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Semi-Analytical Estimation for the Escape of Solutions of Linear Differential Equations with Slowly Varying Coefficients
Full text linkEmerging vibrations are a major and often unpredictable limitation in engineering design. The engineering description of these vibrations is usually based on linear models with constant parameters. However, in reality, this is rarely the case and parameters do vary in engineering systems during operation. Among these systems, we consider the simplest case: a scalar system with two time scales, one fast describing the process dynamics, and another slow for parameter change. It has been noted that at a bifurcation point, going from stable to unstable, dynamical systems are resilient and do not immediately lose stability. However, traditional numerical schemes are not able to handle this situation, and numerical solutions cannot be trusted.
We present and test a method for approximating solutions, and we name these approximations \textit{semi-analytical solutions};
we rigorously prove the efficacy of the method. Our main result shows that semi-analytical solutions exist beyond the bifurcation time
The arc-Floer conjecture and the embedded Nash problem in singular spaces
Full text linkIn this thesis, we address the conjectural relation between contact loci in arc
spaces and the symplectic properties of the Milnor fibration. We will mainly deal
with the arc-Floer conjecture, first recorded in the literature in [12]. It states
that given a smooth algebraic variety X and an algebraic function f : X →C
which defines an isolated hypersurface singularity, the cohomology with compact
support of the m-contact locus of f coincides (up to a shift) with the Floer
homology of the m-th iterate of a symplectic monodromy of f. In essence,
this problem follows the same philosophy as the famous monodromy conjecture,
aiming to relate the same algebraic and topological invariants of the hypersurface
singularity defined by f.
The primary goal of this thesis is to show that the arc-Floer conjecture holds
when X is of dimension two, providing the first piece of evidence supporting
the conjecture. This is the main result of Eduardo de Lorenzo Poza and the
author in [19]. The proof is done by computing both invariants in terms of an
m-separating embedded resolution of f, so comparing both results yields the
desired conclusion.
In the topological side, a crucial tool to analyse the Floer homology of the
iterates of the monodromy in terms of an embedded resolution is the symplectic
A’Campo fibration constructed in [28]. Moreover, in dimension two, Seidel
showed in [59] that Floer homology is essentially a purely topological object.
Bringing this idea to our setting, we are able to compute the Floer homology
of the iterates of the monodromy explicitly, a rather unusual achievement in
symplectic topology.
On the contrary, in the algebraic side, the first difficulty that arises when
studying the geometry of contact loci is that they are typically not irreducible.
In fact, in [24] the question of determining the irreducible components of contact
loci in terms of an embedded resolution of f was posed. This problem is now
known as the embedded Nash problem, and it was addressed in depth by Nero
iii
Budur, Eduardo de Lorenzo Poza, Javier Fernández de Bobadilla, Tomasz Pełka
and the author in [11]. In particular, the case when X is two-dimensional was
completely solved. This result allows understanding the topology of contact
loci through its irreducible components, obtaining its cohomology with compact
support.
The techniques used in the proof of the arc-Floer conjecture in dimension two
suggest that the problem could have a natural extension by allowing singularities
in X. This is an ongoing project, but the first steps towards this new goal
have already been made. In particular, in [18] the author posed the embedded
Nash problem in greater generality by allowing singularities in X, and solved it
again in the case of dimension two. Also, under some topological conditions
on the singularity, the computation of the Floer homology of the iterates of
the monodromy can be carried out similarly to the smooth-ambient-space case.
This thesis presents this computation for the first time in this generality
Acoustical features as knee health biomarkers: A critical analysis
No full textAcoustical knee health assessment has long promised an alternative to clinically available medical imaging tools, but this modality has yet to be adopted in medical practice. The field is currently led by machine learning models processing acoustical features, which have presented promising diagnostic performances. However, these methods overlook the intricate multi-source nature of audio signals and the underlying mechanisms at play. By addressing this critical gap, the present paper introduces a novel causal framework for validating knee acoustical features. We argue that current machine learning methodologies for acoustical knee diagnosis lack the required assurances and thus cannot be used to classify acoustic features as biomarkers. Our framework establishes a set of essential theoretical guarantees necessary to validate this claim. We apply our methodology to three real-world experiments investigating the effect of researchers’ expectations, the experimental protocol, and the wearable employed sensor. We reveal latent issues such as underlying shortcut learning and performance inflation. This study is the first independent result reproduction study in acoustical knee health evaluation. We conclude by offering actionable insights that address key limitations, providing valuable guidance for future research in knee health acoustics.RYC2021-032853-
Adaptive multi-stage integration schemes for Hamiltonian Monte Carlo
Full text linkHamiltonian Monte Carlo (HMC) is a powerful tool for Bayesian statistical inference due to its potential to rapidly explore high dimensional state space, avoiding the random walk behavior typical of many Markov Chain Monte Carlo samplers. The proper choice of the integrator of the Hamiltonian dynamics is key to the efficiency of HMC. It is becoming increasingly clear that multi-stage splitting integrators are a good alternative to the Verlet method, traditionally used in HMC. Here we propose a principled way of finding optimal, problem-specific integration schemes (in terms of the best conservation of energy for harmonic forces/Gaussian targets) within the families of 2- and 3-stage splitting integrators. The method, which we call Adaptive Integration Approach for statistics, or s-AIA, uses a multivariate Gaussian model and simulation data obtained at the HMC burn-in stage to identify a system-specific dimensional stability interval and assigns the most appropriate 2-/3-stage integrator for any user-chosen simulation step size within that interval. s-AIA has been implemented in the in-house software package HaiCS without introducing computational overheads in the simulations. The efficiency of the s-AIA integrators and their impact on the HMC accuracy, sampling performance and convergence are discussed in comparison with known fixed-parameter multi-stage splitting integrators (including Verlet). Numerical experiments on well-known statistical models show that the adaptive schemes reach the best possible performance within the family of 2-, 3-stage splitting schemes.PID2019-104927GB-C21
MCIN/AEI/10.13039/501100011033
ERDF (“A way of making Europe”)
KK-2022/00006
KK-2021/00022
KK-2021/00064
LCF/BQ/DI20/1178002
Cost-sensitive ordinal classification methods to predict SARS-CoV-2 pneumonia severity
Full text linkObjective: To study the suitability of cost-sensitive ordinal artificial intelligence-machine learning (AI-ML) strategies in the prognosis of SARS-CoV-2 pneumonia severity.
Materials & methods: Observational, retrospective, longitudinal, cohort study in 4 hospitals in Spain. Information regarding demographic and clinical status was supplemented by socioeconomic data and air pollution exposures. We proposed AI-ML algorithms for ordinal classification via ordinal decomposition and for cost-sensitive learning via resampling techniques. For performance-based model selection, we defined a custom score including per-class sensitivities and asymmetric misprognosis costs. 260 distinct AI-ML models were evaluated via 10 repetitions of 5×5 nested cross-validation with hyperparameter tuning. Model selection was followed by the calibration of predicted probabilities. Final overall performance was compared against five well-established clinical severity scores and against a ‘standard’ (non-cost sensitive, non-ordinal) AI-ML baseline. In our best model, we also evaluated its explainability with respect to each of the input variables.
Results: The study enrolled =1548 patients: 712 experienced low, 238 medium, and 598 high clinical severity. =131 variables were collected, becoming =148 features after categorical encoding. Model selection resulted in our best-performing AI-ML pipeline having: a) no imputation of missing data, b) no feature selection (i.e. using the full set of features), c) ‘Ordered Partitions’ ordinal decomposition, d) cost-based reimbalance, and e) a Histogram-based Gradient Boosting classifier. This best model (calibrated) obtained a median accuracy of 68.1% [67.3%, 68.8%] (95% confidence interval), a balanced accuracy of 57.0% [55.6%, 57.9%], and an overall area under the curve (AUC) 0.802 [0.795, 0.808]. In our dataset, it outperformed all five clinical severity scores and the ‘standard’ AI-ML baseline.
Discussion & conclusion: We conducted an exhaustive exploration of AI-ML methods designed for both ordinal and cost-sensitive classification, motivated by a real-world application domain (clinical severity prognosis) in which these topics arise naturally. Our model with the best classification performance exploited successfully the ordering information of ground truth classes, coping with imbalance and asymmetric costs. However, these ordinal and cost-sensitive aspects are seldom explored in the literature
High-quality smooth finishing of blade-like geometries via multi-pass 5-axis flank CNC machining using conical cutting tools
Full text linkExisting multi-pass planning methods often result in undesirable gaps or overlaps between adjacent paths of a cutter. These gaps and/or overlaps in the path-planning stage cause artifacts in the physical machining and these locations must be polished as a post-process using either a tiny ball-end cutter and/or by hand-polishing. While highly curved or convex geometries are impossible to be flank CNC-machined with conical tools, certain hyperbolic geometries, like blades of blisks, admit a new path-planning strategies that aim at smooth surface finish by joining neighboring flank paths of the tool with G1 continuity.
In this paper, we further develop the approach of flank milling with conical tools such that the surface finish is as smooth as
possible in terms of the continuity of the neighboring paths, verify the effectiveness of the method by physical machining of a
particular testcase blade geometry and show that our machining paths reduce the machining error by up to 85% while maintaining the same machining time when compared to a state-of-the-art commercial software. Moreover, in the vicinity of the boundaries of the paths, the proposed approach basically eliminates the approximation error caused by the transition from one path to another, providing smooth surface finish, and consequently avoiding the necessity of post-process polishing.This work was supported by the Basque Government via the AURRERA project (Elkartek KK-2024/00024), by the Spanish Ministry of Science, Innovation and Universities, grant No PID2019-104488RB-I00, by BCAM “SeveroOchoa” accreditation CEX2021-001142-S, and by the European Union’s Horizon 2020 program under grant agreement No
862025. Michael Bartoˇn was supported by the Ram´on y Cajal fellowship RYC-2017-22649. Thanks are expressed to Grant
PDC2021-121792-I00 HCR Taylor funded by MCIN/AEI/10.13039/501100011033 and, as appropriate, by “ERDF A way
of making Europe”, or by the “European Union NextGeneration EU/PRTR” and also to the Basque Government university
Group grant IT1573-22
Concave Grain Boundaries Stabilized by Boron Segregation for Efficient and Durable Oxygen Reduction
Full text linkThe oxygen reduction reaction (ORR) is a critical process that limits the efficiency of fuel cells and metal-air batteries due to its slow kinetics, even when catalyzed by platinum (Pt). To reduce Pt usage, enhancing both the specific activity and electrochemically active surface area (ECSA) of Pt catalysts is essential. Here, ultrafine, grain boundary (GB)-rich Pt nanoparticle assemblies are proposed as efficient ORR catalysts. These nanowires offer a large ECSA and a high density of concave GB sites, which improve specific activity. Atoms at these GB sites exhibit increased coordination and lattice distortion, leading to a favorable reduction in oxygen binding energy and enhanced ORR performance. Furthermore, boron segregation stabilizes these GBs, preserving active sites during catalysis. The resulting boron-stabilized Pt nanoassemblies demonstrate ORR specific and mass activities of 9.18 mA cm−2 and 6.40 A mg−1Pt (at 0.9 V vs. RHE), surpassing commercial Pt/C catalysts by over 35-fold, with minimal degradation after 60 000 potential cycles. This approach offers a versatile platform for optimizing the catalytic performance of a wide range of nanoparticle systems
Obtaining patient phenotypes in SARS-CoV-2 pneumonia, and their association with clinical severity and mortality
Full text linkBackground There exists consistent empirical evidence in the literature pointing out ample heterogeneity in terms
of the clinical evolution of patients with COVID-19. The identifcation of specifc phenotypes underlying in the popula‑
tion might contribute towards a better understanding and characterization of the diferent courses of the disease. The
aim of this study was to identify distinct clinical phenotypes among hospitalized patients with SARS-CoV-2 pneumo‑
nia using machine learning clustering, and to study their association with subsequent clinical outcomes as severity
and mortality.
Methods Multicentric observational, prospective, longitudinal, cohort study conducted in four hospitals in Spain. We
included adult patients admitted for in-hospital stay due to SARS-CoV-2 pneumonia. We collected a broad spectrum
of variables to describe exhaustively each case: patient demographics, comorbidities, symptoms, physiological status,
baseline examinations (blood analytics, arterial gas test), etc.
For the development and internal validation of the clustering/phenotype models, the dataset was split into train‑
ing and test sets (50% each). We proposed a sequence of machine learning stages: feature scaling, missing data
imputation, reduction of data dimensionality via Kernel Principal Component Analysis (KPCA), and clustering
with the k-means algorithm. The optimal cluster model parameters –including k, the number of phenotypes– were
chosen automatically, by maximizing the average Silhouette score across the training set.
Results We enrolled 1548 patients, each of them characterized by 92 clinical attributes (d=109 features after variable
encoding). Our clustering algorithm identifed k=3 distinct phenotypes and 18 strongly informative variables: Pheno‑
type A (788 cases [50.9% prevalence] – age∼57, Charlson comorbidity∼1, pneumonia CURB-65 score∼0 to 1, respira‑
tory rate at admission∼18 min-1
, FiO2∼21%, C-reactive protein CRP∼49.5 mg/dL [median within cluster]); phenotype
B (620 cases [40.0%] – age∼75, Charlson∼5, CURB-65∼1 to 2, respiration∼20 min-1
, FiO2∼21%, CRP∼101.5 mg/dL);
and phenotype C (140 cases [9.0%] – age∼71, Charlson∼4, CURB-65∼0 to 2, respiration∼30 min-1
, FiO2∼38%, CRP∼
152.3 mg/dL).
Hypothesis testing provided solid statistical evidence supporting an interaction between phenotype and each clini‑
cal outcome: severity and mortality. By computing their corresponding odds ratios, a clear trend was found for highe
Semi-analytical framework for the study of finite-time stability of forced dynamical systems with time varying parameters
Full text linkWe present a framework to analytically approximate the solution of forced dynamical systems with time varying parameters and to analyse their finite-time stability. The work was inspired by an example in robotic machining, where the mechanical parameters of the system can vary over a wide range during the process, and where there are large forces due to an assumed cutting operation. The simplest possible non-autonomous linear system undergoing dynamic stability loss is studied which serves as a solid foundation to explore the mathematical intricacy behind such systems. After defining the differential equation corresponding to this simple system, the complementary function is studied using a frozen-time approach. The particular integral can be evaluated for this system by the asymptotic expansion of error functions. We present a new approach for the approximation of particular integrals, the iterative integration by parts (IIBP) method, which is then extensively studied and compared to the equations describing the exact analytic solution. The convergence and sensitivity of the IIBP method are discussed. The method is extended to multiple degrees of freedom mechanical systems with time varying parameters. It is shown that standard numerical schemes are not suitable for predicting finite-time stability properties even in the simplest case, because small errors accumulate causing large differences from the analytical solution
Imaginary past of a quantum particle moving on imaginary time
Full text linkThe analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration |\mathcal{T}|. Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |\mathcal{T}| being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it remembers very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why