Thurston's influence on French math traced and problems solved.
problem Major problems in French math traced back to Thurston.
method Overview and survey of Thurston's influence and results.
result French math problems rooted in Thurston's work.
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48 results for “mathematical tradition”
Math and dance blend in choreographer's research.
problem Exploring intersections between dance and mathematics.
method Choreographic practice and mathematical concepts.
result Examples of fractals, braids in choreography.
This paper deals with a semi-classical limit (Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussion of it may be found in [7].
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.
Overview of Thurston's work in math.
problem None explicitly stated, focuses on Thurston's contributions.
method Presentation of significant results.
result Impact of Thurston's work on mathematics.
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension …
This article reviews mathematical insights into neural networks and machine learning.
problem Understanding the success and subtleties of neural network-based machine learning.
method Rigorous mathematical analysis, numerical experiments, and simplified models.
result Identification of open problems in the field.
Extends Bayesian theory to handle complex interdependencies in multidimensional event spaces.
problem Complex interdependencies between events and hypotheses sets in real-world systems.
method Developed a mathematical formalism for modeling complex relationships through rigorous derivation and validated using analytical proofs, simulations, and case studies.
result MDSE theory improves prediction accuracy by 15-20% compared to standard Bayesian methods in high interdimensionality datasets.
Machine learning impacts computational math, offering new functions approximations.
problem Machine learning's black box nature hinders further progress in computational math.
method Analyzes machine learning's impact on computational math and vice versa.
result Integrating computational math with machine learning can enhance both fields.
The goal of this note is to illustrate the impact of a self-financing condition recently introduced by the authors. We present the analyses of two specific applications usually considered in more traditional models in financial mathematics. They include hedging European options with limit orders and the optimal behavio…
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.
Blockchain protocol improves traditional mutual funds with performance fees and investor protection.
problem Operational issues and performance fees in traditional mutual funds.
method Developed a blockchain protocol that integrates features of mutual funds and hedge funds.
result Blockchain can simplify performance fee calculations and protect investors.
This paper explores the interactions between knot theory and quantum computing. On one side, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical backgr…
Adversarial deep hedging learns to hedge without specifying asset price models.
problem Lack of effective underlying asset models for deep hedging.
method Adversarial learning framework where a hedger and a generator compete to improve hedging performance.
result Adversarial deep hedging achieves competitive performance without explicit asset process modeling.
Study proposes a new approach for deep hedging using artificial market simulations.
problem Challenges in selecting the best model for underlying asset simulations in deep hedging.
method Proposes artificial market simulations to replicate financial market stylized facts.
result Achieves similar performance to traditional approaches without mathematical finance models.
This paper provides a mathematical framework for time-delay reservoir computing.
problem Lack of rigorous mathematical foundations for reservoir computing properties.
method Control-theoretic framework, formal definitions of separation and fading memory, explicit lower bound derivation.
result Established formal definitions and connections to stability notions for time-delay systems.
Estimating fundamental matrices is a classic problem in computer vision. Traditional methods rely heavily on the correctness of estimated key-point correspondences, which can be noisy and unreliable. As a result, it is difficult for these methods to handle image pairs with large occlusion or significantly different cam…
Interest Rates and Inflationecon.GN
This article is an extension of the work of one of us (Coopersmith, 2011) in deriving the relationship between certain interest rates and the inflation rate of a two component economic system. We use the well-known Fisher relation between the difference of the nominal interest rate and its inflation adjusted value to e…
This paper uses crypto derivatives data to estimate yield curves for cryptocurrencies.
problem Estimating yield curves for cryptocurrencies without bond markets.
method Using mathematical tools and data from cryptocurrency derivatives markets.
result Yield curves can be constructed for cryptocurrencies using derivative data.
AI without networkscs.LG
New AI framework without networks outperforms traditional models.
problem The role of artificial neural networks (ANNs) in AI is unclear and raises ethical and legal concerns.
method Developed a parameter-free, statistically consistent data interpolation method for AI.
result Framework outperforms traditional mathematical models and ANN-based models in various applications.
In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related t…
VERAFI improves financial AI by verifying calculations and compliance.
problem Financial AI systems generate errors and violations during reasoning.
method VERAFI combines dense retrieval, reranking, and automated reasoning policies.
result VERAFI achieves 94.7% factual correctness, 81% relative improvement.
Exact Matrix Seriation through Mathematical Optimization: Stress and Effectiveness-Based Modelsmath.OC
Unified optimization framework for matrix seriation.
problem Discovering latent structure in relational data.
method Mathematical optimization models for seriation.
result Optimization models enhance solution quality and interpretability.
This paper provides a PAC-Bayesian bound for CVaR in machine learning.
problem Learning algorithms minimizing CVaR of empirical loss.
method Generalization bound of PAC-Bayesian type, reducing CVaR estimation to expectation estimation.
result The bound is small when empirical CVaR is small, providing concentration inequalities for CVaR.
Bayesian model predicts sequences better than LSTMs by identifying underlying rules.
problem Current RNNs struggle to generalize from limited training data and identify underlying rules in sequences.
method Bayesian model that learns underlying concepts from sequences and generalizes to new data.
result Bayesian model predicts sequences better than traditional LSTMs.
New -algebra approach unifies machine learning strategies.
problem Lack of diverse and information-rich data models in machine learning.
method Integrates -algebra into machine learning frameworks.
result Unified learning strategies and new data models.
Traditional anatomical analyses captured only a fraction of real phenomic information. Here, we apply deep learning to quantify total phenotypic similarity across 2468 butterfly photographs, covering 38 subspecies from the polymorphic mimicry complex of and . E…
Combines model-based and model-free RL for better financial market performance.
problem Challenges of Reinforcement Learning in volatile financial markets.
method Adapts model-based RL with model-free RL, incorporating contextual signals and walk-forward analysis.
result Outperforms traditional financial models in various metrics.
Modern machine learning algorithms have been adopted in a range of signal-processing applications spanning computer vision, natural language processing, and artificial intelligence. Many relevant problems involve subspace-structured features, orthogonality constrained or low-rank constrained objective functions, or sub…
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity are not only manifestly Lorentz covariant but also gauge covariant or diffeomorp…
Improved model predicts wildfire spread on slopes.
problem Accurate prediction of wildfire spread on slopes.
method Combines Rothermel model, Huygens' principle, and advanced techniques.
result More precise model of wildfire propagation.
Towards a mathematical framework to inform Neural Network modelling via Polynomial Regressionstat.ML
A mathematical framework connects neural networks and polynomial regression for better model understanding.
problem Neural networks are black boxes with challenges in dimensioning and prediction error evaluation.
method Developed a mathematical framework using Taylor expansion to relate neural networks and polynomial regression.
result Polynomial approximations from neural networks trained on polynomial data are accurate locally.
Gauge Flow Modelscs.LG
Gauge Flow Models use a learnable Gauge Field in Generative Flow Models.
problem Improving generative model performance.
method Integrates a learnable Gauge Field into Flow ODEs.
result Gauge Flow Models outperform traditional Flow Models in Flow Matching experiments.
We introduce various quantitative and mathematical definitions for price momentum of financial instruments. The price momentum is quantified with velocity and mass concepts originated from the momentum in physics. By using the physical momentum of price as a selection criterion, the weekly contrarian strategies are imp…
Predictive coding networks use inference learning for efficient AI modeling.
problem Traditional AI methods struggle with complex neural patterns.
method Inference learning for hierarchical Bayesian inference models.
result PCNs outperform traditional BP methods in efficiency and flexibility.
Neural SVEs model complex systems with memory, outperforming traditional methods.
problem Modeling systems with memory effects and irregular behavior.
method Introducing neural stochastic Volterra equations as a physics-inspired architecture.
result Neural SVEs outperform neural SDEs and DeepONets in various applications.
Topological data analysis aims to extract topological quantities from data, which tend to focus on the broader global structure of the data rather than local information. The Mapper method, specifically, generalizes clustering methods to identify significant global mathematical structures, which are out of reach of man…
Paper proposes ExsdHawkes to model LOBs, capturing volatility dynamics.
problem Modeling volatility signature plots in LOBs with high-frequency trading dynamics.
method Extended State-Dependent Hawkes Process (ExsdHawkes) with relaxed constraints.
result ExsdHawkes uniquely reproduces volatility signature plots, identifying MLOs as catalysts.
Neural differential equations combine deep learning and differential equations for modeling complex systems.
problem Modeling complex systems with high capacity and efficiency.
method Combining neural networks and differential equations, focusing on neural ordinary, controlled, and stochastic differential equations.
result NDEs offer high-capacity function approximation, strong priors, and handle irregular data efficiently.
NeuralChaos efficiently approximates complex stochastic processes.
problem Representing and computing square-integrable predictable processes over time.
method Introduces NeuralChaos, a neural operator architecture for -valued predictable processes.
result NeuralChaos achieves best -term chaoslet approximation rates and is dense in .
Alternative wavelet analysis method for financial signals.
problem Analyzing oscillations in financial signals with noise.
method Modeling financial signals as isolated events producing ripples of various frequencies.
result Element analysis distinguishes between noise and logically matched generators.
The optimal momentum mapmath.SG
The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden--Weinstein reduction procedure takes advantage of this and associates to the…
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.
In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. …
Interacting particle methods are increasingly used to sample from complex and high-dimensional distributions. These stochastic particle integration techniques can be interpreted as an universal acceptance-rejection sequential particle sampler equipped with adaptive and interacting recycling mechanisms. Practically, the…
A restricted Boltzmann machine (RBM) is a two-layer neural network with shared weights and has been extensively studied for dimensionality reduction, data representation and recommendation systems in the literature. The traditional RBM requires a probabilistic interpretation of the values on both layers and a Markov ch…
New framework explains leading digit patterns without probabilistic assumptions.
problem Explaining leading digit distributions without relying on probabilistic models.
method Shift-invariant functional equation and affine-plus-periodic formulas.
result Unified mathematical foundation for understanding digit distributions.
Graph clustering improved using Boltzmann machine heuristics.
problem Graph clustering to form densely connected clusters.
method Two mathematical programming formulations, two variations of Boltzmann machine heuristic.
result Boltzmann machine provides superior solutions and faster computation times.