Paper discusses the Fisher metric and differentiability in statistical models.
problem Understanding the relationship between Fisher metric and differentiability in statistical models.
method Comparison of different concepts and models in Information Geometry, mathematical statistics, and measure theory.
result Discussion of various models and their differentiability properties.
The paper explains knowledge distillation by analyzing visual concepts in DNNs.
problem Understanding how knowledge distillation affects the learning of visual concepts in deep neural networks.
method The paper proposes three hypotheses and designs mathematical metrics to evaluate feature representations of DNNs.
result The hypotheses were verified through experiments on various DNNs.
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.
Ray-Singer torsion is a mathematical concept with applications in physics.
problem No specific problem stated in the abstract.
method No specific method stated in the abstract.
result No specific key result stated in the abstract.
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.
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.
In this paper, we briefly discuss a mathematical concept that can be used in economics.
Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.
problem Historical use of Prytz planimeter to approximate areas.
method Sub-Riemannian geometry and connections/horizontal lifts.
result Mathematical description and analysis of Prytz planimeter.
The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using t…
The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presente…
We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as the supergeometric setting. The review is supplemented by a few new concepts and examples.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
Paper shows equivalence between MM and PH for n-D Morse functions.
problem Relationship between Mathematical Morphology and Persistent Homology.
method Examined pairing of extrema in Morse functions using dynamics and persistence.
result Equivalence proven between dynamics and persistence on n-D Morse functions.
Equivalence shown between two mathematical concepts for hyperbolic surfaces.
problem None explicitly stated, but related to mathematical equivalence of concepts.
method Benjamini-Schramm convergence and zeta functions equivalence demonstration.
result Equivalence of Benjamini-Schramm convergence and zeta functions for compact hyperbolic surfaces.
Proposes a simple method to represent and manipulate concepts using polynomials and moment statistics.
problem Lack of a mathematical framework to define and operate on concepts.
method Characterizes concepts as zero sets of polynomials and uses moment statistics for representation; proposes a dictionary-based method to learn hierarchical structures.
result Signature of concepts can be used to discover common structures and recursively produce higher-level concepts.
Survey on advanced gauge theory concepts.
problem Understanding higher gauge theory structures.
method Introduction to higher structures and connections on higher principal bundles.
result Summarized applications and principles of higher gauge theories.
The paper models and prices cyber insurance risks, distinguishing idiosyncratic, systematic, and systemic risks.
problem Modeling and pricing cyber insurance policies, especially for systemic risks.
method Distinguishes three types of cyber risks and proposes methods for their valuation.
result Complex methods are needed for systemic cyber risks, including risk-neutral valuation and monetary risk measures.
This paper is a tutorial on Formal Concept Analysis (FCA) and its applications. FCA is an applied branch of Lattice Theory, a mathematical discipline which enables formalisation of concepts as basic units of human thinking and analysing data in the object-attribute form. Originated in early 80s, during the last three d…
Mathematical framework for transfer learning feasibility and transfer risk.
problem Theoretical analysis of transfer learning.
method Reformulated transfer learning as an optimization problem, introduced transfer risk concept.
result Demonstrated the potential and benefits of incorporating transfer risk in transfer learning evaluation.
Paper formalizes Simon's satisficing through FFSD, proving its equivalence to expected utility theory.
problem Formalizing Herbert Simon's bounded rationality concept in economic decision-making.
method Developed FFSD framework using Lean 4 theorem prover, proving equivalence to expected utility theory.
result Equivalence theorem linking FFSD to expected utility maximization for approximate indicator functions.
The paper applies math and physics to language models, introducing entropy and geometric concepts.
problem Understanding and improving language models to approximate intelligent language.
method Formal definitions, functional analysis, topology, thermodynamics, and set theory.
result Entropy function reveals key obstacles for LLMs and offers insights into language models.
This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in fact organized in the language topological quantum field theory.
We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called…
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.
Paper defines mathematical framework for neural network explainability.
problem Neural network explainability and equivariant operators.
method Mathematical framework based on Group Equivariant Non-Expansive Operators (GENEOs) and complexity measures.
result Formal properties and interpretability of Group Equivariant Operators (GEOs) defined.
The paper proves stability of the positive mass theorem using intrinsic flat convergence.
problem Stability of the positive mass theorem in mathematical relativity.
method Intrinsic flat convergence of points and applications to stability.
result Revisits and strengthens the stability results for graphical hypersurfaces of Euclidean space.
This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…
Study Gram determinants in knot theory, focusing on a Möbius band determinant.
problem Closed formula for the Gram determinant of type (Mb)1. method Survey of Gram determinants, focusing on a Möbius band determinant.
result Speculation on closed formula for (Mb)1 Gram determinant. The so-called great divergence in the income per capita is described in the Unified Growth Theory as the mind-boggling and unresolved mystery about the growth process. This mystery has now been solved: the great divergence never happened. It was created by the manipulation of data. Economic growth in various regions is…
This paper studies how knots combine using Alexander Polynomials.
problem How knots combine and their determinants behave.
method Basic knot theory, Alexander Polynomials, and composition techniques.
result Generalized solution for knot determinants in compositions.
The abstract reviews financial concepts using physics.
problem Financial pricing and risk management.
method Discrete time formalism, path integral, Green's function formulas.
result Formulas for pricing and risk mitigation methods.
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…
Euler derived elastica equation using modern mathematical concepts.
problem Euler's original derivation of elastica equation has not been properly interpreted.
method Euler used Noether's theorem and the Goldstein-Petrich scheme.
result Euler's equation is the static modified KdV equation.
The main purpose of this paper is to formalize the modelling process, analysis and mathematical definition of corruption when entering into a contract between principal agent and producers. The formulation of the problem and the definition of concepts for the general case are considered. For definiteness, all calculati…
Defines tensor eigenvalues and singular values without basis, simplifying analysis.
problem Defines tensor eigenvalues and singular values without basis.
method Intrinsic definition of tensor eigenvalues and singular values using concepts from pure mathematics.
result Shows the relationship between tensor analysis and pure mathematics.
New concept of illiquidity linked to credit risk, using Jarrow & Turnbull's analogy.
problem Understanding illiquidity in financial markets, especially with credit risk.
method Introduces a constraint-based notion of illiquidity, using Jarrow & Turnbull's foreign exchange analogy.
result A new mathematical framework for understanding illiquidity in financial markets.
This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.
problem Understanding the historical context and philosophical implications of angles and solid angles.
method Historical review and analysis of mathematical and philosophical works.
result Questions raised by Euler about angles and solid angles are timeless and relevant to modern mathematics.
Study particle dynamics in non-differentiable fractal spaces.
problem Understanding motion in non-smooth, probabilistic geometries.
method Use fiber bundle theory to characterize multivalued geodesic trajectories.
result Developed a hybrid theory combining surface and stochastic process theories.
This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces fundamental concepts and algorithms by building on first principles, while also e…
Study clarifies Bayesian generalization error in CBM for 3-layered linear neural networks.
problem Understanding the generalization error in concept bottleneck models.
method Mathematical analysis of Bayesian generalization error and free energy in CBM for 3-layered linear neural networks.
result CBM significantly alters the parameter region and Bayesian generalization error compared to standard models.
Extends V-IP framework to use LLMs for generating task-relevant concepts, improving interpretability and performance.
problem Limited applicability of V-IP to small-scale tasks due to manual data annotation.
method Integrates Foundational Models with Large Language and Multimodal Models to generate and annotate concepts.
result FM+V-IP achieves better test performance with fewer concepts/queries compared to other frameworks.
Paper defines XAI concepts using category theory.
problem Lack of precise mathematical definitions for XAI.
method Uses Category theory to define XAI concepts rigorously.
result Establishes a theoretical foundation for XAI.
In this pedagogical study, carried out by adopting standard mathematical methods of nonlinear dynamics, we have presented some simple analytical models to understand terminal behaviour in industrial growth. This issue has also been addressed from a dynamical systems perspective, with especial emphasis on the concept of…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way. It starts off with a review o…
LLMs will inevitably hallucinate due to their mathematical structure.
problem The inherent limitations of Large Language Models (LLMs).
method Analysis of LLMs using computational theory and Godel's Incompleteness Theorem.
result Hallucinations in LLMs are an inevitable feature, not just errors.
LLMs can be tricked into recalling facts based on context clues.
problem Manipulation of LLMs' factual recall through context changes.
method Mathematical exploration of transformers' associative memory properties.
result Transformers use self-attention and value matrix for associative memory.
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …