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Context. Mass is the most critical physical parameter in the evolution of a star. Since stars form in clusters their Initial Mass Function (IMF) is decisive in their evolution. Aims. Use Gaia DR3-based stellar masses_mass flame and the stellar members found for fifteen nearby open clusters from Paper I, to estimate their mass segregation and distribution. Methods. For each cluster, the single stars' main sequence was fitted with a moving straight line weighted fit to the Color-Magnitude Diagram, stars brighter than the residuals dispersion were taken as binaries. Single stars masses were obtained from a cubic spline fit to the mass_flame vs. G magnitude data. For binary stars, the individual masses of each component were estimated using simulated-based inference. We used the minimum spanning tree concept to measure the mass segregation of each cluster. From the stellar mass distribution, an estimate of the power-law coefficient that best describes it was used to characterize the IMF. Results. Mass segregation is visible in all the clusters, the older ones have about 50% of their most massive stars segregated, while younger ones extend from about 30% to 55%. The IMF of the studied clusters is well described by a power-law of index 2.09. Conclusions. Significant mass segregation, from one-third to one-half of its most massive population is present in open clusters as young as > 10 Myr. Mass segregation may be strong for only a few of the most massive stars or less intense but extended to a larger fraction of those, it may start as early as 0.20 of the relaxation time of the cluster and progress over time by increasing both the number of the most massive stars affected and their amount of segregation. Older open clusters show evidence of binary disruption as time progresses., Comment: Submitted to A&A. 11 pages, 6 figures

TThe Period-Luminosity (PL) relation for Cepheid variable stars in the Large Magellanic Cloud (LMC) is crucial for distance measurements in astronomy. This study analyzes the impact of using the median rather than the mean on the PL relation's slope and zero point. It also examines the persistence of the break at approximately 10 days and addresses specification issues in the PL relation model. Using $VI$-band median and mean magnitudes from the OGLE-IV survey, corrected for extinction, we fit the PL relation employing robust $MM$-regression, which features a high breakdown point and robust standard errors. Statistical tests and residual analysis are conducted to identify and correct model deficiencies. Our findings indicate a significant change in the PL relation for Cepheids with periods of 10 days or longer, regardless of whether median or mean magnitudes are used. A bias in the zero point and slope estimators is observed when using median magnitudes instead of mean magnitudes, especially in the $V$-band. By identifying and correcting regression issues and considering the period break, our estimators for slope and zero point are more accurate for distance calculations. Comparative analysis of the models for each band quantifies the bias introduced by using median magnitudes, highlighting the importance of considering the Cepheids' period for accurate location measure results, similar to those obtained using mean magnitudes.

Context. Since the first publication of the Gaia catalogue a new view of our Galaxy has arrived. Its astrometric and photometric information has improved the precision of the physical parameters of open star clusters obtained from them. Aims. Using the Gaia DR3 catalogue, we aim to find physical stellar members including faint stars for 370 Galactic open clusters located within 1 kpc. We also estimate the age, metallicity, distance modulus and extinction of these clusters. Methods. We employ the HDBSCAN algorithm on both astrometric and photometric data to identify members in the open clusters. Subsequently, we refine the samples by eliminating outliers through the application of the Mahalanobis metric utilizing the chi-square distribution at a confidence level of 95%. Furthermore, we characterize the stellar parameters with the PARSEC isochrones. Results. We obtain reliable star members for 370 open clusters with an average parallax error of 0.16 mas. We identify about 40% more stars in these clusters compared to previous work using the Gaia DR2 catalogue, including faint stars as new members with G > 17. Before the clustering application we correct the parallax zero-point bias to avoid spatial distribution stretching that may affect clustering results. Our membership lists include merging stars identified by HDBSCAN with astrometry and photometry. We note that the use of photometry in clustering can recover up to 10% more stars in the fainter limit than clustering based on astrometry only, this combined with the selection of stars filtering them out by quality cuts significantly reduces the number of stars with huge parallax error. After clustering, we estimate age, Z, and AV from the photometry of the membership lists., Comment: Submitted to A&A. 11 pages, 10 figures

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems. Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization. [1] On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl.9 (1988), no 4, 905-918

Machine learning (ML) is a promising approach for predicting small molecule properties in drug discovery. Here, we provide a comprehensive overview of various ML methods introduced for this purpose in recent years. We review a wide range of properties, including binding affinities, solubility, and ADMET (Absorption, Distribution, Metabolism, Excretion, and Toxicity). We discuss existing popular datasets and molecular descriptors and embeddings, such as chemical fingerprints and graph-based neural networks. We highlight also challenges of predicting and optimizing multiple properties during hit-to-lead and lead optimization stages of drug discovery and explore briefly possible multi-objective optimization techniques that can be used to balance diverse properties while optimizing lead candidates. Finally, techniques to provide an understanding of model predictions, especially for critical decision-making in drug discovery are assessed. Overall, this review provides insights into the landscape of ML models for small molecule property predictions in drug discovery. So far, there are multiple diverse approaches, but their performances are often comparable. Neural networks, while more flexible, do not always outperform simpler models. This shows that the availability of high-quality training data remains crucial for training accurate models and there is a need for standardized benchmarks, additional performance metrics, and best practices to enable richer comparisons between the different techniques and models that can shed a better light on the differences between the many techniques., Comment: 46 pages, 1 figure

Context. Near open clusters as Pleiades, Praesepe and Blanco 1 have been extensively studied due to their proximity to the Sun. The Gaia data brings the opportunity to investigate these clusters, since it contains valuable astrometric and photometric information which can be used to update their kinematic and stellar properties. Aims. Our goal is to carry out a star membership study in these nearby open clusters employing an astrometric model with proper motions and an unsupervised clustering machine learning algorithm using positions, proper motions and parallaxes. The star members are selected from the cross-matching between both methods. Methods. We use the Gaia DR3 catalogue to determine star members using two approaches: a classical Bayesian model and the unsupervised machine learning algorithm DBSCAN. For star members we build the radial density profiles, the spatial distributions and compute the King parameters. The ages and metallicities were estimated using the BASE-9 Bayesian software. Results. We identified 958, 744 and 488 star members for the Pleiades, Praesepe and Blanco 1 respectively. We corrected the distances and built the spatial distributions, finding that Praesepe and Blanco 1 have elongated shape structures. The distances, ages and metallicities obtained were consistent with the reported in the literature. Conclusions. We obtained catalogues of star members, updated kinematic and stellar parameters for these open clusters. We found that the proper motions model can find a similar number of members as the unsupervised clustering algorithm does when the cluster population form an overdensity in the vector point diagram. It allows to select an adequate size of the proper motions region to run these methods. Our analysis found stars that are being directed towards the outskirts of the Praesepe and Blanco 1, which exhibit elongated shapes., Comment: 11 pages, 7 figures. Submitted to Astronomy & Astrophysics

We define the (convex) joint numerical range for an infinite family of compact operators in a Hilbert space H. We use this set to determine whether a self-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimal respect to the set of diagonals in a fixed basis E of H in the operator norm, that is ||A|| <= ||A+D||, for all diagonal D. We also describe the moment set m_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms of joint numerical ranges and obtain equivalences between the intersection of moments of two subspaces and of its two related joint numerical ranges. Moreover, we relate the condition of minimality of A or the intersection of the moments of the eigenspaces of ||A|| and -||A|| to the intersection of the joint numerical ranges of two finite families of certain finite hermitian matrices. We also study geometric properties of the set m_S such as extremal curves related with the basis E. All these conditions are directly related with the description of minimal self-adjoint compact operators.

For a given complex finite dimensional subspace $S$ of $\mathbb{C}^n$ and a fixed basis, we study the compact and convex subset of $\left(\mathbb{R}_{\geq 0}\right)^n$ that we call the moment of $S$ $m_S=$ convex hull ($\{|s|^2\in\mathbb{R}^n_{\geq 0}: s\in S \wedge \|s\|=1\} )$ $\simeq \{ Diag(Y) \in M_n^h(\mathbb{C}):Y\geq 0, tr(Y)=1, P_S Y P_S=Y\}$ where $|s|^2=(|s_1|^2,|s_2|^2,\dots,|s_n|^2)$. This set is relevant in the determination of minimal hermitian matrices ($M\in M^h_n$ such that $\|M+D\|\leq D$ for every diagonal $D$ and $\| \|$ the spectral norm). We describe extremal points and curves of $m_S$ in terms of principal vectors that minimize the angle between $S$ and the coordinate axes. We also relate $m_S$ to the joint numerical range $W$ of $n$ rank one $n\times n$ matrices constructed with the orthogonal projection $P_S$ and the fixed basis used. This connection provides a new approach to the description of $m_S$ and to minimal matrices. As a consequence the intersection of two of these joint numerical ranges allow the construction or detection of a minimal matrix, a fact that is easier to corroborate than the equivalent condition for moments. It is also proved that $m_S$ is a semi-algebraic set equal to the intersection of the mentioned $W$ with a hyperplane and whose generated positive cone coincides with that of $W$.

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact., Comment: 33 pages

We revisit the subject of Be star candidates towards the Magellanic Clouds, previously studied by the authors using SPM4 proper motions. We obtain GAIA DR2 parallaxes and proper motions for 2357 and 994 LMC and SMC Be candidates, respectively. Parallaxes and proper motions vs. color V-I easily reveal the presence of the redder galactic contaminant foreground, as concluded in our previous work, but this time we do find a few red Be stars candidates consistent with being true Magellanic objects. A membership assessment to each Magellanic Cloud is done for each Be candidate, based on the distribution of their parallaxes and proper motions. From a compilation of published catalogues of spectroscopically confirmed Be stars, we found that 40 (LMC) and 64 (SMC) of these Be candidates, are in fact Be stars. Near infrared IRSF JHKs magnitudes were obtained for about 70% the Be stars candidates with GAIA DR2 astrometric data. Mid-infrared SAGE IRAC 3.6, 4.5, 5.8 and 8.0 $\mu$m magnitudes were obtained for about 85% as well. 6 LMC and 7 SMC confirmed Be stars show optical, near- or mid-infrared colours redder than what has been typically measured for Classical Be stars. Several of the Be candidates follow those redder-than-expected colours distributions suggesting the existence of more red Magellanic Be stars., Comment: Slight change in title, two new more authors, 15 pages, 12 figures, accepted by MNRAS

We study certain pairs of subspaces $V$ and $W$ of $\mathbb{C}^n$ we call supports that consist of eigenspaces of the eigenvalues $\pm\|M\|$ of a minimal hermitian matrix $M$ ($\|M\|\leq \|M+D\|$ for all real diagonals $D$). For any pair of orthogonal subspaces we define a non negative invariant $\delta$ called the adequacy to measure how close they are to form a support and to detect one. This function $\delta$ is the minimum of another map $F$ defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of $F$ in order to approximate $\delta$. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

We study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group U_(K+C) of the unitization of the compact operators K(H)+C, or equivalently, the quotient U_(K+C)/ U_Diag(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

We study the elementary C*-algebra whose elements are the sum of a diagonal plus a compact operator. We describe the structure of the unitary group, the sets of ideals, automorhisms and projections., Comment: 24 pages

We present a proper motion investigation of a sample of Be star candidates towards the Magellanic Clouds, which has resulted in the identification of separate populations, in the Galactic foreground and in the Magellanic background. Be stars are broadly speaking B-type stars that have shown emission lines in their spectra. In this work, we studied a sample of 2446 and 1019 Be star candidates towards the LMC and SMC respectively, taken from the literature and proposed as possible Be stars due to their variability behaviour in the OGLE-II I band. JHKs magnitudes from the IRSF catalog and proper motions from the SPM4 catalog, were obtained for 1188 and 619 LMC and SMC Be stars candidates, respectively. Color-color and vector-point diagrams were used to identify different populations among the Be star candidates. In the LMC sample, two populations with distinctive infrared colours and kinematics were found, the bluer sample is consistent with being in the LMC and the redder one with belonging to the Milky Way disk. This settles the nature of the redder sample which had been described in previous publications as a possible unknown subclass of stars among the Be candidates in the LMC. In the SMC sample, a similar but less evident result was obtained, since this apparent unknown subclass was not seen in this galaxy. We confirm that in the selection of Be stars by their variability, although generally successful, there is a higher risk of contamination by Milky Way objects towards redder B$-$V and V$-$I colors., Comment: 8 pages, 7 figures

Let H be a separable Hilbert space, and D(B(H))^ah the anti-Hermitian bounded diagonals in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators U_kd = {u in U(H): such that u-e^D in K(H) for D in D(B(H))^ah} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob={ e^K b e^{-K} : K in K(H)^ah } of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on Ob defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space K(H)^ah / D(K(H)^ah). Then for every c in Ob and x in T(\ob)_c there exist a minimal lifting Z_0 in B(H)^ah (in the quotient norm, not necessarily compact) such that g(t)=e^{t Z_0} c e^{-t Z_0} is a short curve on Ob in a certain interval., Comment: Minor typos were corrected. The statement of the Lemma 3.14 and its proof were modified

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\{z \in {\mathbb C}: Re(z) \in [1/4;3/4]\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by $s_j(A^tB^{1-t}+B^tA^{1-t})\le s_j(A+B)$, answering in the negative a question made by K. Audenaert and F. Kittaneh., Comment: 9 pages, 1 figure

The universality and linearity of the Leavitt law are hypotheses commonly adopted in studies of galaxy distances using Cepheid variables as standard candles. In order to test these hypotheses, we obtain slopes of the Leavitt law using linear regressions of fundamental-mode Cepheids observed by the Optical Gravitational Lensing Experiment project in the Magellanic Clouds. We find that the slopes in $VI$-bands and in the Wesenheit index behave exponentially, indicating non-linearity. We also find that the slopes obtained using long-period Cepheids can be considered as universal in the $VI$-bands, but not in the Wesenheit index., Comment: 8 pages, 3 figures. Accepted for publication in MNRAS

We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such that ||C|| is less or equal than || C + D ||, for all D in D(K(H)). This property is equivalent to || C || = min{||C+D||: D in D(K(H))} = dist (C,D(K(H))), where D(K(H)) denotes the space of compact diagonal operators in a fixed base of H and ||.|| is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal., Comment: 12 pages

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained., Comment: 23 pages

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval $[a,b]\subset\mathbb R$, we study the action defined in the Lie group of $n\times n$ unitary matrices $\mathcal{U}(n)$ by $$ S(\alpha)=\int_a^b L(\dot\alpha(t))\,dt\,, $$ where $\alpha:[a,b]\to\mathcal{U}(n)$ is a rectifiable curve. We prove that the one-parameter subgroups of $\mathcal{U}(n)$ are the optimal paths, provided the spectrum of the exponent is bounded by $\pi$. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in $\mathcal{U}(n)$ as well as angular metrics in the Grassmann manifold, Comment: 20 pages

Let X and Y be two nxn Hermitian matrices. In the article "Proof of a conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and V such that $$ e^{i X}e^{i Y}=e^{i (UXU^*+VBV^*)}. $$ In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II$_1$ factor.

We describe properties of a Hermitian square matrix M in M_n(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in M_n(C) and || || the operator norm. These matrices are related to some particular positive matrices with their range included in the eigenspaces of the eigenvalues +||M|| and -||M|| of M. We show how a constructive method can be used to obtain minimal matrices of any dimension relating this problem with majorization results in R^n., Comment: 9 pages

For a given symmetrically normed ideal I on an infinite dimensional Hilbert space H, we study the rectifiable distance in the classical Banach-Lie unitary group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that one-parameter subgroups of U_I are short paths, provided the spectrum of the exponent is bounded by $\pi$, and that any two elements of U_I can be joined with a short path, thus obtaining a Hopf-Rinow theorem in this infinite dimensional setting, for a wide and relevant class of (non necessarily smooth) metrics. Then we prove that the one-parameter groups are the unique short paths joining given endpoints, provided the symmetric norm considered is strictly convex., Comment: This preprint has been replaced by arXiv:1107.2439v1, which deals with the far more general case of symmetric Lagrangians in the unitary group

Let $B$ be a C$^*$-algebra and $X$ a C$^*$ Hilbert $B$-module. If $p\in B$ is a projection, denote by $S_p =\{x\in X : < x,x> =p\}$, the $p$-sphere of $X$. For $\phi$ a state of $B$ with support $p$ in $B$ and $x\in S_p$, consider the state $\phi_x$ of $L_B(X)$ given by $\phi_x(t)= \phi(< x,t(x)>)$. In this paper we study certain sets associated to these states, and examine their topologic properties. As an application of these techniques, we prove that the space of states of the hyperfinite II$_1$ factor $R_0$, with support equivalent to a given projection $p\in R_0$, regarded with the norm topology (of the conjugate space of $R_0$), has trivial homotopy groups of all orders. The same holds for the space $$ S_p(R_0)=\{v\in R_0:v^*v=p\}\subset R_0 $$ of partial isometries with initial space $p$, regarded with the ultraweak topology., Comment: 23 pages, Latex