ShantanuT01/DACTYL-Pretraining huggingface dataset:
A conjecture due to Cilleruelo states that for an irreducible polynomial $f$ with integer coefficients of degree $d\geq 2$, the least common multiple $L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log L_f(N)\sim (d-1)N\log N$ as $N\to \infty$. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of $N$ depending on the range of shifts.
We present coordinated multiwavelength observations of the bright, nearby BL Lac object Mrk 421 taken in 2013 January-March, involving GASP-WEBT, Swift, NuSTAR, Fermi-LAT, MAGIC, VERITAS, and other collaborations and instruments, providing data from radio to very-high-energy (VHE) gamma-ray bands. NuSTAR yielded previously unattainable sensitivity in the 3-79 keV range, revealing that the spectrum softens when the source is dimmer until the X-ray spectral shape saturates into a steep power law with a photon index of approximately 3, with no evidence for an exponential cutoff or additional hard components up to about 80 keV. For the first time, we observed both the synchrotron and the inverse-Compton peaks of the spectral energy distribution (SED) simultaneously shifted to frequencies below the typical quiescent state by an order of magnitude. The fractional variability as a function of photon energy shows a double-bump structure which relates to the two bumps of the broadband SED. In each bump, the variability increases with energy which, in the framework of the synchrotron self-Compton model, implies that the electrons with higher energies are more variable. The measured multi-band variability, the significant X-ray-to-VHE correlation down to some of the lowest fluxes ever observed in both bands, the lack of correlation between optical/UV and X-ray flux, the low degree of polarization and its significant (random) variations, the short estimated electron cooling time, and the significantly longer variability timescale observed in the NuSTAR light curves point toward in-situ electron acceleration, and suggest that there are multiple compact regions contributing to the broadband emission of Mrk 421 during low-activity states.
We investigate a symmetric set of three quantum states in three dimensions having interesting properties, which we call the lifted trine states. We show that for the ensemble consisting of the three lifted trine states taken with equal probabilities, the POVM measurement realizing the accessible information must contain six projectors, giving a counterexample to a conjecture of Levitin.
We study the question of existence of time-periodic, spatially periodic solutions for dispersive evolution equations, and in particular, we introduce a framework for demonstrating the nonexistence of such solutions. We formulate the problem so that doubly periodic solutions correspond to fixed points of a certain operator. We prove that this operator is locally contracting, for almost every temporal period, if the Duhamel integral associated to the evolution exhibits a weak smoothing property. This implies the nonexistence of nontrivial, small-amplitude time-periodic solutions for almost every period if the smoothing property holds. This can be viewed as a partial analogue of scattering for dispersive equations on periodic intervals, since scattering in free space implies the nonexistence of small coherent structures. We use a normal form to demonstrate the smoothing property on specific examples, so that it can be seen that there are indeed equations for which the hypotheses of the general theorem hold. The nonexistence result is thus established through the novel combination of small divisor estimates and dispersive smoothing estimates. The examples treated include the Korteweg-de Vries equation and the Kawahara equation.
We study the magnetic excitation spectrum of the S=1 quantum Heisenberg spin chain with Hamiltonian : H = sum_i cos(theta) S_i S_i+1 + sin(theta) (S_i S_i+1)^2. We focus on the range -pi/4 < theta < +pi/4 where the spin chain is in the gapped Haldane phase. The excitation spectrum and static structure factor is studied using direct Lanczos diagonalization of small systems and density-matrix renormalization group techniques combined with the single-mode approximation. The magnon dispersion has a minimum at q=pi until a critical value theta_c = 0.38 is reached at which the curvature (velocity) vanishes. Beyond this point, which is distinct from the VBS point and the Lifshitz point, the minimum lies at an incommensurate value that goes smoothly to 2pi/3 when theta approaches pi/4, the Lai-Sutherland point. The mode remains isolated from the other states: there is no evidence of spinon deconfinement before the point theta =+pi/4. These findings explain recent observation of the magnetization curve M approx (H -H_c)^1/4 for theta =theta_c.
We revisit the protocols to create maximally entangled states between two Josephson junction (JJ) charge phase qubits coupled to a microwave field in a cavity as a quantum data bus. We devote to analyze a novel mechanism of quantum decoherence due to the adiabatic entanglement between qubits and the data bus, the off-resonance microwave field. We show that even through the variable of the data bus can be adiabatically eliminated, the entanglement between the qubits and data bus remains and can decoher the superposition of two-particle state. Fortunately we can construct a decoherence-free subspace of two-dimension to against this adiabatic decoherence.To carry out the analytic study for this decoherence problem, we develop Fr\H{o}hlich transformation to re-derive the effective Hamiltonian of these system, which is equivalent to that obtained from the adiabatic elimination approach .
This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation $c(x)\partial_{t}^{2}u - \Delta u = 0$ in a bounded smooth domain in $\R^{d}$ from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset $\Gamma$ of the boundary determines explicitly the coefficient $c$ provided that $c$ is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient $c$ from the knowledge of the difference between the local Dirichlet to Neumann maps.
The use of time-delay gravitational lenses to examine the cosmological expansion introduces a new standard ruler with which to test theoretical models. The sample suitable for this kind of work now includes 12 lens systems, which have thus far been used solely for optimizing the parameters of $\Lambda$CDM. In this paper, we broaden the base of support for this new, important cosmic probe by using these observations to carry out a one-on-one comparison between {\it competing} models. The currently available sample indicates a likelihood of $\sim 70-80%$ that the $R_{\rm h}=ct$ Universe is the correct cosmology versus $\sim 20-30%$ for the standard model. This possibly interesting result reinforces the need to greatly expand the sample of time-delay lenses, e.g., with the successful implementation of the Dark Energy Survey, the VST ATLAS survey, and the Large Synoptic Survey Telescope. In anticipation of a greatly expanded catalog of time-delay lenses identified with these surveys, we have produced synthetic samples to estimate how large they would have to be in order to rule out either model at a $\sim 99.7%$ confidence level. We find that if the real cosmology is $\Lambda$CDM, a sample of $\sim 150$ time-delay lenses would be sufficient to rule out $R_{\rm h}=ct$ at this level of accuracy, while $\sim 1,000$ time-delay lenses would be required to rule out $\Lambda$CDM if the real Universe is instead $R_{\rm h}=ct$. This difference in required sample size reflects the greater number of free parameters available to fit the data with $\Lambda$CDM.
We have explored the possibility of using low dimensional phosphorene based nanoscale device for the detection of nitroaromatic based explosives. In this work, we have investigated the structural, electronic, adsorption, and quantum transport properties of phosphorene nanoribbon APNR in the presence of different nitroaromatic compounds NACs using the state-of-the-art first principle density functional theory DFT calculations. Our results reveal that the adsorption of explosive molecules is far from the APNR surface, which neither affects the structure of the explosive molecule nor the APNR surface. However, it changes the electronic energy gap due to the charge transfer between the APNR and explosive molecule. Furthermore, we have examined the transmission function and the current voltage IV characteristic curves for the APNR+explosive systems with the APNR device as a reference employing the nonequilibrium Greens function NEGFs combined with DFT approach. The different IV characteristics compared to pristine APNR device of the system in the presence of explosive molecules indicate that such APNR based device can be very much sensitive and selective towards certain explosive molecules. Hence, our study indicates that APNR material may be an attractive nanodevice for the detection of explosives.
We study ground-state and thermodynamic properties of an $S=1$ Kitaev model. We first clarify the existence of global parity symmetry in addition to the local symmetry on each plaquette, which enables us to perform the large scale calculations up to 24 sites. It is found that the ground state should be singlet and its energy is estimated as $E/N\sim -0.65J$, where $J$ is the Kitaev exchange coupling. We find that a lowest excited state belongs to the same subspace as the ground state and the gap monotonically decreases with increasing system size, which suggests that the ground state of the $S=1$ Kitaev model is gapless. By means of the thermal pure quantum states, we clarify finite temperature properties characteristic of the Kitaev models with $S\le 2$.
Considering a quantized chaotic system, we analyze the evolution of its eigenstates as a result of varying a control parameter. As the induced perturbation becomes larger, there is a crossover from a perturbative to a non-perturbative regime, which is reflected in the structural changes of the local density of states. For the first time the {\em full} scenario is explored for a physical system: an Aharonov-Bohm cylindrical billiard. As we vary the magnetic flux, we discover an intermediate twilight regime where perturbative and semiclassical features co-exist. This is in contrast with the {\em simple} crossover from a Lorentzian to a semicircle line-shape which is found in random-matrix models.
Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erd\H{o}s-Burgess constant} of $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the sum of whose terms is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the polynomial ring over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a quotient ring of $\F_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erd\H{o}s-Burgess constant of the multiplicative semigroup of the ring $R$, in particular, we determined the Erd\H{o}s-Burgess constant in the case when $K$ is the power of a prime ideal or a product of pairwise distinct prime ideals in $\F_q[x]$.
Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--G\"ollnitz--Gordon continued fraction and a level-13 analog of the Roger-Ramanujan continued fraction. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients of the quadratic equations to important formulas obtained by Ramanujan, examine the pattern of the relation between a continued fraction and its parameters, and use the reciprocity formulas to obtain close approximations for all values of the continued fraction. We highlight patterns in the expressions for certain explicit values of the Rogers-Ramanujan continued fraction by expressing them in terms of the golden ratio. We extend the results to reciprocity formulas for Ramanujan's cubic continued fraction and the Ramanujan-Selberg continued fraction.
Magnetars and many of the magnetar-related objects are summarized together and discussed. It is shown that there is an abuse of language in the use of "magnetar". Anomalous X-ray pulsars and soft gamma-ray repeaters are well-known magnetar candidates. The current so called anti-magnetar (for central compact objects), accreting magnetar (for superslow X-ray pulsars in high mass X-ray binaries), and millisecond magnetar (for the central engine of some gamma-ray bursts), they may not be real magnetars in present understandings. Their observational behaviors are not caused by the magnetic energy. Many of them are just neutron stars with strong surface dipole field. A neutron star plus strong dipole field is not a magnetar. The characteristic parameters of the neutron stars for the central engine of some gamma-ray bursts are atypical from the neutron stars in the Galaxy. Possible signature of magnetic activities in accreting systems are discussed, including repeated bursts and a hard X-ray tail. China's future hard X-ray modulation telescope may contribute to finding some magnetic activities in accreting neutron star systems.
A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there are no non-trivially unstable triangle-free graphs of diameter 2. An interesting construction of non-trivially unstable graphs is given and several open problems are posed.
In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HC's. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characterizations of contractible HC's, from which we construct digraphs for enumerating the contractible HC's. Given the impression which the computational data for $m \leq 9$ convey, we conjecture that the asymptotic domination of the contractible HC's versus the non-contractible HC's, among the total number of HC's, depends on the parity of $m$.}
Particle creation at the expense of gravitational field might be sufficient to explain the cosmic evolution history, without the need of dark energy at all. This phenomena has been investigated in a recent work by Lima et-al, assuming particle creation at the cost of gravitational energy in the late Universe. However, the model does not satisfy the WMAP constraint on matter-radiation equality. Here, we have suggested a model, in the same framework, which fits perfectly with SNIa data at low redshift as well as early Integrated Sachs-Wolfe effect on matter-radiation equality determined by WMAP at high redshift. Such a model, requires the presence of nearly 26% primeval matter in the form of baryons and CDM.
For every positive integer $n$, an infinite family of positive integral solutions of the diophantine equation $x^n - y^n = z^{n+1}$ is constructed.
We study vortex solutions in a holographic model of Herzog, Hartnoll, and Horowitz, with a vanishing external magnetic field on the boundary, as is appropriate for vortices in a superfluid. We study relevant length scales related to the vortices and how the charge density inside the core of the vortex behaves as a function of temperature or chemical potential. We extract the critical superfluid velocity from the vortex solutions, study how it behaves as a function of the temperature, and compare it to earlier studies and to the Landau criterion. We also comment on the possibility of a Berezinskii-Kosterlitz-Thouless vortex confinement-deconfinement transition.
Adelic quantum mechanics is form invariant under an interchange of real and p-adic number fields as well as rings of p-adic integers. We also show that in adelic quantum mechanics Feynman's path integrals for quadratic actions with rational coefficients are invariant under changes of their entries within nonzero rational numbers.
Let $q\geqslant2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)\leqslant q.$ In this paper the authors prove some estimates for character sums of the form \[\mathcal{W}(\chi,H;q)=\sum_{n\in\mathscr{F}(H)}\chi(n),\] where \[\mathscr{F}(H)=\left\{n\in\mathbb{Z}|(n,q)=1,1\leqslant n,\bar{n}\leqslant q, |n-\bar{n}|\leqslant H\},\] $\bar{n}$ is defined by $n\bar{n}\equiv1\pmod q.$
Neutron scattering is a powerful probe of strongly correlated systems. It can directly detect common phenomena such as magnetic order, and can be used to determine the coupling between magnetic moments through measurements of the spin-wave dispersions. In the absence of magnetic order, one can detect diffuse scattering and dynamic correlations. Neutrons are also sensitive to the arrangement of atoms in a solid (crystal structure) and lattice dynamics (phonons). In this chapter, we provide an introduction to neutrons and neutron sources. The neutron scattering cross section is described and formulas are given for nuclear diffraction, phonon scattering, magnetic diffraction, and magnon scattering. As an experimental example, we describe measurements of antiferromagnetic order, spin dynamics, and their evolution in the La(2-x)Ba(x)CuO(4) family of high-temperature superconductors.
A detailed study of chemical freeze-out in nucleus-nucleus collisions at beam energy 158A GeV is presented. By analyzing hadronic multiplicities within the statistical hadronization approach, the chemical equilibration of p-p, C-C, Si-Si and Pb-Pb systems is studied as a function of the number of participating nucleons in the system. Additionally, Two Component statistical hadronization model is applied to the data and is found to be able to explain the observed strangeness hadronic phase space under-saturation.
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.
The theory of weak measurement, proposed by Aharonov and coworkers, has been applied by Steinberg to the long-discussed traversal time problem. The uncertainty and ambiguity that characterize this concept from the perspective of von Neumann measurement theory apparently vanish, and joint probabilities and conditional averages become meaningful concepts. We express the Larmor clock and some other well-known methods in the weak measurement formalism. We also propose a method to determine higher moments of the traversal time distribution in terms of the outcome of a gedanken experiment, by introducing an appropriate operator. Since the weak measurement approach can sometimes lead to unphysical results, for example average negative reflection times and higher moments, the interpretation of the results obtained remains an open problem.
For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected graph without isolated vertices. This paper provides bounds for the super connectivity and $k_{1}$-connectivity of the lexicographic product of two graphs.
We study boundary traces of shift-invariant diffusions: two-dimensional diffusions in the upper half-plane $\mathbb{R} \times [0, \infty)$ (or in $\mathbb{R} \times [0, R)$) invariant under horizontal translations. We prove that the corresponding trace processes are L\'evy processes with completely monotone jumps, and, conversely, every L\'evy process with completely monotone jumps is a boundary trace of some shift-invariant diffusion. Up to some natural transformations of space and time, this correspondence is bijective. We also reformulate this result in the language of additive functionals of the Brownian motion in $[0, \infty)$ (or in $[0, R)$), and Brownian excursions. Our main tool is the recent extension of Krein's spectral theory of strings, due to Eckhardt and Kostenko.
We consider the contact process with infection rate $\lambda$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $\tau_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding $\tau_{\mathbb{T}_n^d}$. Let $\lambda_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $\lambda < \lambda_2$, then $\tau_{\mathbb{T}_n^d}$ divided by the height of the tree converges in probability, as $n \to \infty$, to a positive constant. Second, if $\lambda > \lambda_2$, then $\log \mathbb{E}[\tau_{\mathbb{T}_n^d}]$ divided by the volume of the tree converges in probability to a positive constant, and $\tau_{\mathbb{T}_n^d}/\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ converges in distribution to the exponential distribution of mean 1.
We report ``infall asymmetry'' in the HCO$^+$ (1--0) and (3--2) lines toward NGC 1333, extended over $\sim 0.39 {\rm pc}^2$, a larger extent than has been reported be fore, for any star-forming region. The infall asymmetry extends over a major portion of the star-forming complex, and is not limited to a single protostar, or to a single dense core, or to a single spectral line. It seems likely that the infall asymmetry represents inward motions, and that these motions are physically associated with the complex. Both blue-asymmetric and red-asymmetric lines are seen, but in both the (3--2) and (1--0) lines of HCO$^+$ the vast majority of the asymmetric lines are blue, indicating inward motions. The (3--2) line, tracing denser gas, has the spectra with the strongest asymmetry and these spectra are associated with the protostars IRAS 4A and 4B, which most likely indicates a warm central source is affecting the line profiles. The (3--2) and (1--0) lines usually have the same sense of asymmetry in common positions, but their profiles differ significantly, and the (1--0) line appears to trace motions on much larger spatial scales than does the (3--2) line. Line profile models fit the spectra well, but do not strongly constrain their parameters. The mass accretion rate of the inward motions is of order 10$^{-4}$ M$_\odot$/yr, similar to the ratio of stellar mass to cluster age.