The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorit…
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48 results for “Mathematics Achievement”
BCF models estimate causal effects on multiple outcomes in TIMSS data.
problem Estimating causal effects on multiple outcomes in educational data.
method Bayesian Additive Regression Trees (BART) for multivariate causal inference.
result Positive and negative effects of home study conditions and school absence on student achievement.
New analysis shows transformer models can't learn effectively.
problem Transformer models learning in context can't achieve general predictive accuracy.
method Empirical evidence and mathematical analysis of transformer architecture limitations.
result Transformers cannot achieve general predictive accuracy due to architectural limitations.
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.
The mathematical features of a string theory compactification determine the physics of the effective four-dimensional theory. For this reason, understanding the mathematical structure of the possible compactification spaces is of profound importance. It is well established that the compactification space for M-Theory m…
Bayesian Causal Forests model assesses part-time work's impact on student growth.
problem Estimating causal effects of part-time work on student growth in mathematics achievement.
method Longitudinal Bayesian Causal Forests model combining non-parametric and difference-in-differences methods.
result Negative impact of part-time work for most students, potential benefits for those with low school belonging, widening achievement gap identified.
Transformer models can solve complex math problems with less data.
problem Solving complex symbolic mathematics problems with limited data.
method Pretrain transformer models on language translation tasks and fine-tune for symbolic math.
result Pretrained transformer models achieve comparable accuracy to state-of-the-art models with less data.
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
Develops new techniques for learning from sequential data groups.
problem Learning from groups of inputs rather than individual inputs.
method Introduces feature-based and kernel-based learning techniques for sequential data.
result Achieves state-of-the-art performance on various real-world examples.
Prob2Vec embeds problems for adaptive tutoring, achieving high similarity accuracy.
problem Retrieve problems with similar mathematical concepts for adaptive tutoring.
method Hierarchical problem embedding algorithm (Prob2Vec) combining abstraction and embedding steps.
result 96.88% accuracy on problem similarity test, significantly outperforming state-of-the-art sentence embedding methods.
While it has become common to perform automated translations on natural language, performing translations between different representations of mathematical formulae has thus far not been possible. We implemented the first translator for mathematical formulae based on recursive neural networks. We chose recursive neural…
Global critical points of the Standard Model on four-dimensional spacetimes of expanding typemath.DG
Study proves existence of global solutions for Standard Model on expanding spacetimes.
problem Existence of global solutions for the Standard Model on expanding spacetimes.
method Gauge-invariant energy estimate for the Euler-Lagrange equations.
result Existence of global solutions for the Standard Model under specific conditions.
In this paper we outline methodology to efficiently simulate (jump) diffusion bridge sample paths without discretisation error. We achieve this by considering the simulation of conditioned (jump) diffusion bridge sample paths in light of recent work developing a mathematical framework for simulating finite dimensional …
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
We study the influence of an additional scalar potential on various geometric and analytic properties of Dirac-harmonic maps. We will create a mathematical wish list of the possible benefits from inducing the potential term and point out that the latter cannot be achieved in general. Finally, we focus on several potent…
We connect Causal inference and low-rank recovery via RDT and free probability theory.
problem Determining the applicability of causal inference via low-rank recovery.
method Random Duality Theory, free probability theory, and mathematical rigor.
result Exact closed-form worst case phase transitions for causal inference.
Rokhlin's signature theoremsmath.HO
Rokhlin's work simplified signature theorems for mathematicians.
problem Complex signature theorems in topology.
method Accessible explanation of Rokhlin's achievements.
result Simplified understanding of signature theorems.
In this study we introduce a new technique for symbolic regression that guarantees global optimality. This is achieved by formulating a mixed integer non-linear program (MINLP) whose solution is a symbolic mathematical expression of minimum complexity that explains the observations. We demonstrate our approach by redis…
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.
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.
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.
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.
New method tackles inexact bilevel optimization for faster parameter learning.
problem Nested optimization problems in bilevel learning with computationally difficult exact solutions.
method Inexact derivative-free optimization algorithms for approximate lower-level solutions.
result Global convergence and worst-case complexity for the proposed approach.
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.
Ant colonies and boosting algorithms both reduce bias and variance through adaptive mechanisms.
problem Understanding the mathematical principles behind ensemble learning and ant colony behavior.
method Developed a formal mapping between AdaBoost's adaptive reweighting and ant recruitment dynamics.
result Proved that the fundamental theorem of weak learnability has a direct analog in colony decision-making.
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.
Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain. Typical examples include undersampled magnetic resonance imaging, interior tomography, and sparse-view computed tomography, where deep lea…
GCN and GPCA are mathematically connected, leading to improved node classification performance.
problem Improving node classification performance in semi-supervised settings.
method Established a mathematical connection between GCN and GPCA, demonstrating their equivalence and using this to design an effective initialization strategy.
result GPCA paired with a simple MLP achieves similar or better performance than GCN on semi-supervised node classification tasks.
New method uses model comparison signals to improve LLM evaluation accuracy.
problem Limited benchmark sizes and model stochasticity in evaluating LLMs' mathematical reasoning.
method Combines standard labeled outcomes with model comparison signals to design a statistically efficient evaluation framework.
result Semiparametric estimator achieves the semiparametric efficiency bound and substantially improves ranking accuracy.
Method guarantees coherent factuality for language model outputs in reasoning tasks.
problem Ensuring correctness of language model outputs in reasoning tasks.
method Developed a conformal-prediction-based method applied to subgraphs within a deducibility graph.
result Achieved coherent factuality across target coverage levels, 90% on stricter definition.
ESPO optimizes LLMs for complex tasks by balancing fine-grained updates and stability.
problem Gradient underutilization in sequence-level optimization.
method Entropy Grouping Importance Sampling and Entropy Adaptive Clipping.
result ESPO accelerates convergence and achieves state-of-the-art performance.
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable…
We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries a…
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.
TNet combines DL with physics models to solve inverse problems efficiently.
problem Solving inverse problems with limited data and physics constraints.
method Model-constrained deep learning approach using TNet.
result TNet solutions are as accurate as traditional methods but faster.
Paper combines QRM and CNN for better stock option price forecasting.
problem Forecasting stock option prices in a complex market.
method Solves Black-Scholes equation using QRM, trains CNN models on data.
result CNN models improve option price prediction accuracy.
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.
SGPT-PINNs solve PDEs with sparse, small models.
problem Efficiently solving parametric PDEs with minimal resources.
method Sparse and small architecture, mathematically rigorous greedy algorithm, knowledge distillation, down-sampling.
result Achieves high efficiency with significantly fewer parameters.
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.
We attempt to set a mathematical foundation of immunology and amino acid chains. To measure the similarities of these chains, a kernel on strings is defined using only the sequence of the chains and a good amino acid substitution matrix (e.g. BLOSUM62). The kernel is used in learning machines to predict binding affinit…
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…
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…
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.