ShantanuT01/DACTYL-Pretraining huggingface dataset:
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of the system gives only classical correlations. Quantification of entanglement for these systems is given by the use of HS decompositions.
The reconstruction of a smooth field onto a fixed grid is a necessary step for direct comparisons to various real-world observations. Projecting SPH data onto a fixed grid becomes challenging in adaptive environments, where some particles may have smoothing lengths far below the grid size, whilst others are resolved by thousands of pixels. In this paper we show how the common approach of treating particles below the grid size as Monte Carlo tracers of the field leads to significant reconstruction errors, and despite good convergence properties is unacceptable for use in synthetic observations in astrophysics. We propose a new method, where particles smaller than the grid size are `blitted' onto the grid using a high-resolution pre-calculated kernel, and those close to the grid size are subsampled, that allows for converged predictions for projected quantities at all grid sizes.
In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify the theory are described.
We use invasion percolation to compute highly-accurate numerical values for bond and site percolation thresholds p_c on the hypercubic lattice Z^d for d = 4,,,,,13. We also compute the Fisher exponent tau governing the cluster size distribution at criticality. Our results support the claim that the mean-field value tau = 5/2 holds for d >= 6, with logarithmic corrections to power-law scaling at d=6.
We derive the approximate solution for the pairing Hamiltonian in the Berggren ensemble of single particle states including bound, resonance and non-resonant scattering states. We show that this solution is reliable in the limit of a weak pairing interaction.
Experimental and theoretical results for transport through crossed metallic single-wall nanotubes are presented. We observe a zero-bias anomaly in one tube which is suppressed by a current flowing through the other nanotube. The phenomenon is shown to be consistent with the picture of strongly correlated electrons within the Luttinger liquid model. The most relevant coupling between the nanotubes is the electrostatic interaction generated via crossing-induced backscattering processes. Explicit solution of a simplified model is able to describe qualitatively the observed experimental data with only one adjustable parameter.
Aims: We investigate the structure of the circumstellar disk of the T Tauri star S CrA N and test whether the observations agree with the standard picture proposed for Herbig Ae stars.   Methods: Our observations were carried out with the VLTI/AMBER instrument in the H and K bands with the low spectral resolution mode. For the interpretation of our near-infrared AMBER and archival mid-infrared MIDI visibilities, we employed both geometric and temperature-gradient models.   Results: To characterize the disk size, we first fitted geometric models consisting of a stellar point source, a ring-shaped disk, and a halo structure to the visibilities. In the H and K bands, we measured ring-fit radii of 0.73 +- 0.03 mas (corresponding to 0.095 +- 0.018 AU for a distance of 130 pc) and 0.85 +- 0.07 mas (0.111 +- 0.026 AU), respectively. This K-band radius is approximately two times larger than the dust sublimation radius of ~0.05 AU expected for a dust sublimation temperature of 1500 K and gray dust opacities, but approximately agrees with the prediction of models including backwarming (namely a radius of ~0.12 AU). The derived temperature-gradient models suggest that the disk is approximately face-on consisting of two disk components with a gap between star and disk. The inner disk component has a temperature close to the dust sublimation temperature and a quite narrow intensity distribution with a radial extension from 0.11 AU to 0.14 AU.   Conclusions: Both our geometric and temperature-gradient models suggest that the T Tauri star S CrA N is surrounded by a circumstellar disk that is truncated at an inner radius of ~0.11 AU. The narrow extension of the inner temperature-gradient disk component implies that there is a hot inner rim.
We consider the possibility that most of the reflection component, observed in the hard X-ray spectra of Seyfert galaxies, could be formed on an extended medium, at large distance from the central source of primary radiation (e.g. on a torus). Then, the reflector cannot respond to the rapid fluctuations of the primary source. The observed reflected flux is controlled by the time-averaged primary spectrum rather than the instantaneous (observed) one. We show that this effect strongly influences the spectral fit parameters derived under the assumption of a reflection component consistent with the primary radiation. We find that a pivoting primary power-law spectrum with a nearly constant Comptonised luminosity may account for the reported correlation between the reflection amplitude R and the spectral index Gamma, and simultaneously produces an iron line EW that is nearly independent of Gamma. We emphasize the effects of the modelling of the primary component on the determination of the reflection amplitude, and show that in NGC 5548, when these effects are taken into account, the RXTE data are consistent with the reflection features being produced mainly at large distance from the central source.
Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \rightarrow \mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \geq 7$, we work out the splitting type of the locally free sheaf $\Psi^{*}_f T_{{\overline{M}}_g}$, where $\Psi_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\mathcal{P}_g \rightarrow \overline{\mathcal{M}}_g$, where $\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \in |L|$ a stable curve. Moreover, we work out conditions on a fibration $f$ to induce a modular morphism $\Psi_f$ such that the normal sheaf $N_{\Psi_f}$ is locally free.
The generalization of the ARMA time series model to the multidimensional index set $\mathbb{Z}^d$, $d\ge2$, is called spatial ARMA model. The purpose of the following is to specify necessary conditions and sufficient conditions for the existence of strictly stationary solutions of the ARMA equations when the driving noise is i.i.d. Two different classes of strictly stationary solutions are studied, solutions of causal and non-causal models. For the special case of a first order model on $\mathbb{Z}^2$ conditions are obtained, which are simultaneously necessary and sufficient.
We describe a general algorithm for the computation of the remainder function for n-gluon scattering in multi-Regge kinematics for strongly coupled planar N=4 super Yang-Mills theory. This regime is accessible through the infrared physics of an auxiliary quantum integrable system describing strings in AdS5xS5. Explicit formulas are presented for n=6 and n=7 external gluons. Our results are consistent with expectations from perturbative gauge theory. This paper comprises the technical details for the results announced in arXiv:1405.3658 .
The formation of the Milky Way stellar halo is thought to be the result of merging and accretion of building blocks such as dwarf galaxies and massive globular clusters. Recently, Deason et al. (2015) suggested that the Milky Way outer halo formed mostly from big building blocks, such as dwarf spheroidal galaxies, based on the similar number ratio of blue straggler (BS) stars to blue horizontal-branch (BHB) stars. Here we demonstrate, however, that this result is seriously biased by not taking into detailed consideration on the formation mechanism of BHB stars from helium enhanced second-generation population. In particular, the high BS-to-BHB ratio observed in the outer halo fields is most likely due to a small number of BHB stars provided by GCs rather than to a large number of BS stars. This is supported by our dynamical evolution model of GCs which shows preferential removal of first generation stars in GCs. Moreover, there are a sufficient number of outer halo GCs which show very high BS-to-BHB ratio. Therefore, the BS-to-BHB number ratio is not a good indicator to use in arguing that more massive dwarf galaxies are the main building blocks of the Milky Way outer halo. Several lines of evidence still suggest that GCs can contribute a significant fraction of the outer halo stars.
We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{\'e} approximants of the mean cluster number $S(p)$ for site and bond percolation on general $d$-dimensional lattices are upper bounds on this quantity in any Euclidean dimension $d$, where $p$ is the occupation probability. These results lead to certain lower bounds on the percolation threshold $p_c$ that become progressively tighter as $d$ increases and asymptotically exact as $d$ becomes large. These lower-bound estimates depend on the structure of the $d$-dimensional lattice and whether site or bond percolation is being considered. We obtain explicit bounds on $p_c$ for both site and bond percolation on five different lattices: $d$-dimensional generalizations of the simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as well as the $d$-dimensional generalizations of the diamond and kagom{\'e} (or pyrochlore) non-Bravais lattices. These analytical estimates are used to assess available simulation results across dimensions (up through $d=13$ in some cases). It is noteworthy that the tightest lower bound provides reasonable estimates of $p_c$ in relatively low dimensions and becomes increasingly accurate as $d$ grows. We also derive high-dimensional asymptotic expansions for $p_c$ for the ten percolation problems and compare them to the Bethe-lattice approximation. Finally, we remark on the radius of convergence of the series expansion of $S$ in powers of $p$ as the dimension grows.
The clique removal lemma says that for every $r \geq 3$ and $\varepsilon>0$, there exists some $\delta>0$ so that every $n$-vertex graph $G$ with fewer than $\delta n^r$ copies of $K_r$ can be made $K_r$-free by removing at most $\varepsilon n^2$ edges. The dependence of $\delta$ on $\varepsilon$ in this result is notoriously difficult to determine: it is known that $\delta^{-1}$ must be at least super-polynomial in $\varepsilon^{-1}$, and that it is at most of tower type in $\log \varepsilon^{-1}$.   We prove that if one imposes an appropriate minimum degree condition on $G$, then one can actually take $\delta$ to be a linear function of $\varepsilon$ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super-polynomial, as in the unrestricted removal lemma.   We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.
This note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as propagation of chaos, coupling, functional inequalities, provide a good picture of the classical methods that furnish quantitative rates of convergence to equilibrium.
We consider the energy modeling a two component Bose-Einstein condensate in the limit of strong coupling and strong segregation. We prove the $\Gamma$-convergence to a perimeter minimization problem, with a weight given by the density of the condensate. In the case of equal mass for the two components, this leads to symmetry breaking for the ground state. The proof relies on a new formulation of the problem in terms of the total density and spin functions, which turns the energy into the sum of two weighted Cahn-Hilliard energies. Then, we use techniques coming from geometric measure theory to construct upper and lower bounds. In particular, we make use of the slicing technique introduced in Ambrosio-Tororelli (CPAM, 1990).
We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time equivalent to solving a system of polynomial equations over this integral domain. Our result gives a complete description of the algorithmic complexity of tensor rank and allows one to solve several known open problems. In particular, the tensor rank over $\mathbb{Z}$ turns out to be undecidable, which answers the question posed by Gonzalez and Ja'Ja' in 1980. We generalize our result and prove that the symmetric rank admits a similar description of computational complexity as the one we give for usual rank. In particular, computing the symmetric rank of a rational tensor is shown to be NP-hard, which proves a recent conjecture of Hillar and Lim. As a byproduct of our approach, we get a similar characterization of the algorithmic complexity of the minimal rank matrix completion problem, which gives a complete answer to the question discussed in 1999 by Buss, Frandsen, and Shallit.
In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.
We derive a general integral formula on an embedded hypersurface for general relativistic space-times. Suppose the hypersurface is foliated by two-dimensional compact ``sections'' $S_s$. Then the formula relates the rate of change of the divergence of outgoing light rays integrated over $S_s$ under change of section to geometric (convexity and curvature) properties of $S_s$ and the energy-momentum content of the space-time. We derive this formula using the Sparling-Nester-Witten identity for spinor fields on the hypersurface by appropriate choice of the spinor fields. We discuss several special cases which have been discussed in the literature before, most notably the Bondi mass loss formula.
Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies and the Engset process, one of the early (1918) stochastic models of communication networks. This paper investigates the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. The paper relies on martingale methods, a key ingredient is an important family of simple non-negative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.
In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.
A definition of metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance is discussed. In particular, we identify a crucial condition that distinguishes genuine metastable states from other types of slowly decaying modes and which leads to properties similar to those postulated in the restricted ensemble approach \cite{pen71}. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible. As a concrete application of our formalism we present preliminary results on a 2D kinetic Ising model.
We propose some class of statistics suitable for estimation of the Hurst index of the fractional Brownian motion based on the second order increments of an observed discrete trajectory.
We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral $I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds)$, where $N$ is a Poisson random measure with control measure $n$ and $\krnl$ is a suitable kernel function. We do so by combining a Kolmogorov-Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.
We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves.   One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition, with no reference to homology theory, that resembles graph expansion. Yet it is a "limit" of Betti numbers, and hence has a short/long exact sequence theory and resembles the $L^2$ Betti numbers of Atiyah. Also, the maximum excess is defined via a supermodular function, which gives the maximum excess much stronger properties than one has of a typical Betti number. The maximum excess gives a simple interpretation of an important graph invariant, which will be used to study the Hanna Neumann Conjecture in a future paper.   Our sheaf theory can be viewed as a vast generalization of algebraic graph theory: each sheaf has invariants associated to it---such as Betti numbers and Laplacian matrices---that generalize those in classical graph theory.
The bilayer ruthenate Sr_3Ru_2O_7 has been cited as a textbook example of itinerant metamagnetic quantum criticality. However, recent studies of the ultra-pure system have revealed striking anomalies in magnetism and transport in the vicinity of the quantum critical point. Drawing on new experimental data, we propose that the anomalous phase behavior can be accommodated by a mechanism which relies on the effects of strong magnetoelastic coupling. We argue that the phenomenological theory identified here is more generic and raises the possibility of quantum tricritical behaviour.
Let S be a complex Enriques surface; it is the quotient of a K3 surface X by a fixed-point-free involution. The Brauer group Br(S) has a unique nonzero element. We describe its pull-back in Br(X), and show that the surfaces S for which it is trivial form a countable union of hypersurfaces in the moduli space of Enriques surfaces.
Magnetic fields and mass accretion processes create dark and bright spots on the surface of young stars. These spots manifest as surface thermal inhomogeneities, which alter the global temperature measured on the stars. To understand the effects and implications of these starspots, we conducted a large iSHELL high-resolution infrared spectroscopic survey of T Tauri stars in Taurus-Auriga and Ophiuchus star-forming regions. From the K band spectra, we measured stellar temperatures and magnetic field strengths using a magnetic radiative transfer code. We compared our infrared-derived parameters against literature optical temperatures and found a) a systematic temperature difference between optical and infrared observations, and b) a positive correlation between the magnetic field strengths and the temperature differences. The discrepant temperature measurements imply significant differences in the inferred stellar masses from stellar evolutionary models. To discern which temperature better predicts the mass of the star, we compared our model-derived masses against dynamical masses measured from ALMA and PdBI for a sub-sample of our sources. From this comparison we conclude that, in the range of stellar masses from 0.3 to 1.3 $\rm M_\odot$, neither infrared nor optical temperatures perfectly reproduce the stellar dynamical masses. But, on average, infrared temperatures produce more precise and accurate stellar masses than optical ones.
Knowledge distillation compacts deep networks by letting a small student network learn from a large teacher network. The accuracy of knowledge distillation recently benefited from adding residual layers. We propose to reduce the size of the student network even further by recasting multiple residual layers in the teacher network into a single recurrent student layer. We propose three variants of adding recurrent connections into the student network, and show experimentally on CIFAR-10, Scenes and MiniPlaces, that we can reduce the number of parameters at little loss in accuracy.
We use multi-scale modeling to analyze laser-induced fluorescence (LIF) measurements of the CO oxidation reaction over Pd(100) at near-ambient reaction conditions. Integrating density-functional theory based kinetic Monte Carlo simulations of the active catalyst into fluid-dynamical simulations of the mass transport inside the reactor chamber we calculate the reaction product concentration directly above the catalyst surface. Comparing corresponding data calculated for different surface models against the measured LIF signals we can discriminate the one that predominantly actuates the experimentally measured catalytic activity. For the probed CO oxidation reaction conditions the experimental activity is due to pristine Pd(100), possibly coexisting with other (oxidic) domains on the surface.