Mathematical Foundations Research Articles

Predicting treatment plan approval probability for high-dose-rate brachytherapy of cervical cancer using adversarial deep learning

Objective. Predicting the probability of having the plan approved by the physician is important for automatic treatment planning. Driven by the mathematical foundation of deep learning that can use a deep neural network to represent functions accurately and flexibly, we developed a deep-learning framework that learns the probability of plan approval for cervical cancer high-dose-rate brachytherapy (HDRBT). Approach. The system consisted of a dose prediction network (DPN) and a plan-approval probability network (PPN). DPN predicts organs at risk (OAR) D 2cc and CTV D 90% of the current fraction from the patient’s current anatomy and prescription dose of HDRBT. PPN outputs the probability of a given plan being acceptable to the physician based on the patients anatomy and the total dose combining HDRBT and external beam radiotherapy sessions. Training of the networks was achieved by first training them separately for a good initialization, and then jointly via an adversarial process. We collected approved treatment plans of 248 treatment fractions from 63 patients. Among them, 216 plans from 54 patients were employed in a four-fold cross validation study, and the remaining 32 plans from other 9 patients were saved for independent testing. Main results. DPN predicted equivalent dose of 2 Gy for bladder, rectum, sigmoid D 2cc and CTV D 90% with a relative error of 11.51% ± 6.92%, 8.23% ± 5.75%, 7.12% ± 6.00%, and 10.16% ± 10.42%, respectively. In a task that differentiates clinically approved plans and disapproved plans generated by perturbing doses in ground truth approved plans by 20%, PPN achieved accuracy, sensitivity, specificity, and area under the curve 0.70, 0.74, 0.65, and 0.74. Significance. We demonstrated the feasibility of developing a novel deep-learning framework that predicts a probability of plan approval for HDRBT of cervical cancer, which is an essential component in automatic treatment planning.

Foundations of the wald space for phylogenetic trees

AbstractEvolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterisation of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fréchet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.

Mathematical Foundations of Digital Twins (NSF)

Mathematical Foundations of Digital Twins (NSF)

A Practical Study on Enhancing Students’ Mathematical Literacy by “Integration of Doing, Learning and Teaching”

In face of the mathematical foundation of students in higher vocational colleges, Tao Xingzhi’s educational thought is of great guiding significance. “Integration of doing, learning and teaching” is not only the theoretical basis of Tao Xingzhi’s educational thought, but also his methodology and action guideline. Tao Xingzhi emphasizes the unity of knowledge and action, among which action is greater than knowledge. This concept is isomorphic with the mathematical culture curriculum. Drawing on the idea of “integration of doing, learning and teaching”, we explore the math culture curriculum and expand the four forms of teaching and learning in math culture: theoretical training, practical exploration, research-based learning, and reflective learning. In the implementation of the school-based curriculum, local teaching resources should be fully exploited, constructing the mathematics teaching mode of “Integration of doing, learning and teaching”, effective teaching methods and classroom evaluation algorithms, ensuring the relevance of vocational education and improving mathematical literacy.

Correlated Decision Fusion Accompanied with Quality Information on a Multi-Band Pixel Basis for Land Cover Classification.

Decision fusion plays a crucial role in achieving a cohesive and unified outcome by merging diverse perspectives. Within the realm of remote sensing classification, these methodologies become indispensable when synthesizing data from multiple sensors to arrive at conclusive decisions. In our study, we leverage fully Polarimetric Synthetic Aperture Radar (PolSAR) and thermal infrared data to establish distinct decisions for each pixel pertaining to its land cover classification. To enhance the classification process, we employ Pauli's decomposition components and land surface temperature as features. This approach facilitates the extraction of local decisions for each pixel, which are subsequently integrated through majority voting to form a comprehensive global decision for each land cover type. Furthermore, we investigate the correlation between corresponding pixels in the data from each sensor, aiming to achieve pixel-level correlated decision fusion at the fusion center. Our methodology entails a thorough exploration of the employed classifiers, coupled with the mathematical foundations necessary for the fusion of correlated decisions. Quality information is integrated into the decision fusion process, ensuring a comprehensive and robust classification outcome. The novelty of the method is its simplicity in the number of features used as well as the simple way of fusing decisions.

Enough Model Theory to Understand Skolem’s “Paradox”

This article seeks to describe some of the basic foundations of mathematics from a model-theoretic perspective with the goal of obtaining the “Downward Löwenheim–Skolem” theorem. After the theorem is introduced the seemingly paradoxical nature of it is discussed.

Mathematical Foundations of the Non-Hermitian Skin Effect

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

Correcting for collider effects and sample selection bias in psychological research.

Colliders, variables that serve as a common outcome of an independent and dependent variable, pose a major challenge in psychological research. Collider variables can induce bias in the estimation of a population relationship of interest when (a) the composition of a research sample is restricted by scores on a collider variable or (b) researchers adjust for a collider variable in their statistical analyses, as they might do for confounder variables. Both cases interfere with the accuracy and generalizability of statistical results. Despite their importance, however, collider effects remain relatively unknown in psychology. This tutorial article summarizes both the conceptual and the mathematical foundation for collider effects and their relevance to psychological research, and then proposes a method to correct for collider bias in cases of restrictive sample selection based on Thorndike's Case III adjustment (1982). Two simulation studies demonstrated Thorndike's correction as a viable solution for correcting collider bias in research studies, even when restriction on the collider variable was extreme and the selected sample size was as low as N = 100. Bias and relative bias results are reported to evaluate how well the correction equation approximates targeted population correlations under a variety of parameter conditions. We illustrate the application of the correction method to a hypothetical study of intelligence and conscientiousness, discuss the applicability of the method to more complex statistical models as a means of detection for collider bias, and provide code for researchers to apply to their own research. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Тождество теоретического и экспериментального подходов при прототипировании региональных проектов в сфере здравоохранения.

The formalization of processes for finding the parameters of a prototype model of a medical organization implementing a regional project is based on the unity of theoretical and experimental approaches. The purpose of the study was an attempt to create a unified and harmonious theoretical foundation for conducting an experiment in order to find a prototype of a medical organization implementing projects at the regional level. In order to reveal the potential for solving technological problems facing project stakeholders, formalization of approaches was carried out for experimentally finding the parameters of a prototype project for a medical organization implementing a regional project. Materials and methods. The research materials were the works of Russian and foreign scientists related to the issues of mathematical modeling and prototyping. In order to study the research question, more than 30 literature sources were analyzed, directly or indirectly addressing the problems of practical application of object prototyping methods. Methods of system analysis, comparison, systemic and integrated approaches, as well as mathematical modeling methods were used. Results. Complex, multi-level systems when implementing projects in the regions are poorly structured and, accordingly, not fully formalized, so the experimental approach is especially relevant when searching for prototyping tools. The degree of formalization of approaches to searching for a prototype of a project object depends on the completeness (saturation) of processes, their structure and the degree of correlation of process data in a medical organization to process performers, operations, and management objects. Prototyping is designed to optimize the procedures for planning, analyzing and processing the results of research data based on a mathematical foundation. The prototyping technique is primarily aimed at saving resources (labor, financial, time) as much as possible, therefore, justification of the need and mastery of mathematical foundations in formalizing approaches to prototyping is a fundamental factor and the purpose of this discussion. Findings. System analysis using the unity of theoretical and experimental approaches, in the author’s opinion, will allow for a deeper synthesis of knowledge about the potential possibilities for forming decisions through a prototype of a medical organization implementing projects in the healthcare industry.

الفضاءات القابلة للتفكك والفضاءات غير القابلة للتفكك عبر فضاء القرب

اهتم كثيرا من الباحثين في دراسة اغلب المشكلات بعد تحويلها الى صيغ رياضية داخل فضاء القرب لسهولة توظيف هذاالفضاء مع المشكلات الحياتية المختلفة التي تواجهنا. لذلك يصب اهتمام البحث في ايجاد صفة جديدة الى فضاءات القرب باستخدام خواص القرب وخواص العناقيد وربطها معا للحصول على نقاط تراكمية جديدة اطلقنا عليها اسم نقاط كثه ومجموعة كل هذه النقاط اطلقنا عليها اسم مجموعة كثه حيث تم تكوين اساس رياضي متماسك الى الصيغة التبلوجية التي نبحث عنها والتي اطلقنا عليها التفكك واذا كانالفضاء لا يحمل هذه الصفة نطلق عليه فضاء غير قابل للتفكك.

Fractional Reasoning with Representation: Insights from Malaysia

Objective - Fractional reasoning is a crucial aspect of mathematical understanding fundamental in various mathematical concepts, real-world applications, and higher-level mathematical skills. Comprehending and working with fractions through various strategies, such as representation, is essential for students to develop a solid foundation in mathematics. However, fractional reasoning remains challenging in classroom teaching and learning since it requires deep understanding. Methodology/Technique – The current issue is a more comprehensive and conceptually grounded approach to foster a deeper acquisition of fractional reasoning strategies. Hence, this study aims to investigate to what extent primary school pupils develop fractional reasoning strategies to solve related problems, specifically for fractions of an area and fractions of a set of objects. A case study was conducted to interview eight primary school pupils from Perak (in Malaysia) for the data collection. Finding – The participants' solutions were observed to triangulate the interview data. In the content analysis, the identification of codes was carried out. Their findings revealed that the participants relied on strategies of representation methods of enactive and symbolic representations when working on fractions of an area. Novelty – This study introduces a novel perspective by emphasising that the identified fractional reasoning strategies are not isolated skills. The primary school pupils predominantly employed enactive and symbolic representations for fractions of an area, while favouring symbolic representations when reasoning fractions for a set of objects. These insights offer valuable guidance to educators, suggesting that a varied instructional approach, incorporating real-world contexts, can contribute to a more profound and versatile comprehension of fractions across diverse mathematical scenarios. Type of Paper: Empirical JEL Classification: I26, I29 Keywords: Representation, Enactive, Symbolic, Fractions, Fractional Reasoning Reference to this paper should be referred to as follows: Kamaruddin, N.I.H; Hoon, T.S; Hong, J.B.Z. (2024). Fractional Reasoning with Representation: Insights from Malaysia, GATR-Global J. Bus. Soc. Sci. Review, 12(1), 01–14. https://doi.org/10.35609/gjbssr.2024.12.1(1)

Set Theory and Third Mathematical Crisis: A Series of Events Reflected Upon the Mathematical Foundation

Set Theory and Third Mathematical Crisis: A Series of Events Reflected Upon the Mathematical Foundation

The Application of Knowledge Engineering via the Use of a Biomimetic Digital Twin Ecosystem, Phenotype-Driven Variant Analysis, and Exome Sequencing to Understand the Molecular Mechanisms of Disease

Applied Artificial Intelligence, particularly Large Language Models, in biomedical research is accelerating, but effective discovery and validation requires a toolset without limitations or bias. On January 30, 2023, the National Academies of Sciences, Engineering, and Medicine (NAS) appointed an ad hoc committee to identify needs and opportunities to advance the mathematical, statistical, and computational foundations of digital twins in applications across science, medicine, engineering, and society. On December 15, 2023, the NAS released a 164 page report, "Foundational Research Gaps and Future Directions for Digital Twins". This report described the importance of using digital twins in biomedical research. We developed an innovative method that incorporated phenotype ranking algorithms with knowledge engineering via a biomimetic digital twin ecosystem. This ecosystem applied real-world reasoning principles to non-normalized, raw data to identify hidden or "dark data". We performed a clinical exome sequencing study on patients with endometriosis and were able to identify four VUSs potentially associated with endometriosis-related disorders in nearly all patients analyzed. One VUS was identified in all patient samples and could be a biomarker for diagnostics. To the best of our knowledge, this is the first study to incorporate the recomandations of the NAS to biomedical research. This method can be used to understand the mechanisms of any disease, for virtual clinical trials, and to identify effective new therapies.

Lognormal distribution of daily travel time and a utility model for its emergence

Understanding the nature of human movement is important in many fields, such as transportation management, urban planning, and epidemiology. In this study, we focus on travelers who are grouped by their travel modes in the Tokyo Metropolitan Area. Notably, for each group of travelers, the distribution of daily travel time follows a lognormal distribution. For any travel mode, the distributions of normalized travel time for different years (1968–2008) and distances to the city center converge to a curve, although the average travel time and share of travel modes vary. By employing the concept of utility, we developed a disaggregate model wherein an individual maximizes utility from traveling (=benefit - cost) by modifying daily travel time. This model theoretically explained the observed distributions of travel time. The model was further used to demonstrate travel mode choices. The model parameters were estimated using the distribution of travel time, travel mode share, and manageable assumptions. These results provided interesting interpretations of the perceptual utility of the daily travel time of a population. Moreover, the obtained curves of utility versus time are similar to the shapes of the overall benefits assumed in previous studies. Overall benefit is a concept that has been developed and examined since the 1980s, and our study provides a mathematical foundation for this idea.

The Development Of Mathematics In Philosophy And The School Of Formalism Contained In The Philosophy Of Mathematics.

The philosophy of mathematics, with its many contradictions and paradoxes regarding systematic truths, contributes to the creation of a systematic and absolute foundation of mathematical knowledge, with the aim of preserving that foundation. This research aims to answer the question of what efforts are needed to overcome the crisis, contradictions and paradoxes in mathematics, as well as introduce the philosophical foundations of mathematics that are currently often used in the teaching and development of mathematics. The research method used is descriptive research using Systematic Literature Review (SLR). In particular, the emergence of formalism in mathematics education is also a reaction to formalism's attempt to impose the thinking of mathematicians on all humanity, thus forcing mathematics to be presented deductively. Therefore, as a suggestion, mathematics education should pay more attention to the diversity of abilities and potential of individual learners, realize that learners are not as smart as teachers, and that undergraduate and graduate students are not as smart as teachers.

Secure Messaging Using Post-Quantum Key Sharing Based on CSIDH and Fujisaki-Okamoto Transform

Preserving the confidentiality of information exchanges relies fundamentally on an end-to-end encryption system, involving the use of a secret key to secure the entire communication. However, with the imminent emergence of quantum computing, threats to traditional encryption systems are multiplying. This is where post-quantum key sharing, in particular the Commutative Super Isogenies Diffie Hellman (CSIDH) algorithm, comes in. The CSIDH uses the ideal of some class number to a morphism of elliptic curve for calculating the shared key. It offers an innovative solution for secure key generation between two users, while providing a robust defense against potential attacks from quantum computers, whose computing power is redefining the limits of cryptographic security, based on the mathematical foundations of elliptic curves and isogeny. This technological advance represents an essential pillar in preserving the confidentiality of communications, in a context where security challenges are constantly evolving. Combining with Fujisaki-Okamoto transform, socket, and Linux; a chat application over python could be created for transmitting secure messages.

Robustness Verification of Deep Reinforcement Learning Based Control Systems Using Reward Martingales

Deep Reinforcement Learning (DRL) has gained prominence as an effective approach for control systems. However, its practical deployment is impeded by state perturbations that can severely impact system performance. Addressing this critical challenge requires robustness verification about system performance, which involves tackling two quantitative questions: (i) how to establish guaranteed bounds for expected cumulative rewards, and (ii) how to determine tail bounds for cumulative rewards. In this work, we present the first approach for robustness verification of DRL-based control systems by introducing reward martingales, which offer a rigorous mathematical foundation to characterize the impact of state perturbations on system performance in terms of cumulative rewards. Our verified results provide provably quantitative certificates for the two questions. We then show that reward martingales can be implemented and trained via neural networks, against different types of control policies. Experimental results demonstrate that our certified bounds tightly enclose simulation outcomes on various DRL-based control systems, indicating the effectiveness and generality of the proposed approach.

A Review on PolSAR Decompositions for Feature Extraction.

Feature extraction plays a pivotal role in processing remote sensing datasets, especially in the realm of fully polarimetric data. This review investigates a variety of polarimetric decomposition techniques aimed at extracting comprehensive information from polarimetric imagery. These techniques are categorized as coherent and non-coherent methods, depending on their assumptions about the distribution of information among polarimetric cells. The review explores well-established and innovative approaches in polarimetric decomposition within both categories. It begins with a thorough examination of the foundational Pauli decomposition, a key algorithm in this field. Within the coherent category, the Cameron target decomposition is extensively explored, shedding light on its underlying principles. Transitioning to the non-coherent domain, the review investigates the Freeman-Durden decomposition and its extension, the Yamaguchi's approach. Additionally, the widely recognized eigenvector-eigenvalue decomposition introduced by Cloude and Pottier is scrutinized. Furthermore, each method undergoes experimental testing on the benchmark dataset of the broader Vancouver area, offering a robust analysis of their efficacy. The primary objective of this review is to systematically present well-established polarimetric decomposition algorithms, elucidating the underlying mathematical foundations of each. The aim is to facilitate a profound understanding of these approaches, coupled with insights into potential combinations for diverse applications.

Secure transmission and monitoring of ECG signals based on chaotic mapping algorithms

ABSTRACT The rapid advancement of digital healthcare and telemedicine has accentuated the need for robust and secure methods of transmitting sensitive medical data, such as electrocardiogram (ECG) signals. This article presents an innovative approach to ECG signal encryption, leveraging the power of chaotic dynamics through the Henon and Baker maps. Chaotic systems have proven to be formidable tools for encryption due to their inherent unpredictability and sensitivity to initial conditions. In this study, we demonstrate the efficacy of Chaotic Henon Map (CHM) and Chaotic Baker Maps (CBM) in encrypting ECG data, ensuring both data privacy and integrity during transmission. Through a detailed exploration of the CHM and CBM, we elucidate their mathematical foundations and unique properties that make them suitable for ECG encryption. The encryption process involves the generation of chaotic keys, which are utilized to scramble the ECG signal in a way that is virtually impossible to decipher without the correct decryption keys. We also discuss the security features of this encryption method, including resistance against common cryptographic attacks. The results of our experiments demonstrate the robustness of this encryption technique in safeguarding sensitive medical information, making it a valuable addition to the toolkit of secure ECG data transmission methods.

ПРОЄКТУВАННЯ БАГАТОПРОДУКТОВИХ ТРАНСПОРТНИХ МЕРЕЖ З ДИСКРЕТНИМИ ПОТОКАМИ МАГІСТРАЛЬНИХ ДАННИХ

The article considers the importance of transport and communication as key sectors for the development of the economy of Ukraine, as well as the need to create applied mathematical software for the optimization of transport processes and the development of information and analytical systems. Special attention is paid to the development and study of multi-product communication networks with discrete flows of small consignments, which have a significant impact on market transformations in Ukraine, especially in the field of transportation of small consignments. The study focuses on modern and future communication networks, which, as a rule, have a multi-level structure determined by the administrative division of the territory, management hierarchy and technologies for the processing and distribution of goods and information. The authors of the article emphasize the need to design multi-level communication networks, which would take into account the complexity of the system, as well as the impossibility of complete centralization of information processing and management in one link. The article also focuses on the analysis of existing research and publications in this field, paying attention to the contribution of foreign and domestic scientists to the development of models and methods of network resource management. Most research focuses on deterministic models and models based on the mathematical foundations of mass service theories and random Markov processes. The authors indicate the inadequacy of the existing mathematical tools for modeling and analyzing the functioning of multi-product networks with discrete flows, which requires the development of new mathematical models and methods.The purpose of the research is the development and analysis of effective approaches to the design of multi-tracks pipeline transport networks with a special focus on discrete trunk data flows. The authors note that this includes the development of new approaches and methods for effective management and optimization of resource allocation, as well as modeling the specifics of discrete flows in the context of trunk transport systems. An important part of the research is the creation of a hierarchical system structure for automated resource management and flow distribution in large-scale networks. The authors emphasize the need to develop a methodology that integrates already existing achievements in this field and allows to effectively solve practical problems related to long-term development, ongoing planning and operational management in large-scale networks. Such a methodology should include models of different levels of aggregation and be suitable for all levels of the network’s hierarchical structure. The study also includes consideration of mathematical models and algorithms for modeling and analyzing the functioning of multi-product networks with discrete flows.