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Let $\ell$ and $p$ be distinct primes, and let $\Gamma$ be an abelian pro-$p$-group. We study the structure of the algebra $\Lambda:=\mathbb{Z}_\ell[[\Gamma]]$ and of $\Lambda$-modules. In the case $\Gamma\simeq \mathbb{Z}_p^d$, we consider a $\mathbb{Z}_p^d$-extension $K/k$ of a global field $k$ and use the structure theorems to provide explicit formulas for the orders and $\ell$-ranks of certain Iwasawa modules (namely $\ell$-class groups and $\ell$-Selmer groups) associated with the finite subextensions of $K$. We apply this new approach to provide different proofs and generalizations of results of Washington and Sinnott on $\ell$-class groups., Comment: (Very) Preliminary version. Comments are welcome

Neurological disorders, including stroke, spinal cord injuries, multiple sclerosis, and Parkinson's disease, generally lead to diminished upper extremity (UE) function, impacting individuals' independence and quality of life. Traditional assessments predominantly focus on standardized clinical tasks, offering limited insights into real-life UE performance. In this context, this review focuses on wearable technologies as a promising solution to monitor UE function in neurologically impaired individuals during daily life activities. Our primary objective is to categorize the different sensors, review the data collection and understand the employed data processing approaches. After screening over 1500 papers and including 21 studies, what comes to light is that the majority of them involved stroke survivors, and predominantly employed accelerometers or inertial measurement units to collect kinematics. Most analyses in these studies were performed offline, focusing on activity duration and frequency as key metrics. Although wearable technology shows potential in monitoring UE function in real-life scenarios, it also appears that a solution combining non-intrusiveness, lightweight design, detailed hand and finger movement capture, contextual information, extended recording duration, ease of use, and privacy protection remains an elusive goal. These are critical characteristics for a monitoring solution and researchers in the field should try to integrate the most in future developments. Last but not least, it stands out a growing necessity for a multimodal approach in capturing comprehensive data on UE function during real-life activities to enhance the personalization of rehabilitation strategies and ultimately improve outcomes for these individuals., Comment: IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2024

There is a growing interest in using pose estimation algorithms for video-based assessment of Bradykinesia in Parkinson's Disease (PD) to facilitate remote disease assessment and monitoring. However, the accuracy of pose estimation algorithms in videos from video streaming services during Telehealth appointments has not been studied. In this study, we used seven off-the-shelf hand pose estimation models to estimate the movement of the thumb and index fingers in videos of the finger-tapping (FT) test recorded from Healthy Controls (HC) and participants with PD and under two different conditions: streaming (videos recorded during a live Zoom meeting) and on-device (videos recorded locally with high-quality cameras). The accuracy and reliability of the models were estimated by comparing the models' output with manual results. Three of the seven models demonstrated good accuracy for on-device recordings, and the accuracy decreased significantly for streaming recordings. We observed a negative correlation between movement speed and the model's accuracy for the streaming recordings. Additionally, we evaluated the reliability of ten movement features related to bradykinesia extracted from video recordings of PD patients performing the FT test. While most of the features demonstrated excellent reliability for on-device recordings, most of the features demonstrated poor to moderate reliability for streaming recordings. Our findings highlight the limitations of pose estimation algorithms when applied to video recordings obtained during Telehealth visits, and demonstrate that on-device recordings can be used for automatic video-assessment of bradykinesia in PD., Comment: 12 pages, 3 figures, 2 tables

Background: Egocentric video has recently emerged as a potential solution for monitoring hand function in individuals living with tetraplegia in the community, especially for its ability to detect functional use in the home environment. Objective: To develop and validate a wearable vision-based system for measuring hand use in the home among individuals living with tetraplegia. Methods: Several deep learning algorithms for detecting functional hand-object interactions were developed and compared. The most accurate algorithm was used to extract measures of hand function from 65 hours of unscripted video recorded at home by 20 participants with tetraplegia. These measures were: the percentage of interaction time over total recording time (Perc); the average duration of individual interactions (Dur); the number of interactions per hour (Num). To demonstrate the clinical validity of the technology, egocentric measures were correlated with validated clinical assessments of hand function and independence (Graded Redefined Assessment of Strength, Sensibility and Prehension - GRASSP, Upper Extremity Motor Score - UEMS, and Spinal Cord Independent Measure - SCIM). Results: Hand-object interactions were automatically detected with a median F1-score of 0.80 (0.67-0.87). Our results demonstrated that higher UEMS and better prehension were related to greater time spent interacting, whereas higher SCIM and better hand sensation resulted in a higher number of interactions performed during the egocentric video recordings. Conclusions: For the first time, measures of hand function automatically estimated in an unconstrained environment in individuals with tetraplegia have been validated against internationally accepted measures of hand function. Future work will necessitate a formal evaluation of the reliability and responsiveness of the egocentric-based performance measures for hand use.

The videofluoroscopic swallowing study (VFSS) is a gold-standard imaging technique for assessing swallowing, but analysis and rating of VFSS recordings is time consuming and requires specialized training and expertise. Researchers have recently demonstrated that it is possible to automatically detect the pharyngeal phase of swallowing and to localize the bolus in VFSS recordings via computer vision, fostering the development of novel techniques for automatic VFSS analysis. However, training of algorithms to perform these tasks requires large amounts of annotated data that are seldom available. We demonstrate that the challenges of pharyngeal phase detection and bolus localization can be solved together using a single approach. We propose a deep-learning framework that jointly tackles pharyngeal phase detection and bolus localization in a weakly-supervised manner, requiring only the initial and final frames of the pharyngeal phase as ground truth annotations for the training. Our approach stems from the observation that bolus presence in the pharynx is the most prominent visual feature upon which to infer whether individual VFSS frames belong to the pharyngeal phase. We conducted extensive experiments with multiple convolutional neural networks (CNNs) on a dataset of 1245 bolus-level clips from 59 healthy subjects. We demonstrated that the pharyngeal phase can be detected with an F1-score higher than 0.9. Moreover, by processing the class activation maps of the CNNs, we were able to localize the bolus with promising results, obtaining correlations with ground truth trajectories higher than 0.9, without any manual annotations of bolus location used for training purposes. Once validated on a larger sample of participants with swallowing disorders, our framework will pave the way for the development of intelligent tools for VFSS analysis to support clinicians in swallowing assessment.

Let F be a global function field of characteristic p with ring of integers A and let \Phi be a Hayes module on the Hilbert class field H(A) of F. We prove an Iwasawa Main Conjecture for the Z_p^\infty-extension F/F generated by the \mathfrak{p}-power torsion of \Phi (\mathfrak{p} a prime of A). The main tool is a Stickelberger series whose specialization provides a generator for the Fitting ideal of the class group of F. Moreover we prove that the same series, evaluated at complex or p-adic characters, interpolates the Goss Zeta-function or some p-adic L-function, thus providing the link between the algebraic structure (class groups) and the analytic functions, which is the crucial part of Iwasawa Main Conjecture., Comment: to appear in Publicacions Matem\`atiques

Orofacial deficits are common in people with Parkinson's disease (PD) and their evolution might represent an important biomarker of disease progression. We are developing an automated system for assessment of orofacial function in PD that can be used in-home or in-clinic and can provide useful and objective clinical information that informs disease management. Our current approach relies on color and depth cameras for the estimation of 3D facial movements. However, depth cameras are not commonly available, might be expensive, and require specialized software for control and data processing. The objective of this paper was to evaluate if depth cameras are needed to differentiate between healthy controls and PD patients based on features extracted from orofacial kinematics. Results indicate that 2D features, extracted from color cameras only, are as informative as 3D features, extracted from color and depth cameras, differentiating healthy controls from PD patients. These results pave the way for the development of a universal system for automatic and objective assessment of orofacial function in PD., Comment: 4 pages, 1 table

Egocentric vision (a.k.a. first-person vision - FPV) applications have thrived over the past few years, thanks to the availability of affordable wearable cameras and large annotated datasets. The position of the wearable camera (usually mounted on the head) allows recording exactly what the camera wearers have in front of them, in particular hands and manipulated objects. This intrinsic advantage enables the study of the hands from multiple perspectives: localizing hands and their parts within the images; understanding what actions and activities the hands are involved in; and developing human-computer interfaces that rely on hand gestures. In this survey, we review the literature that focuses on the hands using egocentric vision, categorizing the existing approaches into: localization (where are the hands or parts of them?); interpretation (what are the hands doing?); and application (e.g., systems that used egocentric hand cues for solving a specific problem). Moreover, a list of the most prominent datasets with hand-based annotations is provided.

Let $(P_d)$ be any prime of $\mathbb{F}_q[t]$ of degree $d$ and consider the space of Drinfeld cusp forms of level $P_d$, i.e. for the modular group $\Gamma_0(P_d)$. We provide a definition for oldforms and newforms of level $P_d$. Moreover, when the dimension of the vector space of oldforms is one and $P_1=t$ we prove that the space of cuspforms of level $t$ is the direct sum of oldforms and newforms and that the Hecke operator $\mathbf{T}_t$ acting on Drinfeld cusp forms of level 1 is injective, thus providing more evidence for the conjectures presented and stated in [2] and [3].

We define oldforms and newforms for Drinfeld cusp forms of level $t$ and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix $U$ associated to the action of the Atkin operator $\mathbf{U}_t$ on cusp forms of level $t$ and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on the diagonalizability of $\mathbf{U}_t$ and on various other issues. Via the explicit form of the matrix $U$ we are then able to verify our conjectures in various cases (mainly in small weights)., Comment: Final version, to appear in Exp. Math

We generalize some results of Greither and Popescu to a geometric Galois cover $X\rightarrow Y$ which appears naturally for example in extensions generated by $\mathfrak{p}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in \cite{ABBL}) with Goss $\zeta$-function.

We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld cusp forms for $\Gamma_0(t)$: starting with the slopes of eigenvalues and then moving to the space of cusp forms for $\Gamma_1(t)$ to use Teitelbaum's interpretation as harmonic cocycles which makes computations more explicit. We prove $U_t$ is diagonalizable in odd characteristic for (relatively) small weights and explicitly compute the eigenvalues. In even characteristic we show that it is not diagonalizable when the weight is odd (except for the trivial cases) and prove some cases of non diagonalizability in even weight as well. We also formulate a few conjectures, supported by numerical search, about diagonalizability of $U_t$ and the slopes of its eigenforms., Comment: It contains parts of the previous arXiv:1702.08801 [math:NT], which has now been split in two papers with some new results

Let $A$ be an abelian variety defined over a global function field $F$ of positive characteristic $p$ and let $K/F$ be a $p$-adic Lie extension with Galois group $G$. We provide a formula for the Euler characteristic $\chi(G,Sel_A(K)_p)$ of the $p$-part of the Selmer group of $A$ over $K$. In the special case $G=\mathbb{Z}_p^d$ and $A$ a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of $Sel_A(K)_p$ is related to special values of a $p$-adic $\mathcal{L}$-function.

We study the diagonalizability of the Atkin $U$-operator acting on Drinfeld cusp forms for $\Gamma_1(t)$ and $\Gamma(t)$ using Teitelbaum's interpretation as harmonic cocycles. We prove $U$ is diagonalizable for small weights and explicitly compute the eigenvalues. We also formulate a conjecture, supported by numerical search and proofs in some special cases, about non diagonalizability of $U$ in even characteristic., Comment: This paper has been withdrawn and substituted with the two new ones arXiv:1710.01036 and arXiv:1710.01038 containing new developments

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(\theta)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[\theta]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss $\zeta$-function at negative integers)., Comment: Section 3 entirely rewritten

Let $A$ be an abelian variety defined over a global field $F$ of positive characteristic $p$ and let $\calf/F$ be a $\Z_p^{\N}$-extension, unramified outside a finite set of places of $F$. Assuming that all ramified places are totally ramified, we define a pro-characteristic ideal associated to the Pontrjagin dual of the $p$-primary Selmer group of $A$, in order to formulate an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$ (which we also prove for a constant abelian variety). To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a $\Z_p^d$-extension $\calf_d/F$ and for any $\Z_p^{d-1}$-extension contained in $\calf_d\,$, and then use a limit process., Comment: 10 pages, version updated to be compatible with the modifications of arXiv:1310.0680 [math.NT]

Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the number fields obtained by adding to $K$ the coordinates of the $m$-torsion points of $\mathcal{E}$. We look for small (sometimes "minimal") set of generators of $K(\mathcal{E}[m])$. For $m=3$ and $m=4$, we describe explicit generators, degree and Galois groups of the extensions $K(\mathcal{E}[m])/K$.

Let $\L$ be a non-noetherian Krull domain which is the inverse limit of noetherian Krull domains $\L_d$ and let $M$ be a finitely generated $\L$-module which is the inverse limit of $\L_d$-modules $M_d\,$. Under certain hypotheses on the rings $\L_d$ and on the modules $M_d\,$, we define a pro-characteristic ideal for $M$ in $\L$, which should play the role of the usual characteristic ideals for finitely generated modules over noetherian Krull domains. We apply this to the study of Iwasawa modules (in particular of class groups) in a non-noetherian Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$, where $F$ is a function field of characteristic $p$ and $\Gal(\calf/F)\simeq\Z_p^\infty$., Comment: 15 pages, substantial chenges in exposition, new section 2.3

Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $\l$-adic Lie extension ($\l\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $\l$. We shall prove that, under certain conditions, $Sel_A(K)_\l^\vee$ has no nontrivial pseudo-null submodule., Comment: 14 pages

Let F be a global function field of characteristic p>0, K/F an l-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety without complex multiplication. We study Sel_A(K)_l^\vee (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z_l[[\Gal(K/F)]]-module via generalizations of Mazur's Control Theorem. If Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H\simeq Z_l, we are able to give sufficient conditions for Sel_A(K)_l^\vee to be finitely generated as Z_l[[H]]-module and, consequently, a torsion Z_l[[\Gal(K/F)]]-module. We deal with both cases l\neq p and l=p., Comment: 21 pages, revised arguments in sections 2 and 4

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for $\mathbb{Z}_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism., Comment: Final version. To appear in Conference proceedings of the conference t-motives: Hodge structures, transcendence and other motivic aspects at Banff International Research Station (BIRS) by EMS