Mathematical Foundations Research Articles

On the friendship paradox and inversity: A network property with applications to privacy-sensitive network interventions

We provide the mathematical and empirical foundations of the friendship paradox in networks, often stated as "Your friends have more friends than you." We prove a set of network properties on friends of friends and characterize the concepts of ego-based and alter-based means. We propose a network property called inversity that quantifies the imbalance in degrees across edges and prove that the sign of inversity determines the ordering between ego-based or alter-based means for any network, with implications for interventions. Network intervention problems like immunization benefit from using highly connected nodes. We characterize two intervention strategies based on the friendship paradox to obtain such nodes, with the alter-based and ego-based strategy. Both strategies provide provably guaranteed improvements for any network structure with variation in node degrees. We demonstrate that the proposed strategies obtain several-fold improvement (100-fold in some networks) in node degree relative to a random benchmark, for both generated and real networks. We evaluate how inversity informs which strategy works better based on network topology and show how network aggregation can alter inversity. We illustrate how the strategies can be used to control contagion of an epidemic spreading across a set of village networks, finding that these strategies require far fewer nodes to be immunized (less than 50%, relative to random). The interventions do not require knowledge of network structure, are privacy-sensitive, are flexible for time-sensitive action, and only require selected nodes to nominate network neighbors.

Llarull type theorems on complete manifolds with positive scalar curvature

In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold ( M n , g ) (M^{n},g) with scalar curvature R g ≥ 6 R_{g}\geq 6 admits a non-zero degree and 1 1 -Lipschitz map to ( S 3 × T n − 3 , g S 3 + g T n − 3 ) (\mathbb {S}^{3}\times \mathbb {T}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {T}^{n-3}}) , for 4 ≤ n ≤ 7 4\leq n\leq 7 , then ( M n , g ) (M^{n},g) is locally isometric to S 3 × T n − 3 \mathbb {S}^{3}\times \mathbb {T}^{n-3} . Similar results are established for non-compact cases as ( S 3 × R n − 3 , g S 3 + g R n − 3 ) (\mathbb {S}^{3}\times \mathbb {R}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {R}^{n-3}}) being model spaces. We observe that the results differ significantly when n = 4 n=4 compared to n ≥ 5 n\geq 5 . Our results imply that the ϵ \epsilon -gap length extremality of the standard S 3 \mathbb {S}^3 is stable under the Riemannian product with R m \mathbb {R}^m , 1 ≤ m ≤ 4 1\leq m\leq 4 (see D 3 D_{3} . Question in Gromov’s paper [Foundations of mathematics and physics one century after Hilbert, Spring, Cham, 2018, p. 153]).

Applications in Linear Algebra: Bridging Data Science with Mathematical Foundations

Applications in Linear Algebra: Bridging Data Science with Mathematical Foundations

A Neglected Interpretation of Das Kontinuum

Hermann's Weyl Das Kontinuum has inspired several studies in logic and foundations of mathematics over the last century. The book provides a remarkable reconstruction of a large portion of classical mathematics on a predicative basis. However, diverging interpretations of the predicative system formulated by Weyl have been proposed in the literature. In the present work, I analyze an early formalization of Weyl's ideas proposed by [Casari, E. 1964. Questioni di Filosofia Della Matematica, Milano: Feltrinelli] and compare it with other, more well-known, accounts, such as those proposed respectively by Feferman and Avron. In this way, I fill a gap in the literature on Weyl's predicative mathematics by shedding light on an interpretation that has never been studied. Moreover, Casari's work is plausibly the first systematic reconstruction of Weyl's system. In particular, I highlight the different insights about predicativity that can be found in Weyl's work. Casari's reconstruction focuses on the logical and mathematical processes described by Weyl in the first part of his book. Through the analysis of this unexplored perspective, I shed light on fundamental aspects of Weyl's work and elucidate the alleged ambiguities it presents. In particular, in line with Avron's most recent reconstruction of Weyl's ideas, this analysis supports a stronger conception of predicativity than what is commonly attributed to Weyl in the literature. As a result, this work not only provides an analysis of a neglected interpretation of Das Kontinuum, but also contributes to elucidating Weyl's classical conception of predicativity.

Research on Autonomous Manoeuvre Decision Making in Within-Visual-Range Aerial Two-Player Zero-Sum Games Based on Deep Reinforcement Learning

In recent years, with the accelerated development of technology towards automation and intelligence, autonomous decision-making capabilities in unmanned systems are poised to play a crucial role in contemporary aerial two-player zero-sum games (TZSGs). Deep reinforcement learning (DRL) methods enable agents to make autonomous manoeuvring decisions. This paper focuses on current mainstream DRL algorithms based on fundamental tactical manoeuvres, selecting a typical aerial TZSG scenario—within visual range (WVR) combat. We model the key elements influencing the game using a Markov decision process (MDP) and demonstrate the mathematical foundation for implementing DRL. Leveraging high-fidelity simulation software (Warsim v1.0), we design a prototypical close-range aerial combat scenario. Utilizing this environment, we train mainstream DRL algorithms and analyse the training outcomes. The effectiveness of these algorithms in enabling agents to manoeuvre in aerial TZSG autonomously is summarised, providing a foundational basis for further research.

Assessing Machine Learning and Deep Learning Applications Across Diverse Domains

Applications in machine learning (ML) and deep learning (DL) are subjected to a comprehensive analysis of their algorithmic principles, mathematical foundations, and performance indicators as part of the theoretical evaluation process. The purpose of this review is to get an understanding of the various machine learning and deep learning models, including their ability for generalization, convergence characteristics, and computational efficiency, as well as knowing their strengths and shortcomings. Through the investigation of theoretical issues such as the bias-variance tradeoff, overfitting, underfitting, and the influence of hyperparameters, researchers have the ability to enhance model designs and training techniques. As an additional benefit, the theoretical analysis offers insight on the robustness and interpretability of models, which in turn guides the development of applications that are more trustworthy and successful in a range of domains, such as computer vision, natural language processing, and predictive analytics.

The Intersection of Algebra and Cryptography: Enhancing Information Security through Mathematical Foundations

The rapid advancements in digital technologies have necessitated the development of robust information security measures. This paper explores the intersection of algebra and cryptography, focusing on how algebraic principles can enhance cryptographic techniques to provide stronger security foundations. By leveraging mathematical structures such as groups, rings, and fields, we can address critical challenges in encryption, secure communications, and data privacy. This study reviews key algebraic methods used in contemporary cryptographic protocols, including elliptic curve cryptography, homomorphic encryption, and lattice-based cryptography, and demonstrates their practical applications through detailed case studies. Our comparative analysis highlights the superior performance and security of algebra-based cryptographic solutions compared to traditional methods. Finally, we discuss the emerging trends and future directions in algebraic cryptography, emphasizing the potential of these mathematical foundations to address the evolving threats in information security.

Defining a Metric-Driven Approach for Learning Hazardous Situations

Artificial intelligence has brought many innovations to our lives. At the same time, it is worth designing robust safety machine learning (ML) algorithms to obtain more benefits from technology. Reinforcement learning (RL) being an important ML method is largely applied in safety-centric scenarios. In such a situation, learning safety constraints are necessary to avoid undesired outcomes. Within the traditional RL paradigm, agents typically focus on identifying states associated with high rewards to maximize its long-term returns. This prioritization can lead to a neglect of potentially hazardous situations. Particularly, the exploration phase can pose significant risks, as it necessitates actions that may have unpredictable consequences. For instance, in autonomous driving applications, an RL agent might discover routes that yield high efficiency but fail to account for sudden hazardous conditions such as sharp turns or pedestrian crossings, potentially leading to catastrophic failures. Ensuring the safety of agents operating in unpredictable environments with potentially catastrophic failure states remains a critical challenge. This paper introduces a novel metric-driven approach aimed at containing risk in RL applications. Central to this approach are two developed indicators: the Hazard Indicator and the Risk Indicator. These metrics are designed to evaluate the safety of an environment by quantifying the likelihood of transitioning from safe states to failure states and assessing the associated risks. The fact that these indicators are characterized by a straightforward implementation, a highly generalizable probabilistic mathematical foundation, and a domain-independent nature makes them particularly interesting. To demonstrate their efficacy, we conducted experiments across various use cases, showcasing the feasibility of our proposed metrics. By enabling RL agents to effectively manage hazardous states, this approach paves the way for a more reliable and readily implementable RL in practical applications.

Formal Methods for Security Protocol Verification: Model Checking and Theorem Proving Approaches

Formal methods are a key part of making sure that cryptographic systems are safe and reliable. For the purpose of checking security protocols, this paper looks into two well-known formal methods: model checking and theorem proving. When you do model checking, you go through all of a system's possible states to see if it meets certain criteria. On the other hand, theorem proving uses formal evidence and mathematical reasoning to show that protocols are correct. In the beginning of the paper, an outline of security protocols and the need for formal proof methods to make sure they are strong against possible threats is given. Then it goes into the ideas and methods behind model checking and shows how it can be used in security protocol analysis. Model checking lets you do automatic checks, which lets you look into all the possible states of a system and find any security holes. The study then looks at theorem proving as an alternative way to check protocols. Theorem proving uses mathematical thinking to make formal proofs that show protocols are right. Although theorem proving usually needs help from a person with the right knowledge, it gives more security by making sure everything is right using strong mathematical foundations. It is talked about what the pros and cons of each method are, taking things like scale, automation, and expression into account. It also talks about important improvements and new tools in each method, showing how people are still working to make security protocol testing techniques more efficient and effective. The last part of the paper compares model checking and theorem proving, focusing on how they work together and how important it is to use both to make sure the security of cryptographic protocols. Also, it shows how important it is to choose the right formal methods based on the needs and features of the protocol in question.

Different negations and fuzziness in Web resources and resource description and their mathematical foundation

In Web resources and resource descriptions, many resources and descriptions of resources have different negations and fuzziness. In this paper, we investigate and analyze the cognition and characteristics of “negation” in philosophy, logic, linguistics and other knowledge fields, and based on the distinction between the concept of “contradiction” and the concept of “opposition”, and “clear concept” and “fuzzy concept”, we propose that there are three different kinds of negation in knowledge from the conceptual level, namely “contradictory negation”, “opposite negation” and “intermediary negation”. In order to establish a mathematical basis that can fully reflect the internal relations, properties and laws of these different negations and fuzziness, we propose a “Scoi: sets with contradictory negation, opposite negation and intermediary negation” and “Lcoi&PLcoi: logic with contradictory negation, opposite negation and intermediary negation”, and the operational properties of Scoi and the formal deduction properties of Lcoi have been proven. The soundness theorem, completeness theorem and compactness theorem for Lcoi are proved under the three-valued semantic interpretation of Lcoi, and the soundness and completeness for Lcoi&PLcoi are discussed under the given an infinite-valued semantics interpretation of Lcoi&PLcoi. Furthermore, we also demonstrated that descriptions of objective things such as “x is neither a positive nor a negative integer” and “he is neither rich nor poor” typle of statements are “intermediary negation” in Scoi and Lcoi&PLcoi. It shows that “intermediary negation” in Scoi and Lcoi&PLcoi has objectively.In order to enable RDF(Resource Description Framework) to describe Web resources with different negations and fuzziness, we extend RDF based on Scoi and Lcoi&PLcoi, and propose a resource description framework RDFcoi which can fully handle fuzzy attribute values and their different negations in attributes of resources. How to reflect the uncertainty of the relationship between Web resources and its attribute values, we believe that the relationship between a resource and fuzzy attribute value in one of its attributes should be a composite mapping from the resource domain to the fuzzy attribute value set and then to [0, 1], and a mathematical representation is given in this paper. Thus, RDFcoi differ from other RDF extensions. RDFcoi maintains the basic syntax and normative of RDF, and extends the basic syntax and semantics of RDF. Where, the concepts of RDFcoi triple with three kinds of negation, RDFcoi formula and RDFcoi graph are defined. RDFcoi vocabulary is determined and an interpretation is given. RDFcoi-entailment is defined and its properties are discussed. In addition, we also study the reasoning regimes of RDFcoi. By using the given logical translation method, a well-typed and interoperable RDFcoi graph can be transformed into a set of formulas in Lcoi&PLcoi, thus establishing “interoperability” between RDFcoi and Lcoi&PLcoi. From the perspective of reasoning mechanism of RDFcoi-entailment, RDFcoi reasoning can be carried out in Lcoi&PLcoi, and RDFcoi has the same reasoning mechanism as formal inference in Lcoi&PLcoi.

Upaya Meningkatkan Kemampuan Numerasi Anak dalam Menulis Angka 8 melalui Media Vituleka

The learning strategies and media used by educators will influence the success of achieving learning objectives. The use of appropriate media that suits the character of early childhood can support the achievement of learning goals effectively. Good numeracy skills at an early age are a strong foundation for further mathematics learning in the future. This research is based on children's ability to write the number eight (number 8) which is still imperfect. In general, children make the number eight by stacking circles from bottom to top. The aim of this research is to improve the numeracy skills of special children in writing the number 8. This classroom action research was carried out in group A at UNESA Labschool 1 Kindergarten. This classroom action research uses the Kemmis and Mc.Taggart method with two cycles. Each cycle includes planning, implementation, observation and reflection. This research uses video tutorials and children's worksheets (Vituleka). The subjects of this research were 18 group A students. In cycle I, students' numeracy ability in writing the number 8 was 50%. In cycle II, numeracy ability in writing the number 8 became 83.3%, an increase of 33.3% from the previous cycle. Based on research data, Vituleka media can improve students' ability to write the number 8.

A comparative evaluation of measures to assess randomness in human-generated sequences

Whether and how well people can behave randomly is of interest in many areas of psychological research. The ability to generate randomness is often investigated using random number generation (RNG) tasks, in which participants are asked to generate a sequence of numbers that is as random as possible. However, there is no consensus on how best to quantify the randomness of responses in human-generated sequences. Traditionally, psychologists have used measures of randomness that directly assess specific features of human behavior in RNG tasks, such as the tendency to avoid repetition or to systematically generate numbers that have not been generated in the recent choice history, a behavior known as cycling. Other disciplines have proposed measures of randomness that are based on a more rigorous mathematical foundation and are less restricted to specific features of randomness, such as algorithmic complexity. More recently, variants of these measures have been proposed to assess systematic patterns in short sequences. We report the first large-scale integrative study to compare measures of specific aspects of randomness with entropy-derived measures based on information theory and measures based on algorithmic complexity. We compare the ability of the different measures to discriminate between human-generated sequences and truly random sequences based on atmospheric noise, and provide a systematic analysis of how the usefulness of randomness measures is affected by sequence length. We conclude with recommendations that can guide the selection of appropriate measures of randomness in psychological research.

Deconvolution of dynamic heterogeneity in protein structure.

Heterogeneity is intrinsic to the dynamic process of a chemical reaction. As reactants are converted to products via intermediates, the nature and extent of heterogeneity vary temporally throughout the duration of the reaction and spatially across the molecular ensemble. The goal of many biophysical techniques, including crystallography and spectroscopy, is to establish a reaction trajectory that follows an experimentally provoked dynamic process. It is essential to properly analyze and resolve heterogeneity inevitably embedded in experimental datasets. We have developed a deconvolution technique based on singular value decomposition (SVD), which we have rigorously practiced in diverse research projects. In this review, we recapitulate the motivation and challenges in addressing the heterogeneity problem and lay out the mathematical foundation of our methodology that enables isolation of chemically sensible structural signals. We also present a few case studies to demonstrate the concept and outcome of the SVD-based deconvolution. Finally, we highlight a few recent studies with mechanistic insights made possible by heterogeneity deconvolution.

HOW ARE STUDENTS' CREATIVE REASONING ABILITIES IN SOLVING STRAIGHT-LINE EQUATION PROBLEMS?

Creative reasoning is a person's ability to solve mathematical problems in a manner of thinking that deviates from the norm, yet remains logical and grounded in a strong mathematical foundation. While crucial for students, many have not yet attained proficiency in creative reasoning, necessitating comprehensive research on the subject. This study aims to elucidate students' creative reasoning abilities in tackling straight-line equation problems, focusing on indicators such as creativity, plausibility, and anchoring. Employing a qualitative approach with a case study design, the research involved three students from Madrasah Tsanawiyah who had studied straight-line equations. The primary instrument was the researcher, supported by a creative reasoning ability description test and interviews. Data collection utilized triangulation techniques, encompassing tests and interviews. Analysis of the research findings entailed describing students' creative mathematical reasoning abilities for each creative reasoning indicator in straight-line equation material. Results indicated that while students could provide correct and reasonable arguments in solving mathematical problems, they struggled to devise alternative methods and employ strategies based on mathematical concepts. Consequently, they only met the plausibility criterion, falling short in creativity and anchoring. Thus, it can be concluded that students have not yet achieved creative reasoning proficiency in solving straight-line equation problems.

Cover-time Gumbel fluctuations in finite-range, symmetric, irreducible random walks on torus

In this paper, we provide the mathematical foundation for an explicit and universal feature of cover time for a large class of random work processes, which was previously observed by Chupeau et al (2015 Nat. Phys. 11 844–7). Specifically, we rigorously establish that the fluctuations of the cover time, normalized by the mean first passage time, follow a Gumbel distribution, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. The result contributes to a better understanding of cover-time behavior in random search processes, especially on the efficiency of exhaustive searches. Our approach builds upon the work of Belius (2013 Probab. Theory Relat. Fields 157 635–89) on cover times for simple random walks, leveraging a strong coupling between the random walk and random interlacements.

Quantum Cryptography: Mathematical Foundations and Practical Applications for Secure Communication Protocols

Quantum cryptography is a new and exciting area that uses quantum physics to protect communication lines from being spied on or intercepted. Fundamental ideas in this field, like the uncertainty principle and the fact of quantum entanglement, are used to achieve levels of security that have never been seen before. Our in-depth study, "Quantum Cryptography: Mathematical Foundations and Practical Applications for Secure Communication Protocols," looks at both the math behind quantum cryptographic protocols and how they are used in the real world. Our study goes into great detail about the theories behind quantum cryptography. It explains ideas like quantum key distribution (QKD), quantum teleportation, and quantum secure direct communication (QSDC). One of the main ideas behind quantum cryptography is the idea of qubits, which are like regular bits but in quantum mechanics. They can be in more than one state at the same time because of superposition. Quantum cryptographic methods use this property to make sure communication is safe by putting data in quantum states and taking advantage of the fact that quantum measures are inherently unpredictable. The study we're doing looks at how to use quantum cryptography in typical everyday situations. We look at the problems that come up when you try to build infrastructure for quantum transmission, such as noise, decoherence, and scale. We make a plan for making strong and trustworthy quantum cryptographic systems by giving details about how experiments are set up and how technology is improving. Our study looks into how quantum cryptography could be used for things other than just keeping communications safe. We look into what it means for new technologies like quantum networks, quantum computing, and safe multi-party processing. We hope that by explaining the bigger effects of quantum cryptography, we can encourage more study and new ideas in this ground-breaking area.

Wittgenstein and the liar

Ludwig Wittgenstein’s remarks on contradictions and paradoxes have been met with incomprehension and have fueled the widespread and long-standing prejudice that his later thoughts on the foundations of logic and mathematics are the “surprisingly insignificant product of a sparkling mind” (Kreisel, British Journal for the Philosophy of Science 9:135–158, 1959, p. 158). This paper disagrees; it argues that Wittgenstein’s remarks on semantic paradoxes suggest an account of the Liar and its kin that is not only of historical interest but also represents a hitherto unnoticed paraconsistent alternative to established approaches to the Liar. In what follows, a reading of Wittgenstein’s remarks will be offered according to which Wittgenstein subscribes to a form of dialetheism (that is, the view that there are sentences that are both true and false). In contrast to modern dialetheist approaches to the Liar, however, some of Wittgenstein’s remarks suggest combining a dialetheist position with what is called ‘logical nihilism’ (that is, the view that there are no universally valid inference rules).

Controlled remote implementation of operations for many systems

Quantum communication plays a key role in the next generation of information transfer and security schemes, by using quantum entanglement and measurements from quantum mechanics. We introduce the mathematical and physical foundations of quantum communication, such as the CNOT gate and Kronecker product. Then we propose two quantum communication protocols, namely dense coding and stealth coding. These protocols offer unique advantages that are theoretically unbreakable. Then we extend the protocols to the quantum communication protocol allowing for third-party supervision to ensure information security. Based on this, a communication protocol involving four parties is designed, enabling them to exchange information while being supervised.

A randomized controlled study of a second grade numeracy intervention with Swedish students at-risk of mathematics difficulties.

Early numeracy interventions including basic arithmetic are crucial for young students at risk for early mathematics difficulties (MDs), yet few studies have evaluated numeracy interventions in second grade with a randomized controlled design. This pre- and post-test randomized controlled study evaluated the effects of an intensive 9-week numeracy and arithmetic programme for second-grade students at risk for early MDs. The focus of the programme was students' foundational understanding of numbers and mathematical concepts and procedural fluency with arithmetic tasks. A total of 753 first-grade students from 21 schools in Sweden were screened for low achievement in number knowledge and arithmetic. Students considered at risk for MDs (≤25 percentile on two consecutive first-grade mathematics screenings) were individually randomized to an intervention group (n = 32) or control group (n = 30) at the beginning of second grade (7-8 years old). Trained teachers administered the one-to-one, explicit programme to intervention group students in elementary school settings. The intervention group received numeracy instruction emphasizing foundational mathematics concepts and procedures. Controls received teaching as usual with potential special education support provided by their schools. The intervention group demonstrated significantly greater improvements in conceptual knowledge, arithmetic calculations and problem-solving compared to the control group, with medium size effects observed. A supplemental and intensive programme, with explicit instruction emphasizing numeracy, substantially improved knowledge and skills essential for arithmetic learning. Instruction in conceptual number knowledge and procedures also shows a significant impact on basic arithmetic problem-solving.

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.