Mathematical Insight Research Articles

Mathematical insights into the original Retinex algorithm for image enhancement.

The Retinex theory, originally developed by Land and McCann as a computation model of the human color sensation, has become, with time, a pillar of digital image enhancement. In this area, the Retinex algorithm is widely used to improve the quality of any input image by increasing the visibility of its content and details, enhancing its colorfulness, and weakening, or even removing, some undesired effects of the illumination. The algorithm was originally described by its creators in terms of a sequence of image processing operations and was not fully formalized mathematically. Later, works focusing on aspects of the original formulation and adopting some of its principles tried to frame the algorithm within a mathematical formalism: this yielded every time a partial rendering of the model and resulted in several interesting model variants. The purpose of the present work is to fill a gap in the Retinex-related literature by providing a complete mathematical formalization of the original Retinex algorithm. The overarching goals of this work are to provide mathematical insights into the Retinex theory, promote awareness of the use of the model within image enhancement, and enable better appreciation of differences and similarities with later models based on Retinex principles. For this purpose, we compare our model with others proposed in the literature, paying particular attention to the work published in 2005 by Provenzi and others.

The theoretical framework on humanist ethno-metaphorical mathematics learning model: An impactful insight in learning mathematics

The educational revolution has posed an immense challenge to the world of education. It demands the development of a generation that can take on the challenges and changes bred by the ever-rapid revolution. It is thus inevitable that education must enable improvements of individual hard skills and soft skills that are required to keep up with such changes, including mathematical hard skills and soft skills. The problem is that not all mathematics learning approaches, particularly in the case of Indonesia, are capable of such improvements and of answering to such demands, challenges, and changes that are posed by the revolution. This research seeks to build a theoretical framework out of a systematic analysis based on various pieces of literature that are relevant and fitting to the theoretical framework under development. In this study, a theoretical framework on humanist ethno-metaphorical mathematics learning is developed as a theoretical foundation. This theory is designed for creating a humanist mathematics learning approach based on ethnomathematics and metaphorical thinking to develop students’ mathematical hard skills and soft skills and thus enable them to deal with the current and future problems and changes.

Drawing muscles with diagrams: how a novel dissection cut inspired Nicolaus Steno's mathematical myology (1667)

In 1667, twenty years before Isaac Newton published his mathematization of physics, and more than ten years before the publication of Giovanni Borelli's De motu animalium , the Danish anatomist Nicolaus Steno published an entirely new geometrical theory of muscle motion in the book Elementorum myologiæ specimen . Historians of science have studied this book in recent decades, but the recent rediscovery of a seventeenth-century muscle atlas at the Bibliothèque interuniversitaire de Santé in Paris sheds new light on the largely overlooked origin of Steno's mathematical theory of muscles. In this article, we show that Steno's muscle diagrams result from a tension that Steno faced when combining his interest in illustrations with presenting his mathematical insights about the inner structure of muscle fibres. Furthermore, we argue that Steno's diagrams are deeply connected to observations through a new method of dissecting the muscles. The observational origins of Steno's mathematical insight are further confirmed by the strong correlation between Steno's depictions of the structure and function of skeletal muscles and the results of current biomechanical investigations.

MIRA: a multi-physics approach to designing a fusion power plant

Fusion systems codes (SCs) are deployed to produce the baseline of the European fusion power reactor (DEMO) within its conceptual design. A DEMO baseline is mostly defined by a radial/vertical reactor sketch and major reactor parameters, such as fusion and net electric power, magnetic fields, and plasma burn time. A baseline shall also meet a set of prescribed reactor requirements, constraints, and architectural features. According to the conceptual design workflow implemented within the EU-DEMO programme, the output from the SC is transferred to the detailed physics and engineering design codes. Presently-available fusion SCs rely on rather basic physics and engineering models (mostly at zero or one-dimensional level). The design codes, instead, are very detailed but run on much longer computing times. To fill the gap between systems and design codes, the multi-fidelity systems/design tool modular integrated reactor analysis (MIRA)—has been recently developed. MIRA incorporates the physics and the engineering insights of the utmost domains of tokamak reactors and relies on a higher spatial resolution, spanning from 1D up to 3D modelling frames. The MIRA approach has been applied to the DEMO 2017 baseline, generated by the EU reference SC PROCESS and used as input to MIRA. In the paper, the architectural and mathematical insights of the MIRA package are described, along with an EU-DEMO 2017 baseline analysis.

Beyond the Wenzel and Cassie–Baxter world: Mathematical insight into contact angles

The Wenzel and Cassie–Baxter wetting states are treated as basis states producing a continuous spectrum of intermediate states. The model considering them as a linear combination of the basis states is proposed and successfully applied to independent data converted within this framework from the terms of absolute vertical deviations as roughness characteristics into the Wenzel r‐based roughness concept. Roughness dependences of apparent contact angles are reproduced therewith. The degree of symmetry is shown to influence measurements, calculations and possibly definition of the contact angle. The cotangent formula linking the drop size and apparent contact angle is derived from the well‐known non‐linear differential model for a drop shape. This enables one to pre‐estimate changes in contact angles due to drop size changes what is important for a more precise definition of superhydrophobicity and superhydrophilicity. The analysis of an oval‐shaped curve as the most general contact line on inclines is performed. The dependence of apparent contact angle on incline tilt and azimuthal angle is derived for the regime of pinning the drop onto a flat surface having been tilted before, unlike the well‐known case of tilting the flat surface with a sessile drop whose contact line is circle‐shaped. The contact angle is found to be a non‐monotonous function of azimuthal angle because the contact line possesses now a reduced symmetry and, therefore, a varying curvature with extrema. Drop ensembles on incline are discussed, and their modeling with application to macroscopic processes requires treatment of the contact line in its most general case, that is, oval shaped.

Action-based embodied design for mathematics learning: A decade of variations on a theme

Embodied cognition theory emphasizes that bodily interaction with the environment is important for all forms of learning, including mathematics. This theoretical trend coincides well with developments in motion responsive technology, and has resulted in numerous embodied technologies for mathematics learning. This review aims to contribute to clarifying theoretically and empirically grounded design principles of action-based embodied designs for mathematics learning. We analyzed 79 publications between 2010 and 2019, containing 15 studies assessing 15 sensorimotor problems for five mathematical domains (proportion, angle, area, parabola, and sine function), and explicated the characteristics of the technologies, their learning sequences and elicited learning processes, and the influence of within-topic ask variations on students’ learning. We found that action-based designs pose motor control problems using continuous motion feedback to facilitate learners to discover and practice a challenging new ways of moving their hand(s) in which to ground mathematical cognition. The state of discovery of the sensorimotor solution is important, and passive and readymade designs are cautioned. The learning sequence in which these technologies are embedded, elicit mathematical knowing through necessary and sequential phases in which personal idiosyncratic experiences increasingly converge into a culturally shared mathematical discourse. In the qualitative stage, an acting step elicits students to actively establish new motor coordination-patterns through the emergence of new perceptual structures known as attentional anchors. In the subsequent reflecting step, students’ personal sensorimotor experiences and attentional anchors become the ground for referencing in (a shared) mathematical discourse through multimodal (words, gestures) collaboration with a tutor. In the quantitative stage, measuring artifacts (grids, protractors, numbers, variables) are included in students’ field of promoted action, which discretize and formalize students’ actions and subsequent reflections into culturally recognizable quantitative forms. Critically, task factors such as the type of objects students manipulate (cursors icons, bars, rectangle), and the direction these objects are moved (parallel, orthogonal), affect students’ attentional anchors and subsequent reflections in the qualitative stage, but converge to similar mathematical insights in the quantitative stage. These insights help to better use (new) motion responsive technology in eliciting child–computer interaction that can lead to mathematical cognition and beyond.

Machine Learning Advances in Microbiology: A Review of Methods and Applications

Microorganisms play an important role in natural material and elemental cycles. Many common and general biology research techniques rely on microorganisms. Machine learning has been gradually integrated with multiple fields of study. Machine learning, including deep learning, aims to use mathematical insights to optimize variational functions to aid microbiology using various types of available data to help humans organize and apply collective knowledge of various research objects in a systematic and scaled manner. Classification and prediction have become the main achievements in the development of microbial community research in the direction of computational biology. This review summarizes the application and development of machine learning and deep learning in the field of microbiology and shows and compares the advantages and disadvantages of different algorithm tools in four fields: microbiome and taxonomy, microbial ecology, pathogen and epidemiology, and drug discovery.

MATHEMATICAL INSIGHTS INTO THE DYNAMICS OF INNATE IMMUNE RESPONSE DURING INFLAMMATION

Innate immune system cells activate in response to infection and trigger an acute inflammatory reaction to restore tissue homeostasis and promote subsequent tissue repair. Their activation and functions must be very well regulated to avoid tissue damage, organ dysfunction, or even death. In this work, a new set of mathematical models is presented to examine the dynamics of the innate immune system response to tissue damage and provide further understanding of the role of the innate immune system during the early stages of an inflammatory response. Different damaged cells production functions are proposed to represent the effect of secondary tissue damage by the innate immune system. The stability and bifurcation analyses of the model reveal that there is an important threshold parameter that can be controlled in order to avoid sustained chronic inflammation and secure a successful healing outcome. A set of numerical simulations is also performed to support the presented theoretical results and demonstrate the medical applicability of the new mathematical model.

Mathematical Model Insights into EEG Origin under Transcranial Direct Current Stimulation (tDCS) in the Context of Psychosis.

Schizophrenia is a psychotic disease that develops progressively over years with a transition from prodromal to psychotic state associated with a disruption in brain activity. Transcranial Direct Current Stimulation (tDCS), known to alleviate pharmaco-resistant symptoms in patients suffering from schizophrenia, promises to prevent such a psychotic transition. To understand better how tDCS affects brain activity, we propose a neural cortico-thalamo-cortical (CTC) circuit model involving the Ascending Reticular Arousal System (ARAS) that permits to describe major impact features of tDCS, such as excitability for short-duration stimulation and electroencephalography (EEG) power modulation for long-duration stimulation. To this end, the mathematical model relates stimulus duration and Long-Term Plasticity (LTP) effect, in addition to describing the temporal LTP decay after stimulus offset. This new relation promises to optimize future stimulation protocols. Moreover, we reproduce successfully EEG-power modulation under tDCS in a ketamine-induced psychosis model and confirm the N-methyl-d-aspartate (NMDA) receptor hypofunction hypothesis in the etiopathophysiology of schizophrenia. The model description points to an important role of the ARAS and the -rhythm synchronicity in CTC circuit in early-stage psychosis.

Nonstandard Finite Element Methods

Finite element methodologies dominate the computational approaches for the solution to partial differential equations and nonstandard finite element schemes most urgently require mathematical insight in their design. The hybrid workshop vividly enlightened and discussed innovative nonconforming and polyhedral methods, discrete complex-based finite element methods for tensor-problems, fast solvers and adaptivity, as well as applications to challenging ill-posed and nonlinear problems.

Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks.

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.

Intrinsic dimension of geometric data sets

The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from computer science. Among the mathematical insights into data dimensionality, there is an intimate link between the dimension curse and the phenomenon of measure concentration, which makes the former accessible to methods of geometric analysis. The present work provides a comprehensive study of the intrinsic geometry of a data set, based on Gromov's metric measure geometry and Pestov's axiomatic approach to intrinsic dimension. In detail, we define a concept of geometric data set and introduce a metric as well as a partial order on the set of isomorphism classes of such data sets. Based on these objects, we propose and investigate an axiomatic approach to the intrinsic dimension of geometric data sets and establish a concrete dimension function with the desired properties. Our model for data sets and their intrinsic dimension is computationally feasible and, moreover, adaptable to specific ML/KD-algorithms, as illustrated by various experiments.

Deployment Optimization of Reconfigurable Intelligent Surface for Relay Systems

To harness the benefits of both relay and reconfigurable intelligent surface (RIS), we consider in this paper an RIS-aided wireless relaying system, in which a decode-and-forward relay (R) forwards data from a source (S) to a destination (D) with the assistance of an RIS. We first derive the closed-form of the upper bound of the achievable rate of the system. Under the homogeneous transmit power and noise, we derive the optimal RIS deployment in closed-form expressions to draw insights for practical design. Then, for general system setups, we formulate the achievable rate maximization problem by optimizing the RIS deployment. Though the problem is non-convex, we propose an solution to find the optimal RIS deployment and provide some mathematical and theoretical insights. It is unveiled that the optimal strategy is to deploy the RIS near the relay, rather than near S or D in the traditional RIS-aided systems without relay. The numerical results validate our theoretical analysis and show that the RIS and relay can complement each other, and the rate performance is significantly improved over the benchmark schemes.

Effects of Acute Exercise on Verbal, Mathematical, and Spatial Insight Creativity

Effects of Acute Exercise on Verbal, Mathematical, and Spatial Insight Creativity

Hierarchical timescales in the neocortex: Mathematical mechanism and biological insights

A cardinal feature of the neocortex is the progressive increase of the spatial receptive fields along the cortical hierarchy. Recently, theoretical and experimental findings have shown that the temporal response windows also gradually enlarge, so that early sensory neural circuits operate on short timescales whereas higher-association areas are capable of integrating information over a long period of time. While an increased receptive field is accounted for by spatial summation of inputs from neurons in an upstream area, the emergence of timescale hierarchy cannot be readily explained, especially given the dense interareal cortical connectivity known in the modern connectome. To uncover the required neurobiological properties, we carried out a rigorous analysis of an anatomically based large-scale cortex model of macaque monkeys. Using a perturbation method, we show that the segregation of disparate timescales is defined in terms of the localization of eigenvectors of the connectivity matrix, which depends on three circuit properties: 1) a macroscopic gradient of synaptic excitation, 2) distinct electrophysiological properties between excitatory and inhibitory neuronal populations, and 3) a detailed balance between long-range excitatory inputs and local inhibitory inputs for each area-to-area pathway. Our work thus provides a quantitative understanding of the mechanism underlying the emergence of timescale hierarchy in large-scale primate cortical networks.

Pembimbingan Tes Intelegensi Umum Calon Pegawai Negeri Sipil di Kota Sorong

This service activity aims to improve the skills of test-takers in solving general intelligence test questions for prospective civil servants in 2021. The method used in this service is participatory action research which involves collaboration with participants through actions given by the trainer as well as a facilitator. The action is in the form of delivering material, giving information on how to quickly solve problems, and tiered exercises. An important finding in this service is that students who are guided in following explanations and exercises well, meanwhile through pre-test and post-test data, it is concluded that there is a gap in mathematical insight and case analysis for civil servant enthusiasts from the natural sciences department or allied with graduates from majoring in social sciences or a cognate. This difference in responses and learning outcomes becomes a follow-up in the next service with different treatments.

Pendulum: the partial and global approach

The pendulum was an important scientific instrument in the 17th century. It became a typical textbook problem in the 18th century. After the introduction of vectors in physics in the 1890s, the pendulum problem started to be progressively solved in the manner we know nowadays from introductory mechanics courses. Starting from F = ma and using the tangential component of weight, we arrive at the equation of motion. The other part of F = ma, which concerns the weight radial component, tension and centripetal force, has in no way deserved the same attention from textbook writers, if any. Physics education research has, however, pointed out relevant aspects of the phenomenon related with the latter part of the equation. The present paper presents an experiment, which reflects those relevant aspects, and provides a mathematical insight into the phenomenon suitable for the high school student.

Scaling High-Quality Pairwise Link-Based Similarity Retrieval on Billion-Edge Graphs

SimRank is an attractive link-based similarity measure used in fertile fields of Web search and sociometry. However, the existing deterministic method by Kusumoto et al. [ 24 ] for retrieving SimRank does not always produce high-quality similarity results, as it fails to accurately obtain diagonal correction matrix D . Moreover, SimRank has a “connectivity trait” problem: increasing the number of paths between a pair of nodes would decrease its similarity score. The best-known remedy, SimRank++ [ 1 ], cannot completely fix this problem, since its score would still be zero if there are no common in-neighbors between two nodes. In this article, we study fast high-quality link-based similarity search on billion-scale graphs. (1) We first devise a “varied- D ” method to accurately compute SimRank in linear memory. We also aggregate duplicate computations, which reduces the time of [ 24 ] from quadratic to linear in the number of iterations. (2) We propose a novel “cosine-based” SimRank model to circumvent the “connectivity trait” problem. (3) To substantially speed up the partial-pairs “cosine-based” SimRank search on large graphs, we devise an efficient dimensionality reduction algorithm, PSR # , with guaranteed accuracy. (4) We give mathematical insights to the semantic difference between SimRank and its variant, and correct an argument in [ 24 ] that “if D is replaced by a scaled identity matrix (1-Ɣ)I, their top-K rankings will not be affected much”. (5) We propose a novel method that can accurately convert from Li et al. SimRank ~{S} to Jeh and Widom’s SimRank S . (6) We propose GSR # , a generalisation of our “cosine-based” SimRank model, to quantify pairwise similarities across two distinct graphs, unlike SimRank that would assess nodes across two graphs as completely dissimilar. Extensive experiments on various datasets demonstrate the superiority of our proposed approaches in terms of high search quality, computational efficiency, accuracy, and scalability on billion-edge graphs.

Political districting without geography

Geographical considerations such as contiguity and compactness are necessary elements of political districting in practice. Yet an analysis of the problem without such constraints yields mathematical insights that can inform real-world model construction. In particular, it clarifies the sharp contrast between proportionality and competitiveness and how it might be overcome in a properly formulated objective function. It also reveals serious weaknesses of the much-discussed efficiency gap as a criterion for gerrymandering.

A Critical Evaluation of NCERT Mathematics Textbook of Senior Secondary Stage

The aim of the mathematics curriculum at the senior secondary stage is to provide students with an appreciation of the wide variety of the application of mathematics and equip them with the basic tools that enable such application. A careful choice between the often conflicting demands of depth versus breadth needs to be made at this stage. The rapid explosion of mathematics as a discipline, and of its range of application, favors an increase in the breadth of coverage. Such an increase must be dictated by mathematical considerations of the importance of topics to be included. Topics that are more naturally the province of other disciplines may be left out of the mathematics curriculum. The treatment of topics must have an objective, that is, the communication of mathematical insights and concepts, which naturally arouse the interest and curiosity of students.