ShantanuT01/DACTYL-Pretraining huggingface dataset:
The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In this note, we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra.
A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,\nu)$ where $\nu$ is an infinitesimally excessive measure. Then a weak solution is constructed.
Microlensing observations towards M31 are a powerful tool for the study of the dark matter population in the form of MACHOs both in the Galaxy and the M31 halos, a still unresolved issue, as well as for the analysis of the characteristics of the M31 luminous populations. In this work we present the second year results of our pixel lensing campaign carried out towards M31 using the 152 cm Cassini telescope in Loiano. We have established an automatic pipeline for the detection and the characterisation of microlensing variations. We have carried out a complete simulation of the experiment and evaluated the expected signal, including an analysis of the efficiency of our pipeline. As a result, we select 1-2 candidate microlensing events (according to different selection criteria). This output is in agreement with the expected rate of M31 self-lensing events. However, the statistics are still too low to draw definitive conclusions on MACHO lensing.
Let s_k/t_k, k>= 0, be the convergents of the continued fraction expansion of a real number x. We investigate the sequence of Jacobi symbols (s_k/t_k), k>= 0. We show that this sequence is purely periodic with shortest possible period length 24 for x=e=2.718281... and shortest possible period length 40 for x=e^2. Further, we make the first steps towards a general theory of such sequences of Jacobi symbols. For instance, we show that there are uncountably many numbers x such that this sequence has the period 1 (of length 1), and that every natural number L actually occurs as the shortest possible period length of some x.
Kondo effect is a prominent quantum phenomenon describing the many-body screening of a local magnetic impurity. Here, we reveal a new type of non-magnetic Kondo behavior generated by gauge fluctuations in strongly-correlated baths. We show that a non-magnetic bond defect not only introduces the potential scattering but also locally enhances the gauge fluctuations. The local gauge fluctuations further mediate a pseudospin exchange interaction that produces an asymmetric Kondo fixed point in low-energy. The gauge-fluctuation-induced Kondo phenomena do not exhibit the characteristic resistivity behavior of conventional Kondo effect, but display a non-monotonous temperature dependence of thermal conductivity as well as an anisotropic pseudospin correlation. Moreover, with its origin from gauge fluctuations, the Kondo features can be regarded as promising indicators for identifying quantum spin liquids. Our work advances fundamental knowledge of novel Kondo phenomena in strongly-correlated systems, which have no counterparts in thermal baths within the single-particle description.
We calculate the leading divergences at NNLO for the octet part of the nonleptonic weak sector of chiral perturbation theory, using renormalization group methods. The role of counterterms which vanish at the equation of motion and their use to simplify the calculation is shown explicitly. The obtained countertem Lagrangian can be employed to calculate the chiral double log contributions of quantities in this sector, most notably the K --> pi pi amplitude. The double log contribution of the latter is discussed in a separate paper.
We examine static perfect fluid spheres in the presence of a cosmological constant. New exact matter solutions are discussed which require the Nariai metric in the vacuum region. We generalize the Einstein static universe such that neither its energy density nor its pressure is constant throughout the spacetime. Using analytical techniques we derive conditions depending on the equation of state to locate the vanishing pressure surface. This surface can in general be located in regions with decreasing area group orbits. We use numerical methods to integrate the field equations for realistic equations of state and find consistent results.
We investigate the evolution of a population of 100 dark matter satellites orbiting in the gravitational potential of a realistic model of M31. We find that after 10 Gyr, seven subhalos are completely disrupted by the tidal field of the host galaxy. The remaining satellites suffer heavy mass loss and overall, 75% of the mass initially in the subhalo system is tidally stripped. Not surprisingly, satellites with pericentric radius less than 30 kpc suffer the greatest stripping and leave a complex structure of tails and streams of debris around the host galaxy. Assuming that the most bound particles in each subhalo are kinematic tracers of stars, we find that the halo stellar population resulting from the tidal debris follows an r^{-3.5} density profile at large radii. We construct B-band photometric maps of stars coming from disrupted satellites and find conspicuous features similar both in morphology and brightness to the observed Giant Stream around Andromeda. An assumed star formation efficiency of 5-10% in the simulated satellite galaxies results in good agreement with the number of M31 satellites, the V-band surface brightness distribution, and the brightness of the Giant Stream. During the first 5 Gyr, the bombardment of the satellites heats and thickens the disk by a small amount. At about 5 Gyr, satellite interations induce the formation of a strong bar which, in turn, leads to a significant increase in the velocity dispersion of the disk.
We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R   We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local well-posedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifth-order equation in the KdV hierarchy is globally well-posed in the energy space H^2.
Knowledge Distillation (KD) has been used in image classification for model compression. However, rare studies apply this technology on single-stage object detectors. Focal loss shows that the accumulated errors of easily-classified samples dominate the overall loss in the training process. This problem is also encountered when applying KD in the detection task. For KD, the teacher-defined hard samples are far more important than any others. We propose ADL to address this issue by adaptively mimicking the teacher's logits, with more attention paid on two types of hard samples: hard-to-learn samples predicted by teacher with low certainty and hard-to-mimic samples with a large gap between the teacher's and the student's prediction. ADL enlarges the distillation loss for hard-to-learn and hard-to-mimic samples and reduces distillation loss for the dominant easy samples, enabling distillation to work on the single-stage detector first time, even if the student and the teacher are identical. Besides, ADL is effective in both the supervised setting and the semi-supervised setting, even when the labeled data and unlabeled data are from different distributions. For distillation on unlabeled data, ADL achieves better performance than existing data distillation which simply utilizes hard targets, making the student detector surpass its teacher. On the COCO database, semi-supervised adaptive distillation (SAD) makes a student detector with a backbone of ResNet-50 surpasses its teacher with a backbone of ResNet-101, while the student has half of the teacher's computation complexity. The code is avaiable at https://github.com/Tangshitao/Semi-supervised-Adaptive-Distillation
We consider the polar curves $\PSO$ arising from generic projections of a germ $(S,0)$ of complex surface singularity onto $\C^2$. Taking $(S,0)$ to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the $\delta$-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of $(S,0)$. These two (equisingularity)-data for $\PSO$ are described in term, on the one side of the geometry of the tangent cone of $(S,0)$ and on the other side of the limit-trees introduced by T. de Jong and D. van Straten for the deformation theory of these minimal singularities. These trees give a combinatorial device for the description of the polar curve which makes it much clearer than in our previous Note on the subject. This previous work mainly relied on a result of M. Spivakovsky. Here we give a geometrical proof via deformations (on the tangent cone, and what we call Scott deformations) and blow-ups, although we need Spivakovsky's result at some point, extracting some other consequences of it along the way.
We study here the Gamma-Ray Burst (GRB) environment through the analysis of the optical absorption features due to the gas surrounding the GRB. In particular, we analyze high resolution spectroscopic observations of GRB080319B and GRB080330 taken with UVES at the VLT, starting 8m30s and 1.5hr after the GRB trigger, respectively. The spectra show that the ISM of the GRB host galaxies are complex, with several components contributing to the host absorption system. In addition, we detect strong excited absorption lines, from which we derive information on the gas distance from the site of the GRB explosion. Under the assumption that the excited features are produced by indirect UV pumping, we found that this distance results to be 2-6 kpc for GRB080319B and 280+/-50 pc for GRB080330, meaning that the power of the GRB radiation can influence the conditions of the interstellar medium up to a distance of several hundred pc.
We present a simplified analytic approach to the problem of the spiraling of a massive body orbiting within the dark halo of a dwarf galaxy. This dark halo is treated as the core region of a King distribution of dark matter particles, in consistency with the observational result of dwarf galaxies having solid body rotation curves. Thus we derive a simple formula which provides a reliable and general first order solution to the problem, totally analogous to the one corresponding to the dynamical friction problem in an isothermal halo. This analytic approach allows a clear handling and a transparent understanding of the physics and the scaling of the problem. A comparison with the isothermal case shows that in the core regions of a King sphere, dynamical friction proceeds at a different rate, and is sensitive to the total core radius. Thus, in principle, observable consequences may result. In order to illustrate the possible effects, we apply this formula to the spiraling of globular cluster orbits in dwarf galaxies, and show how present day globular cluster systems could in principle be used to derive better limits on the structure of dark halos around dwarf galaxies, when the observational situation improves. As a second application, we study the way a massive black hole population forming a fraction of these dark halos would gradually concentrate towards the centre, with the consequent deformation of an originally solid body rotation curve. This effect allows us to set limits on the fraction/mass of any massive black hole minority component of the dark halos of dwarf galaxies. In essence, we take advantage of the way the global matter distribution fixes the local distribution function for the dark matter particles, which in turn determines the dynamical friction problem.
Quantum metrology utilizes quantum effects to reach higher precision measurements of physical quantities compared with their classical counterparts. However the ubiquitous decoherence obstructs its application. Recently, non-Markovian effects are shown to be effective in performing quantum optical metrology under locally dissipative environments\cite{PhysRevLett.123.040402}. However, the mechanism is still rather hazy. Here, we uncover the reason why forming a bound state can protect the quantumness against a dissipative ambient via the quantum Fisher information of entangled coherent states. An exact analytical expression of the quantum Fisher information in the long-encoding-time condition is derived, which reveals that the dynamics of precision can asymptotically reach the ideal-case-promised one easily when the average photon number is small. Meanwhile, the scaling exhibits a transition from the weak Heisenberg limit to the sub-classical limit with the increase of average photon number. Our work provides a recipe to realize ultrasensitive measurements in the presence of noise by utilizing non-Markovian effects.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S \to \bar{E_u} has a dense image.   The "unramified outside S" number fields we use are cut out from the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main global theorem of Harris-Taylor, and the construction of automorphic representations with prescribed local behaviours.   We explain how stronger results would follow from the knowledge of some expected properties of Siegel modular forms, and we discuss the case of the Galois group of a maximal algebraic extension of Q unramified outside a single prime p and infinity.
We investigate the growth of needles from a flat substrate. We focus on the situation when needles suddenly begin to grow from the seeds randomly distributed on the line. The width of needles is ignored and we additionally assume that (i) the growth rate is the same for all needles; (ii) the direction of the growth of each needle is randomly chosen from the same distribution; (iii) whenever the tip of a needle hits the body of another needle, the former needle freezes, while the latter continues to grow. We elucidate the large time behavior by employing an exact analysis and the Boltzmann equation approach. We also analyze the evolution when seeds are located on a half-line, on a finite interval. Needles growing from the two-dimensional substrate are also examined.
We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\'{e}vy measure and the third method uses the analytic continuation of the L\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.
The dispersion of the A-valence-band in GaN has been deduced from the observation of high-index magneto-excitonic states in polarised interband magneto-reflectivity and is found to be strongly non-parabolic with a mass in the range 1.2-1.8 m_{e}. It matches the theory of Kim et al. [Phys. Rev. B 56, 7363 (1997)] extremely well, which also gives a strong k-dependent A-valence-band mass. A strong phonon coupling leads to quenching of the observed transitions at an LO-phonon energy above the band gap and a strong non-parabolicity. The valence band was deduced from subtracting from the reduced dispersion the electron contribution with a model that includes a full treatment of the electron-phonon interaction.
We model quantum Hall skyrmions in graphene monolayer at quarter filling by a theory of CP3 fields and study the energy minimizing skyrmions in presence of valley pseudospin anisotropy and Zeeman coupling. We present a diagram of all types of skyrmions in a wide range of the anisotropy parameters. For each type of skyrmion, we visualize it on three Bloch spheres, and present the profiles of its texture on the graphene honeycomb lattice, thus providing references for the STM/STS imaging of spin-pseudospin textures in graphene monolayer in quantum Hall regime. Besides the spin and pseudospin skyrmions for the corresponding degrees of freedom of an electron in the N=0 Landau level, we discuss two unusual types -- the "entanglement skyrmion" whose texture lies in the space of the entanglement of spin and pseudospin, as well as the "deflated pseudospin skyrmion" with partial entanglement. For all skyrmion types, we study the dependence of the energy and the size of a skyrmion on the anisotropy parameters and perpendicular magnetic field. We also propose three ways to modify the anisotropy energy, namely the sample tilting, the substrate anisotropy and the valley pseudospin analogue of Zeeman coupling.
Magnetic skyrmions are stable nanosized spin structures that can be displaced at low electrical current densities. Because of these properties, they have been proposed as building blocks of future electronic devices with unprecedentedly high information density and low energy consumption. The electrical detection of skyrmions via the Topological Hall Effect (THE), has so far been demonstrated only at cryogenic temperatures. Here, we report the observation of a skyrmion Topological Hall Effect near room temperature (276 K) in a mesoscopic lamella of FeGe. This region unambiguously coincides with the skyrmion lattice location revealed by neutron scattering. We provide clear evidence of a reentrant helicoid magnetic phase adjacent to the skyrmion phase, and discuss the large THE amplitude (5 n$\Omega$.cm) in view of the ordinary Hall Effect.
Different from salient object detection methods for still images, a key challenging for video saliency detection is how to extract and combine spatial and temporal features. In this paper, we present a novel and effective approach for salient object detection for video sequences based on 3D convolutional neural networks. First, we design a 3D convolutional network (Conv3DNet) with the input as three video frame to learn the spatiotemporal features for video sequences. Then, we design a 3D deconvolutional network (Deconv3DNet) to combine the spatiotemporal features to predict the final saliency map for video sequences. Experimental results show that the proposed saliency detection model performs better in video saliency prediction compared with the state-of-the-art video saliency detection methods.
We study some divisibility properties of quasiperfect numbers. We show that if $N=(p_1 p_2 \cdots p_t)^{2a}=m^2$ is quasiperfect, then $2a+1$ is divisible by $3$ and $N$ has at least one prime factor smaller than $\exp 716.7944$. Moreover, we find some lower bounds concerning quasiperfect numbers of the form $N=m^2$ with $m$ squarefree.
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we study two notions of invariance in parking sequences and present various characterizations and enumerative results.
We present a catalogue of velocities for 551 galaxies (and give the coordinates of 39 stars misclassified as galaxies in our photometric plate catalogue) in a region covering about 100'$\times$100' (0.94$\times$0.94 Mpc for an average redshift of 0.0555, assuming H$_\circ$=50 km s$^{-1}$ Mpc$^{-1}$) in the direction of the rich cluster Abell 85. This catalogue includes previously published redshifts by Beers et al. (1991) and Malumuth et al. (1992), together with our 367 new measurements. A total of 305 galaxies have velocities in the interval 13350-20000 km s$^{-1}$, and will be considered as members of the cluster. Abell 85 therefore becomes one of the clusters with the highest number of measured redshifts; its optical properties are being investigated in a companion paper.
The off-equilibrium probability distribution of the heat exchanged by a ferromagnet in a time interval after a quench below the critical point is calculated analytically in the large-N limit. The distribution is characterized by a singular threshold Qc < 0, below which a macroscopic fraction of heat is released by the k = 0 Fourier component of the order parameter. The mathematical structure producing this phenomenon is the same responsible of the order parameter condensation in the equilibrium low temperature phase. The heat exchanged by the individual Fourier modes follows a non trivial pattern, with the unstable modes at small wave vectors warming up the modes around a characteristic finite wave vector kM. Two internal temperatures, associated to the k = 0 and k = kM modes, rule the heat currents through a fluctuation relation similar to the one for stationary systems in contact with two thermal reservoirs.
For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as $\pi_{\lambda}(H)=\sup \{r! \lambda(G) : G \;\text{is an}\; H\text{-free} \;r\text{-uniform hypergraph}\}$, where $\lambda(G)$ is the Lagrangian of $G$. For an $r$-uniform hypergraph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. Let $P_t=\{e_1, e_2, \dots, e_t\}$ be the linear $3$-uniform path of length $t$, that is, $|e_i|=3$, $|e_i \cap e_{i+1}|=1$ and $e_i \cap e_j=\emptyset$ if $|i-j|\ge 2$. We show that $P_3$ and $P_4$ are perfect, this supports a conjecture in \cite{yanpeng} proposing that all $3$-uniform linear hypergraphs are perfect. Applying the results on Lagrangian densities, we determine the Tur\'an numbers of their extensions.