Hasil untuk "Mathematics"

B. Thompson, John V. Tucker, J. Zucker

A. Sfard

B. Mandelbrot, Michael Aizenman

P. G. Ciarlet

F. H. Adler

A. Berman, R. Plemmons

Lolav Ahmed Khalil

This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed mathematical formula. Additionally, the study found that even numbers greater than or equal to 8, with six or more factors, produce complex numbers. Thus, this research provides two main contributions: firstly, a mathematical formula for the distribution of prime numbers, and secondly, a formula for the distribution of complex numbers. These findings have potential applications in various mathematical fields, including cryptography and problem-solving in number theory.

H. Margenau, G. M. Murphy, V. Twersky

Diana T. Stoeva

This paper was motivated by the worldwide May 12 initiative that aims to celebrate, encourage, and inspire women in mathematics. It presents in short how the May 12 initiative has arisen, what are some of the events in the first years, in particular the Generalized functions online workshop that started in 2021 in this context (and has continued as an annual event ever since), and a brief overview of some female mathematicians who have significant scientific contributions and who are the first women in some aspects: Maryam Mirzakhani (the first female mathematician who was awarded with the prestigious Fields medal; the May 12 initiative appeared in her honour), Hypatia (considered to be the earliest known female mathematician), Sofia Kovalevskaya (the first woman who has been awarded a doctorate in mathematics and considered to be the first woman who got a full professorship in mathematics in the modern academic sense), Emmy Noether (the first woman who gave a plenary lecture at the International Congress of Mathematicians), Karen Uhlenbeck (the first woman awarded with the prestigious Abel Prize), and Ingrid Daubechies (the first woman who became a full professor in mathematics at Princeton University, the first woman elected for President of the International Mathematical Union, the first woman who received the National Academy of Sciences Award in Mathematics). The paper was an invited contribution for the Akademie Intakt of the Austrian Academy of Sciences. It appeared in Akademie Intakt 2021, English Edition, Austrian Academy of Sciences, Vienna, 2021, 10-13, and it is submitted to arXiv with a permission from the publisher.

Maissam Barkeshli, Michael R. Douglas, Michael H. Freedman

Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we further consider how AI may open a grand perspective on mathematics by forging a new route, complementary to mathematical\textbf{ logic,} to understanding the global structure of formal \textbf{proof}\textbf{s}. We begin by providing a sketch of the formal structure of mathematics in terms of universal proof and structural hypergraphs and discuss questions this raises about the foundational structure of mathematics. We then outline the main ingredients and provide a set of criteria to be satisfied for AI models capable of automated mathematical discovery. As we send AI agents to traverse Platonic mathematical worlds, we expect they will teach us about the nature of mathematics: both as a whole, and the small ribbons conducive to human understanding. Perhaps they will shed light on the old question: "Is mathematics discovered or invented?" Can we grok the terrain of these \textbf{Platonic worlds}?

Xiaoyang Chen, Xiaoyang Chen

This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through clear explanations, the text explores how AI can discover hidden mathematical patterns, assist in proving complicated theorems, and even construct counterexamples to challenge conjectures.

Yang-Hui He

We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines in the mathematical sciences. At the heart is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.

Jakub Fusiak, Andreas Wolkenstein, Verena S. Hoffmann

BackgroundPatient preferences are a critical component of shared decision-making (SDM), particularly when choosing between treatment options with differing risks and outcomes. Many methods exist to elicit these preferences, but their complexity, usability, and acceptance vary.ObjectiveWe aim to gain insight into the acceptance, effort and preferences of participants regarding five different methods of preference assessment. Additionally, we investigate the influence of health status, experiences within the health system and of demographic factors on the results.MethodsWe conducted a cross-sectional online survey including five preference elicitation Methods: best-worst scaling, direct weighting, PAPRIKA (Potentially All Pairwise Rankings of all Possible Alternatives), time trade-off, and standard gamble. The questionnaire was distributed via academic and patient advocacy mailing lists, reaching both healthy individuals and those with acute or chronic illnesses. Participants rated each method using six standardized statements on a 5-point Likert scale. Additional items assessed general acceptance of algorithm-assisted preference assessments and the clarity of the questionnaire.ResultsOf 258 initiated questionnaires, 123 (48%) were completed and included in the analysis. Participants were diverse in age, gender, and health status, but predominantly highly educated and digitally literate. Across all measures, the PAPRIKA method received the highest ratings for clarity, usability, and perceived ability to express preferences. Simpler methods (best-worst scaling, direct weighting) were rated as less useful for capturing nuanced preferences, while abstract utility-based methods (standard gamble, time trade-off) were seen as cognitively demanding. Subgroup analyses showed minimal variation across demographic groups. Most participants (82%) could imagine using at least one of the presented methods in real clinical settings, but also emphasized the importance of physician involvement in interpreting results.ConclusionThe interactive PAPRIKA method best balanced cognitive demand and expressiveness and was preferred by most participants. Structured methods for preference elicitation may enhance SDM when integrated into clinical workflows and supported by healthcare professionals. Further research is needed to evaluate their use in real-world decisions and among more diverse patient populations.

Harbir Antil

The report describes the discussions from the Workshop on Mathematical Opportunities in Digital Twins (MATH-DT) from December 11-13, 2023, George Mason University. It illustrates that foundational Mathematical advances are required for Digital Twins (DTs) that are different from traditional approaches. A traditional model, in biology, physics, engineering or medicine, starts with a generic physical law (e.g., equations) and is often a simplification of reality. A DT starts with a specific ecosystem, object or person (e.g., personalized care) representing reality, requiring multi -scale, -physics modeling and coupling. Thus, these processes begin at opposite ends of the simulation and modeling pipeline, requiring different reliability criteria and uncertainty assessments. Additionally, unlike existing approaches, a DT assists humans to make decisions for the physical system, which (via sensors) in turn feeds data into the DT, and operates for the life of the physical system. While some of the foundational mathematical research can be done without a specific application context, one must also keep specific applications in mind for DTs. E.g., modeling a bridge or a biological system (a patient), or a socio-technical system (a city) is very different. The models range from differential equations (deterministic/uncertain) in engineering, to stochastic in biology, including agent-based. These are multi-scale hybrid models or large scale (multi-objective) optimization problems under uncertainty. There are no universal models or approaches. For e.g., Kalman filters for forecasting might work in engineering, but can fail in biomedical domain. Ad hoc studies, with limited systematic work, have shown that AI/ML methods can fail for simple engineering systems and can work well for biomedical problems. A list of `Mathematical Opportunities and Challenges' concludes the report.

Michel Caffarel, Pierre del Moral, Luc de Montella

The Diffusion Monte Carlo method with constant number of walkers, also called Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used Monte Carlo methodology for computing the ground-state energy and wave function of quantum systems. In this study, we present the first mathematically rigorous analysis of this class of stochastic methods on non necessarily compact state spaces, including linear diffusions evolving in quadratic absorbing potentials, yielding what seems to be the first result of this type for this class of models. We present a novel and general mathematical framework with easily checked Lyapunov stability conditions that ensure the uniform-in-time convergence of Diffusion Monte Carlo estimates towards the top of the spectrum of Schrödinger operators. For transient free evolutions, we also present a divergence blow up of the estimates w.r.t. the time horizon even when the asymptotic fluctuation variances are uniformly bounded. We also illustrate the impact of these results in the context of generalized coupled quantum harmonic oscillators with non necessarily reversible nor stable diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force.

Martin Klazar

We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only hereditarily at most countable (HMC) sets.

Journal of Control Science and Engineering

Anirban Chaudhuri, Graham Pash, David A. Hormuth et al.

We develop a methodology to create data-driven predictive digital twins for optimal risk-aware clinical decision-making. We illustrate the methodology as an enabler for an anticipatory personalized treatment that accounts for uncertainties in the underlying tumor biology in high-grade gliomas, where heterogeneity in the response to standard-of-care (SOC) radiotherapy contributes to sub-optimal patient outcomes. The digital twin is initialized through prior distributions derived from population-level clinical data in the literature for a mechanistic model's parameters. Then the digital twin is personalized using Bayesian model calibration for assimilating patient-specific magnetic resonance imaging data. The calibrated digital twin is used to propose optimal radiotherapy treatment regimens by solving a multi-objective risk-based optimization under uncertainty problem. The solution leads to a suite of patient-specific optimal radiotherapy treatment regimens exhibiting varying levels of trade-off between the two competing clinical objectives: (i) maximizing tumor control (characterized by minimizing the risk of tumor volume growth) and (ii) minimizing the toxicity from radiotherapy. The proposed digital twin framework is illustrated by generating an in silico cohort of 100 patients with high-grade glioma growth and response properties typically observed in the literature. For the same total radiation dose as the SOC, the personalized treatment regimens lead to median increase in tumor time to progression of around six days. Alternatively, for the same level of tumor control as the SOC, the digital twin provides optimal treatment options that lead to a median reduction in radiation dose by 16.7% (10 Gy) compared to SOC total dose of 60 Gy. The range of optimal solutions also provide options with increased doses for patients with aggressive cancer, where SOC does not lead to sufficient tumor control.

XIONG Zhongmin, ZENG Qi, LU Peng, WANG Zhenhua, ZHENG Zongsheng

Logical reasoning is the ability to perceive patterns and connections between visual elements. Endowing computers with human-like reasoning ability is a critical area of research；state-of-the-art deep neural networks have achieved superhuman performance in image processing and other fields.However，the concept of logical reasoning through images requires further research.To address the problems of insufficient feature extraction and generalization of Multi-scale Relation Network（MRNet），an improved logical reasoning method，called Residual Attention Multi-scale Relation Network（ResAMRNet），is proposed. In the backbone network，shallow features are integrated into the deep network training process by utilizing residual structures and combining jump and long jump. This reduces the loss of feature information and improves the feature extraction capability of the model. In the reasoning module，the channel attention mechanism and residuals are combined to detect the relationship features between each image line.It can differentiate the significance of each feature channel，learn the attention weight adaptively，and extract key features.In this study，a Double-pooled Efficient Channel Attention（DECA） mechanism is proposed to combine global maximum pooling to further obtain feature information regarding objects and to improve generalization.Experimental results on representative logical reasoning datasets，Relational and Analogical Visual rEasoNing（RAVEN） and Improved RAVEN（I-RAVEN），show that the accuracy of the proposed method using these datasets is higher by 8.3 and 18.1 percentage points，respectively，than that of MRNet. Therefore，it demonstrates strong logical reasoning capabilities.

R. Fateman, S. Wolfram