The Blueprints of Intelligence: A Functional-Topological Foundation for Perception and Representation

Eduardo Di Santi

arxiv: 2512.05089 · v7 · submitted 2025-12-04 · 💻 cs.LG · math.OC

The Blueprints of Intelligence: A Functional-Topological Foundation for Perception and Representation

Eduardo Di Santi This is my paper

Pith reviewed 2026-05-17 01:16 UTC · model grok-4.3

classification 💻 cs.LG math.OC

keywords functional topologycompact manifoldsperceptual functionalempirical radiusHausdorff stabilitygeneralization from sparse dataphysical signal variabilityBanach space

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The pith

Real-world signals concentrate on compact, low-variability subsets of function space, forming stable perceptual manifolds that enable generalization from sparse evidence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that physical processes do not produce signals with completely arbitrary variability. Instead, the valid realizations concentrate on compact subsets within a space of functions. These subsets carry stable invariants and a finite empirical radius that together induce a continuous perceptual functional. The resulting geometry constrains how signals can vary, makes processes more identifiable, and explains why generalization from few examples is possible. Data from five domains including railway machines, batteries, heart signals, solar irradiance, and tidal cycles show the key properties stabilize after surprisingly few samples.

Core claim

The set of valid realisations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite empirical radius, and an induced continuous perceptual functional. This geometry provides structural constraints on variability, conditions for identifiability, and support for generalisation from sparse evidence. Across the examined domains the empirical radius and internal Hausdorff stability saturate after few samples.

What carries the argument

Compact subset of a Banach space with stable invariants, finite empirical radius, and induced continuous perceptual functional

If this is right

Variability of signals is structurally constrained by the geometry of the compact set.
Conditions for identifiability of the underlying physical process are provided by the stable invariants.
Generalisation from sparse evidence is directly supported by the finite radius and continuity of the perceptual functional.
Perceptual manifold properties such as radius and Hausdorff stability reach saturation after only a few samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Machine learning systems might achieve higher sample efficiency by explicitly searching for or assuming such compact manifolds in their representations.
The framework could be tested on synthetic signals engineered to have high arbitrary variability to check whether rapid saturation occurs only for real physical processes.
Similar compactness might be observable in other domains such as neural population activity or economic time series, offering a route to broader validation.

Load-bearing premise

Real-world phenomena generate signals that concentrate on compact, low-variability subsets of functional space rather than arbitrary variability.

What would settle it

Measuring signals from any physical process and finding that the empirical radius keeps growing without bound as more samples are added, instead of saturating after a small number, would disprove the compactness claim.

Figures

Figures reproduced from arXiv: 2512.05089 by Eduardo Di Santi.

**Figure 1.** Figure 1: Different physical systems (PM, battery, ECG) generate signals that lie on compact, low-variability manifolds. This schematic illustrates the universal geometric structure shared by such deterministic perceptual manifolds: each domain has its own manifold in C 0 ([0, T]), but all exhibit the same compact, boundedvariability geometry. 2 Related Work Joint Embedding Predictive Architectures (JEPA) show tha… view at source ↗

**Figure 2.** Figure 2: illustrates this progression. The empirical radius ˆrn grows rapidly at first and then stabilizes once the extremal variations of the phenomenon have been observed. This behavior provides a simple operational criterion for manifold completion. To formalize this estimation procedure, we compute ˆrn incrementally as new samples arrive. The pseudocode below summarizes the algorithm used in all experiments, i… view at source ↗

**Figure 3.** Figure 3: Empirical saturation curve of the perceptual radius. The radius grows rapidly during early sampling as new variability is discovered, then gradually stabilizes as additional realizations fall within the established boundary. This empirical pattern is consistent across all domains and provides a practical criterion for identifying when the perceptual manifold has been fully explored. of the functional space… view at source ↗

**Figure 4.** Figure 4: Saturation of the real point-machine manifold. All geometric metrics stabilize after [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Saturation of the simulated point-machine manifold. The geometry is stable and compact [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Representative RMS instantaneous power envelopes for railway point machines. In green: randomly sampled real manoeuvres from an in-service machine, showing the idle–inrush–locking–idle organisation with a terminal impact transient. In grey: Monte Carlo waveforms generated by the physics-aware AC simulator, which reproduce the same qualitative morphology while exploring admissible parametric variability.… view at source ↗

**Figure 7.** Figure 7: Saturation of the battery discharge manifold. The Hausdorff distance between Xn and Xn/2, as well as global radius estimates, stabilize rapidly, indicating that additional discharge curves do not introduce new geometric structure beyond a small number of samples. Importantly, this saturation is observed despite the non-stationary nature of battery aging across cycles. The result indicates that while aging … view at source ↗

**Figure 8.** Figure 8: Saturation of the real ECG perceptual manifold. Geometric metrics stabilize after approximately 20–40 samples, demonstrating extreme compactness of physiological signal spaces. 7.3.2 Synthetic ECG Generators To assess whether geometric saturation depends on accurate physiological modeling, we analyze two synthetic ECG generators with markedly different levels of realism. These generators are not intended t… view at source ↗

**Figure 9.** Figure 9: Synthetic beats generated by the McSharry dynamical model. Gaussian morphological emulator. As a deliberately simplified baseline, we also construct a purely morphological ECG emulator based on a superposition of Gaussian components representing the P, QRS, and T waves, with bounded parameters and smooth perturbations. This generator ignores electrophysiological dynamics entirely and produces visibly ideal… view at source ↗

**Figure 10.** Figure 10: Synthetic beats produced by the Gaussian morphological emulator. Remark 7.1. The synthetic generators are visibly imperfect and do not reproduce fine physiological detail, yet produce bounded families of continuous waveforms. 7.3.3 Saturation of the McSharry ECG Manifold We apply the same Monte Carlo radius estimation pipeline to the ECG signals generated by the McSharry model. For increasing subsets XMcS… view at source ↗

**Figure 11.** Figure 11: Saturation of the ECG manifold generated by the McSharry dynamical model. Geometric metrics stabilize after a small number of samples, indicating a compact functional manifold despite imperfect physiological realism. tinuous by design. The empirical perceptual radius again saturates extremely rapidly: dH(XGauss n , XGauss n/2 ) ≈ 10−2 , rmax(XGauss n ) ≈ constant for n ≳ 20–40. This confirms that saturati… view at source ↗

**Figure 12.** Figure 12: Saturation of the ECG manifold generated by a Gaussian morphological emulator. Even with a highly simplified and non-physiological generator, the perceptual manifold remains compact and exhibits rapid radius convergence. 7.3.5 Summary: Saturation Across Real and Synthetic ECG Manifolds Across real ECG recordings, the McSharry dynamical generator, and the simplified Gaussian morphological emulator, we obse… view at source ↗

read the original abstract

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalisation from few examples. We formalise this principle through a deterministic functional-topological framework in which the set of valid realisations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite empirical radius, and an induced continuous perceptual functional. This geometry provides structural constraints on variability, conditions for identifiability, and support for generalisation from sparse evidence. We develop this framework and examine its empirical relevance across five real-world domains: electromechanical railway point machines, electrochemical battery discharge, physiological ECG signals, atmospheric solar irradiance, and geophysical tidal cycles. Where available, we also compare real data with corresponding deterministic simulators. Across these domains, the empirical radius and internal Hausdorff stability of the perceptual manifold saturate after surprisingly few samples, indicating that admissible signal families occupy compact, low-variability regions of function space. These results suggest that compact perceptual manifolds provide a useful organising principle for both physical processes and learned representations, and support deterministic functional topology as a promising framework for understanding perception and representation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Physical signals concentrate in compact low-variability regions of function space with quick empirical saturation across domains, but the supporting definitions and controls stay thin.

read the letter

Colleague, the main takeaway is that this paper argues real physical processes generate signals that occupy compact, low-variability subsets of a Banach space, complete with stable invariants and a finite empirical radius that induces a continuous perceptual functional. The author tests the idea on five domains—railway machines, battery discharge, ECG, solar irradiance, and tidal cycles—and reports that both the empirical radius and Hausdorff stability level off after only a few samples, with some simulator comparisons where available. That cross-domain pattern is the concrete part worth noting. It gives a consistent empirical hint that admissible signals really do cluster rather than spread arbitrarily, which lines up with the intuition about efficient generalization from sparse data. The work applies the same topological framing to quite different signal types and checks real versus simulated cases, which adds a bit of grounding. The soft spots sit in the execution. The key quantities like empirical radius and perceptual functional lack explicit definitions or derivation steps in the provided summary, making it hard to judge independence from the compactness claim itself. There are no error bars, null-model comparisons, or statistical controls mentioned, so the saturation result could look stronger than the data strictly support. If those measures end up depending on choices that reinforce the low-variability story, the circularity risk is real and needs separation. This is aimed at readers who already think about functional analysis or topological constraints in perception and representation learning. Someone working on generalization in physical sensing or efficient modeling might extract useful organizing ideas from the empirical observations, even if the formal side needs tightening. It deserves a serious referee to check the math and push for clearer methods and controls rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a deterministic functional-topological framework in which valid realisations from physical processes form a compact subset of a Banach space endowed with stable invariants, a finite empirical radius, and an induced continuous perceptual functional. This geometry is claimed to constrain variability and enable generalisation from sparse evidence. The framework is developed and tested empirically across five domains (electromechanical railway point machines, electrochemical battery discharge, physiological ECG signals, atmospheric solar irradiance, and geophysical tidal cycles), with comparisons to simulators where available; the key observation is that empirical radius and internal Hausdorff stability saturate after surprisingly few samples.

Significance. If the central quantities are rigorously defined and the saturation results hold under appropriate controls, the work would supply a geometric organising principle for low-variability concentration in real-world signals, offering structural constraints on identifiability and generalisation that could inform both theoretical models of perception and practical representation learning.

major comments (2)

[Abstract] Abstract and framework section: the abstract asserts saturation of empirical radius and Hausdorff stability after few samples but supplies neither the mathematical definition of these quantities nor any statistical controls, error bars, or comparison against null models; the central claim therefore rests on unshown derivations.
[Empirical Results] Empirical results section: the empirical radius and internal stability are described as saturating in the five domains; without explicit computation details it is impossible to determine whether these quantities are computed independently or are effectively fitted parameters whose values are then used to support the compactness claim.

minor comments (2)

[Framework] Clarify the precise relationship between the perceptual functional and the underlying Banach-space embedding; the current description leaves open whether the functional is derived from the geometry or postulated separately.
[Discussion] Add a brief discussion of how the chosen domains were selected and whether the saturation behaviour generalises beyond the reported cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the opportunity to strengthen the clarity of our presentation. We address each major comment below and have revised the manuscript to incorporate additional definitions, computational details, and controls as requested.

read point-by-point responses

Referee: [Abstract] Abstract and framework section: the abstract asserts saturation of empirical radius and Hausdorff stability after few samples but supplies neither the mathematical definition of these quantities nor any statistical controls, error bars, or comparison against null models; the central claim therefore rests on unshown derivations.

Authors: The definitions of empirical radius (the radius of the minimal enclosing ball in the chosen Banach-space metric) and internal Hausdorff stability (Hausdorff distance between nested subsets of increasing cardinality) are formally introduced in Section 2 of the framework. The abstract is kept concise by design, yet we agree that a brief inline definition improves readability. We have therefore expanded the abstract with one-sentence definitions of both quantities. On statistical controls, the original manuscript already reports comparisons against deterministic simulators in three domains; we have now added bootstrap-derived error bars on the saturation curves and a null-model baseline (uniform sampling in the ambient function space) to quantify that the observed early saturation is statistically distinguishable from unstructured variability. revision: yes
Referee: [Empirical Results] Empirical results section: the empirical radius and internal stability are described as saturating in the five domains; without explicit computation details it is impossible to determine whether these quantities are computed independently or are effectively fitted parameters whose values are then used to support the compactness claim.

Authors: Both quantities are computed directly from the observed samples without parameter fitting: the empirical radius is obtained by solving the smallest-ball problem on the finite set of signal realizations, and internal stability is the Hausdorff distance between the manifolds induced by the first k and first n samples. To eliminate ambiguity we have inserted an explicit algorithmic subsection (new Algorithm 1) together with the precise metric and discretization choices used in each domain. These additions make clear that the reported saturation values are independent data-derived statistics rather than fitted parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework is self-contained organising principle

full rationale

The paper introduces the functional-topological framework by definition as a formalisation of the observed principle that real-world signals concentrate on compact low-variability subsets, then directly measures saturation of the empirical radius and Hausdorff stability after few samples across five independent domains with simulator comparisons where available. Compactness is treated as an a-priori organising principle rather than a derived theorem, and the empirical saturation results constitute independent external validation against real data rather than a fit or self-citation that reduces to the input claim. No load-bearing step equates a prediction to its own fitted parameter or relies on unverified self-citation chains.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on the domain assumption that signals are non-arbitrary and introduces the empirical radius and perceptual functional as central constructs whose definitions and independence from data fitting cannot be verified from the abstract.

free parameters (1)

empirical radius
Described as finite and saturating; appears to be a data-derived quantity whose exact computation is not given.

axioms (1)

domain assumption Real-world phenomena do not generate arbitrary variability; their signals concentrate on compact, low-variability subsets of functional space.
Opening premise of the abstract that underpins the entire formalization.

invented entities (1)

perceptual functional no independent evidence
purpose: Continuous functional induced by the compact geometry to support perception and generalization.
New construct introduced in the framework with no independent evidence or falsifiable prediction supplied in the abstract.

pith-pipeline@v0.9.0 · 5502 in / 1450 out tokens · 44088 ms · 2026-05-17T01:16:32.603281+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the set of valid realisations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite empirical radius, and an induced continuous perceptual functional
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

If the family {f(s, θ)} is uniformly bounded and equicontinuous on [0, T], then the perceptual set M is compact in C⁰([0, T])

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

A path towards autonomous machine intelligence

Yann LeCun. A path towards autonomous machine intelligence. Technical report, Meta AI Research,

work page
[2]

White paper / technical report

work page
[3]

McGraw-Hill, 1991

Walter Rudin.Functional Analysis. McGraw-Hill, 1991. 22

work page 1991
[4]

Royden and Patrick Fitzpatrick.Real Analysis

Halsey L. Royden and Patrick Fitzpatrick.Real Analysis. Pearson, 4 edition, 2010

work page 2010
[5]

Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals, and Systems, 2(4):303–314, 1989

George Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals, and Systems, 2(4):303–314, 1989

work page 1989
[6]

Approximation capabilities of multilayer feedforward networks.Neural Networks, 4(2):251–257, 1991

Kurt Hornik. Approximation capabilities of multilayer feedforward networks.Neural Networks, 4(2):251–257, 1991

work page 1991
[7]

Vl-jepa: Joint embedding predictive architecture for vision- language, 2025

Delong Chen, Mustafa Shukor, Th´ eo Moutakanni, Willy Chung, Jade Yu, Tejaswi Kasarla, Allen Bolourchi, Yann LeCun, and Pascale Fung. Vl-jepa: Joint embedding predictive architecture for vision- language, 2025

work page 2025
[8]

Scalable, technology-agnostic diagnosis and predictive main- tenance for point machine using deep learning.arXiv preprint arXiv:2508.11692, 2025

Ruixiang Ci, Eduardo Di Santi, Cl´ ement Lefebvre, Nenad Mijatovic, Michele Pugnaloni, Jonathan Brown, Victor Mart´ ın, and Kenza Saiah. Scalable, technology-agnostic diagnosis and predictive main- tenance for point machine using deep learning.arXiv preprint arXiv:2508.11692, 2025

work page arXiv 2025
[9]

Edgar.Measure, Topology, and Fractal Geometry

Gerald A. Edgar.Measure, Topology, and Fractal Geometry. Springer, 2008

work page 2008
[10]

A public railway point machine operating current dataset for fault diagnosis.Data in Brief, 32:106123, 2020

Jian Li, Kai Zhang, and Yufei Wang. A public railway point machine operating current dataset for fault diagnosis.Data in Brief, 32:106123, 2020. Dataset accessible through publisher supplementary material or mirrored repositories

work page 2020
[11]

Railway point machine operating current dataset.https:// data.mendeley.com/datasets/v43h2m7s4v/1, 2020

Jian Li, Kai Zhang, and Yufei Wang. Railway point machine operating current dataset.https:// data.mendeley.com/datasets/v43h2m7s4v/1, 2020. Canonical public dataset for PM current signals (China)

work page 2020
[12]

Nasa ames prognostics center of excellence: Li-ion battery aging dataset.https://ti.arc.nasa.gov/tech/dash/pcoe/prognostic-data-repository/, 2007

Bhaskar Saha and Kai Goebel. Nasa ames prognostics center of excellence: Li-ion battery aging dataset.https://ti.arc.nasa.gov/tech/dash/pcoe/prognostic-data-repository/, 2007. NASA PCoE Li-ion battery prognostics dataset

work page 2007
[13]

Prognostics methods for battery health monitoring using a bayesian framework

Bhaskar Saha and Kai Goebel. Prognostics methods for battery health monitoring using a bayesian framework. In2007 IEEE Aerospace Conference, pages 1–8. IEEE, 2007

work page 2007
[14]

Modeling li-ion battery capacity depletion in a particle filter framework

Bhaskar Saha and Kai Goebel. Modeling li-ion battery capacity depletion in a particle filter framework. Proceedings of the Annual Conference of the Prognostics and Health Management Society, 2011

work page 2011
[15]

Moody and Roger G

George B. Moody and Roger G. Mark. The mit-bih arrhythmia database.IEEE Engineering in Medicine and Biology Magazine, 20(3):45–50, 2001. Original dataset released in 1980

work page 2001
[16]

Moody and R.G

G.B. Moody and R.G. Mark. Mit-bih arrhythmia database.https://physionet.org/content/mitdb/ 1.0.0/, 1980. Canonical ECG dataset used in signal analysis and medical AI

work page 1980
[17]

Goldberger, Luis A

Ary L. Goldberger, Luis A. Nunes Amaral, Leon Glass, Jeffrey M. Hausdorff, Plamen Ch. Ivanov, Roger G. Mark, Joseph E. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. Physiobank, physiotoolkit, and physionet. Circulation 101(23):e215–e220, 2000. General reference for MIT-BIH and related datasets

work page 2000
[18]

A dynamical model for generating synthetic electrocardiogram signals.IEEE Transactions on Biomedical Engineering, 50(3):289–294, 2003

Patrick E McSharry, Gari D Clifford, Lionel Tarassenko, and Leonard A Smith. A dynamical model for generating synthetic electrocardiogram signals.IEEE Transactions on Biomedical Engineering, 50(3):289–294, 2003. 23

work page 2003
[19]

Euipo esearch plus: Ai-based image search for trade marks and designs.https://euipo.europa.eu/eSearch/, 2019

European Union Intellectual Property Office (EUIPO). Euipo esearch plus: Ai-based image search for trade marks and designs.https://euipo.europa.eu/eSearch/, 2019. Accessed: 2026-01-11. Appendices Appendix A Mathematical Proofs This appendix provides complete proofs for the theorems and propositions stated in Sections 2–4. All results are stated in the Ban...

work page 2019
[20]

The family isuniformly bounded, meaning there existsM >0 such that∥f(s, θ)∥ ∞ ≤Mfor all (s, θ)

work page
[21]

for everyε >0 there existsδ >0 such that for allt 1, t2 ∈[0, T]: |t1 −t 2|< δ⇒ |f(s, θ)(t 1)−f(s, θ)(t 2)|< ε

The family isequicontinuous, i.e. for everyε >0 there existsδ >0 such that for allt 1, t2 ∈[0, T]: |t1 −t 2|< δ⇒ |f(s, θ)(t 1)−f(s, θ)(t 2)|< ε. By the Arzel` a–Ascoli Theorem [2, 3], any uniformly bounded and equicontinuous family of functions has compact closure inC 0([0, T]). ThusMis compact. Appendix A.2 Proof of Proposition 3.2 (Finiteness of the Per...

work page

[1] [1]

A path towards autonomous machine intelligence

Yann LeCun. A path towards autonomous machine intelligence. Technical report, Meta AI Research,

work page

[2] [2]

White paper / technical report

work page

[3] [3]

McGraw-Hill, 1991

Walter Rudin.Functional Analysis. McGraw-Hill, 1991. 22

work page 1991

[4] [4]

Royden and Patrick Fitzpatrick.Real Analysis

Halsey L. Royden and Patrick Fitzpatrick.Real Analysis. Pearson, 4 edition, 2010

work page 2010

[5] [5]

Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals, and Systems, 2(4):303–314, 1989

George Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals, and Systems, 2(4):303–314, 1989

work page 1989

[6] [6]

Approximation capabilities of multilayer feedforward networks.Neural Networks, 4(2):251–257, 1991

Kurt Hornik. Approximation capabilities of multilayer feedforward networks.Neural Networks, 4(2):251–257, 1991

work page 1991

[7] [7]

Vl-jepa: Joint embedding predictive architecture for vision- language, 2025

Delong Chen, Mustafa Shukor, Th´ eo Moutakanni, Willy Chung, Jade Yu, Tejaswi Kasarla, Allen Bolourchi, Yann LeCun, and Pascale Fung. Vl-jepa: Joint embedding predictive architecture for vision- language, 2025

work page 2025

[8] [8]

Scalable, technology-agnostic diagnosis and predictive main- tenance for point machine using deep learning.arXiv preprint arXiv:2508.11692, 2025

Ruixiang Ci, Eduardo Di Santi, Cl´ ement Lefebvre, Nenad Mijatovic, Michele Pugnaloni, Jonathan Brown, Victor Mart´ ın, and Kenza Saiah. Scalable, technology-agnostic diagnosis and predictive main- tenance for point machine using deep learning.arXiv preprint arXiv:2508.11692, 2025

work page arXiv 2025

[9] [9]

Edgar.Measure, Topology, and Fractal Geometry

Gerald A. Edgar.Measure, Topology, and Fractal Geometry. Springer, 2008

work page 2008

[10] [10]

A public railway point machine operating current dataset for fault diagnosis.Data in Brief, 32:106123, 2020

Jian Li, Kai Zhang, and Yufei Wang. A public railway point machine operating current dataset for fault diagnosis.Data in Brief, 32:106123, 2020. Dataset accessible through publisher supplementary material or mirrored repositories

work page 2020

[11] [11]

Railway point machine operating current dataset.https:// data.mendeley.com/datasets/v43h2m7s4v/1, 2020

Jian Li, Kai Zhang, and Yufei Wang. Railway point machine operating current dataset.https:// data.mendeley.com/datasets/v43h2m7s4v/1, 2020. Canonical public dataset for PM current signals (China)

work page 2020

[12] [12]

Nasa ames prognostics center of excellence: Li-ion battery aging dataset.https://ti.arc.nasa.gov/tech/dash/pcoe/prognostic-data-repository/, 2007

Bhaskar Saha and Kai Goebel. Nasa ames prognostics center of excellence: Li-ion battery aging dataset.https://ti.arc.nasa.gov/tech/dash/pcoe/prognostic-data-repository/, 2007. NASA PCoE Li-ion battery prognostics dataset

work page 2007

[13] [13]

Prognostics methods for battery health monitoring using a bayesian framework

Bhaskar Saha and Kai Goebel. Prognostics methods for battery health monitoring using a bayesian framework. In2007 IEEE Aerospace Conference, pages 1–8. IEEE, 2007

work page 2007

[14] [14]

Modeling li-ion battery capacity depletion in a particle filter framework

Bhaskar Saha and Kai Goebel. Modeling li-ion battery capacity depletion in a particle filter framework. Proceedings of the Annual Conference of the Prognostics and Health Management Society, 2011

work page 2011

[15] [15]

Moody and Roger G

George B. Moody and Roger G. Mark. The mit-bih arrhythmia database.IEEE Engineering in Medicine and Biology Magazine, 20(3):45–50, 2001. Original dataset released in 1980

work page 2001

[16] [16]

Moody and R.G

G.B. Moody and R.G. Mark. Mit-bih arrhythmia database.https://physionet.org/content/mitdb/ 1.0.0/, 1980. Canonical ECG dataset used in signal analysis and medical AI

work page 1980

[17] [17]

Goldberger, Luis A

Ary L. Goldberger, Luis A. Nunes Amaral, Leon Glass, Jeffrey M. Hausdorff, Plamen Ch. Ivanov, Roger G. Mark, Joseph E. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. Physiobank, physiotoolkit, and physionet. Circulation 101(23):e215–e220, 2000. General reference for MIT-BIH and related datasets

work page 2000

[18] [18]

A dynamical model for generating synthetic electrocardiogram signals.IEEE Transactions on Biomedical Engineering, 50(3):289–294, 2003

Patrick E McSharry, Gari D Clifford, Lionel Tarassenko, and Leonard A Smith. A dynamical model for generating synthetic electrocardiogram signals.IEEE Transactions on Biomedical Engineering, 50(3):289–294, 2003. 23

work page 2003

[19] [19]

Euipo esearch plus: Ai-based image search for trade marks and designs.https://euipo.europa.eu/eSearch/, 2019

European Union Intellectual Property Office (EUIPO). Euipo esearch plus: Ai-based image search for trade marks and designs.https://euipo.europa.eu/eSearch/, 2019. Accessed: 2026-01-11. Appendices Appendix A Mathematical Proofs This appendix provides complete proofs for the theorems and propositions stated in Sections 2–4. All results are stated in the Ban...

work page 2019

[20] [20]

The family isuniformly bounded, meaning there existsM >0 such that∥f(s, θ)∥ ∞ ≤Mfor all (s, θ)

work page

[21] [21]

for everyε >0 there existsδ >0 such that for allt 1, t2 ∈[0, T]: |t1 −t 2|< δ⇒ |f(s, θ)(t 1)−f(s, θ)(t 2)|< ε

The family isequicontinuous, i.e. for everyε >0 there existsδ >0 such that for allt 1, t2 ∈[0, T]: |t1 −t 2|< δ⇒ |f(s, θ)(t 1)−f(s, θ)(t 2)|< ε. By the Arzel` a–Ascoli Theorem [2, 3], any uniformly bounded and equicontinuous family of functions has compact closure inC 0([0, T]). ThusMis compact. Appendix A.2 Proof of Proposition 3.2 (Finiteness of the Per...

work page