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We prove that every finite Borel measure $\mu$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $\mu|_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}^s_\infty$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

Portage of MG5AMC to use SIMD and GPU

We prove that each Borel function $V\colon \Omeg \to [-\infty,+\infty]$ defined on an open subset $\Omega \subset \mathbb{R}^N$ induces a decomposition $\Omega=S\cup\Bigcup_i D_i$ such that every function in $W{1.2}_0 (\Omega)\capL^2(\Omega;V^+dx) $ is zero almost everywhere on $S$ and existence of nonnegative supersolutions of $-\Delta+V$ on each component $D_i$ yields nonnegativity of the associated quadratic form $/int_{D_i} (|\nabla \xi|^2+V\xi^2)$.

Rééducation robotisée des membres supérieurs : introduction - évidence scientifique

Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.

Dear Editor, Hypoxaemia is a typical feature of the coronavirus disease 2019 (COVID-19). The rapid rise in the number of patients requiring oxygen therapy during the pandemic may cause a sharp increase in oxygen demands and potential threats of supply disruption, particularly in developing countries1 or nursing homes. Moreover, the use of elevated oxygen flows via nasal cannula raises concerns about exhaled air dispersion distance and the potential risk of generating aerosols [...]

The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurzweil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonautonomous ODE. By means of the correspondence between impulse differential equations (shortly IDEs) and GODEs, we translate the result to IDEs. Instead of the classical hypotheses that the functions on the righthand side of an IDE are piecewise continuous, it is enough to require that they are integrable in the sense of Lebesgue, allowing such functions to have many discontinuities. Our second main result provides conditions for the existence of a bifurcation point with respect to the trivial solution of a periodic boundary value problem for a GODE depending upon a parameter, and, again, we apply such result to IDEs. The machinery employed to obtain the main results are the topological degree theory, tools from the theory of compact operators and an Arzelà-Ascoli-type theorem for regulated functions.

Consider a locally compact group G=Q⋉V such that V is abelian and the action of Q on the dual abelian group Vˆ has a free orbit of full measure. We show that such a group G can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of G on L2(V); (2) by twisting the coproduct on the group von Neumann algebra of G by a dual 2-cocycle obtained from the G-equivariant Kohn–Nirenberg quantization of V×Vˆ; (3) by considering the bicrossed product defined by a matched pair of subgroups of Q⋉Vˆ both isomorphic to Q. In the simplest case of the ax+b group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum ax+b group of Baaj–Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups G. Using recent results of De Commer we show that a class of G-Galois objects is parametrized by certain cohomology classes in H2(G;T). This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in H2(G;T) can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group G with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual 2-cocycles such that the corresponding quantization of G is neither commutative nor cocommutative.

Service contracts characterise the desired behavioural compliance of a composition of services, typically defined by the fulfilment of all service requests through service offers. Contract automata are a formalism for specifying behavioural service contracts. Based on the notion of synthesis of the most permissive controller from Supervisory Control Theory, a safe orchestration of contract automata can be computed that refines a composition into a compliant one. This short paper summarises the contributions published in [8], where we endow contract automata with two orthogonal layers of variability: (i) at the structural level, constraints over service requests and offers define different configurations of a contract automaton, depending on which requests and offers are selected or discarded; and (ii) at the behavioural level, service requests of different levels of criticality can be declared, which induces the novel notion of semi-controllability. The synthesis of orchestrations is thus extended to respect both the structural and the behavioural variability constraints. Finally, we show how to efficiently compute the orchestration of all configurations from only a subset of these configurations. A recently redesigned and refactored tool supports the developed theory.

Over the development of modern cryptography, often, alternative cryptographic schemes are developed to achieve goals that in some important respect are orthogonal. Thus, we have to choose either a scheme which achieves the first goal and not the second, or vice versa. This results in two types of schemes that compete with each other. In the basic area of user privacy, specifically in anonymous (multi-use credentials) signing, such an orthogonality exists between anonymity and accountability. The conceptual contribution of this work is to reverse the above orthogonality by design, which essentially typifies the last 25 years or so, and to suggest an alternative methodology where the opposed properties are carefully folded into a single scheme. The schemes will support both opposing properties simultaneously in a bifurcated fashion, where: First, based on rich semantics expressed over the message’s context and content, the user, etc., the relevant property is applied point-wise per message operation depending on a predicate; andSecondly, at the same time, the schemes provide what we call “branch-hiding;” namely, the resulting calculated value hides from outsiders which property has actually been locally applied. Specifically, we precisely define and give the first construction and security proof of a “Bifurcated Anonymous Signature” (BiAS): A scheme which supports either absolute anonymity or anonymity with accountability, based on a specific contextual predicate, while being branch-hiding. This novel signing scheme has numerous applications not easily implementable or not considered before, especially because: (i) the conditional traceability does not rely on a trusted authority as it is (non-interactively) encapsulated into signatures; and (ii) signers know the predicate value and can make a conscious choice at each signing time. Technically, we realize BiAS from homomorphic commitments for a general family of predicates that can be represented by bounded-depth circ

Inspired by recent formulæ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of P1 as an isomonodromic tau function associated with a difference equation. As a byproduct we obtain an analogue of the Kontsevich matrix model for this tau function. A connection with the Charlier ensemble is also considered.

We consider special families of orthogonal polynomials satisfying differential equations. Besides known hypergeometric cases, we look especially for Heun's differential equations. We show that such equations are satisfied by orthogonal polynomials related to some classical weight functions modified by Dirac weights or by division of powers of binomials. An appropriate set of biorthogonal rational functions, or 2-point Padé approximations, is also described.

Domain-specific quantitative modeling and analysis approaches are fundamental in scenarios in which qualitative approaches are inappropriate or unfeasible. In this paper, we present a tool-supported approach to quantitative graph-based security risk modeling and analysis based on attack-defense trees. Our approach is based on QFLan, a successful domain-specific approach to support quantitative modeling and analysis of highly configurable systems, whose domain-specific components have been decoupled to facilitate the instantiation of the QFLan approach in the domain of graph-based security risk modeling and analysis. Our approach incorporates distinctive features from three popular kinds of attack trees, namely enhanced attack trees, capabilities-based attack trees and attack countermeasure trees, into the domain-specific modeling language. The result is a new framework, called RisQFLan, to support quantitative security risk modeling and analysis based on attack-defense diagrams. By offering either exact or statistical verification of probabilistic attack scenarios, RisQFLan constitutes a significant novel contribution to the existing toolsets in that domain. We validate our approach by highlighting the additional features offered by RisQFLan in three illustrative case studies from seminal approaches to graph-based security risk modeling analysis based on attack trees.

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext-category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann's cohomology vanishing theorem for semi-simple Leibniz algebras.

We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, and we examine some examples.

In the category of groups, split extensions have a lot of interesting properties. One of them is the fact that the category of split extensions is equivalent to the category of group actions. Unfortunately, this does not hold in any category, for example the category of monoids does not have this property. Nevertheless, it was proved in [1,2], that there exists such an equivalence if the split extensions of monoids are “Schreier extensions”. In this talk, I will explain this construction of Schreier split extensions and their properties. Also motivated by a recent result on split extensions of unitary magmas [3], I will then consider split extensions in the category of (non-)associative bialgebras. Which conditions on the split extensions of bialgebras do we need to have an equivalence with the category of actions of bialgebras? This presentation, which is based on the article [4], will eventually answer this question. References [1] N. Martins-Ferreira, A. Montoli, M. Sobral, Semidirect products and crossed modules in monoids with operations, J. Pure Appl. Algebra 217 (2013), 334–347. [2] D. Bourn, N. Martins-Ferreira, A. Montoli, M. Sobral, Schreier split epimorphisms in monoids and semirings, Textos de Matematica S´erie B, Departamento de Matematica da Universidade de Coimbra 45 (2014). [3] M. Gran, G. Janelidze and M. Sobral, Split extensions and semidirect products of unitary magmas, Comment. Math. Univ. Carolin. 60 (2019), 509–52. [4] F. Sterck, Split extensions of bialgebras and Hopf algebras, accepted in Applied Categorical Structures (2021).

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group G(d,1,n). The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table S of the cyclic group of order d. The representations of the quantum universal enveloping algebra of the general linear Lie algebra glm, with quantum parameter an even root of unity of order 2d, provide a categorical interpretation of the matrix ⋀mS. We also prove some positivity conjectures of Cuntz at the decategorified level.

Gravitational waves provide a unique tool for observational astronomy. While the first LIGO--Virgo catalogue of gravitational-wave transients (GWTC-1) contains eleven signals from black hole and neutron star binaries, the number of observations is increasing rapidly as detector sensitivity improves. To extract information from the observed signals, it is imperative to have fast, flexible, and scalable inference techniques. In a previous paper, we introduced BILBY: a modular and user-friendly Bayesian inference library adapted to address the needs of gravitational-wave inference. In this work, we demonstrate that BILBY produces reliable results for simulated gravitational-wave signals from compact binary mergers, and verify that it accurately reproduces results reported for the eleven GWTC-1 signals. Additionally, we provide configuration and output files for all analyses to allow for easy reproduction, modification, and future use. This work establishes that BILBY is primed and ready to analyse the rapidly growing population of compact binary coalescence gravitational-wave signals

We use 47 gravitational-wave sources from the Third LIGO-Virgo-KAGRA Gravitational-Wave Transient Catalog (GWTC-3) to estimate the Hubble parameter H(z), including its current value, the Hubble constant H0. Each gravitational-wave (GW) signal provides the luminosity distance to the source and we estimate the corresponding redshift using two methods: the redshifted masses and a galaxy catalog. Using the binary black hole (BBH) redshifted masses, we simultaneously infer the source mass distribution and H(z). The source mass distribution displays a peak around 34M⊙, followed by a drop-off. Assuming this mass scale does not evolve with redshift results in a H(z) measurement, yielding H0=68+12−7kms−1Mpc−1 (68% credible interval) when combined with the H0 measurement from GW170817 and its electromagnetic counterpart. This represents an improvement of 17% with respect to the H0 estimate from GWTC-1. The second method associates each GW event with its probable host galaxy in the catalog GLADE+, statistically marginalizing over the redshifts of each event's potential hosts. Assuming a fixed BBH population, we estimate a value of H0=68+8−6kms−1Mpc−1 with the galaxy catalog method, an improvement of 42% with respect to our GWTC-1 result and 20% with respect to recent H0 studies using GWTC-2 events. However, we show that this result is strongly impacted by assumptions about the BBH source mass distribution; the only event which is not strongly impacted by such assumptions (and is thus informative about H0) is the well-localized event GW190814.

Figure 1 Corrected version of Fig. 1 in the original article Phys. Rev. D 100, 122002 (2019)

After the detection of gravitational waves from compact binary coalescences, the search for transient gravitational-wave signals with less well-defined waveforms for which matched filtering is not well-suited is one of the frontiers for gravitational-wave astronomy. Broadly classified into "short" ≲1 \,s and "long" ≳1 \,s duration signals, these signals are expected from a variety of astrophysical processes, including non-axisymmetric deformations in magnetars or eccentric binary black hole coalescences. In this work, we present a search for long-duration gravitational-wave transients from Advanced LIGO and Advanced Virgo's third observing run from April 2019 to March 2020. For this search, we use minimal assumptions for the sky location, event time, waveform morphology, and duration of the source. The search covers the range of 2 -- 500~s in duration and a frequency band of 24−2048 Hz. We find no significant triggers within this parameter space; we report sensitivity limits on the signal strength of gravitational waves characterized by the root-sum-square amplitude hrss as a function of waveform morphology. These hrss limits improve upon the results from the second observing run by an average factor of 1.8

This paper presents the results of a search for generic short-duration gravitational-wave transients in data from the third observing run of Advanced LIGO and Advanced Virgo. Transients with durations of milliseconds to a few seconds in the 24--4096 Hz frequency band are targeted by the search, with no assumptions made regarding the incoming signal direction, polarization or morphology. Gravitational waves from compact binary coalescences that have been identified by other targeted analyses are detected, but no statistically significant evidence for other gravitational wave bursts is found. Sensitivities to a variety of signals are presented. These include updated upper limits on the source rate-density as a function of the characteristic frequency of the signal, which are roughly an order of magnitude better than previous upper limits. This search is sensitive to sources radiating as little as ∼10−10M⊙c2 in gravitational waves at ∼70 Hz from a distance of 10~kpc, with 50\% detection efficiency at a false alarm rate of one per century. The sensitivity of this search to two plausible astrophysical sources is estimated: neutron star f-modes, which may be excited by pulsar glitches, as well as selected core-collapse supernova models

We report the observation of gravitational waves from two compact binary coalescences in LIGO's and Virgo's third observing run with properties consistent with neutron star–black hole (NSBH) binaries. The two events are named GW200105_162426 and GW200115_042309, abbreviated as GW200105 and GW200115; the first was observed by LIGO Livingston and Virgo and the second by all three LIGO–Virgo detectors. The source of GW200105 has component masses $8.{9}_{-1.5}^{+1.2}$ and $1.{9}_{-0.2}^{+0.3}\,{M}_{\odot }$, whereas the source of GW200115 has component masses $5.{7}_{-2.1}^{+1.8}$ and $1.{5}_{-0.3}^{+0.7}\,{M}_{\odot }$ (all measurements quoted at the 90% credible level). The probability that the secondary's mass is below the maximal mass of a neutron star is 89%–96% and 87%–98%, respectively, for GW200105 and GW200115, with the ranges arising from different astrophysical assumptions. The source luminosity distances are ${280}_{-110}^{+110}$ and ${300}_{-100}^{+150}\,\mathrm{Mpc}$, respectively. The magnitude of the primary spin of GW200105 is less than 0.23 at the 90% credible level, and its orientation is unconstrained. For GW200115, the primary spin has a negative spin projection onto the orbital angular momentum at 88% probability. We are unable to constrain the spin or tidal deformation of the secondary component for either event. We infer an NSBH merger rate density of ${45}_{-33}^{+75}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ when assuming that GW200105 and GW200115 are representative of the NSBH population or ${130}_{-69}^{+112}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ under the assumption of a broader distribution of component masses.

We present a search for dark photon dark matter that could couple to gravitational-wave interferometers using data from Advanced LIGO and Virgo's third observing run. To perform this analysis, we use two methods, one based on cross-correlation of the strain channels in the two nearly aligned LIGO detectors, and one that looks for excess power in the strain channels of the LIGO and Virgo detectors. The excess power method optimizes the Fourier Transform coherence time as a function of frequency, to account for the expected signal width due to Doppler modulations. We do not find any evidence of dark photon dark matter with a mass between mA∼10−14−10−11 eV/c2, which corresponds to frequencies between 10-2000 Hz, and therefore provide upper limits on the square of the minimum coupling of dark photons to baryons, i.e. U(1)B dark matter. For the cross-correlation method, the best median constraint on the squared coupling is ∼1.31×10−47 at mA∼4.2×10−13 eV/c2; for the other analysis, the best constraint is ∼1.2×10−47 at mA∼5.7×10−13 eV/c2. These limits improve upon those obtained in direct dark matter detection experiments by a factor of ∼100 for mA∼[2−4]×10−13 eV/c2.

Rapidly spinning neutron stars are promising sources of persistent, continuous gravitational waves. Detecting such a signal would allow probing of the physical properties of matter under extreme conditions. A significant fraction of the known pulsar population belongs to binary systems. Searching for unknown neutron stars in binary systems requires specialized algorithms to address unknown orbital frequency modulations. We present a search for continuous gravitational waves emitted by neutron stars in binary systems in early data from the third observing run of the Advanced LIGO and Advanced Virgo detectors using the semicoherent, GPU-accelerated, BinarySkyHough pipeline. The search analyzes the most sensitive frequency band of the LIGO detectors, 50 - 300 Hz. Binary orbital parameters are split into four regions, comprising orbital periods of 3 - 45 days and projected semimajor axes of 2 - 40 light-seconds. No detections are reported. We estimate the sensitivity of the search using simulated continuous wave signals, achieving the most sensitive results to date across the analyzed parameter space.

SModelS is an automatized tool enabling the fast interpretation of simplified model results from the LHC within any model of new physics respecting a ℤ_2 symmetry. In this contribution, we report on two important updates of SModelS during 2020: the extension of the SModelS' database with 13 ATLAS and 10 CMS analyses, including 5 ATLAS and 1 CMS analyses at full Run 2 luminosity, and the ability to use full likelihoods now provided by ATLAS in the form of pyhf JSON files. Moreover, we briefly explain how to use SModelS and give an overview of ongoing developments.

This paper describes the first all-sky search for long-duration, quasi-monochromatic gravitational-wave signals emitted by ultralight scalar boson clouds around spinning black holes using data from the third observing run of Advanced LIGO. We analyze the frequency range from 20~Hz to 610~Hz, over a small frequency derivative range around zero, and use multiple frequency resolutions to be robust towards possible signal frequency wanderings. Outliers from this search are followed up using two different methods, one more suitable for nearly monochromatic signals, and the other more robust towards frequency fluctuations. We do not find any evidence for such signals and set upper limits on the signal strain amplitude, the most stringent being ≈10−25 at around 130~Hz. We interpret these upper limits as both an "exclusion region" in the boson mass/black hole mass plane and the maximum detectable distance for a given boson mass, based on an assumption of the age of the black hole/boson cloud system.

In 2015, Chen and Yang proposed a volume conjecture that stated that certain Turaev-Viro invariants of an hyperbolic 3-manifold should grow exponentially with a rate equal to the hyperbolic volume. Since then, this conjecture has been proven or numerically tested for several hyperbolic 3-manifolds, either closed or with boundary, the boundary being either a family of tori or a family of higher genus surfaces. The current paper now provides new numerical checks of this volume conjecture for 3-manifolds with one toroidal boundary component and one geodesic boundary component. More precisely, we study a family of hyperbolic 3-manifolds $M_g$ introduced by Frigerio. Each $M_g$ can be seen as the complement of a knot in an handlebody of genus $g$. We provide an explicit code that computes the Turaev-Viro invariants of these manifolds $M_g$, and we then numerically check the Chen-Yang volume conjecture for the first six members of this family. Furthermore, we propose an extension of the volume conjecture, where the second coefficient of the asymptotic expansion only depends on the topology of the boundary of the manifold. We numerically check this property for the manifolds $M_2$ to $M_7$ and we also observe that the second coefficient grows linearly in the Euler characteristic $\chi(\partial M_g)$.

The paper presents the gravitational-wave measurement of the Hubble constant (H0) using the detections from the first and second observing runs of the Advanced LIGO and Virgo detector network. The presence of the transient electromagnetic counterpart of the binary neutron star GW170817 led to the first standard-siren measurement of H0. Here we additionally use binary black hole detections in conjunction with galaxy catalogs and report a joint measurement. Our updated measurement is H0 = ${69}_{-8}^{+16}$ km s−1 Mpc−1 (68.3% of the highest density posterior interval with a flat-in-log prior) which is an improvement by a factor of 1.04 (about 4%) over the GW170817-only value of ${69}_{-8}^{+17}$ km s−1 Mpc−1. A significant additional contribution currently comes from GW170814, a loud and well-localized detection from a part of the sky thoroughly covered by the Dark Energy Survey. With numerous detections anticipated over the upcoming years, an exhaustive understanding of other systematic effects are also going to become increasingly important. These results establish the path to cosmology using gravitational-wave observations with and without transient electromagnetic counterparts.

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

We establish sharp global rigidity upper bounds for universal determinantal point processes describing edge eigenvalues of random matrices. For this, we first obtain a general result which can be applied to general (not necessarily determinantal) point processes which have a smallest (or largest) point: it allows to deduce global rigidity upper bounds from the exponential moments of the counting function of the process. By combining this with known exponential moment asymptotics for the Airy and Bessel point processes, we improve on the best known upper bounds for the global rigidity of the Airy point process, and we obtain new global rigidity results for the Bessel point process. Secondly, we obtain exponential moment asymptotics for the Wright's generalized Bessel process and the Meijer-G process, up to and including the constant term. As a direct consequence, we obtain new results for the expectation and variance of the associated counting functions. Furthermore, by combining these asymptotics with our general rigidity theorem, we obtain new global rigidity upper bounds for these point processes.

Cosmic-ray antiprotons are a powerful tool for astroparticle physics. While the bulkof measured antiprotons is consistent with a secondary origin, the precise data of theAMS-02 experiment provides us with encouraging prospects to search for a subdominantprimary component,e.g.from dark matter. In this brief review, we discuss recent limitson heavy dark matter as well as a tentative signal from annihilation of dark matter witha mass.100 GeV. We emphasize the special role of systematic errors that can affect thesignal. In particular, we discuss recent progress in the modeling of secondary productioncross sections and correlated errors in the AMS-02 data, the dominant ones originatingfrom uncertainties in the cross sections for cosmic-ray absorption in the detector.

When C is a semi-abelian category, it is well known that the category Grpd(C) of internal groupoids in C is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category Grpd(C) is a semi-abelian action representable algebraically coherent category with normalizers if and only if C is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.