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The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann's geometry in the process leading to its discovery. Our principal aim consists in challenging this historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus. [ABSTRACT FROM AUTHOR]

This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students' ability to perform Actions on their graphing derivative Schema, that is, of its thematization. It also contributes to the literature on the learning of differential calculus by showing how advanced students use their knowledge to construct relations between concepts when facing complex situations. The work of three graduate students on transforming complex graphs and determining their properties and their relation to the domain structure is analyzed to determine their solution approaches. Their graphing derivative Schema is analyzed in depth in terms of the construction of relations among the Schema structures and assimilation and accommodation mechanisms involved in thematization in APOS theory. These findings are important in informing and developing didactic strategies to foster university students' understanding of derivatives, which can smoothe the transition to the study of advanced mathematics courses. [ABSTRACT FROM AUTHOR]

In the early 1980s, the theory of didactic situations put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the epistemological programme. This paper describes and interprets some of the contributions of the anthropological theory of the didactic (ATD) to the development of this programme. These contributions come from the new institutional and ecological perspective provided by the ATD to approach the didactic problem and from the significance it assigns to macrodidactic phenomena characterised by didactic transposition. They are illustrated with research works involving different fields of school mathematics in which the ATD team led by Spanish researchers played a central role: research in school algebra and its relations with functional modelling, differential calculus, and real numbers. The paper concludes by briefly describing the contributions of the ATD to the dialogue between didactic theories, another domain that has notable Spanish contributions. [ABSTRACT FROM AUTHOR]

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γ ℓ (.) ). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ − fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. [ABSTRACT FROM AUTHOR]

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM. [ABSTRACT FROM AUTHOR]

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs. [ABSTRACT FROM AUTHOR]

In the context of mathematics education, the integration of artificial intelligence (AI) in teaching calculus is revolutionizing instructional methodologies and enhancing learning experiences both inside and outside the classroom. This study explores the use of specific AI tools, including ChatGPT, MathGPT, Gemini, and Wolfram Alpha, to deepen students' understanding of key mathematical concepts such as derivatives and rates of change through continuous interaction with a virtual tutor. By employing well-designed prompts, these tools facilitated problem-solving exercises that were verified and refined by AI, fostering both precision in calculations and conceptual clarity. Observations from the classroom implementation reveal that students not only improved their accuracy in performing derivative calculations but also developed a clear understanding of the distinctions between average and instantaneous rates of change. The AI tools created a dynamic, adaptive learning environment, providing immediate feedback and simulations that significantly boosted student engagement and motivation. These findings underscore the potential of AI to transform mathematics education by making learning more personalized and accessible, ultimately enhancing educational outcomes and preparing students for future academic and professional challenges. Furthermore, this study introduces an innovative approach to refining AI prompts and interactions, highlighting the importance of iterative improvement to enhance the quality of AI feedback. This approach is crucial for developing better problem-solving skills and ensuring a comprehensive understanding of mathematical concepts. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the positivity of the real part of the logarithmic derivative of the Riemann &#163l; -function when 1/2 <σ < 1 and t is sufficiently large. We consider explicit upper and lower bounds of R/(α-σ) where the summation runs over the zeros of £ (s) on the line 1/2+it. We also examine the positivity of R £/£ (s) in the strip 1/2 < σ < 1 assuming that there occur non-trivial zeros of £ (s) off the critical line. [ABSTRACT FROM AUTHOR]

This paper studies the engineering students' understanding of the concept of derivative with a discrete approach using sequences introduced by Weigand [21]. In this approach, a step-by-step method of difference sequence for functions defined on Z and Q is proposed. This concept was taught as part of a mathematics course in an engineering college in an Iranian university. HomeWorks with questions based on derivative with a discrete approach were constructed and performed for the participants. Their written answers, which were used to explore the students' mental structures of these mentioned concepts, were analyzed using APOS (Action-Process-Object-Schema) theory and we performed interviews so that the students could explain about their written answers. The results show that students tend to adopt an algorithmic approach when solving derivative problems and students' understanding of the meaning of derivative with a discrete approach was mainly procedural. Also according to the observed mental structures, a suggested genetic decomposition for the derivative with a discrete approach is presented. [ABSTRACT FROM AUTHOR]

This study is concerned with the analysis of mathematical textbook tasks at the second-level in Ireland, in the context of the introduction of the revised curriculum entitled 'Project Maths'. A total of 7635 tasks on the topics of Pattern, Sequences and Series, and Differential Calculus were analysed; these tasks were selected from editions of three textbook series available before and after the curriculum change. The analysis was carried out using a framework specially formulated for the purposes of this study, based on an amended version of the Project Maths objectives centred on its representation/characterisation of 'synthesis and problem-solving skills'. Our findings suggest that the post-Project Maths textbook tasks offer more problem-solving opportunities compared to those in the older textbooks, but that there is still much scope for further development. Based on our analysis, it would appear that all three textbook series have neglected important objectives of the Project Maths curriculum such as providing opportunities for students to devise models/techniques and translate information from written to mathematical form. [ABSTRACT FROM AUTHOR]

This article presents an instructional technology-based cycle model intended to support and facilitate the teaching and learning of mathematics, particularly calculus. The study used quantitative methods with quasi-experimental research that uses non-randomised assignments of the study group that are categorised into experimental and control groups including 36 and 30 students in the control and experimental groups, respectively, at a university in Ethiopia. A pre-test was administered to the experimental and control groups before the intervention (statistical control over the groups) to identify students' abilities in the two groups. Based on nine steps, Vygotsky's theory of learning model was implemented in the classroom using GeoGebra software. The findings suggest that the GeoGebra classroom-oriented approach to learning differential calculus, using the cycle model, had a positive effect on students' conceptual understanding and a very positive effect on their procedural understanding. The article recommends that using the cycle model of instruction in the teaching and learning of calculus can bring important benefits to different stages of schooling in the Ethiopian context and beyond. [ABSTRACT FROM AUTHOR]

This study examines how fractional derivatives affect the theory of curves and related surfaces, an application area that is expanding daily. There has been limited research on the geometric interpretation of fractional calculus. The present study applied the Caputo fractional calculation method, which has the most suitable structure for geometric computations, to examine the effect of fractional calculus on differential geometry. The Caputo fractional derivative of a constant is zero, enabling the geometric solution and understanding of many fractional physical problems. We examined flux tubes, which are magnetic surfaces that incorporate these lines of magnetic fields as parameter curves. Examples are visualized using mathematical programs for various values of Caputo fractional analysis, employing theory‐related examples. Fractional derivatives and integrals are widely utilized in various disciplines, including mathematics, physics, engineering, biology, and fluid dynamics, as they yield more numerical results than classical solutions. Also, many problems outside the scope of classical analysis methods can be solved using the Caputo fractional calculation approach. In this context, applying the Caputo fractional analytic calculation method in differential geometry yields physically and mathematically relevant findings, particularly in the theory of curves and surfaces. [ABSTRACT FROM AUTHOR]

Calculations by aerospace and automotive engineers in the 1950s for industrial applications are usually viewed as the beginnings of the Finite Element Method (FEM). However, some of its basic ideas can already be found on earlier occasions. One of the examples already discussed in the literature is the treatment of the brachistochrone problem by Gottfried Wilhelm Leibniz (1646–1716) in 1697, even if he did not rigorously prove his ideas there. The mathematics educator Karl Schellbach (1804–1892) appears to have provided the first proof for the brachistochrone problem using FEM‐like methods in an article of 1851. However, one has to look at the details: Schellbach certainly first solves some elementary problems of the calculus of variations in this article by discretizing the situation. But then he generalizes this method in order to derive the general Euler‐Lagrange equation. And only on the basis of this differential equation and by use of standard methods from differential calculus he deals with the problem of the brachistochrone. In the final sections of his article, Schellbach generalizes his method to higher dimensions, which makes his considerations appear even more FEM‐like. For example, in his treatment of Plateau's problem he begins by dividing the domain of definition into triangles, which is even shown in an illustration. [ABSTRACT FROM AUTHOR]

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions. [ABSTRACT FROM AUTHOR]

The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno's formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno's formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000). [ABSTRACT FROM AUTHOR]

For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested. [ABSTRACT FROM AUTHOR]

A mixed boundary-value problem for a poroelastic semi-infinite strip is formulated using Biot's theory. Two cases of boundary conditions are considered depending on the permeability of the longer sides. By using the integral transforms and matrix differential calculus, the original boundary value problem was reduced to a one-dimensional problem, which was then solved analytically. The explicit formulae are derived for the effective stress, pore pressure, and displacements. These functions are analyzed depending on the permeability of the boundary, the properties of the poroelastic material, and the compression profile along the sides. [ABSTRACT FROM AUTHOR]

This work is underpinned by the anthropological theory of the didactic (ATD). The aim is to formulate, in a coordinated way, an outline of a reference epistemological model for functional modelling, REM(FM), which gives to the elementary differential calculus (EDC) a new raison d'être. After describing the general criteria to be met by the REM, a redefinition of what is meant by functional modelling is proposed by means of an activity diagram that integrates the role assigned to EDC in this mathematical activity. In coherence with this way of interpreting the role played by the EDC in mathematical activity, new educational ends associated with its study are proposed, as well as didactic means that we consider appropriate to achieve these ends. In short, we propose a new modality for the study of EDC in the transition from Secondary education to University, that is, a new reference didactical paradigm (RDP) for the study of EDC. Taking this DP as a reference, a brief analysis is made of the current mode of study of EDC, highlighting the didactic phenomena that emerge when the study of EDC in the transition from Secondary education to University is governed by this DP. [ABSTRACT FROM AUTHOR]

Our aim in this article is to present ideas linked to some objects of school mathematics that reveal transpositional connections pertinent to the development of epistemological reference models, linking them to the notions of mathematical objects of Differential and Integral Calculus. The theoretical framework is centered on Didactic Transposition and the Anthropological Theory of the Didactic. We show how some of the objects of school mathematics are the focus of the notions of the objects of Differential and Integral Calculus, as well as highlighting the transpositional connections between school mathematics and higher education mathematics. In these transpositive connections lie possibilities for the development of epistemological models of reference that make the teaching of the objects of Differential and Integral Calculus more comprehensible. [ABSTRACT FROM AUTHOR]

The objective of this article is to highlight the potential of Conway's theory compared to the classical concept of number with a view to contributing to the development of Epistemological Reference Models for teaching Differential and Integral Calculus. The search for a single answer to the epistemological question "What is a number?" has mobilized Mathematics epistemologists for centuries, considered essential for the foundation of this concept. John Horton Conway, an English mathematician from Princeton University, dedicated himself to researching this issue and resulted in a theory presented in the 1970s. In this article we bring elements about this theory highlighting its contributions to the evolution of the foundation of the concept of number. Conway's definition of number meets the complementarity of the intensional and extensional aspects of this concept, bringing advantages to Mathematics teaching. Scientific investigations and results of teaching practices in the field of teaching have encouraged questions about the importance of the role that the concept of real numbers has for learning Calculus and Real Analysis. Add to this question, and for Mathematics in general, and for the formation of analytical thinking, and for mathematical thinking? The reflections carried out in this article aim to raise epistemological and cognitive aspects about the classical construction of number, seeking to have an impact on current epistemology. [ABSTRACT FROM AUTHOR]

In this article, we resume discussions about the application of Differential and Integral Calculus in the Teaching of Chemical Kinetics in Teacher Training. Aspects aimed at demonstrating statements through the resolution of tasks in a course of studies and research as a strategy aimed at appropriating concepts in the domain of reaction kinetics, in the context of activities based specifically on the Anthropological Theory of the Didactic (ATD), based on epistemological models of reference and the dominant one around the object speed laws of chemical reaction. Along the way, the importance and reason for being of Integral and Differential Calculus techniques for teacher training and the formulation of didactic organizations that can serve as support for tasks in the exercise of the profession in high school were highlighted, highlighting the use of instruments of ICT, such as Excel, in the construction of these tasks. It was observed that throughout the course the students understood the steps proposed in the reference epistemological model by completing a crucial task, associated with the question that generated the course, which necessarily applies integration and differentiation techniques to obtain ordinary differential equations, associated with the technique analysis of the least squares method, for the kinetic characterization of a chemical system, a medicine in the simulation of the experimental determination of its expiration date. [ABSTRACT FROM AUTHOR]

This article aims to present a method of intervention in the classroom, specifically in the Differential and Integral Calculus course of a Mathematics Teaching degree, when working with students on the Intermediate Value Theorem. We base our approach on the studies by Bicudo (1991), Garnica (1992), and Garnica and Bicudo (1994) regarding a teaching methodology based on hermeneutic work with mathematical texts in the classroom. We describe the experience conducted and our understanding of the activity carried out. We present the questions posed by the students and the indications of their understanding of the ideas introduced in the studied theorem. [ABSTRACT FROM AUTHOR]

Differential and Integral Calculus (CDI) is an essential subject for teaching Mathematics and other sciences. Despite this importance, we observed student failure and high rates of failure or dropout, which justifies the relevance of considering epistemological aspects that make it possible to understand different phenomena in the teaching of this discipline. In this sense, we propose a study of epistemological elements of density and mass knowledge, through multivariational integrals, since integral is essential knowledge for the area of exact sciences. The objective of this study is to investigate the development and implementation of an intervention proposal, centered on study and research activities, that offers CDI students opportunities to explore the concept of the integral of one and more variables. To this end, we carried out an investigation through the creation and implementation of an intervention based on work with task-solving episodes, in order to analyze generalization movement(s) that students performed to define a multivariational defined integral from defined integrals of a variable. The results showed that the students mobilized a set of knowledge of multiple integrals, from the context of calculating mass in one, two and three dimensions. Expansive generalization was used to expand procedural issues in the calculation of an integral, while reconstructive generalization was used to understand structural aspects of the Riemann integral of more than one variable. [ABSTRACT FROM AUTHOR]

The objective of this article is at identifying epistemological obstacles manifested in learning Limit of real functions with a real variable and associate them with categories proposed by previous studies. Thereunto, we collect productions from the Theses and Dissertations Catalog (CTD in Portuguese acronym) by Capes, with filters for academic master's and doctorate productions published in the last ten years. We developed a theoretical framework based on researchers who categorize epistemological obstacles within the content of Limit, which were also a reference for the works selected in CTD. Categories proposed by the authors were associated with each other, and based on this joint discussion, a specific categorization was created for grouping epistemological obstacles discussed by these authors. This categorization supported the analysis of difficulties reported in five dissertations and one thesis, which described students' difficulties regarding various aspects involving Limit. From the analysis carried out, it was concluded that the most common difficulties are associated with obstacles categorized into E1, E2 and E4: Complexity of basic mathematical objects, Notion and formalization of Limit and Calculus Ruptures. [ABSTRACT FROM AUTHOR]

The normalized fractional derivatives (NFDs) are a normalization of the existing fractional derivatives (FDs) which admit geometrical meanings. They have finite ordinary derivatives at the initial point, which cause smoothness of solutions for related fractional differential equations (FDEs). For FDs with non-singular kernels, the NFDs do not vanish at the starting point, in general, and therefore related FDEs admit solutions without the need to impose extra conditions. In this paper, we consider FDEs with the normalized Caputo–Fabrizio derivative of Caputo type. We show in closed forms the solutions of related FDEs and show that the Cauchy problem with the NFD admits a nontrivial solution. We also define the higher-order NFDs and show that related FDEs can be solved by transforming them to integro-differential equations with integer orders. [ABSTRACT FROM AUTHOR]

The aim of this research article is to share a study conducted within the epistemological phase of mathematical theory in the context of the sciences. This study, developed by Patricia Camarena Gallardo, involves the development, presentation, and analysis of a didactic intervention aimed at revisiting the study of functions in an introductory course on differential and integral calculus. This approach is based on a contextualized problem in civil engineering, specifically a situation related to the dynamic analysis of a frame. The development of the intervention, with an expected duration of 12 hours per class, was methodologically supported by the analysis of a textbook on structural mechanics, as well as two calculus textbooks. Additionally, the study considered the curriculaof calculus courses, topics related to the historical development of the notion of function, the epistemological obstacles encountered in this process, as well as cognitive issues related to it. [ABSTRACT FROM AUTHOR]

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

Mustafa Salim [Tunakan] was a significant mathematician who flourished during the final years of the Ottoman Empire. He imparted knowledge of algebra, integration and machine theory at the universities of this period. In addition to his textbooks in these fields, he published one article in Osmanlı Mühendis ve Mimar Cemiyeti Mecmuası, one article in Mühendis Mektebi Mecmuası, two articles in Genç Mühendis and seven articles in Darülfünun Fünun Fakültesi Mecmuası. The aim of this study is to make a mathematical analysis of all of Mustafa Salim's articles published in the field of mathematics in the journals named Osmanlı Mühendis ve Mimar Cemiyeti Mecmuası, Mühendis Mektebi Mecmuası and Genç Mühendis. For this purpose, Mustafa Salim's four articles published in these journals were initially translated into Latin letters, after which mathematical analyses were conducted on the topics covered in the articles. In these articles; Mustafa Salim defined and solved problems at undergraduate levels in the subjects of integral, differential calculus, linear algebra, functions and geometric location. Mustafa Salim, who adopted a very simple and concise narrative style, did not make any mathematical mistakes in his articles, except for a few minor errors. [ABSTRACT FROM AUTHOR]

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This paper introduces and investigates a class of noncommutative spacetimes that I will call "T-Minkowski," whose quantum Poincaré group of isometries exhibits unique and physically motivated characteristics. Notably, the coordinates on the Lorentz subgroup remain commutative, while the deformation is confined to the translations (hence the T in the name), which act like an integrable set of vector fields on the Lorentz group. This is similar to Majid's bicrossproduct construction, although my approach allows the description of spacetimes with commutators that include a constant matrix as well as terms that are linear in the coordinates (the resulting structure is that of a centrally extended Lie algebra). Moreover, I require that one can define a covariant braided tensor product representation of the quantum Poincaré group, describing the algebra of N-points. This also implies that a 4D bicovariant differential calculus exists on the noncommutative spacetime. The resulting models can all be described in terms of a numerical triangular R-matrix through RTT relations (as well as RXX, RXY, and RXdX relations for the homogeneous spacetime, the braiding, and the differential calculus). The R-matrices I find are in one-to-one correspondence with the triangular r-matrices on the Poincaré group without quadratic terms in the Lorentz generators. These have been classified, up to automorphisms, by Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a series, focusing on the identification of all the quantum Poincaré groups that are allowed by my assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi, and braiding constructions. [ABSTRACT FROM AUTHOR]

This comprehensive investigation explores the application and significance of q-differential calculus in the realm of vector functions of several variables, addressing critical aspects such as q-Rolle’s theorem, the q-Mean-value theorem, and q-chain rule for vector functions. Additionally, we investigate the q-gradient, q-Jacobian, and q-Hessian operators, elucidating their roles in quantifying rates of change, determining directional derivatives, and characterizing critical points of multivariate functions. Furthermore, this research provides a rigorous treatment of Multivariate and Bivariate Taylor theorems in the context of q-differential calculus, presenting analytical expansions of functions around specific points and showcasing their utility in approximating functions in higher dimensions. The q-Maximum and Minimum are demonstrated and discussed as well. [ABSTRACT FROM AUTHOR]

In this article, we focus on the application of fractional differential calculus in image processing, specifically in the context of image decoding techniques. The article presents different methods of image decoding based on Riemann–Liouville fractional integration and compares their performance with established methods in the denoising of noisy images. Here we create new fractional masks by using the Middle Point approach method, the Grumwald–Letnikov scheme, the Toufik–Riemann scheme, and the Euler method with the help of the Riemann–Liouville fractional integral operator. The results demonstrate that the suggested algorithms outperform the compared techniques. The paper also suggests that the concepts presented can be applied to other definitions of fractional calculus. The future research goal is to determine the optimal fractional order for a given set of algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we construct a bi-covariant (h,h′)-deformed differential calculus on the Hopf superalgebra of functions on the quantum supergroup GL h.h ′ (1|1) and obtain an extended differential calculus (Cartan calculus) by including inner derivations and Lie derivatives. In doing so, we use left-Cartan-Maurer 1-forms and left-vector fields. [ABSTRACT FROM AUTHOR]

We propose a sparse algebra for samplet compressed kernel matrices to enable efficient scattered data analysis. We show that the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. The compression can be performed in cost and memory that scale essentially linearly with the number of data points for kernels of finite differentiability. The same holds true for the addition and multiplication of S-formatted matrices. We prove that the inverse of a kernel matrix, given that it exists, is compressible in the S-format as well. The use of selected inversion allows to directly compute the entries in the corresponding sparsity pattern. Moreover, S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as A α or exp (A) of a matrix A . The matrix algebra is justified mathematically by pseudo differential calculus. As an application, we consider Gaussian process learning algorithms for implicit surfaces. Numerical results are presented to illustrate and quantify our findings. [ABSTRACT FROM AUTHOR]

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function's derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow's rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. [ABSTRACT FROM AUTHOR]

The late Professor Heinz Neudecker (1933–2017) made significant contributions to the development of matrix differential calculus and its applications to econometrics, psychometrics, statistics, and other areas. In this paper, we present an insightful overview of matrix-oriented findings and their consequential implications in statistics, drawn from a careful selection of works either authored by Professor Neudecker himself or closely aligned with his scientific pursuits. The topics covered include matrix derivatives, vectorisation operators, special matrices, matrix products, inequalities, generalised inverses, moments and asymptotics, and efficiency comparisons within the realm of multivariate linear modelling. Based on the contributions of Professor Neudecker, several results related to matrix derivatives, statistical moments and the multivariate linear model, which can literally be considered to be his top three areas of research enthusiasm, are particularly included. [ABSTRACT FROM AUTHOR]

The fact that the graph of the exponential function exp is always at or above the straight line through the origin with slope exp⁡(1) is well-known and can be easily proved using differential calculus. We provide a simple algebraic proof of that fact and use that fact to construct a template for maximizing the likelihood function that avoids the log-likelihood function and differentiation in a number of examples, including the two-parameter normal distribution family.

A novel version of the Sumudu transform is introduced by using the notion of •-integral, which is called non-Newtonian Sumudu transform. The fundamental characteristics of the non-Newtonian Sumudu transform are demonstrated. The obtained results are applied to the solving of non-Newtonian differential equations and growth models. [ABSTRACT FROM AUTHOR]

Motivated by the conspicuous use of momentum-based algorithms in deep learning, we study a nonsmooth nonconvex stochastic heavy ball method and show its convergence. Our approach builds upon semialgebraic (definable) assumptions commonly met in practical situations and combines a nonsmooth calculus with a differential inclusion method. Additionally, we provide general conditions for the sample distribution to ensure the convergence of the objective function. Our results are general enough to justify the use of subgradient sampling in modern implementations that heuristically apply rules of differential calculus on nonsmooth functions, such as backpropagation or implicit differentiation. As for the stochastic subgradient method, our analysis highlights that subgradient sampling can make the stochastic heavy ball method converge to artificial critical points. Thanks to the semialgebraic setting, we address this concern showing that these artifacts are almost surely avoided when initializations are randomized, leading the method to converge to Clarke critical points. [ABSTRACT FROM AUTHOR]

For vanishing fidelity between initial and final states, two important quantum speed limits, the Mandelstam–Tamm limit (involving energy dispersion) and Margolus–Levitin one (involving excitation energy expectation value), have been derived. While the generalization of the former limit to the case of arbitrary fidelity is straightforward, the relevant generalization of the latter given in the seminal paper by Giovanetti et al. (Phys Rev A67:052109, 2003) was based on the conjectured equality of lower and upper bounds on the right-hand side of generalized Margolus–Levitin inequality, verified numerically up to seven digits. Only recently there appear two proofs of the conjecture. We provide below a very elementary new proof, based on the simplest tools from differential calculus. Thus, the generalized Margolus–Levitin speed limit can be derived much in the spirit of the original one valid for vanishing fidelity. [ABSTRACT FROM AUTHOR]

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered "anomalous". It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ , to create a non-minimal interaction theory with the coupling, ξ R ϕ 2 , between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism. [ABSTRACT FROM AUTHOR]

In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli–Arzelà and Kolmogorov–Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions. [ABSTRACT FROM AUTHOR]

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Copyright of Revista Foco (Interdisciplinary Studies Journal) is the property of Revista Foco and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we first created a fractal Date–Jimbo–Kashiwara–Miwa (FDJKM) long ripple wave model in a non-smooth boundary or microgravity space recorded. Using fractal semi-inverse skill (FSIS) and fractal traveling wave transformation (FTWT), the fractal variational principle (FVP) was derived, and the strong minimum necessary circumstance was attested with the He Wierstrass function. We have discovered two distinct solitary wave solutions, the square form of the hyperbolic secant function and the hyperbolic secant function form. Then, soliton solutions are cultivated through FVP and the minimum steady state condition. Finally, the influences of non-smooth boundaries on solitons were tackled, and the properties of the solution were demonstrated through three-dimensional contour lines. Fractal dimension can impact waveforms, but cannot affect their vertex values. The presentation of soliton solutions (SWS) using techniques is not only laudable but also noteworthy. The technique employed can also be used to investigate solitary wave solutions of other local fractional calculus partial differential equations. [ABSTRACT FROM AUTHOR]