ShantanuT01/DACTYL-Pretraining huggingface dataset:
The Carmichael lambda function $\lambda(n)$ is defined to be the smallest positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$ $\lambda_k(n)$ is defined to be the $k$th iterate of $\lambda(n).$ Let L(n) be the smallest $k$ for which $\lambda_k(n)=1.$ It's easy to show that $L(n) \ll \log n.$ It's conjectured that $L(n)\asymp \log\log n,$ but previously it was not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n) \ll (\log n)^{\delta}$ for almost all $n,$ for some $\delta <1.$ We will also show $L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for L(n).
The quantum Ising chain with the interaction decaying as a power law $1/r^{1+\sigma}$ of the distance between spins $r$ was investigated numerically. A particular attention was paid to the low-energy spectrum, namely, the single-magnon and two-magnon-bound-state masses, $m_{1,2}$, respectively, in the ordered phase. It is anticipated that for each $\sigma$, the scaled bound-state mass $m_2/m_1$ should take a universal constant (critical amplitude ratio) in the vicinity of the critical point. In this paper, we calculated the amplitude ratio $m_2/m_1$ with the exact diagonalization method, which yields the spectral information such as $m_{1,2}$ directly. As a result, we found that the scaled mass $m_2/m_1$ exhibits a non-monotonic dependence on $\sigma$; that is, the bound state is stabilized by an intermediate value of $\sigma$. Such a feature is accordant with a recent observation based on the non-perturbative-renormalization-group method.
We investigate the transverse polarization of the $\Lambda$ hyperon in the processes $e^+e^-\to\Lambda^\uparrow \pi^\pm X$ and $e^+e^-\to\Lambda^\uparrow K^\pm X$ within the framework of the transverse momentum dependent~(TMD) factorization. The transverse polarization is contributed by the convolution of the transversely polarizing fragmentation function~(PFF) $D_{1T}^\perp$ of the lambda hyperon and the unpolarized fragmentation function $D_1$ of pion/kaon. We adopt the spectator diquark model result for $D_{1T}^{\perp}$ to numerically estimate the transverse polarization in $e^+e^-\to\Lambda^\uparrow h X$ process at the kinematical region of Belle Collaboration. To implement the TMD evolution formalism of the fragmentation functions, we apply two different parametrizations on the nonperturbative Sudakov form factors associated with the fragmentation functions of the $\Lambda$, pion and kaon. It is found that our prediction on the polarization in the $\Lambda \pi^+$ production and $\bar{\Lambda} \pi^-$ is consistent with the recent Belle measurement in size and sign, while the model predictions on the polarizations in $\Lambda\pi^-$ and $\Lambda K^\pm$ productions show strong disagreement with the Belle data. The reason for the discrepancies is discussed and possible approaches to improve the calculation in the future are also discussed.
Ultra-short orbital period contact binaries (Porb < 0.26 d) host some of the smallest and least massive stars. These systems are faint and rare, and it is believed that they have reached a contact configuration after several Gyrs of evolution via angular momentum loss, mass transfer and mass loss through stellar wind processes. This study is conducted in the frame of Contact Binaries Towards Merging (CoBiToM) Project and presents the results from light curve and orbital analysis of 30 ultra-short orbital period contact binaries, with the aim to investigate the possibility of them being red nova progenitors, eventually producing merger events. Approximately half of the systems exhibit orbital period modulations, as a result of mass transfer or mass loss processes. Although they are in contact, their fill-out factor is low (less than 30 per cent), while their mass ratio is larger than the one in longer period contact binaries. The present study investigates the orbital stability of these systems and examines their physical and orbital parameters in comparison to those of the entire sample of known and well-studied contact binaries, based on combined spectroscopic and photometric analysis. It is found that ultra-short orbital period contact binaries have very stable orbits, while very often additional components are gravitationally bound in wide orbits around the central binary system. We confirmed that the evolution of such systems is very slow, which explains why the components of ultra-short orbital period systems are still Main Sequence stars after several Gyrs of evolution.
In order to confirm the existence of the dibaryon state $d^*(2380)$ observed at WASA@COSY, we estimate the production of the possible dibaryon and anti-dibaryon pair $d^*\bar{d}^*$ at the energy region of the upcoming experiments at $\bar{\rm{P}}$anda. Based on some qualitative properties of $d^*$ extracted from the analysez in the non-relativistic quark model, the production cross section for this spin-3 particle pair are calculated with the help of an phenomenological effective relativistic and covariant Lagrangian approach.
In this paper we give a new proof to the energy conservation for the weak solutions of the incompressible Navier-Stokes equations. This result was first proved by Shinbrot. The new proof relies on a lemma introduced by Lions.
We introduce a model of information packet transport on networks in which the packets are posted by a given rate and move in parallel according to a local search algorithm. By performing a number of simulations we investigate the major kinetic properties of the transport as a function of the network geometry, the packet input rate and the buffer size. We find long-range correlations in the power spectra of arriving packet density and the network's activity bursts. The packet transit time distribution shows a power-law dependence with average transit time increasing with network size. This implies dynamic queueing on the network, in which many interacting queues are mutually driven by temporally correlated packet streams.
In this paper we show that a system of three fermions is exactly solvable for the case of a single-$j$ in the presence of an angular momentum-$J$ pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-$j$ symbols. It is also found that the "non-integer" eigenvalues of three fermions with angular momentum $I$ around the maximum appear as "non-integer" eigenvalues of four fermions when $I$ is around (or larger than) $J_{\rm max}$ and the Hamiltonian contains only an interaction between pairs of fermions coupled to spin $J=J_{\rm max}=2j-1$. This pattern is also found in five and six fermion systems. A boson system with spin $l$ exhibits a similar pattern.
The Hesse potential is constructed for a class of four-dimensional N=2 supersymmetric effective actions with S- and T-duality by performing the relevant Legendre transform by iteration. It is a function of fields that transform under duality according to an arithmetic subgroup of the classical dualities reflecting the monodromies of the underlying string compactification. These transformations are not subject to corrections, unlike the transformations of the fields that appear in the effective action which are affected by the presence of higher-derivative couplings. The class of actions that are considered includes those of the FHSV and the STU model. We also consider heterotic N=4 supersymmetric compactifications. The Hesse potential, which is equal to the free energy function for BPS black holes, is manifestly duality invariant. Generically it can be expanded in terms of powers of the modulus that represents the inverse topological string coupling constant, $g_s$, and its complex conjugate. The terms depending holomorphically on $g_s$ are expected to correspond to the topological string partition function and this expectation is explicitly verified in two cases. Terms proportional to mixed powers of $g_s$ and $\bar g_s$ are in principle present.
New data for the two-dimensional J1-J2 model shows that the critical points for singlet and triplet excitations near J2/J1=0.38 are very close together, or coincident. We propose that this is a more general result: in strongly frustrated spin systems the critical dynamics of singlet and triplet excitations are closely related. An effective field theory for such a critical point, or points, is developed, and its implications discussed. The analysis suggests the existance of gapless or almost gapless singlet excitations in the magnetically disordered phase.
A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.
We consider the 2D $J_1-J_2$ classical XY model on a square lattice. In the frustrated phase corresponding to $J_2>J_1/2$, an Ising like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete $Z_2$ symmetry plus the U(1) global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the non-universal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, XY models.
Automatic eye gaze estimation is an important problem in vision based assistive technology with use cases in different emerging topics such as augmented reality, virtual reality and human-computer interaction. Over the past few years, there has been an increasing interest in unsupervised and self-supervised learning paradigms as it overcomes the requirement of large scale annotated data. In this paper, we propose RAZE, a Region guided self-supervised gAZE representation learning framework which leverage from non-annotated facial image data. RAZE learns gaze representation via auxiliary supervision i.e. pseudo-gaze zone classification where the objective is to classify visual field into different gaze zones (i.e. left, right and center) by leveraging the relative position of pupil-centers. Thus, we automatically annotate pseudo gaze zone labels of 154K web-crawled images and learn feature representations via `Ize-Net' framework. `Ize-Net' is a capsule layer based CNN architecture which can efficiently capture rich eye representation. The discriminative behaviour of the feature representation is evaluated on four benchmark datasets: CAVE, TabletGaze, MPII and RT-GENE. Additionally, we evaluate the generalizability of the proposed network on two other downstream task (i.e. driver gaze estimation and visual attention estimation) which demonstrate the effectiveness of the learnt eye gaze representation.
The theory for manifestation of stochastic appearance of embryos in the global decay of metastable phase has been constructed. The regime of droplets growth is supposed to be both free molecular one and diffusion one. The deviation for a mean droplets number from the value predicted by the theory based on averaged characteristics have been calculated. The value of dispersion for the distribution of the total droplets number in particular attempt has been also calculated analytically. Comparison with results of numerical simulation has been given and the correspondence between simulation and analytical approximate results is rather good.
We realize Surface Code quantum memories for nearest-neighbor qubits with always-on Ising interactions. This is done by utilizing multi-qubit gates that mimic the functionality of several gates. The previously proposed Surface Code memories rely on error syndrome detection circuits based on CNOT gates. In a two-dimensional planar architecture, to realize a two-qubit CNOT gate in the presence of couplings to other neighboring qubits, the interaction of the target qubit with its three other neighbors must cancel out. Here we present a new error syndrome detection circuit utilizing multi-qubit parity gates. In addition to speed up in the error correction cycles, in our approach, the depth of the error syndrome detection circuit does not grow by increasing the number of qubits in the logical qubit layout. We analytically design the system parameters to realize new five-qubit gates suitable for error syndrome detection in nearest-neighbor two-dimensional array of qubits. The five-qubit gates are designed such that the middle qubit is the target qubit and all four coupled neighbors are the control qubits. In our scheme, only one control parameter of the target qubits must be adjusted to realize controlled-unitary operations. The gate operations are confirmed with a fidelity of >99.9% in a simulated system consists of nine nearest-neighbor qubits.
Spectrally-resolved observations of three pure rotational lines of H$_2$, conducted with the EXES instrument on SOFIA toward the classic bow shock HH7, reveal systematic velocity shifts between the S(5) line of ortho-H$_2$ and the two para-H$_2$ lines [S(4) and S(6)] lying immediately above and below it on the rotational ladder. These shifts, reported here for the first time, imply that we are witnessing the conversion of para-H$_2$ to ortho-H$_2$ within a shock wave driven by an outflow from a young stellar object. The observations are in good agreement with the predictions of models for non-dissociative, C-type molecular shocks. They provide a clear demonstration of the chemical changes wrought by interstellar shock waves, in this case the conversion of para-H$_2$ to ortho-H$_2$ in reactive collisions with atomic hydrogen, and provide among the most compelling evidence yet obtained for C-type shocks in which the flow velocity changes continuously.
Symmetry considerations are of extreme importance to the topological properties of crystals. A crystal field splitting {\delta} yields Dirac nodes near the Brillouin zone center in Cd3As2, but its value has yet to be determined with precision. We study the band structure of Cd3As2 using magnetooptical infrared spectroscopy measurements on epilayers with low carrier density grown by molecular beam epitaxy. By combining angular dependent cyclotron resonance with Landau level spectroscopy measurements in the Faraday geometry, we determine that {\delta} is positive and equal to 15+/-5 meV in Cd3As2. Our results lead to a more accurate knowledge of the details of the band structure of this Dirac semimetal such as the position its Dirac nodes in momentum space and their splitting into Weyl nodes under a magnetic field.
We study the motion of independent particles in a dynamical random environment on the integer lattice. The environment has a product distribution. For the multidimensional case, we characterize the class of spatially ergodic invariant measures. These invariant distributions are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment. We also investigate the correlations in this measure. For dimensions one and two, we prove convergence to equilibrium from spatially ergodic initial distributions. In the one-dimensional situation we study fluctuations of the net current seen by an observer traveling at a deterministic speed. When this current is centered by its quenched mean its limit distributions are the same as for classical independent particles.
We investigate the high temperature phase of layered manganites, and demonstrate that the charge-orbital phase transition without magnetic order in La$_{0.5}$Sr$_{1.5}$MnO$_4$ can be understood in terms of the density wave instability. The orbital ordering is found to be induced by the nesting between segments of Fermi surface with different orbital characters. The simultaneous charge and orbital orderings are elaborated with a mean field theory. The ordered orbitals are shown to be $d_{x^2-y^2} \pm d_{3z^2-r^2}$.
We consider the propagation of wave packets for a nonlinear Schr\"odinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.
We present the magnetic structure of $\mathrm{TmMn_3O_6}$, solved via neutron powder diffraction - the first such study of any $R\mathrm{Mn_3O_6}$ A-site columnar-ordered quadruple perovskite to be reported. We demonstrate that long range magnetic order develops below 74 K, and at 28 K a spin-flop transition occurs driven by $f$-$d$ exchange and rare earth single ion anisotropy. In both magnetic phases the magnetic structure may be described as a collinear ferrimagnet, contrary to conventional theories of magnetic order in the manganite perovskites. Instead, we show that these magnetic structures can be understood to arise due to ferro-orbital order, the A, A$'$ and A$''$ site point symmetry, $mm2$, and the dominance of A-B exchange over both A-A and B-B exchange, which together are unique to the $R\mathrm{Mn_3O_6}$ perovskites.
For any graph $G$ with $a,b\in V(G)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V(S(G,a,b))$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper [Asplund et al., \textit{Reconfiguration graphs of shortest paths}, Discrete Mathematics \textbf{341} (2018), no. 10, 2938--2948], it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced $4$-cycles.
We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear index-reduction formula of these numbers. Further, Oresme polynomials that are natural extensions of the $k$-Oresme numbers are introduced and some relations for the derivatives of these polynomials in the form of convolution are proved.
We introduce a new relativistic orthogonal states quantum key distribution protocol which leverages the properties of both quantum mechanics and special relativity to securely encode multiple bits onto the spatio-temporal modes of a single photon. If the protocol is implemented using a single photon source, it can have a key generation rate faster than the repetition rate of the source, enabling faster secure communication than is possible with existing protocols. Further, we provide a proof that the protocol is secure and give a method of implementing the protocol using line-of-sight and fiber optic channels.
In this theoretical study, we investigate the origins of the very low thermal conductivity of tin selenide (SnSe) using ab-initio calculations. We obtained high-temperature lattice thermal conductivity values that are close to those of amorphous compounds. We also found a strong anisotropy between the three crystallographic axes: one of the in-plane directions conducts heat much more easily than the other. Our results are compatible with most of the experimental literature on SnSe, and differ markedly from the more isotropic values reported by a recent study.
BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.
Supernova explosions are inherently asymmetric and can accelerate new-born neutron stars (NSs) to hundreds of km/s. Two prevailing theories to explain NS kicks are ejecta asymmetries (e.g., conservation of momentum between NS and ejecta) and anisotropic neutrino emission. Observations of supernova remnants (SNRs) can give us insights into the mechanism that generates these NS kicks. In this paper, we investigate the relationship between NS kick velocities and the X-ray morphologies of 18 SNRs observed with the Chandra X-ray Observatory and the Rontgen Satellite (ROSAT). We measure SNR asymmetries using the power-ratio method (a multipole expansion technique), focusing on the dipole, quadrupole, and octupole power-ratios. Our results show no correlation between the magnitude of the power-ratios and NS kick velocities, but we find that for Cas A and G292.0+1.8, whose emission traces the ejecta distribution, their NSs are preferentially moving opposite to the bulk of the X-ray emission. In addition, we find a similar result for PKS 1209-51, CTB 109, and Puppis A; however their emission is dominated by circumstellar/interstellar material, so their asymmetries may not reflect their ejecta distributions. Our results are consistent with the theory that NS kicks are a consequence of ejecta asymmetries as opposed to anisotropic neutrino emission. In the future, additional observations to measure NS proper motions within ejecta-dominated SNRs are necessary to constrain robustly the NS kick mechanism.
We study nonclassical correlations in propagating two-mode squeezed microwave states in the presence of noise. We focus on two different types of correlations, namely, quantum entanglement and quantum discord. Quantum discord has various intriguing fundamental properties which require experimental verification, such as the asymptotic robustness to environmental noise. Here, we experimentally investigate quantum discord in propagating two-mode squeezed microwave states generated via superconducting Josephson parametric amplifiers. By exploiting an asymmetric noise injection into these entangled states, we demonstrate the robustness of quantum discord against thermal noise while verifying the sudden death of entanglement. Furthermore, we investigate the difference between quantum discord and entanglement of formation, which can be directly related to the flow of locally inaccessible information between the environment and the bipartite subsystem. We observe a crossover behavior between quantum discord and entanglement for low noise photon numbers, which is a result of the tripartite nature of noise injection. We demonstrate that the difference between entanglement and quantum discord can be related to the security of certain quantum key distribution protocols.