These notes grew out of a lecture course on mathematical methods of classical physics for students of mathematics and mathematical physics at the master's level. Also, physicists with a strong interest in mathematics may find this text useful as a resource complementary to existing textbooks on classical physics. Topic…
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48 results for “mathematical methods”
Improves recommender systems using mathematical principles.
problem Accuracy and speed of recommender systems.
method Explains and analyzes mathematical principles and algorithms.
result Describes the strengths and weaknesses of mathematical distance methods.
Mathematical Foundations of Regression Methods for the approximation of the Forward Initial Marginq-fin.RM
Abundant literature has been published on approximation methods for the forward initial margin. The most popular ones being the family of regression methods. This paper describes the mathematical foundations on which these regression approximation methods lie. We introduce mathematical rigor to show that in essence, al…
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
This paper describes a new method for Symbolic Regression that allows to find mathematical expressions from a dataset. This method has a strong mathematical basis. As opposed to other methods such as Genetic Programming, this method is deterministic, and does not involve the creation of a population of initial solution…
RL methods applied to option pricing using modified QLBS and RLOP models.
problem Applying reinforcement learning to price options accurately.
method Developed modified QLBS and RLOP models, implemented RL learning algorithm with neural networks.
result Optimal hedging strategies learned by RL outperform baseline models.
These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output…
Book introduces deep learning methods with math, theory, and applications.
problem Understanding deep learning algorithms and their mathematical foundations.
method Reviews various ANN architectures and optimization methods, covers theoretical aspects.
result Provides a solid mathematical foundation for deep learning.
FX Market Volatilityq-fin.MF
Paper examines two methods for FX market volatility modeling.
problem FX market volatility modeling problem.
method Classical econometric GCH and mathematical approaches (SSA, dynamical systems stability analysis).
result Both mathematical tools show promising results in FX market volatility modeling.
Develops a mathematical model for automatic differentiation in machine learning.
problem Current automatic differentiation lacks a simple mathematical model for machine learning.
method Articulates relationships between program differentiation and nonsmooth functions, provides a class of functions and nonsmooth calculus.
result Shows how nonsmooth calculus applies to stochastic approximation methods and evidence of artificial critical points.
A new mathematical framework simplifies securitization structuring.
problem Challenges in structuring asset-backed securities.
method PEAL Method: a 10-step mathematical framework.
result Enhances risk characterization and market transparency.
Advanced mathematical relativity course for math and physics students.
problem Mathematical foundations of general relativity.
method Comprehensive lecture notes covering advanced topics.
result Comprehensive coverage of mathematical relativity for advanced students.
The paper surveys mathematical results on filtration enlargement with financial examples.
problem Mathematical finance applications of filtration enlargement theory.
method Exhaustive survey and interpretation of key results from literature.
result Provides a compendium of known mathematical results for mathematical finance researchers.
Mathematical general relativity reviewed.
problem Challenges in mathematical modeling of general relativity.
method Selected topics reviewed.
result Insights into mathematical models of general relativity.
Mathematical study shows post-hoc explanations are better than attention weights alone.
problem Understanding the internal behavior of attention-based models.
method Mathematical analysis of a simple attention-based architecture.
result Post-hoc explanations provide more useful insights than attention weights alone.
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Al…
Survey of mathematical foundations for reinforcement learning.
problem Design and analysis of modern reinforcement learning algorithms.
method Organizes mathematical structures from probability, optimization, and operator theory.
result Unified mathematical entry point for researchers in various fields.
Quantum algorithms speed up financial model calculations.
problem Computing financial model expectations efficiently.
method Quantum-accelerated multilevel Monte Carlo methods.
result Improved speed-up for financial model calculations.
Mathematical theory of super fiber bundles and connections developed.
problem Modeling anticommuting fermionic fields in mathematical physics.
method Detailed introduction to super fiber bundles, relative supermanifolds, and connections; construction of parallel transport map.
result Construction and comparison of parallel transport map with other methods in the literature.
AI aids in mathematics research and problem-solving.
problem Complex mathematical problems and discoveries.
method Explains AI principles and diverse applications in math.
result AI assists in discovering patterns, proving theorems, and challenging conjectures.
The analysis of mathematical structure of the method of operator manifold guides our discussion. The latter is a still wider generalization of the method of secondary quantization with appropriate expansion over the geometric objects. The nature of operator manifold provides its elements with both quantum field and geo…
Mathematical framework for differential machine learning in finance.
problem Theoretical assumptions in financial models and their impact on machine learning algorithms.
method Rigorous mathematical framework for differential machine learning in finance.
result Theoretical grounding enhances the predictive capabilities of neural networks in financial applications.
Classically time is kept fixed for infinitesimal variations in problems in mechanics. Apparently, there appears to be no mathematical justification in the literature for this standard procedure. This can be explained canonically by unveiling the intrinsic mathematical structure of time in Lagrangian mechanics. Moreover…
Mathematical analysis shows Delisle-Euler map methods are optimal.
problem Comparing ancient and modern map drawing methods.
method Analyzing similarities and differences between ancient and modern map drawing methods.
result Delisle-Euler map methods are optimal among conical maps.
AI mirrors modern math's autonomous development, raising interpretive challenges.
problem AI's effectiveness in math mirrors historical autonomy of math.
method Analyzes historical evolution of modern mathematics and AI's role.
result AI's affinity with math's historical autonomy suggests interpretive limits.
Self-supervised skip-tree training improves mathematical reasoning in language models.
problem Improving logical reasoning in language models for formal mathematics.
method Self-supervised language modeling on mathematical formulas, skip-tree task.
result Models trained on skip-tree task outperform standard models in mathematical reasoning tasks.
Mathematical models with time dependent parameters are of great interest in financial Mathematics because they capture real life scenarios in the financial market. In this study, via the Lie group technique, we analyse evolution-type equations with time dependent parameters and give the general symmetry structure of th…
MAD framework learns operators from physics-embedded data efficiently.
problem Data-driven methods require costly labeled datasets and model-driven techniques face efficiency-accuracy trade-offs.
method Integrates physical laws with data-driven learning to generate physics-embedded analytical solutions and synthetic data.
result Eliminates dependence on experimental or simulated training data, enabling efficient operator learning across multi-parameter systems.
Method reveals hidden token embeddings of large language models.
problem Understanding hidden token embeddings in large language models.
method Structured prompts to expose token input embeddings up to homeomorphism.
result Mathematical proof for generic LLMs shows effectiveness of method.
An exploration of the mathematical structure and behavioural biases of 21st century financial crisesq-fin.ST
Study financial crises using mathematical techniques to compare equity performance.
problem Comparing financial crises to understand market dynamics and investor strategies.
method New mathematical techniques including portfolio diversification, linear operator method, and combinatorial portfolio optimisation.
result New methods to quantify and compare equity returns during different market crises.
This paper offers a mathematical introduction to GANs.
problem Understanding GANs from a mathematical perspective.
method Mathematical analysis of GANs.
result Provides clarity for math-oriented students.
Recently, machine learning (ML) has established itself in various worldwide benchmarking competitions in computational biology, including Critical Assessment of Structure Prediction (CASP) and Drug Design Data Resource (D3R) Grand Challenges. However, the intricate structural complexity and high ML dimensionality of bi…
Explains isometric immersions and their applications.
problem Isometric immersions and their applications in math and physics.
method Historical overview and applications.
result Explains the importance and applications of isometric immersions.
Mathematical model predicts international trade and global economy dynamics.
problem Understanding complex international trade and economy interactions.
method Developed a mathematical model for non-equilibrium processes in open systems.
result Predicted model accurately reflects international trade and economy.
There has been a lot of recent interest in adopting machine learning methods for scientific and engineering applications. This has in large part been inspired by recent successes and advances in the domains of Natural Language Processing (NLP) and Image Classification (IC). However, scientific and engineering problems …
Improves neural network search in combinatorial spaces of mathematical symbols.
problem Early commitment and initialization bias limit exploration in neural network search.
method Entropy regularization and distribution initialization methods.
result Improves performance, increases sample efficiency, lowers solution complexity.
The field of medical image reconstruction has seen roughly four types of methods. The first type tended to be analytical methods, such as filtered back-projection (FBP) for X-ray computed tomography (CT) and the inverse Fourier transform for magnetic resonance imaging (MRI), based on simple mathematical models for the …
We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood math…
Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very…
Higher topos theory applied to physics.
problem No specific problem stated.
method Exposition of higher topos theory.
result No specific key result mentioned.
Quantum cohomology connects quantum physics with classical math.
problem Quantum cohomology's relevance in modern mathematics.
method Informal review and discussion of connections.
result Quantum cohomology still has significant relevance in mathematics.
Georg de Buquoy, Lord de Vaux, lived in Nove Hrady, Prague and Cerveny Hradek for most of his productive life. From his extensive scientific contributions, both theoretical and experimental, we expand here the discussion of his contributions to mathematical economy. He is mainly celebrated as the first persons to defin…
We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appe…
Mathematical advances needed for Digital Twins, differing from traditional models.
problem Foundational mathematical advances required for Digital Twins.
method Multi-scale, multi-physics modeling and coupling, different reliability criteria and uncertainty assessments.
result AI/ML methods can perform well in biomedical problems but fail in simple engineering systems.
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
First, we analyze the variance of the Cross Validation (CV)-based estimators used for estimating the performance of classification rules. Second, we propose a novel estimator to estimate this variance using the Influence Function (IF) approach that had been used previously very successfully to estimate the variance of …
Mathematical method based on a direct or indirect analysis of growth rates is described. It is shown how simple assumptions and a relatively easy analysis can be used to describe mathematically complicated trends and to predict growth. Only rudimentary knowledge of calculus is required. Projected trajectories based on …
Lean 4 library formalizes mathematical finance, verifying over 200 theorems.
problem Formal verification of complex financial mathematics.
method Lean 4 proof assistant, Mathlib, BrownianMotion package, formal verification of over 200 theorems.
result Formal verification yields certified unification of known financial results.