Lev Nohimovich Kantorovitch · Research Portal (King's College London) · 2016
This book covers the advanced mathematical techniques useful for physics and engineering students, presented in a form accessible to physics students, avoiding precise mathematical jargon and laborious proofs. Instead, all proofs are given in a simplified form that is clear and convincing for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each chapter.
Cited by 0Open access
B. Vaĭnberg · 2024
This book provides a single source for both students and advanced researchers on asymptotic methods employed in the linear problems of mathematical physics. It opens with a section based on material from special courses given by the author, which gives detailed coverage of classical material on the equations of mathematical physics and their applications, and includes a simple explanation of the Maslov Canonical Operator method. The book goes on to present more advanced material from the author's own research. Topics range from radiation conditions and the principle of limiting absorption for general exterior problems, to complete asymptotic expansion of spectral function of equations over all of space. This book serves both as a manual and teaching aid for students of mathematics and physics and, in summarizing for the first time in a monograph problems previously investigated in journal articles, as a comprehensive reference for advanced researchers.
Cited by 334
Robert P. Gilbert, Roger G. Newton · Defense Technical Information Center (DTIC) · 1970
The event of the Symposium on Analytic Methods in Mathematical Physics, 2 through 6 June 68, marked the end of the third year of an interdisciplinary program in mathematical physics at Indiana University. This joint program was initiated because both departments were aware of mathematical physics as an important discipline in its own right. The advanced seminars in scattering theory, several complex variables, differential equations, elasticity, and general methods of mathematical physics made the participants even more conscious of the central role played by complex variable techniques in theoretical physics. As anticipated, a symposium which would stress the uses of complex analysis and related methods for the solving of problems motivated by physics was of great interest to mathematicians and physicists alike.
Cited by 159
Zheshen Gao, Tianyi Li, Yurii Kvasiuk et al. · ArXiv.org · 2025
Large language models (LLMs) have shown strong capabilities in complex reasoning, and test-time scaling techniques can enhance their performance with comparably low cost. Many of these methods have been developed and evaluated on mathematical reasoning benchmarks such as AIME. This paper investigates whether the lessons learned from these benchmarks generalize to the domain of advanced theoretical physics. We evaluate a range of common test-time scaling methods on the TPBench physics dataset and compare their effectiveness with results on AIME. To better leverage the structure of physics problems, we develop a novel, symbolic weak-verifier framework to improve parallel scaling results. Our empirical results demonstrate that this method significantly outperforms existing test-time scaling approaches on TPBench. We also evaluate our method on AIME, confirming its effectiveness in solving advanced mathematical problems. Our findings highlight the power of step-wise symbolic verification for tackling complex scientific problems.
Cited by 0Open access
K.T. Feng, Yilun Zhao, Yixin Liu et al. · ArXiv.org · 2025
We introduce PHYSICS, a comprehensive benchmark for university-level physics problem solving. It contains 1297 expert-annotated problems covering six core areas: classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, atomic physics, and optics. Each problem requires advanced physics knowledge and mathematical reasoning. We develop a robust automated evaluation system for precise and reliable validation. Our evaluation of leading foundation models reveals substantial limitations. Even the most advanced model, o3-mini, achieves only 59.9% accuracy, highlighting significant challenges in solving high-level scientific problems. Through comprehensive error analysis, exploration of diverse prompting strategies, and Retrieval-Augmented Generation (RAG)-based knowledge augmentation, we identify key areas for improvement, laying the foundation for future advancements.
Cited by 0Open access
Michela Petrini, Gianfranco Pradisi, Alberto Zaffaroni · Advanced textbooks in physics · 2018
This book provides a self-contained and rigorous presentation of the main mathematical tools needed to approach many courses at the last year of undergraduate in Physics and MSc programs, from Electromagnetism to Quantum Mechanics. It complements A Guide to Mathematical Methods for Physicists with advanced topics and physical applications. The different arguments are organised in three main sections: Complex Analysis, Differential Equations and Hilbert Spaces, covering most of the standard mathematical method tools in modern physics. One of the purposes of the book is to show how seemingly different mathematical tools like, for instance, Fourier transforms, eigenvalue problems, special functions and so on, are all deeply interconnected. It contains a large number of examples, problems and detailed solutions, emphasising the main purpose of relating concrete physical examples with more formal mathematical aspects.
Cited by 1
Christof Keebaugh, Emily Marshman, Chandralekha Singh · Physical Review Physics Education Research · 2024
[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] We discuss an investigation of student sensemaking and reasoning in the context of degenerate perturbation theory (DPT) in quantum mechanics. We find that advanced undergraduate and graduate students in quantum physics courses often struggled with expertlike sensemaking and reasoning to solve DPT problems. The sensemaking and reasoning were particularly challenging for students as they tried to integrate physical and mathematical concepts to solve DPT problems. Their sensemaking showed local coherence but lacked global consistency with different knowledge resources getting activated in different problem-solving tasks even if the same concepts were applicable. Depending upon the issues involved in the DPT problems, students were sometimes stuck in the “physics mode” or “math mode” and found it challenging to coordinate and integrate the physics and mathematics appropriately to solve quantum mechanics problems involving DPT. Their sensemaking shows the use of various reasoning primitives. It also shows that some advanced students struggled with self-monitoring and checking their answers to make sure they were consistent across different problems. Some also relied on memorized information, invoked authority, and did not make appropriate connections between their DPT problem solutions and the outcomes of experiments. Advanced students in quantum mechanics often displayed analogous patterns of challenges in sensemaking and reasoning as those that have been found in introductory physics. Student sensemaking and reasoning show that these advanced students are still developing expertise in this novel quantum physics domain as they learn to integrate physical and mathematical concepts. Published by the American Physical Society 2024
Cited by 10Open access
K.T. Feng, Yilun Zhao, Yixin Liu et al. · 2025
We introduce PHYSICS, a comprehensive benchmark for university-level physics problem solving.It contains 1,297 expert-annotated problems covering six core areas: classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, atomic physics, and optics.Each problem requires advanced physics knowledge and mathematical reasoning.We develop a robust automated evaluation system for precise and reliable validation.Our evaluation of leading foundation models reveals substantial limitations.Even the most advanced model, o3-mini, achieves only 59.9% accuracy, highlighting significant challenges in solving high-level scientific problems.Through comprehensive error analysis, exploration of diverse prompting strategies, and Retrieval-Augmented Generation (RAG)-based knowledge augmentation, we identify key areas for improvement, laying the foundation for future advancements.
Cited by 0Open access
Malcolm P. Kennett · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 0
Malcolm P. Kennett · Cambridge University Press eBooks · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 0
Malcolm P. Kennett · Cambridge University Press eBooks · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 4
Malcolm P. Kennett · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 0
WORLD SCIENTIFIC eBooks · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 31
Malcolm P. Kennett · Cambridge University Press eBooks · 2020
This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging topics in the core physics curriculum and provides students with a solid foundation for tackling advanced topics in statistical mechanics.
Cited by 0
Association for Computational Linguistics 2025, Chen, Ling, Fang, Meng et al. · Underline Science Inc. · 2025
Retrieval-augmented generation (RAG) with foundation models has demonstrated notable capabilities across various tasks and domains, showcasing advanced problem-solving skills in natural language understanding, generation, and complex decision-making processes. However, their proficiency in addressing expert-level reasoning, such as solving mathematical physics problems, remains relatively unexplored. This paper investigates the potential of RAG techniques to solve Olympic-level mathematical physics problems, motivated by students' natural inclination to reference past problems when preparing for competitions. We propose PhOPile, a high-quality, multimodal, physics-specific dataset tailored for Olympic-level challenges. In this dataset, problems from 2021 serve as the evaluation set, while data from other years form the knowledge base for RAG. Additionally, we benchmark RAG methods with foundation models to report and analyze results that can inform future research. Additionally, we highlight the significant role of reflection through experimental results. Furthermore, we conduct a high-granularity evaluation of the performance of popular LLMs and large multimodal models (LMMs) on our dataset, assessing their physics reasoning and RAG capabilities.
Cited by 0Open access
Yanxiang Gong, Juan Guo, Chuantao Ma et al. · 2022 IEEE Asia-Pacific Conference on Image Processing, Electronics and Computers (IPEC) · 2022
It is a research direction of educational technology that advanced computer language can be used to promote the modernization of physics teaching. Theoretical physics is characterized by abstract concepts, difficult to understand physical images and complex formula derivation, which makes students feel difficult in learning theoretical physics. Among the more popular scientific computing languages, Maple language has unique advantages in the modern application of theoretical physics teaching, including powerful symbolic computing function, intuitive language, a variety of built-in commands and functions, and solving general mathematical problems without complex programming. Taking two typical theoretical physics problems as examples, this paper demonstrates the application of Maple language in theoretical physics teaching techniques. These teaching techniques enable students to intuitively understand physical images. It uses powerful scientific computing functions to make it easier for college students to participate in research training or solve engineering problems.
Cited by 1
Lawrence Marcotte · FHSU Scholars Repository (Fort Hays State University) · 1950
The object of the author in writing this thesis was to make a compilation of material necessary to work certain problems in mathematical physics with one use of Fourier series. Although whole books have been written on different aspects of this thesis, all had some intervening material which was too advanced for the average student. Any students interested in mathematics or physics cannot fail to see the importance of Fourier series. To mention a few applications, the series is used in finding the solution to various problems in heat distribution, electrical transmission, and aircraft construction.
Cited by 1Open access
Karasev, M V · 2005
This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kählerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.
Cited by 0
Valery Ochkov · 2020
25 Problems for STEM Education introduces a new and emerging course for undergraduate STEM programs called Physical-Mathematical Informatics. This course corresponds with the new direction in education called STE(A)M (Science, Technology, Engineering, [Art] and Mathematics). The book focuses on undergraduate university students (and high school students), as well as the teachers of mathematics, physics, chemistry and other disciplines such as the humanities. This book is suitable for readers who have a basic understanding of mathematics and math software. Features Contains 32 interesting problems (studies) and new and unique methods of solving these physical and mathematical problems using a computer as well as new methods of teaching mathematics and physics Suitable for students in advanced high school courses and undergraduates, as well as for students studying Mathematical Education at the Master’s or PhD level One of the only books that attempts to bring together ST(E)AM techniques, computational mathematics and informatics in a single, unified format
Cited by 8
Knupp, Richard, Qualls, Joshua · ScholarWorks - MoreheadState (Morehead State University) · 2021
Interdisciplinary courses like PHYS481/MATH481 “Mathematics for Scientists and Engineers” play an important role in undergraduate curriculum: physics students get exposure to advanced mathematical ideas that they might otherwise miss in their undergraduate studies, and both mathematics and physics students get experience directly applying mathematical ideas and techniques to real-world problems. Several in-state and peer institutions offer an additional course in mathematical physics. Although the advantages of a second course are plentiful, Morehead State University currently does not. In this talk we report on the progress of creating the cross-listed course “Advanced Mathematics for Physics” and mathematical techniques currently missing from coursework that would be most beneficial to graduating physics majors. As an example, we focus specifically on Schwarz-Christoffel (SC) transformations from complex analysis. SC transformations are functions that map points in the extended upper half of the complex plane to the vertices of a polygonal region. These mappings are quite useful in the realm of physics; in a general sense these mappings can be useful in contexts such as electric fields, magnetic fields, and fluid dynamics. For this project, it is specifically helpful when working in the context of electric field distributions and electric potentials within that electric field distribution. We conclude by discussing our computational project that introduces students to these SC transformations and helps them apply SC transformations in an appropriate physics context. This computational project introduces the SC transformation and implements these transformations in both Python and Mathematica.
Cited by 0