Miracle-dz/UniFAR-FTUs huggingface dataset:
Lie groupoids of mappings taking values in a Lie groupoid

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper \'etale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current Lie groupoid as a current Lie algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators given by postcomposition with a fixed function, between manifolds of $C^\ell$-functions. Under natural hypotheses, these operators turn out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map. These results are new in their generality and of independent interest.
3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have $3$-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -- which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$ -- and the first progress towards the analogous conjecture over number fields other than $\mathbb{Q}$. Our results follow from a computation of the average size of the $\phi$-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny $\phi$.
DISCOVERY OF THE MOST ISOLATED GLOBULAR CLUSTER IN THE LOCAL UNIVERSE

We report the discovery of two new globular clusters in the remote halos of M81 and M82 in the M81 Group based on Hubble Space Telescope archive images. They are brighter than typical globular clusters (MV = −9.34 mag for GC-1 and MV = −10.51 mag for GC-2), and much larger than known globular clusters with similar luminosity in the Milky Way Galaxy and M81. Radial surface brightness profiles for GC-1 and GC-2 do not show any features of tidal truncation in the outer part. They are located much farther from both M81 and M82 in the sky, compared with previously known star clusters in these galaxies. Color–magnitude diagrams of resolved stars in each cluster show a well-defined red giant branch (RGB), indicating that they are metal-poor and old. We derive a low metallicity with [Fe/H] ≈−2.3 and an old age ∼14 Gyr for GC-2 from the analysis of the absorption lines in its spectrum in the Sloan Digital Sky Survey in comparison with the simple stellar population models. The I-band magnitude of the tip of the RGB for GC-2 is 0.26 mag fainter than that for the halo stars in the same field, showing that GC-2 is ∼400 kpc behind the M81 halo along our line of sight. The deprojected distance to GC-2 from M81 is much larger than any other known globular clusters in the local universe. This shows that GC-2 is the most isolated globular cluster in the local universe.
Accelerated First-Order Methods: Differential Equations and Lyapunov Functions

We develop a theory of accelerated first-order optimization from the viewpoint of differential equations and Lyapunov functions. Building upon the work of previous authors, we consider differential equations which model the behavior of accelerated gradient descent. Our main contribution is to provide a general framework for discretizating the differential equations to produce accelerated methods. An important novelty in our approach is the treatment of stochastic discretizations, which introduce randomness at each iteration. This leads to a unified derivation of a wide variety of methods, which include accelerated gradient descent, FISTA, and accelerated coordinate descent as special cases.
Algebraic multigrid methods *

This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$ , such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_{c}$ is the span of the eigenvectors corresponding to the first $n_{c}$ eigenvectors $\bar{R}A$ (with $\bar{R}=R+R^{T}-R^{T}AR$ ). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with $\bar{R}A$ , and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.
The Steinhaus property and Haar‐null sets

It is shown that, if G is an uncountable Polish group and A ⊆ G is a universally measurable set such that A−1A is meager, then the set Tl(A) = {μ ∈ P(G): μ(gA) = 0 for all g ∈ G} is co‐meager. In particular, if A is analytic and not left Haar‐null, then 1 ∈ Int (A−1AA−1A).
The slice spectral sequence of a $C_4$-equivariant height-4 Lubin-Tate theory

We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.
A tour problem on a toroidal board

Let $A$ be an $n\times m$ toroidal array consisting of filled cells and empty cells. Assume that an orientation $R=(r_1,\dots,r_n)$ of each row and $C=(c_1,\dots,c_m)$ of each column of $A$ is fixed. Given an initial filled cell $(i_1,j_1)$ consider the list $ L_{R,C}=((i_1,j_1),(i_2,j_2),\ldots,(i_k,j_k),$ $(i_{k+1},j_{k+1}),\ldots)$ where $j_{k+1}$ is the column index of the filled cell $(i_k,j_{k+1})$ of the row $R_{i_k}$ next to $(i_k,j_k)$ in the orientation $r_{i_k}$, and where $i_{k+1}$ is the row index of the filled cell of the column $C_{j_{k+1}}$ next to $(i_k,j_{k+1})$ in the orientation $c_{j_{k+1}}$. We propose the following "Crazy Knight's Tour Problem": Do there exist $R$ and $C$ such that the list $L_{R,C}$ covers all the filled cells of $A$? Here we provide a complete solution for the case with no empty cells and we obtain partial results for square arrays where the filled cells follow some specific regular patterns.
Adaptive output consensus of nonlinear fractional-order multi-agent systems: a fractional-order backstepping approach

This paper presents the distributed control design for a class of fractional-order strict-feedback nonlinear multi-agent systems in the presence of unknown dynamics by employing backstepping strategy. Considering that the information of followers’ states are not fully measurable for feedback design, the fractional-order infinite-dimension neural-network state observer is introduced to estimate the unavailable states. The infinite-dimension neuroadaptive laws are also proposed to eliminate the undesirable effects of the unknown nonlinear functions. Besides, based on the Lyapunov fractional-order stability approach and graph theory, unlike the existing results, a distributed neural adaptive observer-based control architecture is designed to ensure that all the closed-loop network signals are ultimately bounded. Finally, a simulation example is given to demonstrate the validity of the proposed control method.
Thin monodromy in Sp(4)

Abstract We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.
Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory

This paper studies the large-j asymptotics of the Lorentzian Engle–Pereira–Rovelli–Livine (EPRL) spinfoam amplitude on a 4D simplicial complex with an arbitrary number of simplices. The asymptotics of the spinfoam amplitude is determined by the critical configurations. Here we show that, given a critical configuration in general, there exists a partition of the simplicial complex into three types of regions R Nondeg ,R Deg -A? > and R Deg -B? >, where the three regions are simplicial sub-complexes with boundaries. The critical configuration implies different types of geometries in different types of regions, i.e. (1) the critical configuration restricted to R Nondeg ? > implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted to R Deg -A? > is degenerate of type-A in our definition of degeneracy, but it implies a nondegenerate discrete Euclidean geometry in R Deg -A? >, (3) the critical configuration restricted to R Deg -B? > is degenerate of type-B, and it implies a vector geometry in R Deg -B? >. With the critical configuration, we further make a subdivision of the regions R Nondeg ? > and R Deg -A? > into sub-complexes (with boundaries) according to their Lorentzian/Euclidean oriented 4-volume V4(v) of the 4-simplices, such that sgn(V4(v)) is a constant sign on each sub-complex. Then in each sub-complex R Nondeg ? > or R Deg -A? >, the spinfoam amplitude at the critical configuration gives the Regge action in a Lorentzian signature or an Euclidean signature respectively. The Regge action reproduced here contains a sign prefactor sgn(V4(v)) related to the oriented 4-volume of the 4-simplices. Therefore the Regge action reproduced here can be viewed as a discretized Palatini action with an on-shell connection. Finally, the asymptotic formula of the spinfoam amplitude is given by a sum of the amplitudes evaluated at all possible critical configurations, which are the products of the amplitudes associated with different types of geometries.
The resurgence properties of the incomplete gamma function I

In this paper, we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymtotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.
Distinguished graduates in mathematics of Jagiellonian University in the interwar period. Part II: 1928-1939

In this study, we present profiles of some distinguished graduates in mathematics of the Jagiellonian University from the years 1918-1939. We discuss their professional paths and scholarly achievements, instances of scientific collaboration, connections with other academic centers in Poland and worldwide, involvement in mathematical education and teacher training, as well as their later roles in Polish scientific and academic life. We also try to understand in what way they were shaped by their studies and how much of Krak\'ow scientific traditions they continued. We find strong support for the claim that there was a distinct, diverse and deep mathematical stream in Krak\'ow between the wars, rooted in classical disciplines like differential equations and geometry, but also open to new trends in mathematics. Part I concerns the graduates before the university reform, in 1918-1926.
Henkin Measures for the Drury–Arveson Space

We exhibit Borel probability measures on the unit sphere in $\mathbb C^d$ for $d \ge 2$ which are Henkin for the multiplier algebra of the Drury-Arveson space, but not Henkin in the classical sense. This provides a negative answer to a conjecture of Clouâtre and Davidson.
Masur’s criterion does not hold in the Thurston metric

We construct a counterexample for an analogue of Masur's criterion in the setting of Teichm\"uller space equipped with the Thurston metric. For that, we find a minimal, filling, non-uniquely ergodic lamination $\lambda$ on the seven-times punctured sphere with uniformly bounded annular projection distances. Then we show that a geodesic in the corresponding Teichm\"uller space that converges to $\lambda$, stays in the thick part for the whole time.
Zariski-like topologies for lattices with applications to modules over associative rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).
Evaluation of Probiotic Properties of Lactobacilli in Breast Milk and Their Inhibitory Effect on Pathogenic Bacteria of the Gastrointestinal Tracts

10.30699/jambs.30. 143.536 Background & Objective: Breast milk contains nutrients such as carbohydrates, essential fatty acids, proteins, vitamins, minerals, and a source of communal bacteria with probiotic potential that is very effective in the prevention and treatment of neonatal infections. The aim of this study was the evaluation of probiotic properties of lactobacilli in breast milk and their inhibitory effect on pathogenic bacteria of the gastrointestinal tracts. Materials & Methods: In a cross-sectional descriptive study, during 10 months from January to October 2018, 100 breast milk samples were collected by referring to health centers after isolation. Lactobacilli strains were evaluated based on morphological characteristics, catalase, and hot staining tests, survival tests in acidic conditions, and bile salt tolerance to evaluate probiotic properties. Antibiotic resistance of probiotic strains and ability to inhibit pathogenic bacteria was evaluated by well method and growth inhibition zone. Results: 122 lactobacilli belonging to 12 species were identified from 100 samples of breast milk by phototypical methods. The predominant species belonging to casei and other lactobacilli were Fermentum, Plantarum, and Gasseri, respectively.The highest antibiotic resistance was related to vancomycin (63.15%). The 3 isolates L4, L14 and L16 were able to strongly inhibit all the studied gastrointestinal pathogens. Conclusion: Breast milk is a rich source of beneficial probiotic lactobacilli, which can be useful in breast milk for infants who are not breastfed to prevent neonatal infections.
Subnormal Embedding Theorems for Groups

Some subnormal embeddings of groups into groups with additional properties are established, and in particular, embeddings of countable groups into two‐generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, H. Neumann and G. Higman on embeddings of that type. Through the consideration of subnormal embeddings of finite groups into finite groups with additional properties, a result of H. Heineken and J. Lennox is illustrated.
On the Griffiths Groups of Fano Manifolds of Calabi-Yau Hodge Type

A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold, the fivefold quartic double solid, and the fivefold intersection of a quadric and a cubic. We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a smooth general complete intersection FCY manifolds, is also infinitely generated. In the case of the fivefold quartic double solid, we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for the fivefold intersection of a quadric and a cubic there is a noncommutative CY 3-fold, such that the Griffiths group of the intersection surjects on to the Griffiths group of noncommutative 3-fold. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back to honest algebraic varieties such as products of elliptic curves and K3-surfaces.
Spectrum of perturbations in anisotropic inflationary universe with vector hair

We study both the background evolution and cosmological perturbations of anisotropic inflationary models supported by coupled scalar and vector fields. The models we study preserve the U(1) gauge symmetry associated with the vector field, and therefore do not possess instabilities associated with longitudinal modes (which instead plague some recently proposed models of vector inflation and curvaton). We first intoduce a model in which the background anisotropy slowly decreases during inflation; we then confirm the stability of the background solution by studying the quadratic action for all the perturbations of the model. We then compute the spectrum of the h× gravitational wave polarization. The spectrum we find breaks statistical isotropy at the largest scales and reduces to the standard nearly scale invariant form at small scales. We finally discuss the possible relevance of our results to the large scale CMB anomalies.
Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $\pi_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n$
Complex symmetric Hamiltonians and exceptional points of order four and five

In the broad context of physics ranging from classical experimental optics to quantum mechanics of unitary as well as non-unitary systems there emerge interesting phenomena related to the presence of the so called Kato's exceptional points in the space of parameters. An elementary linear-algebraic method of their localization is proposed and applied to the class of tridiagonal $N$ by $N$ complex symmetric toy-model generators of evolution $H=H(\gamma)$. The implementation of the method is shown to provide new models with the exceptional points $\gamma=\gamma^{(EP)}$ of higher orders. Two distinct areas of applicability are expected to lie (1) in quantum mechanics of non-Hermitian (open as well as closed) systems, and (2) in the experiments using the coupled classical optical waveguides simulating the EP-related effects in the laboratory.
A direct measurement of tomographic lensing power spectra from CFHTLenS

We measure the weak gravitational lensing shear power spectra and their cross-power in two photometric redshift bins from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). The measurements are performed directly in multipole space in terms of adjustable band powers. For the extraction of the band powers from the data we have implemented and extended a quadratic estimator, a maximum likelihood method that allows us to readily take into account irregular survey geometries, masks, and varying sampling densities. We find the 68 per cent credible intervals in the $\sigma_8$-$\Omega_{\rm m}$-plane to be marginally consistent with results from $Planck$ for a simple five parameter $\Lambda$CDM model. For the projected parameter $S_8 \equiv \sigma_8(\Omega_{\rm m}/0.3)^{0.5}$ we obtain a best-fitting value of $S_8 = 0. 768_{-0.039}^{+0.045}$. This constraint is consistent with results from other CFHTLenS studies as well as the Dark Energy Survey. Our most conservative model, including modifications to the power spectrum due to baryon feedback and marginalization over photometric redshift errors, yields an upper limit on the total mass of three degenerate massive neutrinos of $\Sigma m_\nu < 4.53 \, {\rm eV}$ at 95 per cent credibility, while a Bayesian model comparison does not favour any model extension beyond a simple five parameter $\Lambda$CDM model. Combining the shear likelihood with $Planck$ breaks the $\sigma_8$-$\Omega_{\rm m}$-degeneracy and yields $\sigma_8=0.818 \pm 0.013$ and $\Omega_{\rm m} = 0.300 \pm 0.011$ which is fully consistent with results from $Planck$ alone.