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The innovation for entrepreneurial systems and the advocacy to enact policies that institutionalise it had recently flooded the literature. Every system has fundamental principles responsible for their state, progress and change; such is the impact of the Igbo apprenticeship scheme on the entrepreneurial system. This has recently resulted in many scholars calling for government involvement through enacting policies that will make the scheme all-inclusive for sustainable economic growth in developing economies. Amongst the huge body of literature in its advocacy, none has addressed the question of the underlying principles for reproducing the relations in terms of the mathematical phenomena that the system embodies. This work shows that the success of the Ndịgbo apprenticeship hinges on certain defined underlying metaphorical mathematical functional patterns responsible for the strategic informal management behavioural model. The symbolic characteristics derived from the informal discipleship learning mode are activated for commerce, by young males and their hands-on in trading, with persistent cooperation, integrity, communication and collective purposeful attempts to succeed. The evidence of the model described is extracted from the explorative and exploitative trajectories that emerged from their goal-pursuing dedications to satisfy customer needs, with optimal methods for managing resources and building economies of scale, emerge. This is known as Ịgba-ọsọ-ahịa. This work reveals that Ịgba-ọsọ-ahịa embodies attributes of models that have shaped world-class productive systems such as (a) the geometrical Mandelbrot model, (b) Markov Chain variant with a backoff process, (c) stochastic nonlinear models, (d) network spatial model and (e) system computational model. Finally, a simplistic novel stochastic model of the socio-economic processes is developed to solve complex engineering decisions and hyperparameter optimisation problems in this work. [ABSTRACT FROM AUTHOR]

In a previous research effort by this group, pseudo engine dynamometer data in multi-dimensional arrays were combined with dynamic equations to form a crank angle resolved engine model compatible with a real-time simulator. The combination of the real-time simulator and external targets enabled the development of a software-in-the-loop (SIL) environment that enabled near-real-time development of AI/ML and real-time deployment of the resulting AI/ML. Military applications, in particular, are unlikely to possess large quantities of non-sparse operational data that span the full operational range of the system, stove-piping the ability to develop and deploy AI/ML which is sufficient for near-real-time or real-time control. AI/ML has been shown to be well suited for predicting highly non-linear mathematical phenomena and thus military systems could potentially benefit from the development and deployment of AI/ML acting as soft sensors. Given the non-sparse nature of the data, it becomes exceedingly important that AI/ML be developed and deployed in near-real-time or real-time in parallel to a real-time system to overcome the inadequacy of applicable data. This research effort used parallel processing to reduce the training duration of the shallow artificial neural networks (SANN) and forest algorithms forming ensemble models. This research is novel in that the SIL environment enables pre-developed AI/ML to be adapted in near-real-time or develop AI/ML in response to changes within the operation of the applied system, different load torques, engine speeds, and atmospheric conditions to name a few. Over time it is expected that the continued adaptation of the algorithms will lead to the development of AI/ML that is suitable for real-time control and energy management. [ABSTRACT FROM AUTHOR]

This paper explores the integration of STEM activities in teaching and learning, emphasizing the importance of innovative pedagogical approaches in effectively introducing theoretical concepts, such as variables and functions, and merging them with practical applications. Drawing on existing literature, this study investigates the integration of STEM activities with real-world applications to enhance mathematics learning, highlighting intrinsic motivation, selfefficacy beliefs, and goal orientation as key factors in fostering student engagement. This case study explores the integration of a STEM activity to introduce students to variables and functions through a pendulum experiment. The aim is to demonstrate the impact of this approach on students' understanding of abstract mathematical concepts, as well as their problem-solving skills. By combining cognitive and social constructivism with technological modes (virtual labs), the study showcases the transformative potential of innovative techniques in STEM education. The outcomes of the study highlight, to some extent, the positive effects of STEM activities on students' engagement, motivation, understanding of theoretical concepts, and problem-solving skills. The focus on hands-on activities supports practical learning experiences and fosters critical thinking. Additionally, virtual labs enrich students' exploration of complex mathematical phenomena, enhancing their ability to apply prior knowledge to new contexts and transcend the boundaries of traditional lab settings. Overall, the findings underscore the transformative potential of innovative pedagogical approaches and technological modes in creating engaging learning environments within STEM disciplines. [ABSTRACT FROM AUTHOR]

This paper describes how plan-do-study-act cycles engaged a classroom mentor teacher and student teacher in a professional collaboration that resulted in two inquiry activities for high-school geometry classes. The PDSA cycles were carried out in four high school geometry classes, each with 30 to 35 students, in a mid-Atlantic urban school district in the U.S. The four geometry classes were co-taught by the second and third authors of this paper. The data consisted of classroom documents (e.g., activity prompts, tasks), classroom observations, student feedback about activities, and monthly PDSA reports. The PDSA cycles had a direct effect on the professional learning of the teachers. The resultant classroom activities used a data collection approach to engaging students in inquiry to learn about trigonometry functions and density. Student learning behaviors were noticeably improved during these activities compared with traditional mathematics instruction. We concluded that the data collection sequence provided an accessible entry point for students to begin scientific inquiry in mathematics. The process opened the conceptual space for students to develop curiosity about mathematical phenomena and to explore their own research questions. The use of culturally relevant topics was especially compelling to students, and the open-ended nature of these exploratory activities allowed students to see mathematics through their own cultural lenses. [ABSTRACT FROM AUTHOR]

This paper aims to propose a Novel Condition-based maintenance (CBM) decision aid model for optimising the maintenance of complex multi-component systems. As the degradation level of each component is assumed to be independent and stochastic, it follows a specific probability distribution determined from historical data of experimental observations and inspection. The main objective is to optimise the total cost for providing maintenance actions and reducing the excess of spare parts usage. The decision support model consists of determining measurements on components with the aim of estimating the instant of time of removing predictively one or a group of components before they fail. The measurement model includes the mean residual lifetime (MRL) and some extensions developed for this purpose. For demonstrating the pertinency of the proposed model, we use a preventive maintenance strategy for one-component systems and a grouping/opportunistic maintenance for multi-component systems. Besides, a numerical comparative study performing these measurements is carried out using several examples and a case study from Electric energy distribution systems. The solution is illustrated as a decision-making optimal model for optimising the maintenance operations' costs and the total number of spare parts. The numerical results and the comparison show the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles. • We construct and solve new integrable systems affected by white Gaussian noise. • A novel integrable system along with its Lax pair, influenced by Brownian motion and white Gaussian noise, is proposed. • We utilize classical and generalized Darboux transformations to solve the stochastic NLS equation. • We demonstrate that the Darboux transformation is a powerful tool, enabling the generation of new solutions. [ABSTRACT FROM AUTHOR]

The remarkable performance of overparameterized deep neural networks (DNNs) must arise from an interplay between network architecture, training algorithms, and structure in the data. To disentangle these three components for supervised learning, we apply a Bayesian picture based on the functions expressed by a DNN. The prior over functions is determined by the network architecture, which we vary by exploiting a transition between ordered and chaotic regimes. For Boolean function classification, we approximate the likelihood using the error spectrum of functions on data. Combining this with the prior yields an accurate prediction for the posterior, measured for DNNs trained with stochastic gradient descent. This analysis shows that structured data, together with a specific Occam's razor-like inductive bias towards (Kolmogorov) simple functions that exactly counteracts the exponential growth of the number of functions with complexity, is a key to the success of DNNs. Deep neural networks predict well despite having many more parameters than data points. The authors show that deep neural networks prefer simpler solutions due to an inbuilt "Occam's razor". Favouring simplicity avoids overfitting and captures patterns in data, explaining their success. [ABSTRACT FROM AUTHOR]

This research paper examines the data dependence of fixed point sets for pseudo-contractive multifunctions in partial metric spaces using the notion of C -class functions. By building upon previous findings from the literature, this work sheds more light on some new perspectives as well as generalizations on this issue. To illustrate how the C -class function can be applied to study the data dependence of fixed point sets for a certain pseudo-contractive multifunction, an illustrative example is given. [ABSTRACT FROM AUTHOR]

In the development of hybrid special polynomials, it is essential to incorporate the monomiality principle, operational rules, and other related properties. This research study adopts this approach and introduces a novel polynomial type called the two-iterated degenerate Appell polynomials, utilizing the monomiality principle. This study yields new findings that align with previous research efforts. The study presents explicit formulas and key properties of these polynomials, while also establishing connections with other polynomial types such as the Bernoulli, Euler, and Genocchi polynomials. These connections facilitate the derivation of additional results. By employing the monomiality principle and considering the aforementioned polynomials as initial members of the Appell family, this study contributes to the expansion of knowledge in the field of special polynomials. [ABSTRACT FROM AUTHOR]

In nonlinear Schrödinger equations (NLSEs), the third-order generalized NLSE is a significant sculpture which is utilized for modeling ultrashort pulses in fiber optics. In this study, we obtained wave and soliton solutions by using new extended and modified Rational expansion method to get several types of soliton such as bright solitons, dark solitons, perodic solitons and traveling waves. In three-dimensional and two-dimensional plots, we present graphical representations in dissimilar structures of some solutions to understand the phenomena physically. The development and achievements of computing show the power and effectiveness of current technology. In addition, we are able to resolve various other high-order NLSEs with the assistance of effortless and effectual technique. [ABSTRACT FROM AUTHOR]

The aim of this article is to present an overview of the current state of the mathematical and scientific competences in Primary Education in Spain and how their development is assessed in STEAM experiences (with students of 6–12 years of age). To this end, a systematic review based on the PRISMA 2020 protocol was carried out in the Scopus, Web of Sciences, and Dialnet databases, analyzing all articles published from 1993 to 2022 relating to the subject. In total, 53 articles were selected, most published in the last five years and primarily in the Dialnet database. It was found that the majority of the targeted objectives in the experiences were related to STEAM competence, but only 14 evaluated aspects related to mathematical or scientific competence. The mathematical competence was evaluated by the measurement of geometry or number content and their corresponding operations and by how information data (statistics) are graphically represented. In a few cases, the evaluation is also related to computational thinking and orientation skills. In the case of scientific competence, its evaluation was mainly based on the assessment of the understanding of concepts, with those concepts being related to Physics in 80% of the cases. The only evaluated aspect of scientific competence other than concept understanding was hypothesis formulation, which was featured in two articles. These results highlight the need to systematically incorporate assessments that provide a clearer view of mathematical or scientific competence development in implementing STEAM experiences in primary education. [ABSTRACT FROM AUTHOR]

This paper explores the operational principles and monomiality principles that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as A n [ r ] H (l 1 , l 2 , l 3 , ... , l r ; ϑ) . These DMHAP forms essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further. [ABSTRACT FROM AUTHOR]

Mathematics involves thinking and reasoning, both of which should be emphasized to improve students' knowledge and abilities. The main aim of this study was to examine undergraduate students' conceptual and procedural mathematical knowledge after exposure to software-embedded inductive reasoning strategy in cloud-based environments. Purposive sampling techniques were used to select participants--60 undergraduate students enrolled in the Ordinary Differential Equations (ODEs) course. A quasi-experimental approach using a post-test control group was adopted for the study. Research instruments included a test of mathematical knowledge and an inductive reasoning strategy with cloud-based symbolic mathematics software. The data were analyzed using a quantitative approach involving mean, standard deviation, and a t-test. The results revealed that the mathematical knowledge of the experimental group statistically exceeded the 60% criteria at the 0.05 significance level. The mathematical knowledge of students in the experimental group was significantly better than that of the control group at a statistically significant level of 0.05. The results of this study imply that incorporating cloud tools as part of the inductive reasoning strategy had a positive impact on students' mathematical knowledge. [ABSTRACT FROM AUTHOR]

Spanish educational curriculum adopts a mathematical process-based approach, which encompasses problem solving, reasoning and proof, communication, connections and representation. A fundamental role in the integration of these processes in mathematics teaching is played by teachers' professional practice of designing tasks. According to this, our aim is to analyze the ways in which pre-service early childhood teachers understand the mathematical processes in the professional practice of designing early algebra tasks and to identify how they intend to promote these processes through the tasks. Content analysis techniques were used to examine the designed tasks. To illustrate the data analysis and results, six tasks are presented. As a result, pre-service early childhood teachers associate problem solving with challenges and questions. They understand problems as unfamiliar situations but ignore the relationships between students and tasks. Moreover, they do not encourage exploration of phases of problem solving and tend to use strategies more suitable for routine tasks. Communication is identified in all the tasks designed, encouraging interaction and discussion. However, only one task explicitly promotes mathematical language. For reasoning and proof, pre-service teachers begin to use questions to elicit explanations and justifications, but do not encourage verification strategies and various modes of reasoning. The process of connections is only present in one task, reflecting the fragmented nature of mathematics teaching. We conclude that the professional practice of designing mathematical tasks is a powerful in teacher education. However, training programs should place greater emphasis on the meaningful use of mathematical processes. [ABSTRACT FROM AUTHOR]

This study investigates the Atangana-Baleanu time-stochastic fractional neutral integro-differential equation, a complex mathematical model with broad applications in various scientific disciplines. Utilizing Banach's fixed point theory, we rigorously establish the existence and uniqueness of the mild solution to this equation. Our analysis centrally revolves around investigating the Mittag-Leffler non-singular and non-local kernel, emphasizing its crucial significance in elucidating the behavior of the equation. By integrating concepts from fractional differential equations and stochastic differential systems, we contribute to a deeper comprehension of these mathematical phenomena. Our findings not only contribute significantly to advancing theoretical understanding but also establish a solid groundwork for practical applications across various fields.

This paper explores the integration of STEM activities in teaching and learning, emphasizing the importance of innovative pedagogical approaches in effectively introducing theoretical concepts, such as variables and functions, and merging them with practical applications. Drawing on existing literature, this study investigates the integration of STEM activities with real-world applications to enhance mathematics learning, highlighting intrinsic motivation, self-efficacy beliefs, and goal orientation as key factors in fostering student engagement. This case study explores the integration of a STEM activity to introduce students to variables and functions through a pendulum experiment. The aim is to demonstrate the impact of this approach on students' understanding of abstract mathematical concepts, as well as their problem-solving skills. By combining cognitive and social constructivism with technological modes (virtual labs), the study showcases the transformative potential of innovative techniques in STEM education. The outcomes of the study highlight, to some extent, the positive effects of STEM activities on students' engagement, motivation, understanding of theoretical concepts, and problem-solving skills. The focus on hands-on activities supports practical learning experiences and fosters critical thinking. Additionally, virtual labs enrich students' exploration of complex mathematical phenomena, enhancing their ability to apply prior knowledge to new contexts and transcend the boundaries of traditional lab settings. Overall, the findings underscore the transformative potential of innovative pedagogical approaches and technological modes in creating engaging learning environments within STEM disciplines.

Given a linear differential equation with coefficients in \mathbb {Q}(x), an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck–Katz p-curvature conjecture. [ABSTRACT FROM AUTHOR]

This paper shares a flexible activity for engaging people in mathematical inquiry that does not depend on fixed prior knowledge built on a famous property of Möbius strips. The discussion includes the implementation of the activity and its extensions, facilitator moves and common participant thinking, suggestions for adaptation and integration into courses, reflection on ways to leverage the activity for discussion with students at the start of a course about the nature of mathematical inquiry and norms for learning through inquiry. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and λ -Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and thoroughly examine new generalized  ψ -conformable fractional integral and derivative operators associated with the auxiliary function  ψ (t) . We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit  ψ -conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications. [ABSTRACT FROM AUTHOR]

This manuscript examines fuzzy fixed point results using the concepts of S -metric space. We introduce two contractive maps, γ - and γ -weak contractions, within the context of S -metric spaces. These contractive maps form the cornerstone of our research, offering a novel approach to solving mathematical problems. We explore fixed point results derived from the application of these maps, showcasing their utility in finding solutions in diverse mathematical scenarios. Furthermore, we provide concrete examples that illustrate the practical relevance and versatility of our theorems, emphasizing their potential applications across a wide range of scientific and engineering domains. This manuscript presents the novel concepts of γ - and γ -weak contractions and establishes their importance in mathematical research. By demonstrating their effectiveness in solving real-world problems and offering illustrative examples, our work contributes valuable tools and insights to the broader scientific community, enhancing our understanding of contractive maps and their applications. [ABSTRACT FROM AUTHOR]

The Hermite-Hadamard inequality is one of the most interesting inequalities that give lower and upper bounds of the mean value of a convex function in a way that refines the convex characteristic of the function. This paper presents a new reversed version of this outstanding result, with applications toward means of positive numbers, operator inequalities, and the Riemann-Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

We present a novel framework for introducing generalized 3-variable 1-parameter Hermitebased Appell polynomials. These polynomials are characterized by generating function, series definition, and determinant definition, elucidating their fundamental properties. Moreover, utilizing a factorization method, we established recurrence relations, shift operators, and various differential equations, including differential, integrodifferential, and partial differential equations. Special attention is given to exploring the specific cases of 3-variable 1-parameter generalized Hermite-based Bernoulli, Euler, and Genocchi polynomials, offering insights into their unique features and applications. [ABSTRACT FROM AUTHOR]

A good theory of mathematical beauty is more practical than any current observation, as new predictions about physical reality can be self-consistently verified. This belief applies to the current status of understanding deep neural networks including large language models and even the biological intelligence. Toy models provide a metaphor of physical reality, allowing mathematically formulating the reality (i.e., the so-called theory), which can be updated as more conjectures are justified or refuted. One does not need to present all details in a model, but rather, more abstract models are constructed, as complex systems such as the brains or deep networks have many sloppy dimensions but much less sti dimensions that strongly impact macroscopic observables. This type of bottom-up mechanistic modeling is still promising in the modern era of understanding the natural or artificial intelligence. Here, we shed light on eight challenges in developing theory of intelligence following this theoretical paradigm. Theses challenges are representation learning, generalization, adversarial robustness, continual learning, causal learning, internal model of the brain, next-token prediction, and the mechanics of subjective experience. [ABSTRACT FROM AUTHOR]

This paper establishes new results related to geometric function theory by presenting a new subclass of harmonic functions with complex values within the open unit disk, characterized by a second-order differential inequality. The investigation explores the bounds on the coefficients and estimates of the function growth. This paper also demonstrates that this subclass remains stable under the convolution operation applied to its members. In addition, in the last section, images of the unit disk under some functions of this class are given. [ABSTRACT FROM AUTHOR]

We propose and test a novel conjectural relation satisfied by the superconformal index of maximally supersymmetric U(N) gauge theory in four dimensions. Analogous relations appear to be also valid for the superconformal indices of a large collection of other gauge theories, as well as for a broad class of index-like generating functions. The relation expresses the finite N index as a systematic series of corrections to a large N answer. Individual corrections have an holographic interpretation as the analytic continuation of contributions from “giant graviton” branes fixed by a specific symmetry generator. [ABSTRACT FROM AUTHOR]

This article demonstrates that every sufficiently large odd integer can be expressed as the sum of one square of a prime, six cubes of primes, and 23 powers of two. This finding represents an improvement on the previous results of Sinnadurai in 1965 and Hooley in 1981. [ABSTRACT FROM AUTHOR]

In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann's landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. 'fusion') of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of 'bounded localized vertex operators,' which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models. [ABSTRACT FROM AUTHOR]

While learning to rank (LTR) is widely employed in web searches to prioritize pertinent webpages from the retrieved contents based on input queries, traditional LTR models stumble over two principal stumbling blocks leading to subpar performance: (1) the lack of well-annotated query-webpage pairs with ranking scores to cover search queries of various popularity, debilitating their coverage of search queries across the popularity spectrum, and (2) ill-trained models that are incapable of inducing generalized representations for LTR, culminating in overfitting. To tackle above challenges, we proposed a G ̲ e n e r a t i v e S ̲ e m i - S ̲ u p e r v i s e d P ̲ r e -trained (GS 2 P) Learning to Rank model. Specifically, GS 2 P first generates pseudo-labels for the unlabeled samples using tree-based LTR models after a series of co-training procedures, then learns the representations of query-webpage pairs with self-attentive transformers via both discriminative (LTR) and generative (denoising autoencoding for reconstruction) losses. Finally, GS 2 P boosts the performance of LTR through incorporating Random Fourier Features to over-parameterize the models into "interpolating regime", so as to enjoy the further descent of generalization errors with learned representations. We conduct extensive offline experiments on a publicly available dataset and a real-world dataset collected from a large-scale search engine. The results show that GS 2 P can achieve the best performance on both datasets, compared to baselines. We also deploy GS 2 P at a large-scale web search engine with realistic traffic, where we can still observe significant improvement in real-world applications. GS 2 P performs consistently in both online and offline experiments. [ABSTRACT FROM AUTHOR]

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with T-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixedtime terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence. [ABSTRACT FROM AUTHOR]

Information and Communication Technologies (ICTs) can provide rich learning opportunities in classroom teaching, which requires teachers' effective instrumental orchestrations. However, there is a lack of research regarding the development of teachers' instrumental orchestrations. In this study, we designed and implemented a professional development course, aiming to explore whether the course could change preservice mathematics teachers' instrumental orchestrations. To this end, an analytical framework of instrumental orchestrations within a GeoGebra environment was established. Data were collected from Chinese preservice mathematics teachers, including the videotapes of their simulated lessons, GeoGebra files, written documents, self-reflections, online speaking, etc. In the results, three participants' changes in instrumental orchestrations were presented as main cases. It was found that the three preservice mathematics teachers' types of instrumental orchestrations did not change, but the frequencies of those increased significantly. The changes in didactical configurations were mainly reflected in the layout of GeoGebra, and the changes in exploitation modes were mainly manifested in the techniques teachers use and the management of mathematics tasks. In addition, it was revealed that theoretical support (instrumental approach), technical support (knowledge about using GeoGebra), and community support ("Same Content Different Designs" activity and communicating and discussing with others) were useful course elements inviting changes in instrumental orchestrations. In this regard, this study contributes to providing a promising professional development course that can be used in teacher training so as to improve teachers' instrumental orchestrations. [ABSTRACT FROM AUTHOR]

The ability of prospective teachers to design numeracy problems has a crucial role in developing students' mathematical literacy. This study aims to analyze the quality of problems designed by 34 prospective mathematics teachers collaboratively in the context of numeracy. The data was analyzed supervising the solvability aspects, the context used, and the cognitive process level of the problems designed. The findings showed that most of the problems designed by prospective teachers were solvable, with the majority of cognitive processes reaching the applying level and related to personal contexts. The implications of these findings can provide insight into the potential for developing more effective numeracy problem design skills in mathematics teacher education. Suggestions for future research include further research into teaching methods that can improve prospective teachers' numeracy problem design skills, as well as research that explores the direct impact of problem quality on students' mathematical literacy. [ABSTRACT FROM AUTHOR]

The author uses extreme value theory to study time-varying idiosyncratic tail risk for a large panel of US stocks. The author demonstrates a significant performance gain by using forward-looking information extracted from implied volatilities and nonlinear models, compared to linear models that use only backward-looking information. Extreme value theory plays a key role in predicting the distribution of return realizations conditional on the occurrence of a tail event. The author finds that, surprisingly, out-the-money calls (respectively, puts) contain important information about lower (respectively, upper) tails. Furthermore, the author finds evidence that the asymmetric nature of the negative tail distribution in comparison to the positive tail is captured by nonlinear models only. [ABSTRACT FROM AUTHOR]