial sub codes. Eight student uses were 24/7 support, explain difficult concepts, conversational partner, personalized feedback and materials, provide writing support, offer self-assessment, facilitate engagement, and self-determination. In addition to be affordances of the AI, the data from the articles also showed limitations to ChatGPT and misuses, specifically, inaccuracies and hallucinations, potential bias, and tool limitations. Misuses are plagiarism and cheating, privacy issues and spread of false information. This study is a springboard for researchers, practitioners, policy makers and funders in understanding the emerging state of the field of ChatGPT. [ABSTRACT FROM AUTHOR]

Let F q be the finite field of q elements, and F q n its extension of degree n. A normal basis of F q n over F q is a basis of the form { σ i (α) : i = 0 , ⋯ , n - 1 } where σ denotes the Frobenius automorphism of F q n over F q and α ∈ F q n . Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. Some problems on normal bases are characterized in terms of linearized polynomials. We show that such problems can be finally reduced to the determination of the irreducible factors of x n - 1 over F q . Using this approach, we extend the known results with alternative as well as relatively elementary proofs. And we believe that this approach can be used to deal with other similar problems. [ABSTRACT FROM AUTHOR]

The Poisson upward continuation of gravity field functionals, typically given on the Earth's irregular topography with limited spatial extent, necessitates some proper treatments due to the theoretical requirements. This study reviews the Poisson theory and investigates a more rigorous methodology for the numerical solution of the spherical integral equation by addressing many issues such as spherical-Earth model implementation, far-zone effect, Poisson kernel modification, and suitable ground data reduction scheme. We first explore the far-zone effect and search for an optimal near-zone spherical cap radius above which the truncation error is negligible. We then compare different variants of remove-restore technique with a modified Poisson kernel to upwardly continue the ground gravity field data to predefined height levels close to the Earth's surface. Different combinations of long- and short-wavelength contributions are studied extensively. Numerical experiments have been performed using simulated ground gravity anomaly and gravity disturbance data synthesized from ultrahigh-degree global geopotential model (GGM). Numerical results show that the far-zone contribution may reach up to several milligal levels and should be taken into consideration when the cap size radius is less than 1° for the upward continuation height levels between 3000 m and 5000 m above the sea level. Among the various solutions, the best agreement between the Poisson upward-continued ground data and its synthetic counterpart has been obtained when the gravity field data input directly into the Poisson integral is reduced both for GGM and residual terrain model (RTM). [ABSTRACT FROM AUTHOR]

The rapid advancement of technology has resulted in significant changes in the field of education. In recent years, the Internet of Things (IoT) and Bayesian network technology have emerged as promising tools for enhancing operations research and teaching performance evaluation. IoT devices and sensors can provide real-time data on student performance, while Bayesian network technology can analyze and predict the complex relationships between various variables. This paper proposes a performance evaluation model based on IoT and Bayesian network technology for operations research teaching. Operations research is a broad discipline with a high level of practicality and applicability. It is widely used in business administration, production planning, traffic management, engineering construction, and financial economics. Using experiments, analysis, and quantification, operations research organizes and manages all types of resources in the system, such as human resources, funds, and goods, and formulates the best business plan. Operations research teaching must combine intuitive geometry and immaterial theorems to improve students' understanding. The methods used in the teaching stage are simple: the teaching content could be more exciting and unintended, and it is difficult to grasp the teaching focus, making accurate evaluation of teaching performance difficult. Therefore, in the Bayesian network, the incremental results and the performance evaluation model of operations research achieve an accuracy of 87.3%. By thoroughly examining the fundamentals of Bayesian network technology, one can offer technical assistance in building an operational research teaching performance evaluation model. To combine with real-world teaching scenarios to assess the likelihood of risky events occurring during instruction and make a teaching performance evaluation model grounded in operational research. The B5 teaching effect has the highest risk probability of teaching work (0.89), and teachers with high professional titles have higher average values. [ABSTRACT FROM AUTHOR]

A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of non-scalar multiplications required to evaluate it. This quantity is also described as its tensor rank, which is the smallest number of rank-1 matrices whose span contains its first slice space. In this paper we derive upper bounds on the tensor ranks of certain classes of 3-tensors and give explicit constructions of sets of rank-1 matrices containing their first slice spaces. We also show how these results can be applied in coding theory to derive upper bounds on the tensor rank of some rank-metric codes. In particular, we compute the tensor rank of some families of F q m -linear codes and we show that they are extremal with respect to Kruskal's tensor rank bound. [ABSTRACT FROM AUTHOR]

Cortical populations of neurons develop sparse representations adapted to the statistics of the environment. To learn efficient population codes, synaptic plasticity mechanisms must differentiate relevant latent features from spurious input correlations, which are omnipresent in cortical networks. Here, we develop a theory for sparse coding and synaptic plasticity that is invariant to second-order correlations in the input. Going beyond classical Hebbian learning, our learning objective explains the functional form of observed excitatory plasticity mechanisms, showing how Hebbian long-term depression (LTD) cancels the sensitivity to second-order correlations so that receptive fields become aligned with features hidden in higher-order statistics. Invariance to second-order correlations enhances the versatility of biologically realistic learning models, supporting optimal decoding from noisy inputs and sparse population coding from spatially correlated stimuli. In a spiking model with triplet spike-timing-dependent plasticity (STDP), we show that individual neurons can learn localized oriented receptive fields, circumventing the need for input preprocessing, such as whitening, or population-level lateral inhibition. The theory advances our understanding of local unsupervised learning in cortical circuits, offers new interpretations of the Bienenstock-Cooper-Munro and triplet STDP models, and assigns a specific functional role to synaptic LTD mechanisms in pyramidal neurons. Author summary: To understand how our brains carve out meaningful stimuli from a sea of sensory information, experimentalists often focus on individual neurons and their receptive fields; i.e., the input patterns that best stimulate the neurons. The receptive field of a neuron adapts to the statistics of the environment through plastic changes in the structure and strength of its incoming connections. Prevailing models of this adaptation, such as Sparse Coding and BCM theory, suggest that these connections are adjusted in a self-organized, unsupervised manner such that neurons can extract relatively rare, but complex patterns from the sensory input stream. Our theory extends these models, demonstrating that certain variants of the BCM model are capable of selectively ignoring pervasive second-order correlations and homing in on the pertinent, sparse signals in the stream of sensory stimuli. We have identified a carefully calibrated interaction between two aspects of synaptic plasticity: linear long-term depression (LTD) and nonlinear long-term potentiation (LTP). Such a balance enables a learning strategy that is sensitive to the complex statistical properties (higher-order correlations) of sensory inputs. In contrast to earlier approaches, so-called pre-whitening to remove second-order correlations is not necessary. Our findings provide not only a new framework for understanding the diverse roles of synaptic plasticity but also offer insights that may inform the development of artificial neural networks. [ABSTRACT FROM AUTHOR]

Dominique Pradelle's book, "Être et genèse des idéalités: Un ciel sans éternité," explores the nature of mathematical knowledge and the paradox of mathematical objects being both ideal and timeless. Pradelle examines the views of philosophers such as Thomas Aquinas, Bernard Bolzano, and Edmund Husserl, who argue that idealities exist independently of subjective mental acts and have intersubjective and omnitemporal validity. The book also discusses the phenomenological constitution of mathematical objects and their dependence on a horizon structure, as well as the role of axiomatization and formalization in changing the nature of mathematical objects. Pradelle's work challenges naive realism and supports a nuanced understanding of mathematical objects. [Extracted from the article]

The thalamic pulvinar exhibits extensive reciprocal connections with the visual cortex. These connections provide alternative pathways for the transmission of signals between cortical neurons. Although the precise roles of the pulvinar in vision are incompletely understood, research indicates that the pulvinar is involved in target selection among distractors, feature binding, and visual attention. Pulvinar dysfunctions have been associated with sensory deficits observed in individuals with schizophrenia. Computational models suggest that the pulvinar plays a crucial role in establishing persistent activity within the cortical network by regulating the flow of information across visual cortical areas and the synchronization of their activity. We highlight the organization of brain networks involving the pulvinar and propose a pivotal role for the pulvinar within the framework of cortical theories that aim to elucidate the processes of perception and cognitive integration. The pulvinar nucleus of the thalamus is a crucial component of the visual system and plays significant roles in sensory processing and cognitive integration. The pulvinar's extensive connectivity with cortical regions allows for bidirectional communication, contributing to the integration of sensory information across the visual hierarchy. Recent findings underscore the pulvinar's involvement in attentional modulation, feature binding, and predictive coding. In this review, we highlight recent advances in clarifying the pulvinar's circuitry and function. We discuss the contributions of the pulvinar to signal modulation across the global cortical network and place these findings within theoretical frameworks of cortical processing, particularly the global neuronal workspace (GNW) theory and predictive coding. [ABSTRACT FROM AUTHOR]

The problem of building optimal block codes, such as MDS codes, over small fields has been an active area of research that led to several interesting conjectures. In the context of convolutional codes, optimal constructions, such as MDS or MDP, are very rare and all require very large finite fields. In this work, we focus on the problem of constructing optimal convolutional codes with respect to the rank distance, i.e. we study the construction of Maximum Rank Distance (MRD) convolutional codes. Considering convolutional codes within a very general framework, we present concrete novel classes of MRD convolutional codes for a large set of given parameters. [ABSTRACT FROM AUTHOR]

One of the extremely useful branches in graph theory is the labeling of a graph. Graph labeling plays a vital role in many fields such as database management, astronomy, coding theory, X-ray crystallography, communication network addressing and radar. A labeling of a connected simple graph G V , E is a map that assign each element in G with a positive integer number. An edge irregular total λ - -labeling is a map β : V G ∪ E G → 1 , 2 , 3 , ... , λ - such that W β h ≠ W β z where W β h and W β z are weights for any two distinct edges. In this case, G has total edge irregularity strength (TEIS) if λ - is minimum. In this paper, a new family of graphs called square snake graphs is defined and denoted by C 4 , n . Moreover, we define some related graphs of square snake graphs named double square snake graph D C 4 , n , triple square snake graph T C 4 , n and m -multiple square snake graph M m C 4 , n . Finally, we determine TEIS for square snake graphs, double square snake graph, triple square snake graph and m -multiple square snake graph, which have many applications in coding theory and physics. [ABSTRACT FROM AUTHOR]

In the setting of entanglement-assisted quantum error-correcting codes (EAQECCs), the sender and the receiver have access to noiseless pre-shared entanglement. With this assumption, one may obtain codes with better information rates or improved error handling properties. Entanglement incurs costs and must be judiciously calibrated in designing quantum codes, relative to their deployment parameters. Revisiting known constructions, we devise tools from classical coding theory to better understand how the amount of entanglement can be varied. We present three new propagation rules for the parameters of these codes. Tables listing the parameters of qubit and qutrit EAQECCs that we can explicitly construct are supplied for reference and comparison. [ABSTRACT FROM AUTHOR]

Fuzzy graphs as labeled graphs (fuzzy vertex labeling and fuzzy edge labeling) have many applications in real life such as complex networks, coding theory, medical sciences, communication networks, and management sciences. Also, triangular norms as a special class of functions, have many applications in fuzzy set theory, probability and statistics, and other areas. This paper considers the notations of an inverse fuzzy graph and triangular norms to introduce the new type of graphs as valued-inverse Dombi fuzzy graphs. The valued-inverse Dombi fuzzy graphs are a generalization of inverse fuzzy graphs and are dual to Dombi fuzzy graphs. For any given greater than or equal to one real number, we construct a type of Dombi inverse fuzzy graph and investigate some conditions such that the product and union of Dombi inverse fuzzy graphs be a Dombi inverse fuzzy graph. [ABSTRACT FROM AUTHOR]

The predictive coding theory is currently widely accepted as the theoretical basis of perception and chronic perceptual disorders are explained as the maladaptive compensation of the brain to a prediction error. Although this gives us a general framework to work with, it is still not clear who may be more susceptible and/or vulnerable to aberrations in this system. In this paper, we study changes in predictive coding through the lens of tinnitus and pain. We take a step back to understand how the predictive coding system develops from infancy, what are the different neural and bio markers that characterise this system in the acute, transition and chronic phases and what may be the factors that pose a risk to the aberration of this system. Through this paper, we aim to identify people who may be at a higher risk of developing chronic perceptual disorders as a reflection of aberrant predictive coding, thereby giving future studies more facets to incorporate in their investigation of early markers of tinnitus, pain and other disorders of predictive coding. We therefore view this paper to encourage the thinking behind the development of preclinical biomarkers to maladaptive predictive coding. [ABSTRACT FROM AUTHOR]

In order to optimize the implementation complexity and information transmission speed of the fifth generation communication technology, the Polar code based on channel polarization phenomenon gradually receives great attention and research, and forms many industry standards and research conclusions. In recent years, the development and optimization of Polar code encoding and decoding technology promotes the popularity of the fifth generation communication technology, and the rapid application of the new generation communication technology greatly promotes the further iteration of the underlying encoding and decoding technology. By reviewing the research history of Polar code encoding and decoding technology, the research status and context of Polar code encoding and decoding technology in the fifth generation communication are summarized. On this basis, the problems and requirements of current coding and decoding theory are analyzed and discussed in depth, and the research ideas and trends of the underlying coding and decoding theory in the future communication technology are proposed. [ABSTRACT FROM AUTHOR]

Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes that, together with any vector, contain its complement as well. This paper is about binary linear self-complementary codes. A natural goal in coding theory is to find a linear code with a given length n and dimension k such that the minimum distance d is maximal. Codes with these properties are called optimal. Another important issue is classifying the optimal codes, i.e., finding exactly one representative of each equivalence class. In some sense, the classification problem is more general than the minimum distance bounds problem. In this work, we summarize the classification results for self-complementary codes with the maximum possible minimum distance and a length of up to 20. For the classification, we developed a new algorithm that is much more efficient compared to existing ones in some cases. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce the Gaussian Leonardo polynomial sequences {GLen(X)}⃵n=0 and we obtain Binet's formula, generating function of this sequence. Moreover, we define the matrix GL(X) in the form of 3 x 3. Finally, we study on the coding and decoding applications of the Leonardo number by using the Leonardo matrix. [ABSTRACT FROM AUTHOR]

In this technical communication a voltage differencing transconductance amplifier (VDTA)-based unified grounded fractional/integer order negative/positive inductance emulator (IE) has been proposed which can be configured as (i) fractional order negative inductance emulator (FO-PIE) (ii) fractional order positive inductor emulator (FO-PIE) (iii) integer order negative inductance emulator (IO-NIE) and (iv) integer order positive inductor emulator (IO-PIE). The proposed unified IE is designed around a single VDTA and a grounded fractional capacitor (FC) or capacitor only. Non-ideal mathematical formulations of the proposed IE are carried out to enumerate the effect of non-idealities associated with VDTA. The functionality of proposed structure is validated through Virtuoso from Cadence tool suite using 180 nm generic process design kit (gpdk) CMOS technology parameters. Power supplies of ± 0. 7 V are used for VDTA simulation. The continued fraction expansion (CFE)-based rational approximation is used for FC implementation which is realized using RC ladder network truncated to 12th order. Fractional negative and positive inductances of value 78.13 Ω / s 0. 3 , 23.44 Ω / s 0. 5 and 0.78 Ω / s 0. 7 are realized and simulated. The topology is also tested for 1 mH positive and negative integer order inductances. The simulation results corroborate with theoretical propositions. The applicability of the proposed structure is justified through two application examples namely parasitic fractional order inductance cancellation and fractional order high pass filter. [ABSTRACT FROM AUTHOR]

In 2006, Brun, Devetak and Hsieh developed the theory of entanglement-assisted quantum error-correcting codes (EAQECCs), which generalizes that of quantum stabilizer codes. The study of EAQECCs is currently a hot topic in coding theory and quantum information theory. Let q = p e be a prime power and let ℓ be an integer with 0 ≤ ℓ ≤ e - 1 . The ℓ -Galois hulls of MDS codes are an effective tool for constructing MDS EAQECCs. In this paper, we construct twenty-two families of q-ary MDS codes with ℓ -Galois hulls satisfying 2 (e - ℓ) ∣ e , which are not covered by those available in Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021) when ℓ ≠ e 2 . From these families of MDS codes with ℓ -Galois hulls, we obtain twenty-two families of q-ary MDS EAQECCs. Compared with the MDS EAQECCs from Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021), some of the MDS EAQECCs constructed in this paper have much larger dimensions. As an application, we give new families of p-ary entanglement-assisted concatenated quantum codes (EACQCs) by combining an arbitrary p-ary [ [ n ′ , e , d ′ ; c ′ ] ] p EAQECC with the q-ary MDS EAQECCs constructed in this paper. It turns out that the parameters of these EACQCs are very flexible, e.g., the possibility of the lengths of these EACQCs is far greater than the one of the lengths of these MDS EAQECCs. [ABSTRACT FROM AUTHOR]

Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes. In this paper, we study the minimality of linear codes over F q constructed from sunflowers of size s in all cases. For any sunflower, the corresponding linear code is minimal if s ≥ q + 1 , and not minimal if 2 ≤ s ≤ 3 ≤ q . In the case where 3 < s ≤ q , for some sunflowers, the corresponding linear codes are minimal, whereas for some other sunflowers, the corresponding linear codes are not minimal. [ABSTRACT FROM AUTHOR]

The possibility of encoding classical string of information into a quantum string via CNOT operation and Hadamard gate is discussed. It has been shown that maximizing/minimizing the quantum correlation of the generated quantum string depends on the values of the initial classical strings. The kernel of each string, as well as, the total string had been examined, where by applying the Hadamard gate on the classical string, one can decrease its similarity as well as the similarity of the total kernel. However, in the absence of the Hadamard gate, the large similarity of the final quantum string has been observed. The kernel of the final quantum string may be used as an indictor of the security degree of the encoded information, where the large degree of security has depicted at small kernel. The high degree of security can be implemented by applying the Hadamard gate on only one classical string. [ABSTRACT FROM AUTHOR]

In the fields of combinatorics on words and theory of codes, a two-word language { x , y } is a code if and only if x y ≠ y x . But up to now, corresponding characterizations for a three-word language, which forms a code, have not been completely found. Let X = { x , y , z } be a three-word language and | x | , | y | , | z | be their lengths. When | x | = | y | < | z | , a necessary and sufficient condition for X to be a code was obtained in 2018. If | x | < | y | = | z | ≤ 2 | x | , a necessary and sufficient condition for X to be a code is proposed in this paper. [ABSTRACT FROM AUTHOR]

Divisive normalization, a prominent descriptive model of neural activity, is employed by theories of neural coding across many different brain areas. Yet, the relationship between normalization and the statistics of neural responses beyond single neurons remains largely unexplored. Here we focus on noise correlations, a widely studied pairwise statistic, because its stimulus and state dependence plays a central role in neural coding. Existing models of covariability typically ignore normalization despite empirical evidence suggesting it affects correlation structure in neural populations. We therefore propose a pairwise stochastic divisive normalization model that accounts for the effects of normalization and other factors on covariability. We first show that normalization modulates noise correlations in qualitatively different ways depending on whether normalization is shared between neurons, and we discuss how to infer when normalization signals are shared. We then apply our model to calcium imaging data from mouse primary visual cortex (V1), and find that it accurately fits the data, often outperforming a popular alternative model of correlations. Our analysis indicates that normalization signals are often shared between V1 neurons in this dataset. Our model will enable quantifying the relation between normalization and covariability in a broad range of neural systems, which could provide new constraints on circuit mechanisms of normalization and their role in information transmission and representation. Author summary: Cortical responses are often variable across identical experimental conditions, and this variability is shared between neurons (noise correlations). These noise correlations have been extensively studied to understand how they impact neural coding and what mechanisms determine their properties. Here we show how correlations relate to divisive normalization, a mathematical operation widely adopted to describe how the activity of a neuron is modulated by other neurons via divisive gain control. We introduce the first statistical model of this relation. We extensively validate the model and investigate parameter inference in synthetic data. We find that our model, when applied to data from mouse visual cortex, outperforms a popular model of noise correlations that does not include normalization, and it reveals diverse influences of normalization on correlations. Our work demonstrates a framework to measure the relation between noise correlations and the parameters of the normalization model, which could become an indispensable tool for quantitative investigations of noise correlations in the wide range of neural systems that exhibit normalization. [ABSTRACT FROM AUTHOR]

Suppose that you wish to sample a random graph G over n vertices and m edges conditioned on the event that G does not contain a "small" t-size graph H (e.g., clique) as a subgraph. Assuming that most such graphs are H-free, the problem can be solved by a simple rejected-sampling algorithm (that tests for t-cliques) with an expected running time of nO(t). Is it possible to solve the problem in a running time that does not grow polynomially with nt? In this paper, we introduce the general problem of sampling a "random looking" graph G with a given edge density that avoids some arbitrary predefined t-size subgraph H. As our main result, we show that the problem is solvable with respect to some specially crafted k-wise independent distribution over graphs. That is, we design a sampling algorithm for k-wise independent graphs that supports efficient testing for subgraph-freeness in time f(t). nc, where f is a function of t and the constant c in the exponent is independent of t. Our solution extends to the case where both G and H are d-uniform hypergraphs. We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially unbalanced expander graphs whose failure probability is negligible in n (i.e., n (1)). In particular, given constants d > c 1, we output a bipartite graph that has n left nodes and nc right nodes with right-degree of d so that any right set of size at most n (1) expands by factor of (d). This result is extended to the setting of unique expansion as well. We observe that such a negligible-error construction can be employed in many useful settings and present applications in coding theory (batch codes and lowdensity parity-check codes), pseudorandomness (low-bias generators and randomness extractors), and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally computable cryptographic pseudorandom generators based on Goldreich's one-wayness assumption resolving a central open problem in the area of parallel-time cryptography (e.g., Applebaum, Ishai, and Kushilevitz [SIAM J. Comput., 36 (2006), pp. 845-888] and Ishai et al. [Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 433-442]). [ABSTRACT FROM AUTHOR]

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and probabilistically checkable proofs of proximity. Namely, we show that the structure of every algorithm that makes q adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with n1-1/O(q² log{l}o}{g}² q) sample complexity, following the definition of Goldreich and Ron [ACM Trans. Comput. Theory, 8 (2016), 7]. We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacypreserving local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish [SIAM J. Comput., 50 (2021), pp. 788--813]. We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev [Proceedings of the 56th Annual Symposium on Foundations of Computer Science, IEEE, 2015, pp. 1163--1182], bypassing an exponential blowup caused by previous techniques in the case of adaptive testers. We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum [Proceedings of the 8th Innovations in Theoretical Computer Science Conference, 2017, pp. 39:1--39:43; Comput. Complexity, 27 (2018), pp. 99-207] regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal--Szemeedi theorem. [ABSTRACT FROM AUTHOR]

This editorial introduces an invited article by Andreas Eder on a new perceptual control theory of action choice, based on the comparison of real and simulated interoceptive signals generated by action alternatives. Eder extends the cognitive action-control framework, postulating a bi-directional connection between outcomes and actions by introducing "emotional feelings", defined as valued interoceptive signals from the body. An invited commentary by Agnes Moors compares this theory with her own goal-directed theory of action control. While agreeing on the central role of a control cycle and the goal-directed nature of emotional actions, Moors disagrees on the content of the representations involved in the control cycle and the nature of the feelings involved. A second commentary by Bob Bramson and Karin Roelofs discusses the issues of the distinction between perception control vs. action control, the need for biologically plausible implementation alternatives, and potential implications for psychopathology and clinical intervention. Finally, the potential relevance of predictive coding theory and the role of appraisal processes in emotion generation with respect to their bearing on action comparison and choice are discussed. [ABSTRACT FROM AUTHOR]

Background: Given high rates of cancer mortality in Native communities, we examined how urban American Indian and Alaska Native elders talk about colorectal cancer (CRC) and CRC screening. Methods: We conducted seven focus groups with a total of 46 participants in two urban clinics in the Pacific Northwest to assess participant awareness, perceptions, and concerns about CRC and CRC screening. Using speech codes theory, we identified norms that govern when and how to talk about CRC in this population. Results: Our analyses revealed that male participants often avoided screening because they perceived it as emasculating, whereas women often avoided screening because of embarrassment and past trauma resulting from sexual abuse. Both men and women used humor to mitigate the threatening nature of discussions about CRC and CRC screening. Conclusions: We offer our analytic results to assist others in developing culturally appropriate interventions to promote CRC screening among American Indians and Alaska Natives. [ABSTRACT FROM AUTHOR]

Background: Research shows that interventions to protect the sensitive physiological process of birth by improving the birthing room design may positively affect perinatal outcomes. It is, however, crucial to understand the mechanisms and contextual elements that influence the outcomes of such complex interventions. Hence, we aimed to explore care providers' experiences of the implementation of a new hospital birthing room designed to be more supportive of women's birth physiology. Methods: This qualitative study reports on the implementation of the new birthing room, which was evaluated in the Room4Birth randomised controlled trial in Sweden. Individual interviews were undertaken with care providers, including assistant nurses, midwives, obstetricians, and managers (n = 21). A content analysis of interview data was conducted and mapped into the three domains of the Normalisation Process Theory coding manual: implementation context, mechanism, and outcome. Results: The implementation of the new room challenged the prevailing biomedical paradigm within the labour ward context and raised the care providers' awareness about the complex interplay between birth physiology and the environment. This awareness had the potential to encourage care providers to be more emotionally present, rather than to focus on monitoring practices. The new room also evoked a sense of insecurity due to its unfamiliar design, which acted as a barrier to integrating the room as a well-functioning part of everyday care practice. Conclusion: Our findings highlight the disparity that existed between what care providers considered valuable for women during childbirth and their own requirements from the built environment based on their professional responsibilities. This identified disparity emphasises the importance of hospital birthing rooms (i) supporting women's emotions and birth physiology and (ii) being standardised to meet care providers' requirements for a functional work environment. Trial registration: ClinicalTrials.gov: NCT03948815, 14/05/2019. [ABSTRACT FROM AUTHOR]

Identifying and modeling the factors affecting unsafe behaviors (UBs) have been conducted by various methods such as qualitative studies, game theory, and modeling using dynamic system. Therefore, the present study aimed to provide a study protocol in the field of modeling the factors affecting UBs by conducting a case study in Iranian workers. This protocol consists of four phases including identifying the factors affecting UBs using grounded theory, weighting the factors extracted by fuzzy best–worst and mapping relationships with fuzzy cognitive map, modeling effective factors with system dynamic, and examining possible scenarios to reduce UBs; all its stages are described in detail and finally the results of a case study is reported. Since the implementation of policies and adoption of the best method to reduce UBs is based on trial and error and always requires a lot of time, cost, and equipment, using modeling study can help reduce the cost of choosing an intervention method. Also, by using the protocol present in each industry, the best scenarios and models can be identified in accordance with the specified factors and specific to the same industry; we can also select and implement the most appropriate strategy to reduce UBs appropriate for preventing unsafe behavior and reduce industrial accidents. [ABSTRACT FROM AUTHOR]

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual. [ABSTRACT FROM AUTHOR]

"Considerations on the theory based on research" corresponds to an essay whose purpose is to promote the application of theory based on scientific research from the discussion of some considerations that characterize it as a methodology, with the purpose of creating theory, discover new concepts that emerge from the key informants of the study. In this sense, a document containing the introduction, development, final considerations is presented. In the section corresponding to the development, some basic definitions of the grounded theory are expressed, its origin and evolution, conceptualization of specialists in the subject, justification and scope, methodological route with its dimensions, application and process of data collection and analysis. For your better understanding, the open, axial and selective coding process is described until reaching the central category, in addition to the production that can be achieved with the application of this methodology. [ABSTRACT FROM AUTHOR]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

r-circulant matrices have applied in numerical computation, signal processing, coding theory, etc. In this study, our main goal is to investigate the r-circulant matrices of generalized Fermat numbers which are shown by GRa,b,n= a2n +b2n We obtain the eigenvalues, determinants, sum identity of matrices. Also we find upper and lower bounds for the spectral norms of generalized Fermat r-circulant matrices. Beside these, we present GR*a,b,r matrix in the form of the Hadamard product of two matrices as GR*a,b,r - A.B. In addition, we get the right and skewright circulant matrices for r =1 and r = -1. Finally, we examine their different norms (Spectral and Euclidean) and limits for matrix norms. [ABSTRACT FROM AUTHOR]

The mathematics education literature indicates a consensus regarding the importance of developing algebraic thinking in elementary school mathematics. However, the approaches used to implement this concept vary around the world. This study examines how a popular standards-based elementary mathematics textbook series in China provides opportunities to learn about functional thinking, which is a key component of algebraic thinking. Building on the literature, an analytical framework was generated to examine the features of function-related tasks in the textbook series across grades. The results of the fine-grained coding analysis show that four categories of function-related tasks in the textbook provide opportunities to learn about multi-modes of functional thinking. These tasks primarily serve to enhance arithmetic learning, while offering opportunities to learn about functional thinking as an embedded component. Elaborated design and arrangement of the function tasks promote opportunities to learn about multiple modes of functional thinking. In addition, two pathways to support the development of functional thinking are identified. Finally, the implications for task design and textbook development which attempt to develop functional thinking are discussed. [ABSTRACT FROM AUTHOR]