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We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.

We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn-Sobolev and Korn-Hardy inequalities on stratified homogeneous groups.

INTRODUCTION: Despite the fact that joint bleeds (haemarthrosis) frequently occur in people with haemophilia (PwH) with invalidating arthropathies as result, the clinical pain experience has received only limited attention. A sudden increase in pain intensity can be linked to a bleed, but in most cases, no acute bleed is confirmed. Nevertheless, a patient's perception of an acute bleed as cause of the pain might impact the patients' behaviour in response to pain. It is therefore essential to gain more insight into pain coping strategies seen in PwH. AIM: This systematic review aims to identify the range of pain coping behaviour strategies used among PwH and the factors associated with pain coping behaviour. METHODS: This review was reported according to the Preferred Reporting Items for Systematic reviews and Meta-analyses guidelines (PRISMA). PubMed and Web of Science were systematically screened for relevant literature using keyword combinations related to adult PwH, pain and pain coping behaviour strategies. Risk of bias was assessed with the modified Newcastle-Ottowa Scale. RESULTS: Eleven full text articles (nine cross-sectional and two comparative studies) consisting of 1832 PwH met the inclusion criteria. Due to the heterogeneity of the study samples, quality of evaluation instruments and varying risk of bias, it was difficult to draw conclusions regarding the used pain coping behaviour strategies and associated factors. CONCLUSION: Literature on pain coping behaviour strategies and associated factors in PwH is still scarce and describes heterogenous results. Validated haemophilia-specific instruments are warranted to inventory pain coping behaviour in a standardized way.

Stroke results in various interactive cognitive and motor impairments that requires multidisciplinary rehabilitation. For example, preserved cognitive executive functions predict motor recovery success. Despite these links, the evaluation of executive function in hemiparesis patients remains underexplored. Here we developed a virtual reality serious game that measures inhibition based on the visual search paradigm of Treisman (1980). Our objective was to evaluate the serious game feasibility. Twenty individuals with first stroke hemiparesis and twenty age-matched healthy controls were recruited. They were instructed to find a target mole presented amongst distractor moles. The stimulus array consisted of grid of 6 columns and 4 rows of mole hills, from which the target and 11, 17 then 23 distractor moles could randomly appear, in two search conditions (single feature condition and conjunction condition). Responses were made with the ipsilesional hand for individuals with stroke and with the dominant hand for healthy participants, by hitting the target using a virtual hammer. The target appeared in every possible location in the grid (24) for each condition combination (144 trials). Participants were significantly slower in the conjunction compared to single feature condition significantly slower with increased numbers of distractors and significantly slower for individuals with hemiparesis compared to controls. A significant interaction between search condition and distractors showed that reaction time only increased with distractor number for the conjunction condition, with no further interaction between participants. The serious game can successfully evaluate inhibition. However, these preliminary results show no evidence that hemiparesis patients have inhibition impairments.

Background: Since a stroke can impair bimanual activities, enhancing bimanual cooperation through motor skill learning may improve neurorehabilitation. Therefore, robotics and neuromodulation with transcranial direct current stimulation (tDCS) are promising approaches. To date, tDCS has failed to enhance bimanual motor control after stroke possibly because it was not integrating the hypothesis that the undamaged hemisphere becomes the major poststroke hub for bimanual control. Objective: We tested the following hypotheses: (I) In patients with chronic hemiparetic stroke training on a robotic device, anodal tDCS applied over the primary motor cortex of the undamaged hemisphere enhances bimanual motor skill learning compared to sham tDCS. (II) The severity of impairment correlates with the effect of tDCS on bimanual motor skill learning. (III) Bimanual motor skill learning is less efficient in patients than in healthy individuals (HI). Methods: A total of 17 patients with chronic hemiparetic stroke and 7 healthy individuals learned a complex bimanual cooperation skill on the REAplan® neurorehabilitation robot. The bimanual speed/accuracy trade-off (biSAT), bimanual coordination (biCo), and bimanual force (biFOP) scores were computed for each performance. In patients, real/sham tDCS was applied in a crossover, randomized, double-blind approach. Results: Compared to sham, real tDCS did not enhance bimanual motor skill learning, retention, or generalization in patients, and no correlation with impairment was noted. The healthy individuals performed better than patients on bimanual motor skill learning, but generalization was similar in both groups. Conclusion: A short motor skill learning session with a robotic device resulted in the retention and generalization of a complex skill involving bimanual cooperation. The tDCS strategy that would best enhance bimanual motor skill learning after stroke remains unknown.

This thesis deals with Beyond the Standard Model (BSM) scenarios with both displaced and prompt signature. For the displaced signature a Heavy Neutral Lepton analysis in the semileptonic final state is performed. I am looking for right-handed neutrinos with mass between 1 and 15 GeV. Those particles are accommodated in Majoran and Dirac hypothesis, exclusion limits on the mass and on the lepton-neutrino coupling are presented for both models. On the other hand, also BSM scenarios with prompt signature are investigated within the Higgs framework. A ttH multileptonic differential analysis is presented targeting both EFT results and SM unfolding results. Deep Neural Network is largely use for regression and to discriminate SM for EFT signature. Following the stream of investigating the Higgs properties a HH analysis in bb4l final state is presented. This analysis takes the advantage of a very clear signature and a small amount of background that maximise the sensitivity search, limits on the anomalous values of the Higgs trilinear coupling and on the SM hypothesis are shown. All these analyses target the full Run2 dataset of the CMS experiment, (SC - Sciences) -- UCL, 2022

We describe a pretorsion theory in the category Cat of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in Cat that identify pairs of objects: they are faithful and reflect isomorphisms.

This study intended to establish international recommendations for the management of spastic equinovarus foot deformity. Delphi method SETTING: International study PARTICIPANTS: 24 international experts in neuro-orthopedic deformities, from different specialties (Physical and Rehabilitation Medicine physicians, neurologists, geriatricians, orthopaedic surgeons, neurosurgeons, and plastic surgeons) INTERVENTIONS: Experts answered three rounds of questions related to important aspects of diagnosis, assessment and treatment of spastic equinovarus foot deformity. A consensus was established when at least 80% of experts agreed on a statement RESULTS: : A total of 52 items reached consensus. Experts recommend assessing impact of the deformity on functional activities before treatment. Before treatment, it is crucial to differentiate spastic muscle overactivity from soft tissue contractures, identify which muscles are involved in the deformity and evaluate the activity of antagonist muscles. Motor nerve blocks, two-dimensional video analysis and radiological exams are often required to complement a clinical exam. The treatment of equinovarus foot depends on the correctability of the deformity, but also the patient's ability to stand or walk. The preoperative assessment should include an interdisciplinary consultation that must finalize a formal agreement between doctors and the patient, which will define personalized attainable goals before surgery. The establishment of guidelines on managing equinovarus foot will help physicians and surgeons, specialists, and non-specialists to diagnoses and assess the deformity and direct patients to a network of experts, in order to optimize patient functional recovery and improve their autonomy.

We study when and how the norm of a function u in the homogeneous Sobolev spaces W˙s,p(Rn,Rm), with p≥1 and either s=1 or s>1/p, is controlled by the norm of composite function f∘u in the same space.

We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions $s_n$ and $\bar s_n$ introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.

Background and aims Hemineglect is an attentional disorder characterized by spatial and non-spatial (inhibition) attention deficits. Currently, there is no systematic evaluation of both attention components in clinical tests. Our aim was to create an innovative integrative measure combining the assessment of spatial and non-spatial attention. using a virtual reality serious game based on the visual search paradigm of Treisman (1980) Methods: We tested sixty healthy participants. They were asked to perform a target search task where the spatial target was a mole wearing a red helmet. In a feature search condition, the distractors were moles wearing blue helmets (with and without horns). In a conjunction search condition, the distractors were moles wearing blue helmet or a red helmet with horns. Responses were made with the dominant hand by hitting the target with a virtual hammer. Results: Reaction time results demonstrated a significant effect of search tasks and distractors (F(1,57) = 2175.6, p = 0.000; F (2,114) = 97.3, p = 0.000 respectively). Participants were significantly slower in the conjunction compared to feature search task, and they were slower when the number of distractors increased. A significant interaction between search tasks and distractors (F(2,114) = 106.4, p = 0.000) showed that reaction time only increased with distractor number for the conjunction search task. Conclusion: The innovative virtual reality serious game results replicate the Treisman paradigm, providing reliable measures of spatial and non-spatial attention. Future studies will test hemineglect patients in order to deepen the understanding of spatial and non-spatial components of hemineglect.

We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pure braid group.

Background Robot-assisted therapy (RAT) could address an unmet need to relieve the strain on healthcare providers and intensify treatment in the context of an increasing stroke incidence. A comprehensive meta-analysis could provide firmer data about the topic by considering methodology limitations discovered in previous reviews and providing more rigorous evidence. Objective This meta-analysis study identifies RAT's efficacy for patients with stroke. Methods A systematic search of the 7 databases from January 10 to February 1, 2022, located relevant publications. We used the updated Cochrane risk-of-bias checklist for 52 trials to assess the methodologic quality of the included studies. The efficacy of RAT for patients with stroke was estimated using a pooled random-effects model in the Stata 16 software application. Results The final analysis included 2774 patients with stroke from 52 trials. In those patients, RAT was proven to improve quality of movement (mean difference, 0.15; 95% confidence interval, 0.03–0.28) and to reduce balance disturbances (mean difference, −1.28; 95% confidence interval, −2.48 to −0.09) and pain (standardized mean difference, −0.34; 95% confidence interval, −0.58 to −0.09). Conclusions Robot-assisted therapy seems to improve the quality of mobility and reduce balance disturbances and pain for patients with stroke. These findings will help develop advanced rehabilitation robots and could improve health outcomes by facilitating health services for healthcare providers and patients with stroke.

We give a closed formula to evaluate exterior webs (also called MOY webs) and the associated Reshetikhin-Turaev link polynomials.

Background: To plan treatment and measure post-stroke recovery, frequent and time-bounded functional assessments are recommended. With increasing needs for neurorehabilitation, new technological methods such as virtual reality (VR) have emerged. Here, we developed an immersive VR version of the Action Research Arm Test (ARAT-VR) to complement VR neurorehabilitation. Objective: This study aimed to assess the validity, test-retest reliability and usability of the ARAT-VR among individuals with stroke, healthcare professionals and healthy control subjects (HCS). Methods: Eleven healthcare professionals, 30 individuals with stroke, and 25 HCS were recruited. Content validity was assessed by asking healthcare professionals to rate the difficulty of performing each item of the ARAT-VR in comparison to the classical Action Research Arm Test (ARAT). Concurrent validity was measured through correlation between the ARAT-VR and ARAT scores for the individuals with stroke and through scores comparison for the HCS. A third of individuals with stroke and HCS were reseen in a convenient delay to measure test-retest reliability. Usability was measured using the System Usability Scale. Results: On average, healthcare professionals estimated that movement difficulty was comparable in the ARAT-VR to those of the classical ARAT except for the items involving marbles. When assessing paretic hands, the ARAT-VR and classical ARAT were strongly correlated (r=0.84). Test-retest reliability (ICC=0.99; p<0.001) and usability (SUS=82.5[75-90]) were excellent. Conclusion: The ARAT-VR is a valid, reliable and usable tool to assess hand action among individuals with stroke, providing potential to increase assessment frequency and remote evaluation, particularly for use with VR based neurorehabilitation.

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on Rn, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the Lp norm of functions in Lp(Rn).

Serious gaming in virtual reality to assess and rehabilitate motor or cognitive impairments after stroke; The AUTORREVI Project

Background: An increase in dementia prevalence has been accompanied by increasing interest in new rehabilitation methods, such as serious games. Serious games hold the potential to postpone functional and cognitive declines in people with dementia by increasing their independence and engagement; however, the efficacy of serious games remains underexplored. This review was conducted to quantify the effects of serious games in people with dementia, including several newly published trials, with the hopes of contributing to evidence-based practice by offering support for clinical decision-making. Methods: Only randomized controlled trials (RCTs) assessing the impacts of game-based intervention programs compared with conventional therapy on cognitive function, instrumental and non-instrumental activities of daily living, or depression among people with dementia were included in this review. Meta-analyses were performed to determine the pooled standardized mean difference (SMD) of each outcome using a random-effects model. Results: The final search identified 12 studies that met our criteria. Overall, serious games were found to improve cognitive function (pooled SMD: 0.34; 95% CI: 0.07 to 0.61) and alleviated depression (pooled SMD: -0.131; 95% CI: -1.85 to -0.77) in people with dementia. Conclusions: Serious games improve cognitive function and reduce depression in people with dementia. Future studies in this field should aim to evaluate and determine the long-term effect of these games.

For a broad class of point processes, including determinantal point processes, we construct associated marked and conditional ensembles, which allow to study a random configuration in the point process, based on information about a randomly incomplete part of the configuration. We show that our construction yields a well behaving transformation of sufficiently regular point processes. In the case of determinantal point processes, we explain that special cases of the conditional ensembles already appear implicitly in the literature, namely in the study of unitary invariant random matrix ensembles, in the Its-Izergin-Korepin-Slavnov method to analyze Fredholm determinants, and in the study of number rigidity. As applications of our construction, we show that a class of determinantal point processes induced by orthogonal projection operators, including the sine, Airy, and Bessel point processes, satisfy a strengthened notion of number rigidity, and we give a probabilistic interpretation of the Its-Izergin-Korepin-Slavnov method.

Over the last decade, the enormous amount of data gathered by the Large Hadron Collider (LHC) has put the Standard Model (SM) under severe testing, proving it the most successful theory in particle physics. Despite such success, the SM is an inherently incomplete theory. The current challenges facing the SM have led to the proposals that new physics must exist. This new physics beyond the SM may not be necessarily evident in the resonant production of new particles at the LHC. Therefore, the SM effective field theory approach (SMEFT) can be used as a framework to parameterise possible deviations from the SM due to the presence of such new states that otherwise can not be probed directly with the current reach of our colliders. One of many possible probes of new physics is top quark interactions. The motivation behind the vigorous effort put into probing top quark processes is its high mass and strong coupling to the Higgs boson. This thesis presents three SMEFT studies of different top quark processes at hadron colliders. Single top quark production associated with a WZ-bosons pair at next-to-leading-order (NLO) in QCD is non-trivial due to its overlap with other resonant processes at leading-order (LO). In the first study of the thesis, we employed diagram-removal methods to suppress the resonant overlap, and therefore, define tWZ at NLO in a suitably defined phase-space region in which SMEFT predictions can be reliably obtained. This study will allow the inclusion of the tWZ data in future global LHC studies. In the second study, we investigated the four top quark production considering all possible QCD and EW-induced amplitudes for all the relevant dimension-six SMEFT operators. We show that formally subleading terms give rise to significant cross-section contributions, providing sensitivity to a broader class of operators. Inclusive and differential predictions for the LHC and the FCC-hh were obtained. Furthermore, we carried out projection studies by which we s, (SC - Sciences) -- UCL, 2022

Robots in neurorehabilitation

Cosmic inflation is arguably the most favoured paradigm of the very early Universe. It postulates an early phase of fast, nearly exponential, and accelerated expansion. Inflationary models are capable of explaining the overall flatness and homogeneity of today's Universe at large spatial scales. Furthermore, their predictions give the observed statistical features of the primordial density fluctuations imprinted in the cosmic microwave background and the Universe's large scale structure. However, despite the wide acceptance in the physics community, these models are not absent from criticism. In scalar field inflation, a necessary condition to begin inflation is the requirement of a Universe dominated by the field's potential, which implies a subdominant contribution from the scalar field dynamics, i.e. gradients and field velocities. This has given rise to large amounts of scientific debate and literature on the naturalness, and possible fine-tuning of the initial conditions for inflation. Another controversial issue concerns the ending of inflation, and the fact that a preheating mechanism is necessary to transfer the energy of the inflaton to the bath of highly energetic particles of the hot big bang plasma. In this thesis, we present full general relativistic simulations to study these two problems, with a particular focus on the Starobinsky and Higgs models of inflation, being those the most favoured by the latest observations. First, we consider the fine-tuning problem of beginning inflation from a highly dynamical and inhomogeneous "preinflation" epoch in the single-field case. In our second study, we approach the multifield paradigm of inhomogeneous preinflation, together and consistently, with the preheating phase after inflation. These investigations further confirm the robustness of these types of models to highly inhomogeneous initial conditions, while putting in evidence the non-negligible effects of gravitational effects during the (p)reheating. At the, (SC - Sciences) -- UCL, 2022

Recently, in-situ TEM straining experiments on pure copper have unraveled a high stress irradiation creep mechanism. Irradiation induced unpinning of dislocations from defects has been observed and the mean pinning lifetime has been determined. In the present study, molecular dynamics simulations are per- formed on pure copper to investigate the impact of collision cascades on screw dislocations pinned on Frank loops under high-applied stresses at 300 K in order to further quantitatively elucidate this mech- anism. The simulations indicate two possible dislocation unpinning mechanisms. Unpinning can occur through loop destruction when the cascade is generated on the pinning points of the dislocation (type 1 unpinning). Unpinning can also be triggered by the shear stresses building up around a cascade generated in front of the dislocation in the glide plane (type 2 unpinning). Type 2 unpinning generally leads to dis- location repinning on cascade residues, so that it should only marginally contribute to irradiation creep. The mean pinning lifetime due to type 1 unpinning in the conditions of the in-situ TEM experiments is derived from a simple model previously developed for zirconium, and is found in the same orders of magnitude as in the experiments.

The 6-minute walk test (6MWT) is widely used to assess walking capacity among persons with stroke. Whether a shorter and more convenient test, the 2-minute walk test (2MWT) could replace it, was tested. Two retrospective analyses were conducted. The first one was performed on a sample of 20 persons with stroke who performed both the 2MWT and the 6MWT, and the second one, on a group of 82 persons with stroke who performed the 6MWT while measuring the distance covered each minute. Linear regression models were applied to test the validity of 2MWT with regard to 6MWT. In the first group, distances covered during the 2MWT were highly predictive of the distances covered during the 6MWT (estimated adjusted R² = 0.98; P < 0.001). In the second group, distances covered by participants during the first 2 min of the 6MWT were highly and linearly related to the distances they covered during the whole 6MWT (estimated adjusted R² = 0.98; P < 0.001). Furthermore, the distance covered during the first 2 min of the whole 6MWT allowed us to predict 98% of the variance of the 6MWT. Given its good metric properties and its practical advantages, clinicians and researchers could reasonably use the 2MWT when assessing the walking capacity of persons with stroke, instead of the 6MWT.

Background: To plan treatment and measure post-stroke recovery, frequent and time-bounded functional assess- ments are recommended. With increasing needs for neurorehabilitation advances, new technology based methods, such as virtual reality (VR) have emerged. Here, we developed an immersive VR version of the Action Research Arm Test (ARAT-VR) to complement neurorehabilitation. Objective: This study aimed to assess the validity, usability and test–retest reliability of the ARAT-VR among individu- als with stroke, healthcare professionals and healthy control subjects (HCS). Methods: Among the 19 items of the ARAT, 13 items were selected and developed in immersive VR. 11 healthcare professionals, 30 individuals with stroke, and 25 HCS were recruited. Content validity was assessed by asking health- care professionals to rate the difficulty of performing each item of the ARAT-VR in comparison to the classical Action Research Arm Test (ARAT-19). Concurrent validity was first measured using correlation (Spearman tests) between the ARAT-VR and ARAT-19 scores for the individuals with stroke, and second through correlation and comparison between the scores of the ARAT-VR and the reduced version of the ARAT (ARAT-13) for both individuals with stroke and HCS (Wilcoxon signed rank tests and Bland–Altman plots). Usability was measured using the System Usability Scale. A part of individuals with stroke and HCS were re-tested following a convenient delay to measure test–retest reliability (Intra- class correlation and Wilcoxon tests). Results: Regarding the content validity, median difficulty of the 13 ARAT-VR items (0[0 to − 1] to 0[0–1]) evalu- ated by healthcare professionals was rated as equivalent to the classical ARAT for all tasks except those involving the marbles. For these, the difficulty was rated as superior to the real tasks (1[0–1] when pinching with the thumb-index and thumb-middle fingers, and 1[0–2] when pinching with thumb-ring finger). Regarding the concurrent v

We use the octahedron recurrence, which generalizes the quadratic recurrence found by Kuo for standard Aztec diamonds, in order to compute boundary one-refined and two-refined partition functions for two-periodic Aztec diamonds. In a first approach, the geometric tangent method allows to derive the parametric form of the arctic curve, separating the solid and liquid phases. This is done by using the recent reformulation of the tangent method and the one-refined partition functions without extension of the domain. In a second part, we use the two-refined tangent method to rederive the arctic curve from the boundary two-refined partition functions, which we compute exactly on the lattice. The curve satisfies the known algebraic equation of degree 8, of which either tangent method gives an explicit parametrization.

Dexterous manipulation of objects heavily relies on the feedback provided by the tactile afferents innervating the fingertips. Previous studies have suggested that humans might take advantage of partial slip, localized loss of grip between the skin and the object, to gauge the stability of a contact and react appropriately when it is compromised, that is, when slippage is about to happen. To test this hypothesis, we asked participants to perform point-to-point movements using a manipulandum. Through optical imaging, the device monitored partial slip at the contact interface, and at the same time, the forces exerted by the fingers. The level of friction of the contact material was changed every five trials. We found that the level of grip force was systematically adjusted to the level of friction, and thus partial slip was limited to an amount similar across friction conditions. We suggest that partial slip is a key signal for dexterous manipulation and that the grip force is regulated to continuously maintain an upper bound on partial slip across friction conditions.

We prove that the category of cocommutative bialgebras in any locally presentable symmetric monoidal category is an S-protomodular category with respect to a particular class of split extensions of cocommutative bialgebras. We also obtain the “partial” well-known Smith is Huq condition, meaning that two effective S-equivalence relations centralize each other as soon as the normal subobjects associated with them commute in the sense of Huq.

In the context of a tower of (strongly Birkhoff) Galois structures in the sense of categorical Galois theory, we show that the concept of a higher covering admits a characterisation which is at the same time absolute (with respect to the base level in the tower), rather than inductively defined relative to extensions of a lower order; and symmetric, rather than depending on a perspective in terms of arrows pointing in a certain chosen direction. This result applies to the Galois theory of quandles, for instance, where it helps us characterising the higher coverings in purely algebraic terms.

The aim of this talk is to explain how a three-dimensional version of Pedicchio's categorical approach to the binary Smith commutator of Universal Algebra allows us to compare three ternary commutators which occur naturally in mathematics but are a priori very different: the ternary Bulatov commutator~\cite{Bulatov}, a universal-algebraic concept meant to generalise the binary commutator of two congruences to a commutator of three congruences; the ternary Higgins commutator, first introduced in the context of varieties of $\Omega$-groups, of which recently a categorical version was developed; the ternary Smith commutator, which is supposed to allow an interpretation of cohomology via higher extensions as in beyond the case of trivial coefficients = higher central extensions. Our first aim is to capture the Bulatov commutator through a definition which is valid in exact Mal'tsev categories. We then show that in the context of an algebraically coherent semi-abelian category, the ternary Higgins commutator of a triple of normal subobjects corresponds to the normalisation of the ternary Bulatov commutator of the corresponding equivalence relations. This gives a first, indirect answer to the question, in which sense the ternary Higgins commutator expresses that ``three things commute''. On the way, we sketch a general result on commuting triples of arrows, crucial here and in our proposed definition of a ternary Smith commutator. We then explain how this is related to the approach to cohomology in my work with Diana Rodelo. This talk is based on joint work with her and with Cyrille Simeu.

We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context of a homological category. We prove that, if the category is moreover well-powered with (small) joins, then the existence of split extension cores is equivalent to the condition that the change-of-base functors in the fibration of points are geometric. We call a finitely complete category that satisfies this condition an algebraic logos. We give examples of such categories, compare them with algebraically coherent ones, and study equivalent conditions as well as stability under common categorical operations.

The discovery of the Higgs boson (H) ten years ago constitutes the pinnacle of the construction of the standard model (SM) describing the fundamental interactions between elementary particles. While a major success, half a century after its prediction, several important questions are left unanswered by the SM. The study of the Higgs boson is one of the most promising path towards a deeper understanding of the spontaneous symmetry breaking. Its self-interaction, through the simultaneous production of a pair of Higgs bosons (HH), has yet to be measured and any deviation from its SM prediction could be the sign of new physics. The search for HH production in the proton collisions of the Large Hadron Collider (LHC) at CERN is made challenging by the rarity of that process. Machine learning techniques such as deep neural networks (DNN) are the tools of choice to leverage the huge amount of data from the collisions, assisted by the intuition of the physicist. This thesis explores the performance and limitations of these tools on the data generated by the CMS experiment. First as a regression of the matrix element method (MEM) weights. While very powerful to extract information using the knowledge of both the SM and the detector, the MEM suffers from a prohibitive computation time. This can be circumvented by a faster DNN regression of the weights, taking advantage of their interpolation capabilities. Second, they served as the central piece in the search for HH production in the bbWW final state, using the CMS Run-2 data at 13 TeV. Resonant hypotheses were tested, as well as an effective field theory (EFT) non-resonant interpretation into various couplings, including the Higgs boson self-interaction strength, and the data was found to be compatible with SM predictions., (SC - Sciences) -- UCL, 2022

BACKGROUND: Oropharyngeal myofunctional therapy is a multi-component therapy effective to reduce the severity of obstructive sleep apnoea (OSA). However, existing protocols are difficult to replicate in the clinical setting. There is a need to isolate the specific effectiveness of each component of the therapy. OBJECTIVE: To assess the effects of a 6 weeks tongue elevation training programme in patients with OSA. METHODS: We conducted a multicentre randomised controlled trial. Eligible participants were adults diagnosed with moderate OSA who presented low adherence to continuous positive airway pressure therapy (mean use <4 h per night). The intervention group completed a 6 weeks tongue elevation training protocol that consisted in anterior tongue elevation strength and endurance tasks with the Iowa Oral Performance Instrument. The control group completed a 6 weeks sham training protocol that involved expiratory muscle training at very low intensity. Polygraphy data, tongue force and endurance, and OSA symptoms were evaluated pre- and post-intervention. The primary outcome was apneoa-hypopnea index (AHI). RESULTS: Twenty-seven patients (55 ± 11 years) were recruited. According to modified intention-to-treat analysis (n = 25), changes in AHI and c did not significantly differ between groups. Daytime sleepiness (Epworth Sleepiness Scale) and tongue endurance significantly improved in the intervention group compared to the control group (p = .015 and .022, respectively). In the intervention group, 75% of participants had a decrease in daytime sleepiness that exceeded the minimal clinically important difference. CONCLUSION: Six weeks of tongue elevation muscle training had no effect on OSA severity.

Stokes’ manifolds, also known as wild character varieties, carry a natural Poisson structure. Our goal is to provide explicit log-canonical coordinates for this Poisson structure on the Stokes’ manifolds of polynomial connections of rank 2, thus including the second Painlevé hierarchy. This construction provides the explicit linearization of the Poisson structure first discovered by Flaschka and Newell and then rediscovered and generalized by Boalch. We show that, for a connection of degree K, the Stokes’ manifold is a cluster manifold of type $A_{2K}$ with one frozen vertex. The main idea is then applied to express explicitly also the log–canonical coordinates for the Poisson bracket introduced by Ugaglia in the context of Frobenius manifolds and then also applied by Bondal in the study of the symplectic groupoid of quadratic forms.

Lehmer’s famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer’s constants of the group to measure such a gap. In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound 2/√3 for Lehmer’s constants of all torsion-free groups which have non-cyclic free subgroups. Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of Bartholdi and Dasbach-Lalin. Furthermore, via the gluing formula for L2-torsions, we show that the Lehmer’s constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than 2/√3.

We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders in a pretopos. Some examples are provided in the categories of topological spaces and of (small) categories.

We establish nonlocal difference quotient representations for the $L^p$ and total variation norms of distributions $\mathcal{A}u$, where $u$ is a distribution, and $\mathcal{A}$ is a constant coefficient homogeneous first-order linear differential operator. Our results provide a criterion for the boundedness of $\mathcal{A}$-gradients that does not require the use of distributional derivatives.

Let $T_1$, $T_2$ be regular trees of degrees $d_1, d_2 > 3$. Let also $\Gamma \leq$ Aut$(T_1)\times$ Aut$(T_2)$ be a group acting freely and transitively on $V T_1 \times V T_2$. For $i = 1$ and 2, assume that the local action of $\Gamma$ on $T_i$ is 2-transitive; if moreover $d_i \geq 7$, assume that the local action contains Alt$(d_i)$. We show that $\Gamma$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $\Gamma$ that can be checked purely in terms of its local action on a ball of radius 1 in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $\Gamma$ is irreducible, then it is hereditarily just-infinite, provided the[ local action on $T_i$ is not the affine group $\mathbf{F}_5 \rtimes \mathbf{F}^\ast_5$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.

We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.

From the historical figures met in Doctor Who to the scientific hijinks of the kids in the Magic School Bus, most sciences have had their time in the spotlight in fictionalized works. What about math? How can we do mathematics outreach through fiction? But first… should we? Is a western the best setting to present knot theory? What can romance stories tell us about the axiom of choice? Along the way of discussing such questions, we may also review mathematical models of non-linear stories like gamebooks, and we will try to create new math activities. This workshop is a story you cannot miss, a story in which YOU are the (co)protagonist! Some first questions: For a given mathematical subject to popularize is it relevant to present it through fiction? If we don’t know if it is relevant, how could we determine this? If it seems relevant, which type of fiction could be appropriate? Proposed plan of approach : Reviewing forms of fiction: novel, comic book, radio, theatre, movie, board game… Reviewing genres of fiction: science fiction, comedy, historical, mystery, horror… Linearity of the plot and influence of the audience: anything between an un-modifiable classical story and a full improve show with audience input. Non-linear plots and graph theory, with the example of « choose your own adventure » gamebooks. Create a new outreach activity mixing mathematics and creative writing: present basics of graph theory and non-linear stories, then make the audience create a non-linear story associated to a given graph.

We give a simple and constructive extension of Raiţă's result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Raiţă, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincaré lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.

We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in $B D(\Omega)$ and $L D(\Omega)$, which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in $W^{1,p}(\Omega; \mathbb{R}^d)$, which are inspired by the gradient seminorm formulas introduced by Bourgain, Brezis and Mironescu in $W^{1,p}(\Omega)$ and by Dávila in $BV(\Omega)$.

Given the current situation, where no new particles or direct signs for Beyond the Standard Model (BSM) physics have been found, we consider the indirect signs that such states would leave on measurement at LHC and foreseen collider energies. The Standard Model Effective Field Theory (SMEFT) is a powerful framework useful in the search of such deviations. Higher-order SM predictions allow to correctly establish a deviation, while higher-order SMEFT predictions allow to enhance sensitivity by correctly describing correlations, by accounting for new dependencies and allow to precisely interpret deviations (towards the correct family of UV models). It is for this reasons that one-loop contributions should be taken into consideration when making predictions in the SMEFT. Given the fact that the top quark is the heaviest elementary particle known to date, BSM particles are often expected to have sizable couplings to the top. Moreover, studies in the top sector are motivated by the fact that the top is the least known particle from the strong sector and the LHC is a top factory. The top sector is thus of interest in the search of new physics signals. This thesis aims at presenting several theoretical and phenomenological aspects of the SMEFT. We discuss the challenges and subtleties appearing in the computation of the top-pair production in the SMEFT at one-loop in QCD, and the corresponding sensitivity of this process to new physics. As a result, we provide bounds on four-heavy-quark operators from the top-pair production data, which are complementary to other bounds found in the literature. The effects of BSM physics coupling to the top can also be studied in processes without tops in the final state. Because of the large Yukawa coupling of the top, the one-loop induced double Higgs production in $e^+e^-$ colliders offers the possibility of studying such couplings. We perform a sensitivity study for several benchmark values of energy and integrated luminosity foreseen f, (SC - Sciences) -- UCL, 2022

We prove that for a first-order homogeneous linear partial differential operator $\mathcal{A}$ and a map $f$ taking values in the essential range of the operator, there exists a function $u$ of special bounded variation satisfying $\mathcal{A}u(x) = f(x)$ almost everywhere. This extends a result of G. Alberti for gradients on $\mathbb{R}^nN$. In particular, for $m < N$, it is shown that every integrable $m$-form field on $\mathbb{R}^nN$ is the absolutely continuous part of the boundary of some locally normal current with finite mass.