Mathematical Notation Research Articles

Demystifying the effect of receptive field size in U-Net models for medical image segmentation.

Medical image segmentation is a critical task in healthcare applications, and U-Nets have demonstrated promising results in this domain. We delve into the understudied aspect of receptive field (RF) size and its impact on the U-Net and attention U-Net architectures used for medical imaging segmentation. We explore several critical elements including the relationship among RF size, characteristics of the region of interest, and model performance, as well as the balance between RF size and computational costs for U-Net and attention U-Net methods for different datasets. We also propose a mathematical notation for representing the theoretical receptive field (TRF) of a given layer in a network and propose two new metrics, namely, the effective receptive field (ERF) rate and the object rate, to quantify the fraction of significantly contributing pixels within the ERF against the TRF area and assessing the relative size of the segmentation object compared with the TRF size, respectively. The results demonstrate that there exists an optimal TRF size that successfully strikes a balance between capturing a wider global context and maintaining computational efficiency, thereby optimizing model performance. Interestingly, a distinct correlation is observed between the data complexity and the required TRF size; segmentation based solely on contrast achieved peak performance even with smaller TRF sizes, whereas more complex segmentation tasks necessitated larger TRFs. Attention U-Net models consistently outperformed their U-Net counterparts, highlighting the value of attention mechanisms regardless of TRF size. These insights present an invaluable resource for developing more efficient U-Net-based architectures for medical imaging and pave the way for future exploration of other segmentation architectures. A tool is also developed, which calculates the TRF for a U-Net (and attention U-Net) model and also suggests an appropriate TRF size for a given model and dataset.

Ncorpi[formula omitted]N : A [formula omitted] software for N-body integration in collisional and fragmenting systems

NcorpiON is a general purpose N-body software initially developed for the time-efficient integration of collisional and fragmenting systems of planetesimals or moonlets orbiting a central mass. It features a fragmentation model, based on crater scaling and ejecta models, able to realistically simulate a violent impact.The user of NcorpiON can choose between four different built-in modules to compute self-gravity and detect collisions. One of these makes use of a mesh-based algorithm to treat mutual interactions in O(N) time. Another module, much more efficient than the standard Barnes–Hut tree code, is a O(N) tree-based algorithm called FalcON. It relies on fast multipole expansion for gravity computation and we adapted it to collision detection as well. Computational time is reduced by building the tree structure using a three-dimensional Hilbert curve. For the same precision in mutual gravity computation, NcorpiON is found to be up to 25 times faster than the famous software REBOUND.NcorpiON is written entirely in the C language and only needs a C compiler to run. A python add-on, that requires only basic python libraries, produces animations of the simulations from the output files. NcorpiON can communicate with REBOUND’s webGL viewer via MPI for 3D visualization. The name NcorpiON, reminding of a scorpion, comes from the French N-corps, meaning N-body, and from the mathematical notation O(N), due to the running time of the software being almost linear in the total number N of bodies. NcorpiON detects collisions and computes mutual gravity faster than REBOUND, and unlike other N-body integrators, it can resolve a collision by fragmentation. The fast multipole expansions are implemented up to order eight to allow for a high precision in mutual gravity computation.

Geometric diagrams as an effective notation

AbstractIn what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's ‘characteristic visual shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry.

Mathematical communication of preservice mathematics teachers in solving gender-biased higher-order thinking skills problems

Mathematical communication is a crucial 21st-century skill related to achievement and other skills. This study aims to analyze the differences in mathematical communication abilities and describe the difficulties male preservice teachers (MPT) and female preservice teachers (FPT) face in solving gender-biased Higher-Order Thinking Skills (HOTS) problems. This research employs a sequential explanatory mixed-methods design. The respondents, selected through purposive sampling, consisted of 4 MPTs and 10 FPTs. Data on mathematical communication abilities were obtained through a gender-biased HOTS problem-solving test. The respondents' difficulties in solving the problems were gathered through interviews. The results indicate differences between MPT and FPT mathematical communication skills using symbols and mathematical notation in solving problems. Student's difficulties in solving gender-biased HOTS problems are unfamiliarity and disinterest in the context. This study implies that education should emphasize contextual understanding and utilize gender-neutral problems to enhance mathematical communication skills.

Comparative Study of Text Editors for Scientific Paper Writing with Mathematical Notation

Comparative Study of Text Editors for Scientific Paper Writing with Mathematical Notation

The geometry of GTPs and 5d SCFTs

We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over ℙ1 interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers.

Cultural Transitions in Mathematical Discourse: Unveiling Mathematical Writing Hurdles during the Rite of Passage

This qualitative case study delves into the intricate landscape of mathematical writing challenges faced by first-year university students undergoing the critical transition from school-level to university-level mathematical discourse. Conducted at a prominent South African university, the research, employing a purposive sampling technique, engaged thirty first-year mathematics students. Guided by the community of practice theory by Lave and Wenger, alongside van Gennep’s rite of passage analytical lens, the study sought answers to the question: What are the mathematical writing challenges encountered by first-year university students during their rite of passage period? Thematic analysis, informed by Adu's (2019) coding framework, was utilized to systematically examine common themes and patterns within the qualitative data. The findings illuminated key hurdles during this transitional phase, prominently including the inconsistency in working with mathematical notations, erroneous use of universal and existential quantifiers, and a notable confusion between the acts of illustrating and proving in mathematical contexts. In response to these challenges, the study advocates for the explicit incorporation of mathematical writing instruction to scaffold students during this rite of passage. Furthermore, it recommends a shift in the emphasis of first-year mathematics courses—suggesting a redirection from a content-centric approach to one that prioritizes the cultivation of students' new identities. This entails focused attention on teaching the customs, traditions, and adept ways of constructing and articulating mathematical proofs in the university context. The implications of this study extend beyond the immediate challenges identified, offering actionable recommendations to enhance the pedagogical strategies employed in the crucial transition period for first-year mathematics students.

PQ enhancement in grid connected EV charging station using novel GVCR control algorithm for AUPQC device

The rapid increase of environmental impacts together global warming is conquered by substantial selection of electric-vehicles (EV’s) over the internal-combustion engine (ICE) vehicles. The replacement of these vehicles in transportation industry has led to reducing the running cost, ecological emissions, vehicle maintenance. The EV’s are operated by available battery energy and energized through utility-grid integrated EV charging stations. It is noted that, such charging stations may introduce power-quality issues, highly impacting the electric-grid due to presence of power electronic conversion devices in EV charging stations. The primary emphasis of power-quality impacts on electrical distribution grid are counteracted by employing active universal power-quality conditioner (AUPQC) device. The main role of AUPQC has been selected for mitigation of various PQ problems on both electric-grid side and charging station by using feasible control objective. In this work, a novel generalized voltage-current reference (GVCR) control objective has been proposed for extraction of fundamental reference voltage-current signals. The key findings are simple mathematical notations, no transformations, fast response, low dv/dt switch stress, low switching loss and maximum efficiency. The main goal is design, operation and performance of proposed GVCR controlled AUPQC device has been validated under integration of various EV chargers to electric-grid by using MATLAB/Simulink computing tool, simulation results are presented for analysis and interpretation.

IMESH: A DSL for Mesh Processing

Mesh processing algorithms are often communicated via concise mathematical notation (e.g., summation over mesh neighborhoods). However, conversion of notation into working code remains a time-consuming and error-prone process, which requires arcane knowledge of low-level data structures and libraries—impeding rapid exploration of high-level algorithms. We address this problem by introducing a domain-specific language (DSL) for mesh processing called I MESH, which resembles notation commonly used in visual and geometric computing and automates the process of converting notation into code. The centerpiece of our language is a flexible notation for specifying and manipulating neighborhoods of a cell complex, internally represented via standard operations on sparse boundary matrices. This layered design enables natural expression of algorithms while minimizing demands on a code generation backend. In particular, by integrating I MESH with the linear algebra features of the I LA DSL and adding support for automatic differentiation, we can rapidly implement a rich variety of algorithms on point clouds, surface meshes, and volume meshes.

Mathematical Insights into Large Language Models

Purpose: The paper presents an exhaustive examination of the mathematical frameworks that support the creation and operation of large language models. The document commences with an introduction to the core mathematical concepts that are foundational to large language models. It delves into the mathematical algorithms employed in training these models and scrutinizes how various mathematical notions influence their efficacy. Methodology: Furthermore, it dissects the structure of large language models, analyzing the mathematical tenets that dictate their design and functionality. It also considers the mathematical logic underpinning these models' performance and the intricacies involved in their expansion. Additionally, it probes into the mathematical underpinnings of attention mechanisms within large language models, assessing how these mechanisms bolster the models' effectiveness and comprehensibility. Findings: Subsequently, it examines the mathematical bases of attention mechanisms in large language models, considering how these mechanisms augment the models' efficiency and clarity. It also debates the mathematical methods for refining large language models and the hurdles faced in enhancing their interpretability. By understanding the mathematical foundations of LLMs, we can leverage insights from the algorithms and principles driving these models, thus enhancing their inventive output and broadening the horizons of design and artistic expression. Unique contribution to theory, policy and practice: Lastly, it ventures into the ethical considerations surrounding large language models, scrutinizing the mathematical aspects related to these concerns.

Semiotic Mathematics Representation Ability Based on Symbolic in Solving SPLSV Problems in Class VII Students

Mathematical semiotic representation is the ability to analyze and express mathematical ideas or notions of a phenomenon and everyday problem situations into the form of signs, images, symbols, and symbols that represent them and provide meaning and explanation to a package of verbal sign messages. The symbolic stage is the stage where students have understood the symbols and concepts and have ideas that are strongly influenced by language and logic skills and students are able to manipulate symbols or symbols of a particular object. The purpose of this study was to determine the representation of symbolic-based mathematical semiotics in seventh grade students of MTS Al Barokah Ajung Jember on SPLSV material using qualitative descriptive method. The instrument used by researchers in the form of written test questions consisting of two questions tailored to the symbolic representation that has two indicators that use mathematical symbols to solve problems and interpret mathematical symbols. From the results of research that has been done obtained the error of representation on indicators using mathematical symbols to solve the problem of 100% and error of representation on indicators interpreting mathematical symbols of 0%. Several studies have been carried out to explain the mistakes that students make in representation skills. This paper presents the ability of mathematical symbolic semiotic representation of students who are more specific in solving SPLSV problems that can be considered by teachers in designing learning about the ability of mathematical semiotic representation and information for observers of mathematics education.

Music-driven geometric and topologic intuition: a case study with the Klein bottle

The comprehension of mathematical notions can profit from the personal background of a population of students, drawing upon their former knowledge in another field. In particular, the embodiment of mathematics provides us with a bridge to transfer competencies from music to intuitive understanding of non-trivial geometries and topologies. In this study, we analyse an experiment in which music composition students are introduced to the Klein bottle for the first time. Through the Ontological and Semiotic Approach (OSA), a conceptual framework in mathematics education, we evaluate the evolution from an initial intuition of the Klein bottle to a formal understanding of the object via musical renditions and verbal presentations to both neophyte and specialist audiences in mathematics.

STUDENTS’ PRACTICE OF MATHEMATICS AND ITS EFFECT ON THEIR NUMERICAL SKILLS IN PROBLEM-SOLVING IN SECONDARY SCHOOLS IN FAKO DIVISION, SOUTH WEST REGION OF CAMEROON

For several decades mathematics has been called the “critical filter”, because students who are inadequately prepared in mathematics during secondary school lose many career choices that would otherwise be available to them. Also, for those students who pursue post-secondary education in Cameroon, success in O-Level mathematics is a pre-requisite for most degree programmes in business, computing and the sciences. This makes mathematics very pivotal in reshaping the future and students’ attitude towards mathematics an area of concern. Negative attitudes towards mathematics are damaging, leading to disengagement, increased anxiety and lack of confidence, and reluctance to try to improve skills. Cameroon needs graduates with advanced mathematics skills to promote innovation, data synthesis, and technology if it is to solve challenging problems and be competitive in the global scenario. This study examined students’ practice of mathematics and its effect on their numerical skills in problem- solving in secondary schools in Fako Division, South West Region of Cameroon. The objective is to find out whether students’ practice of mathematics has any influence on their numerical skills in problem-solving. The mixed research methodology with convergent parallel design was used. The target population comprised 6350 form five students from secondary schools in Fako Division. The accessible population was made up of 1036 students and the sample size of the study was made of 512 form five students from six colleges using the random sampling technique. There were also 26 teachers and 6 Heads of Mathematics Department. Data was collected using questionnaire and interview guide. Instruments were validated and the reliability coefficients gave satisfactory values of 0.75 and 0.743 for students’ and teachers’ questionnaire respectively. The statistical analyses and findings reveal that students’ practice of mathematics has a positive correlation with their numerical skills in problem-solving. It was recommended that: (1) problem-solving should be part of each curricular unit and begin in kindergarten. For this to be effective, the teaching of problem-solving should not be isolated, instead, it can serve to support and enrich the learning of mathematics concepts and notation. (2) teachers should cultivate students’ interest in mathematics as early as possible. Varying classroom instruction practices could be a remedy to enhance students’ understanding, achievement, and motivation in learning mathematics.

Programming as a mediator of mathematical thinking

We report on three episodes from a case study where upper secondary students numerically explore the definite integral in a Python environment. Our research questions concern how code can mediate and support students' mathematical thinking and what kind of sociomathematical norms emerge as students work together to reach a mutual understanding of a correct solution. The main findings of our investigation are as follows. 1) Students can actively use code as a mediator of their mathematical thinking, and code can even serve as a bridge that helps students to develop their mathematical thinking collaboratively. Further, code can help students to perceive mathematical notions as objects with various properties and to communicate about these properties, even in other semiotic systems than the mathematical language. 2) For the participating students, a common norm was that an acceptable solution is a sufficient condition for the correctness of the solution method although students were aware of a problem in their code, yet also other norms emerged. This demonstrates that learning mathematics with programming can have an effect on what kind of sociomathematical norms emerge in classroom.

On Soft Sets and Formal Concept Analysis (FCA) as mathematical categorization systems - An Engineering Application

This paper deals with an analysis of the reasoning for the cognitive problem of categorization in systems. Such reasoning requires the construction of a formal model of the system that is performed in 3 steps: cognitive description of the system, extraction of system’s meaningful concepts, and design of the mathematical model based on the concepts. Our goal is to explore the cognitive features and their predominance in some old models of categorization well known in literature. We highlight the relation between Formal Concept Analysis (FCA) model and soft sets and we analyze the notion of "parameter" from the cognitive point of view. Based on this analysis, we propose the notion of double soft sets as a mathematical notion more adequate in the engineering applications. All our study is conducted in the framework of material selection in mechanical engineering.

A Review of Fluid Bolus in Critically Ill Patients After Initial Volume Expansion: Bayesian Probability Analysis and Case Studies.

Introduction Fluid resuscitation is a crucial intervention for the management of critically ill patients. However, after initial volume expansion, the advantages of fluid bolus administration remain controversial. Our aim was to investigate the probabilistic reasoning against fluid bolus administration in critically ill patients after initial volume expansion. We then applied this reasoning to two hypothetical case studies that evaluated the benefits and risks associated with a fluid bolus for each patient. Methods We analyzed data from 12 previously published studies, totaling 334 patients, on fluid responsiveness in critically ill patients.Owing to differences in these studies, we used a Monte Carlo simulation based on their parameters to improve our Bayesian prior, generate strong estimates, and address uncertainty. Using the established Bayesian prior for volume responsiveness, we scrutinized two hypothetical case studies employing Bayesian mathematical notation to assess the pre-test probability, posterior probability, and likelihood ratios in patients with septic shock. Results The Monte Carlo simulation yielded a mean response rate of 0.54 (SD = 0.026), suggesting that only approximately 54% of patients were responsive to fluid bolus administration. These results had an effective sample size of 17,204 and an R-hat value of 1, demonstrating the reliability of our results.In our Bayesian case studies, we demonstrate the low probabilities of volume and VO2 responsiveness over time using common bedside testing. Conclusion Our analysis shows that the pretest and posttest probabilities for volume responsiveness following initial fluid resuscitation are low. Additional bedside testing should be pursued before administering additional volume. This approach emphasizes the importance of evidence-based decision-making in the management of critically ill patients to optimize patient outcomes and minimize potential risks.

"It's the most fair thing to do but it doesn't make any sense": Perceptions of Mathematical Fairness Notions by Hiring Professionals

We explore the alignment of organizational representatives involved in hiring processes with five different, commonly proposed fairness notions. In a qualitative study with 17 organizational professionals, for each notion, we investigate their perception of understandability, fairness, potential to increase diversity, and practical applicability in the context of early candidate selection in hiring. In this, we do not explicitly frame our questions as questions of algorithmic fairness, but rather relate them to current human hiring practice. As our findings show, while many notions are well understood, fairness, potential to increase diversity and practical applicability are rated differently, illustrating the importance of understanding the application domain and its nuances, and calling for more interdisciplinary and human-centered research into the perception of mathematical fairness notions.

The human after math: Nnedi Okorafor’s unruly engagements with Cartesian desire

ABSTRACT This article contends that Nnedi Okorafor remakes a European Enlightenment notion of mathematics in her speculative trilogy Binti (2015-2018). The titular character has the power of mathematical thinking, where through the contemplation of mathematical signs, she absents herself from her immediate physical surroundings and enters into a dream-like space. Reading Binti alongside Descartes’ Discourse on the Method, I argue that Okorafor draws on an Enlightenment notion of mathematics as training in reason, where it was believed that through the encounter with mathematical objects, man could learn to think independently of their cultural and religious backgrounds, becoming universal. Her critique of Enlightenment math is not explicit, but rather raises philosophical questions about what exactly math is and how it could have secured the secular human. Binti reveals that math is a linguistic technology for producing imaginative realms detached from the real world that could allow for different experiences of becoming. Okorafor’s work brings mathematics in conversation with literature and also Fanon’s thinking about race and the body. This essay examines how African speculative fiction can intervene in modern liberal notions of the human by interrogating and remaking the seemingly objective discourses that make up our current hegemonic mode of being human.

An efficient approach to parameter extraction of photovoltaic cell models using a new population-based algorithm

This article discusses the problem of accurate and efficient modeling of photovoltaic (PV) panels. It is a highly nonlinear problem. The following models were considered: a single diode model, a double diode model, a triple diode model, a four diode model, a module model (a poly-crystalline Photowatt-PWP201 module and a mono-crystalline STM6-40/36 module). The article presents a mathematical notation of these models, a detailed interpretation of their individual components, and a comparison of obtained results. To increase the effectiveness of modeling, a new population-based algorithm which can handle complex objective functions and a large number of decision variables was developed. This is important for the problem of identifying the parameters of PV cell models because each evaluation of the objective function requires calculating a set of points that determine the current–voltage characteristics. Moreover, in the considered problem a solution is searched with the use of the trial and error method. The proposed algorithm is called Micro Adaptive Fuzzy Cuckoo Search Optimization (μAFCSO). The μAFCSO algorithm uses several new mechanisms that were developed based on our experience with population-based algorithms. The use of these mechanisms has produced very good results in simulations. In the scope of simulation studies, the μAFCSO algorithm was used for parameter extraction in six PV cell models and was also applied to optimize fifteen typical test functions. The test functions were considered in order to demonstrate that our algorithm can be used to solve typical problems processed using population-based algorithms. The results obtained in this study were compared with the results obtained using well-established algorithms. The results obtained in this work are better or comparable to them.

Sosialisasi Penerapan Kontekstual Konsep Matematika dalam Ibadah Salat di SMP Ibnu Khaldun, Banda Aceh

Prayer can serve as a valuable component of mathematical education, especially in Islamic schools, based on the constructivist learning theory. Utilizing community-based research methods, such as action and participatory research, fosters self-awareness and conflict resolution. Through prayer, students can engage in mathematical concepts like counting and multiplication by observing actions like bows and prostrations. Media supplements, such as a ball count, can enhance learning. Contextual notions of school mathematics are evident in prayer, such as adding and multiplying natural numbers, understanding sets, arithmetic sequences, and geometry concepts like lines, angles, and parallelism. For instance, the number of prayer units illustrates adding natural numbers, while the tahajjud prayer exemplifies multiples of two. Moreover, the congregation's rows during group prayer demonstrate line segments and parallel lines, while the standing position showcases various angles. These connections highlight the relationship between prayer and mathematics, offering unique insights into both disciplines. Integrating prayer into mathematical education not only enriches students' understanding but also fosters a holistic approach to learning.