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Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}m^2$ and it has atleast six distinct prime divisors. Also we prove that $N$ must be atleast $2\cdot 10^{12}$. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-5$ and if $\Omega(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $\Omega(n)$ and $\omega(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$., Comment: 24 pages, 5 figures, Suggestions are welcomed

This paper investigates the dynamics of entanglement within the Jaynes-Cummings model (JCM), focusing on the behavior of atoms in both Bell and Werner states when subjected to thermal and squeezed photons. We study the phenomenon of entanglement sudden death (ESD), atomic inversion, and how various interactions (such as Ising-type, dipole-dipole, and Kerr-nonlinearity) affect these dynamics. This work compares the effects of these interactions and photons on the atomic and field subsystems, providing insights into how thermal and quantum noise impact entanglement behavior in these systems., Comment: arXiv admin note: text overlap with arXiv:2405.10564

This paper introduces a robust estimation strategy for the spatial functional linear regression model using dimension reduction methods, specifically functional principal component analysis (FPCA) and functional partial least squares (FPLS). These techniques are designed to address challenges associated with spatially correlated functional data, particularly the impact of outliers on parameter estimation. By projecting the infinite-dimensional functional predictor onto a finite-dimensional space defined by orthonormal basis functions and employing M-estimation to mitigate outlier effects, our approach improves the accuracy and reliability of parameter estimates in the spatial functional linear regression context. Simulation studies and empirical data analysis substantiate the effectiveness of our methods, while an appendix explores the Fisher consistency and influence function of the FPCA-based approach. The rfsac package in R implements these robust estimation strategies, ensuring practical applicability for researchers and practitioners., Comment: 25 pages, 4 figures, 2 tables

A function-on-function regression model with quadratic and interaction effects of the covariates provides a more flexible model. Despite several attempts to estimate the model's parameters, almost all existing estimation strategies are non-robust against outliers. Outliers in the quadratic and interaction effects may deteriorate the model structure more severely than their effects in the main effect. We propose a robust estimation strategy based on the robust functional principal component decomposition of the function-valued variables and $\tau$-estimator. The performance of the proposed method relies on the truncation parameters in the robust functional principal component decomposition of the function-valued variables. A robust Bayesian information criterion is used to determine the optimum truncation constants. A forward stepwise variable selection procedure is employed to determine relevant main, quadratic, and interaction effects to address a possible model misspecification. The finite-sample performance of the proposed method is investigated via a series of Monte-Carlo experiments. The proposed method's asymptotic consistency and influence function are also studied in the supplement, and its empirical performance is further investigated using a U.S. COVID-19 dataset., Comment: 35 pages, 3 tables

The key components of machine learning are data samples for training, model for learning patterns, and loss function for optimizing accuracy. Analogously, unlearning can potentially be achieved through anti-data samples (or anti-samples), unlearning method, and reversed loss function. While prior research has explored unlearning methods and reversed loss functions, the potential of anti-samples remains largely untapped. In this paper, we introduce UnSTAR: Unlearning with Self-Taught Anti-Sample Reasoning for large language models (LLMs). Our contributions are threefold; first, we propose a novel concept of anti-sample-induced unlearning; second, we generate anti-samples by leveraging misleading rationales, which help reverse learned associations and accelerate the unlearning process; and third, we enable fine-grained targeted unlearning, allowing for the selective removal of specific associations without impacting related knowledge - something not achievable by previous works. Results demonstrate that anti-samples offer an efficient, targeted unlearning strategy for LLMs, opening new avenues for privacy-preserving machine learning and model modification.

Predicting software defects early in the development process not only enhances the quality and reliability of the software but also decreases the cost of development. A wide range of machine learning techniques can be employed to create software defect prediction models, but the effectiveness and accuracy of these models are often influenced by the choice of appropriate feature subset. Since finding the optimal feature subset is computationally intensive, heuristic and metaheuristic approaches are commonly employed to identify near-optimal solutions within a reasonable time frame. Recently, the quantum computing paradigm quantum annealing (QA) has been deployed to find solutions to complex optimization problems. This opens up the possibility of addressing the feature subset selection problem with a QA machine. Although several strategies have been proposed for feature subset selection using a QA machine, little exploration has been done regarding the viability of a QA machine for feature subset selection in software defect prediction. This study investigates the potential of D-Wave QA system for this task, where we formulate a mutual information (MI)-based filter approach as an optimization problem and utilize a D-Wave Quantum Processing Unit (QPU) solver as a QA solver for feature subset selection. We evaluate the performance of this approach using multiple software defect datasets from the AEEM, JIRA, and NASA projects. We also utilize a D-Wave classical solver for comparative analysis. Our experimental results demonstrate that QA-based feature subset selection can enhance software defect prediction. Although the D-Wave QPU solver exhibits competitive prediction performance with the classical solver in software defect prediction, it significantly reduces the time required to identify the best feature subset compared to its classical counterpart.

Acoustic haptic technology adds touch sensations to human-machine interfaces by integrating piezoelectric actuators onto touchscreens. Traditional piezoelectric haptic technologies use opaque lead-containing ceramics that are both toxic and visible. We have developed a highly transparent lead-free piezoelectric haptic device using potassium sodium niobate (KNN) and transparent conductive oxide thin films. The KNN film, grown on glass, exhibits a pure perovskite phase and a dense microstructure. This device achieves up to 80% transmittance, surpassing lead zirconate titanate (PZT) thin films. It generates an acoustic resonance at 16.5 kHz and produces a peak-to-peak displacement of 1.0 um at 28 V unipolar, making it suitable for surface rendering applications. This demonstrates the potential of transparent lead-free piezoelectric actuators as an effective alternative to conventional PZT haptic actuators.

In this paper, we study the four-point celestial leaf amplitudes of massless scalar and MHV gluon scattering. These leaf amplitudes are non-distributional decompositions of the celestial amplitudes associated with a hyperbolic foliation of the Klein spacetime. Bulk scale invariance imposes constraints on the total conformal weights of the massless scalars or gluons. Using this constraint we show that the four-point leaf amplitudes have a \textit {simple pole singularity at $ z = \bar z $}, where, $ z,\bar z $ are two real independent conformal cross ratios. The distributional nature of the four-point celestial amplitudes is recovered by adding the leaf amplitudes in the timelike and spacelike wedges of the spacetime. We also verify that the MHV gluon leaf amplitudes satisfy a set of differential equations previously obtained for celestial MHV gluon amplitudes by considering the soft gluon theorems and the subleading terms in the OPE expansion between two positive helicity gluons., Comment: 32 pages, 2 figures

As LLMs are increasingly deployed in global applications, the importance of cultural sensitivity becomes paramount, ensuring that users from diverse backgrounds feel respected and understood. Cultural harm can arise when these models fail to align with specific cultural norms, resulting in misrepresentations or violations of cultural values. This work addresses the challenges of ensuring cultural sensitivity in LLMs, especially in small-parameter models that often lack the extensive training data needed to capture global cultural nuances. We present two key contributions: (1) A cultural harm test dataset, created to assess model outputs across different cultural contexts through scenarios that expose potential cultural insensitivities, and (2) A culturally aligned preference dataset, aimed at restoring cultural sensitivity through fine-tuning based on feedback from diverse annotators. These datasets facilitate the evaluation and enhancement of LLMs, ensuring their ethical and safe deployment across different cultural landscapes. Our results show that integrating culturally aligned feedback leads to a marked improvement in model behavior, significantly reducing the likelihood of generating culturally insensitive or harmful content. Ultimately, this work paves the way for more inclusive and respectful AI systems, fostering a future where LLMs can safely and ethically navigate the complexities of diverse cultural landscapes.

We theoretically investigate emergent topological phases in an extended spin-full Su-Schrieffer-Heeger (SSH) model considering Rashba spin-orbit interaction, all possible complex next to next nearest neighbor (NNNN) hopping preserving Chiral symmetry. Our analysis finds exact condition for which the topological phases of both the spin sectors could be independently varied. We show that it necessarily depends on complex NNNN only. We elaborate in detail the emergent topological phases, its criteria through analytic determination of non-trivial gap-closing condition due to the presence of $\cos 2k$ term. We also find that the profile of topological edge modes for finite winding numbers depend non-monotonously on the value of NNNN hopping elucidating competing effect of model parameters. We extend our study to few coupled chains and show explicitly that depending on the parameters all possible winding number ranging from zero to $2N$ could be obtained, where $N$ is the number of chains considered. Finally we incorporate the study of domain wall and remarkably we find that the location of mid-gap zero energy state by changing the values of model parameters. Our study could be of immensely useful for future applications in quantum technology., Comment: 19 pages, 19 figures

We study quantum-fluctuation-driven fractionalized phases in the vicinity of altermagnetic order. First, the long-range magnetic orders in the vicinity of collinear altermagnetism are identified; these feature a non-coplanar "orbital altermagnet" which has altermagnetic symmetries in spin-rotation invariant observables. We then describe neighboring fractionalized phases with topological order reached when quantum fluctuations destroy long-range spin order, within Schwinger-boson theory and an SU(2) gauge theory of fluctuating magnetism. Discrete symmetries remain broken in some of the fractionalized phases, with the orbital altermagnet becoming an "altermagnetic spin liquid". We compute the electronic spectral function in the doped system, revealing "fractionalized spin-orbit coupling" characterized by split Fermi surfaces, reminiscent of conventional spin-orbit coupling, but with preserved spin-rotation symmetry., Comment: 5+12 pages, 3+1 figures

Using the notion of modulus of continuity at a point of a mapping between metric spaces, we introduce the notion of extensively bounded mappings generalizing that of Lipschitz mappings. We also introduce a metric on it which becomes a norm if the codomain is a normed linear space. We study its basic properties. We also discuss a dilation of an extensively bounded mapping into a Lipschitz mapping as well as into a bounded linear mapping., Comment: 21

Large language models (LLMs) often exhibit subtle yet distinctive characteristics in their outputs that users intuitively recognize, but struggle to quantify. These "vibes" - such as tone, formatting, or writing style - influence user preferences, yet traditional evaluations focus primarily on the single axis of correctness. We introduce VibeCheck, a system for automatically comparing a pair of LLMs by discovering identifying traits of a model ("vibes") that are well-defined, differentiating, and user-aligned. VibeCheck iteratively discover vibes from model outputs, then utilizes a panel of LLM judges to quantitatively measure the utility of each vibe. We validate that the vibes generated by VibeCheck align with those found in human discovery and run VibeCheck on pairwise preference data from real-world user conversations with llama-3-70b VS GPT-4. VibeCheck reveals that Llama has a friendly, funny, and somewhat controversial vibe. These vibes predict model identity with 80% accuracy and human preference with 61% accuracy. Lastly, we run VibeCheck on a variety of models and tasks including summarization, math, and captioning to provide insight into differences in model behavior. VibeCheck discovers vibes like Command X prefers to add concrete intros and conclusions when summarizing in comparison to TNGL, Llama-405b often overexplains its thought process on math problems compared to GPT-4o, and GPT-4 prefers to focus on the mood and emotions of the scene when captioning compared to Gemini-1.5-Flash. Code can be found at https://github.com/lisadunlap/VibeCheck, Comment: added code link, unironic use of the word 'vibe'

Text-To-Image (TTI) Diffusion Models such as DALL-E and Stable Diffusion are capable of generating images from text prompts. However, they have been shown to perpetuate gender stereotypes. These models process data internally in multiple stages and employ several constituent models, often trained separately. In this paper, we propose two novel metrics to measure bias internally in these multistage multimodal models. Diffusion Bias was developed to detect and measures bias introduced by the diffusion stage of the models. Bias Amplification measures amplification of bias during the text-to-image conversion process. Our experiments reveal that TTI models amplify gender bias, the diffusion process itself contributes to bias and that Stable Diffusion v2 is more prone to gender bias than DALL-E 2.

We use machine learning techniques to identify a boosted dark photon jet as a signature of a heavy vector-like fermionic portal matter (PM) connecting the visible and the dark sectors. In this work, the fermionic PM which mixes only with the standard model (SM) third generation up-type quark, dominantly decays to a top quark and dark photon pair. The dark photon then decays $promptly$ to a pair of standard model fermions via the gauge kinetic mixing. We have analyzed two different final states, namely, $(i)~ exactly ~one ~tagged~ dark ~photon ~and ~exactly ~one ~tagged~ top ~quark ~jet$, and $(ii)~ at~ least~ two~tagged ~dark$ $photons~and~at ~least ~one ~tagged ~top ~quark ~jet$ at the 13 and 14 TeV LHC center of mass energies. Both these final states receive significant contributions from the pair and single production processes of the top partner. The rich event topology of the signal processes, $i.e.$, the presence of a boosted dark photon and top quark jet pair, along with the fact that the invariant mass of the system corresponds to the mass of the top partner, help us to significantly suppress the potential SM backgrounds. We have shown that one can set a $2\sigma$ exclusion limit of $3.0$ TeV on the top partner mass with $\sin\theta_L=0.1$ and assuming $100\%$ branching ratio of the top partner in the exactly one tagged dark photon and exactly one tagged top quark jet final state at the 14 TeV LHC center of mass energy assuming 300 fb$^{-1}$ of integrated luminosity., Comment: 26 pages, 12 Figures, 8 Tables

This study uses the Krori-Barua type metric to represent hybrid stars within the $f(Q)$ theory of gravity. We postulate that the hybrid star also contains strange quark matter in addition to regular baryonic matter. To investigate the physical viability of the hybrid star model, we present the graphical behavior of the energy density, radial pressure, and tangential pressure, equation of state parameters, anisotropy, and stability analysis, respectively, by choosing the compact star EXO 1785-248 with a mass of $1.3_{-0.2}^{+0.2}~M_{\odot}$ and radius $8.849_{-0.4}^{+0.4}$ km with five different values of the coupling constants as $a=2$, $a=4$, $a=6$, $a=8$, and $a=10$. The maximum allowed masses and corresponding radii have been calculated using the $M-R$ curve for three different coupling parameter '$a$' values which match the observational data of three distinct compact stars, namely LMC X-4, SMC X-1, and 4U 1538-52. So, the $f(Q)$ theory of gravity can provide results that are plausible for describing the macroscopical characteristics of hybrid star candidates., Comment: Physics of the Dark Universe published version

We compute the non-Drude part of the conductivity tensor in planar Hall set-ups, for tilted Weyl and multi-Weyl semimetals, considering both the type-I and type-II phases. We do so in three distinct set-ups, taking into account the possible relative orientations of the plane spanned by the electric and magnetic fields ($\mathbf E $ and $\mathbf B $) and the direction of the tilt-axis. We derive the analytical expressions for the response tensor, including the effects of the Berry curvature (BC) and the orbital magnetic moment (OMM), both of which arise due to a nontrivial topology of the three-dimensional manifold defined by the Brillouin zone. We exhibit the interplay of the BC-only and the OMM-dependent parts in the nonzero components of the magnetoelctric conductivity, and outline whether the contributions from the former or the latter dominate the overall response. Our results also show that, depending on the configuration of the planar Hall set-up, one may or may not get terms which have a linear-in-$ B$ dependence., Comment: follow-up paper of arXiv:2402.10203 and arXiv:2405.14844

Federated learning (FL) has enabled collaborative model training across decentralized data sources or clients. While adding new participants to a shared model does not pose great technical hurdles, the removal of a participant and their related information contained in the shared model remains a challenge. To address this problem, federated unlearning has emerged as a critical research direction, seeking to remove information from globally trained models without harming the model performance on the remaining data. Most modern federated unlearning methods use costly approaches such as the use of remaining clients data to retrain the global model or methods that would require heavy computation on client or server side. We introduce Contribution Dampening (ConDa), a framework that performs efficient unlearning by tracking down the parameters which affect the global model for each client and performs synaptic dampening on the parameters of the global model that have privacy infringing contributions from the forgetting client. Our technique does not require clients data or any kind of retraining and it does not put any computational overhead on either the client or server side. We perform experiments on multiple datasets and demonstrate that ConDa is effective to forget a client's data. In experiments conducted on the MNIST, CIFAR10, and CIFAR100 datasets, ConDa proves to be the fastest federated unlearning method, outperforming the nearest state of the art approach by at least 100x. Our emphasis is on the non-IID Federated Learning setting, which presents the greatest challenge for unlearning. Additionally, we validate ConDa's robustness through backdoor and membership inference attacks. We envision this work as a crucial component for FL in adhering to legal and ethical requirements.

The dynamical and geometric structures of the Broad Line Region (BLR), along with the origins of continuum time delays in active galaxies, remain topics of ongoing debate. In this study, we aim to reproduce the broadband spectrum, the H$\beta$ line delay, and the continuum time delays available for the source NGC 5548. We employ the standard accretion disk model with the option of an inner hot flow, alongside the lamp-post model to account for disk irradiation and a BLR structure model based on radiation pressure acting on dust. The model is parameterized by the black hole mass (fixed), accretion rate, viewing angle, lamp-post height, cloud density, and cloud covering factor. The resulting continuum delays are calculated as a combination of disk reprocessing and the reprocessing of a fraction of radiation by the BLR. Our model reasonably reproduces the observed broad-band continuum, the H$\beta$ delay, and the continuum inter-band time delays measured during the observational campaign. When the accretion rate is not fixed based on the known distance to the source, we can directly estimate the distance from our model. The resulting value of H$_0$ = $69.03^{+17.81}_{-11.75}$ km s$^{-1}$ Mpc$^{-1}$ represents a noteworthy improvement compared to the findings of Cackett et al. (2007). This pilot study demonstrates that, with sufficient data coverage, it is possible to disentangle the time delays originating from the accretion disk and the BLR. This paves the way for effectively using inter-band continuum time delays in determining the Hubble constant. Additionally, the findings support the adopted model for the formation of the H$\beta$ line., Comment: submitted to A&A, comments welcome

Mixed quantum/classical theory (MQCT) for the treatment of rotationally inelastic transitions during collisions of two identical molecules, described either as indistinguishable or distinguishable partners, is reviewed. The treatment of two molecules as indistinguishable includes symmetrization of rotational wavefunctions, introduces exchange parity, and gives state-to-state transition matrix elements different from those in the straightforward treatment of molecules as distinguishable. Moreover, the treatment of collision partners as indistinguishable is eight times faster. Numerical results, presented for $H_2 + H_2$, CO + CO and $H_2O + H_2O$ systems, indicate good agreement of MQCT calculations with full-quantum calculations from literature and show that an a posteriori correction, applied after the treatment of collision partners as distinguishable, generally produces good results that agree well with the rigorous treatment of collision partners as indistinguishable. This correction for cross section includes either multiplication by 2, or summation over physically indistinguishable processes, depending on the transition type. After this correction, the results of two treatments agree within 5% for most but may reach 10-20% for some transitions. At low collision energies dominated by scattering resonances these differences can be larger, but they tend to decrease as collision energy is increased. It is also shown that if the system is artificially forced to follow the same collision path in the indistinguishable and distinguishable treatments, then all differences between the results of two treatments disappear. This interesting finding gives new insight into the collision process and indicates that indistinguishability of identical collision partners comes into play through the collision path itself, rather than through matrix elements of inelastic transitions.

A search for proton decay into $e^+/\mu^+$ and a $\eta$ meson has been performed using data from a 0.373 Mton$\cdot$year exposure (6050.3 live days) of Super-Kamiokande. Compared to previous searches this work introduces an improved model of the intranuclear $\eta$ interaction cross section, resulting in a factor of two reduction in uncertainties from this source and $\sim$10\% increase in signal efficiency. No significant data excess was found above the expected number of atmospheric neutrino background events resulting in no indication of proton decay into either mode. Lower limits on the proton partial lifetime of $1.4\times\mathrm{10^{34}~years}$ for $p\rightarrow e^+\eta$ and $7.3\times\mathrm{10^{33}~years}$ for $p\rightarrow \mu^+\eta$ at the 90$\%$ C.L. were set. These limits are around 1.5 times longer than our previous study and are the most stringent to date.

Carroll hydrodynamics arises in the $c\to 0$ limit of relativistic hydrodynamics. Instances of its relevance include the Bjorken and Gubser flow models of heavy-ion collisions, where the ultrarelativistic nature of the flow makes the physics effectively Carrollian. In this paper, we explore the structure of hydrodynamics in what can be termed as the Carrollian regime, where instead of keeping only the leading terms in the $c\to 0$ limit of relativistic hydrodynamics, we perform a small-$c$ expansion and retain the subleading terms as well. We do so both for perfect fluids as well as viscous fluids incorporating first order derivative corrections. As apposite applications of the formalism, we utilize the subleading terms to compute modifications to the Bjorken and Gubser flow equations which bring in, in particular, dependence on rapidity., Comment: 31 pages, 1 figure

This paper aims to find a unified approach to studying the cohomology theories of various operators on Leibniz algebras. We first introduce deformation maps in a proto-twilled Leibniz algebra to do this. Such maps generalize various well-known operators (such as homomorphisms, derivations, crossed homomorphisms, Rota-Baxter operators, modified Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators etc) defined on Leibniz algebras and embedding tensors on Lie algebras. We define the cohomology of a deformation map unifying the existing cohomologies of all the operators mentioned above. Then we construct a curved $L_\infty$-algebra whose Maurer-Cartan elements are precisely deformation maps in a given proto-twilled Leibniz algebra. In particular, we get the Maurer-Cartan characterizations of modified Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators on a Leibniz algebra. Finally, given a proto-twilled Leibniz algebra and a deformation map $r$, we construct two governing $L_\infty$-algebras, the first one controls the deformations of the operator $r$ while the second one controls the simultaneous deformations of both the proto-twilled Leibniz algebra and the operator $r$., Comment: This is the first version and having 19 pages

Understanding the origins of massive black hole seeds and their co-evolution with their host galaxy requires studying intermediate-mass black holes (IMBHs) and estimating their mass. However, measuring the mass of these IMBHs is challenging due to the high spatial resolution requirement. A spectrophotometric reverberation monitoring is performed for a low-luminosity Seyfert 1 galaxy NGC 4395 to measure the size of the broad line region (BLR) and black hole mass. The data were collected using the 1.3-m Devasthal fast optical telescope (DFOT) and 3.6-m Devasthal optical telescope (DOT) at ARIES, Nainital, over two consecutive days in March 2022. The analysis revealed strong emission lines in the spectra and light curves of merged 5100{\AA} spectroscopic continuum flux ($f_{\mathrm{5100}}$) with photometric continuum V-band and H$\alpha$, with fractional variabilities of 6.38\% and 6.31\% respectively. In comparison to several previous studies with lag estimation $<$ 90 minutes, our calculated H$\alpha$ lag supersedes by $125.0^{+6.2}_{-6.1}$ minutes using ICCF and {\small JAVELIN} methods. The velocity dispersion ($\sigma_{\mathrm{line}}$) of the broad line clouds is measured to be $544.7^{+22.4}_{-25.1}$ km s$^{-1}$, yielding a black hole mass of $\sim$ $2.2^{+0.2}_{-0.2}\times 10^{4}M_{\mathrm{\odot}}$ and an Eddington ratio of 0.06., Comment: 17 pages, 8 figures, accepted for publication in ApJ

We present near-simultaneous X-ray and optical polarization measurements in the high synchrotron peaked (HSP) blazar Mrk 421. The X-ray polarimetric observations were carried out using {\it Imaging X-ray Polarimetry Explorer} ({\it IXPE}) on 06 December 2023. During {\it IXPE} observations, we also carried out optical polarimetric observations using 104cm Sampurnanand telescope at Nainital and multi-band optical imaging observations using 2m Himalayan Chandra Telescope at Hanle. From model-independent analysis of {\it IXPE} data, we detected X-ray polarization with degree of polarization ($\Pi_X$) of 8.5$\pm$0.5\% and an electric vector position angle ($\Psi_X$) of 10.6$\pm$1.7 degrees in the 2$-$8 keV band. From optical polarimetry on 06 December 2023, in B, V, and R bands, we found values of $\Pi_B$ = 4.27$\pm$0.32\%, $\Pi_V$= 3.57$\pm$0.31\%, and $\Pi_R$= 3.13$\pm$0.25\%. The value of $\Pi_B$ is greater than that observed at longer optical wavelengths, with the degree of polarization suggesting an energy-dependent trend, gradually decreasing from higher to lower energies. This is consistent with that seen in other HSP blazars and favour a stratified emission region encompassing a shock front. The emission happening in the vicinity of the shock front will be more polarized due to the ordered magnetic field resulting from shock compression. The X-ray emission, involving high-energy electrons, originates closer to the shock front than the optical emission. The difference in the spatial extension could plausibly account for the observed variation in polarization between X-ray and optical wavelengths. This hypothesis is further supported by the broadband spectral energy distribution modeling of the X-ray and optical data., Comment: 9 pages, 3 figures. Accepted for publication in ApJ

Discovery of polar skyrmions on the (PbTiO$_3$)$_n$/(SrTiO$_3$)$_n$ superlattice in the absence of any known interaction, which can orient electric polarizations at a point with respect to its neighboring points, is enigmatic. We here show that the coupling of the electric polarization and strain at each layer is responsible for orienting polarizations. We formulate coupled Euler equations for electric polarization and elastic displacement vectors, and solve to find skyrmion solution for a range of electric fields with unique topological number in each layer. The types of skyrmions vary from Neel to Bloch depending on the position of the layers in the superlattice, as observed in the experiments., Comment: 10 pages, 6 figures

For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it., Comment: 19 pages, comments are welcome!

We present a comprehensive analysis of the X-ray spectral properties of 198 newly identified active galactic nuclei (AGNs), leveraging archival data from the {\it Chandra} X-ray Observatory. All these AGNs exhibit a powerlaw spectral signature spanning a broad energy range of $0.5-7.0$ keV, characterized by the photon index ($\Gamma$) values ranging from $0.3^{+0.16}_{-0.14}$ to $2.54^{+0.14}_{-0.13}$. Particularly, 76 of these AGNs display discernible levels of intrinsic absorption, after considering the Galactic absorption. The column densities associated with this local absorption ($n_{\rm H}^{\rm local}$) are within a range of $\sim 10^{19} - 10^{22}\ {\rm cm^{-2}}$. We study the cosmological evolution of AGNs using the variation of $n_{\rm H}^{\rm local}$ and $\Gamma$ with their estimated redshift. The intrinsic spectral signature did not reveal any significant cosmological evolution; however, a deficit of hard sources at high redshift is possibly intrinsic. Our sample covers several decades of broadband intrinsic luminosity ($L_{\rm B}^{\rm intr}$) ranging from $4.59^{+0.41}_{-0.41} \times 10^{42}$ to $2.4^{+0.12}_{-0.12} \times 10^{46}\, {\rm erg~s}^{-1}$ with peak at 1.84 redshift. We also investigate the hardness-luminosity diagram (HLD) to further probe the AGNs. We conduct a sanity check by applying our findings to known AGNs, and the results are consistent with our observations., Comment: 15 pages, 12 figures, 1 table, accepted for publication in Monthly Notices of the Royal Astronomical Society

The optical generation of nonequilibrium spin magnetization plays a crucial role in advancing spintronics, providing ultrafast control of magnetization dynamics without the need for magnetic fields. Here, we demonstrate the feasibility of light-induced nonlinear spin magnetization (LNSM), which becomes a dominant effect in centrosymmetric materials. We reveal the quantum geometric origins of various LNSM contributions in both metallic and insulating systems. Through detailed symmetry analysis, we predict significant LNSM in the antiferromagnetic material CuMnAs. Notably, under circularly polarized light, the spin magnetization exhibits helicity-dependent behavior, reversing with opposite light helicity. These findings open up new possibilities for generating LNSM-driven nonlinear spin-orbit torques and developing innovative opto-spintronic devices., Comment: Comments are welcome. 7 pages, 3 figures, 2 tables

The hitting set problem is one of the fundamental problems in combinatorial optimization and is well-studied in offline setup. We consider the online hitting set problem, where only the set of points is known in advance, and objects are introduced one by one. Our objective is to maintain a minimum-sized hitting set by making irrevocable decisions. Here, we present the study of two variants of the online hitting set problem depending on the point set. In the first variant, we consider the point set to be the entire $\mathbb{Z}^d$, while in the second variant, we consider the point set to be a finite subset of $\mathbb{R}^2$. If you use points in $\mathbb{Z}^d$ to hit homothetic hypercubes in $\mathbb{R}^d$ with side lengths in $[1,M]$, we show that the competitive ratio of any algorithm is $\Omega(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $\Omega(\log M)$ by a factor of $d$. Then, we present an almost tight randomized algorithm with a competitive ratio $O(d^2\log M)$ that significantly improves the best-known competitive ratio of $25^d\log M$. Next, we propose a simple deterministic ${\lfloor\frac{2}{\alpha}+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$\alpha$-fat objects} in $\mathbb{R}^d$ having diameters in the range $[1, M]$ using points in $\mathbb{Z}^d$. This improves the current best-known upper bound by a factor of at least $5^d$. Finally, we consider the hitting set problem when the point set consists of $n$ points in $\mathbb{R}^2$, and the objects are homothetic regular $k$-gons having diameter in the range $[1, M]$. We present an $O(\log n\log M)$ competitive randomized algorithm for that. Whereas no result was known even for squares. In particular, our results answer some of the open questions raised by Khan et al. (SoCG'23) and Alefkhani et al. (WAOA'23)., Comment: 30 pages and 10 figures. arXiv admin note: text overlap with arXiv:2304.06780

The Be/X-ray binary pulsar RX J0440.9+4431 went through a giant outburst in December 2022 with a peak flux of $\sim$2.3 Crab in 15--50 keV. We studied the broad-band timing and spectral properties of RX J0440.9+4431 using four $AstroSat$ observations, where the source transited between subcritical and supercritical accretion regimes. Pulsations were detected significantly above 100 keV. The pulse profiles were found to be highly luminosity- and energy-dependent. A significant evolution in the pulse profile shape near the peak of the outburst indicates a possible change in the accretion mode and beaming patterns of RX J0440.9+4431. The rms pulsed fraction was luminosity- and energy-dependent, with a concave-like feature around 20--30 keV. The depth of this feature varied with luminosity, indicating changes in the accretion column height and proportion of reflected photons. The broad-band continuum spectra were best fitted with a two-component Comptonization model with a blackbody component or a two-blackbody component model with a thermal Comptonization component. A quasi-periodic oscillation at 60 mHz was detected at a luminosity of $2.6 \times 10^{37}$ erg s$^{-1}$, which evolved into 42 mHz at $1.5 \times 10^{37}$ erg s$^{-1}$. The QPO rms were found to be energy dependent with an overall increasing trend with energy. For the first time, we found the QPO frequency varying with photon energy in an X-ray pulsar, which poses a challenge in explaining the QPO with current models such as the Keplarian and beat frequency model. Hence, more physically motivated models are required to understand the physical mechanism behind the mHz QPOs., Comment: Accepted for publication in MNRAS

We investigated the Re-based kagome superconductor Re$_2$Zr through various measurements, including resistivity, magnetization, specific heat, and muon spin rotation and relaxation spectroscopy. These results suggest that Re$_2$Zr is a moderately coupled potential two-gap superconductor. Zero-field muon relaxation data indicate the possible presence of a time-reversal symmetry-breaking state in the superconducting ground state. Our investigation identifies Re$_{2}$Zr as a new unconventional superconductor with a potential complex order parameter that warrants considerable experimental and theoretical interest., Comment: 9 pages, 6 figures

Survival regression is widely used to model time-to-events data, to explore how covariates may influence the occurrence of events. Modern datasets often encompass a vast number of covariates across many subjects, with only a subset of the covariates significantly affecting survival. Additionally, subjects often belong to an unknown number of latent groups, where covariate effects on survival differ significantly across groups. The proposed methodology addresses both challenges by simultaneously identifying the latent sub-groups in the heterogeneous population and evaluating covariate significance within each sub-group. This approach is shown to enhance the predictive accuracy for time-to-event outcomes, via uncovering varying risk profiles within the underlying heterogeneous population and is thereby helpful to device targeted disease management strategies.

This paper explores the intricate relationships between Lucas numbers and Diophantine equations, offering significant contributions to the field of number theory. We first establish that the equation regarding Lucas number $L_n = 3x^2$ has a unique solution in positive integers, specifically $(n, x) = (2, 1)$, by analyzing the congruence properties of Lucas numbers modulo $4$ and Jacobi symbols. We also prove that a Fibonacci number $F_n$ can be of the form $F_n=5x^2$ only when $(n,x)=(5,1)$. Expanding our investigation, we prove that the equation $L_n^2+L_{n+1}^2=x^2$ admits a unique solution $(n,x)=(2,5)$. In conclusion, we determine all non-negative integer solutions $(n, \alpha, x)$ to the equation $L_n^\alpha + L_{n+1}^\alpha = x^2$, where $L_n$ represents the $n$-th term in the Lucas sequence.

Observation of filling factor 6/13 is one of the surprising fractional quantum Hall states in the second Landau level because, in contrast to the standard wisdom, the fractions ($\nu < 1/2$) with lower numerators, namely 4 and 5, have not yet been observed. We find that a state indeed forms at $\nu=6/13$ as an intermittent topological state between two prominent states at $\nu =1/2$ and $\nu = 2/5$ with lower numerators. Also, we predict that a state forms at $\nu=5/13$ as an intermittent to $\nu = 2/5$ and $\nu =3/8$. Our proposed wave functions for $\nu =6/13$ and $5/13$ have excellent overlaps with the corresponding exact ground state wave functions. The Chern-Simons coupling matrices deduced from the form of these wave functions are analyzed to predict the topological properties, which may be experimentally verified.

While witnessing the exceptional success of machine learning (ML) technologies in many applications, users are starting to notice a critical shortcoming of ML: correlation is a poor substitute for causation. The conventional way to discover causal relationships is to use randomized controlled experiments (RCT); in many situations, however, these are impractical or sometimes unethical. Causal learning from observational data offers a promising alternative. While being relatively recent, causal learning aims to go far beyond conventional machine learning, yet several major challenges remain. Unfortunately, advances are hampered due to the lack of unified benchmark datasets, algorithms, metrics, and evaluation service interfaces for causal learning. In this paper, we introduce {\em CausalBench}, a transparent, fair, and easy-to-use evaluation platform, aiming to (a) enable the advancement of research in causal learning by facilitating scientific collaboration in novel algorithms, datasets, and metrics and (b) promote scientific objectivity, reproducibility, fairness, and awareness of bias in causal learning research. CausalBench provides services for benchmarking data, algorithms, models, and metrics, impacting the needs of a broad of scientific and engineering disciplines.

In this work we investigate anharmonic vibrational polaritons formed due to strong light-matter interactions in an optical cavity between radiation modes and anharmonic vibrations beyond the long-wavelength limit. We introduce a conceptually simple description of light-matter interactions, where spatially localized cavity radiation modes couple to localized vibrations. Within this theoretical framework, we employ self-consistent phonon theory and vibrational dynamical mean-field theory to efficiently simulate momentum-resolved vibrational-polariton spectra, including effects of anharmonicity. Numerical simulations in model systems demonstrate the accuracy and applicability of our approach., Comment: 10 pages, 6 figures

Recent research has seen significant interest in methods for concept removal and targeted forgetting in diffusion models. In this paper, we conduct a comprehensive white-box analysis to expose significant vulnerabilities in existing diffusion model unlearning methods. We show that the objective functions used for unlearning in the existing methods lead to decoupling of the targeted concepts (meant to be forgotten) for the corresponding prompts. This is concealment and not actual unlearning, which was the original goal. The ineffectiveness of current methods stems primarily from their narrow focus on reducing generation probabilities for specific prompt sets, neglecting the diverse modalities of intermediate guidance employed during the inference process. The paper presents a rigorous theoretical and empirical examination of four commonly used techniques for unlearning in diffusion models. We introduce two new evaluation metrics: Concept Retrieval Score (CRS) and Concept Confidence Score (CCS). These metrics are based on a successful adversarial attack setup that can recover forgotten concepts from unlearned diffusion models. The CRS measures the similarity between the latent representations of the unlearned and fully trained models after unlearning. It reports the extent of retrieval of the forgotten concepts with increasing amount of guidance. The CCS quantifies the confidence of the model in assigning the target concept to the manipulated data. It reports the probability of the unlearned model's generations to be aligned with the original domain knowledge with increasing amount of guidance. Evaluating existing unlearning methods with our proposed stringent metrics for diffusion models reveals significant shortcomings in their ability to truly unlearn concepts. Source Code: https://respailab.github.io/unlearning-or-concealment

In this study, we explore a mathematical model for tissue growth focusing on the interplay between multiple cell subpopulations with distinct phenotypic characteristics. The model addresses the dynamics of tissue growth influenced by phenotype-dependent growth rates and collective population pressure, governed by Brinkman's law. We examine two primary objectives: the joint limit where viscosity tends to zero while the number of species approaches infinity, yielding an inviscid Darcy-type model with a continuous phenotype variable, and the continuous phenotype limit where the number of species becomes infinite with a fixed viscosity, resulting in a novel viscoelastic tissue growth model. In this sense, this paper provides a comprehensive framework that elucidates the relationships between four different modelling paradigms in tissue growth.

The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of these ten terms very fast by multiplying the 7th term by 11. This trick is based on the divisibility properties of partial sums of Fibonacci-like sequences. We find the maximum Fibonacci number that divides the sum of the Fibonacci numbers 1 through $n$. We discuss the generalization of the trick for other second-order recurrences. We show that a similar trick exists for Pell-like sequences and does not exist for Jacobhstal-like sequences., Comment: 26 pages, 9 tables

Automatic diagnosis techniques have evolved to identify age-related macular degeneration (AMD) by employing single modality Fundus images or optical coherence tomography (OCT). To classify ocular diseases, fundus and OCT images are the most crucial imaging modalities used in the clinical setting. Most deep learning-based techniques are established on a single imaging modality, which contemplates the ocular disorders to a specific extent and disregards other modality that comprises exhaustive information among distinct imaging modalities. This paper proposes a modality-specific multiscale color space embedding integrated with the attention mechanism based on transfer learning for classification (MCGAEc), which can efficiently extract the distinct modality information at various scales using the distinct color spaces. In this work, we first introduce the modality-specific multiscale color space encoder model, which includes diverse feature representations by integrating distinct characteristic color spaces on a multiscale into a unified framework. The extracted features from the prior encoder module are incorporated with the attention mechanism to extract the global features representation, which is integrated with the prior extracted features and transferred to the random forest classifier for the classification of AMD. To analyze the performance of the proposed MCGAEc method, a publicly available multi-modality dataset from Project Macula for AMD is utilized and compared with the existing models., Comment: 27th International Conference on Pattern Recognition (ICPR) 2024

A quasi-twilled associative algebra is an associative algebra $\mathbb{A}$ whose underlying vector space has a decomposition $\mathbb{A} = A \oplus B$ such that $B \subset \mathbb{A}$ is a subalgebra. In the first part of this paper, we give the Maurer-Cartan characterization and introduce the cohomology of a quasi-twilled associative algebra. In a quasi-twilled associative algebra $\mathbb{A}$, a linear map $D: A \rightarrow B$ is called a strong deformation map if $\mathrm{Gr}(D) \subset \mathbb{A}$ is a subalgebra. Such a map generalizes associative algebra homomorphisms, derivations, crossed homomorphisms and the associative analogue of modified {\sf r}-matrices. We introduce the cohomology of a strong deformation map $D$ unifying the cohomologies of all the operators mentioned above. We also define the governing algebra for the pair $(\mathbb{A}, D)$ to study simultaneous deformations of both $\mathbb{A}$ and $D$. On the other hand, a linear map $r: B \rightarrow A$ is called a weak deformation map if $\mathrm{Gr} (r) \subset \mathbb{A}$ is a subalgebra. Such a map generalizes relative Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators, left-averaging operators and right-averaging operators. Here we define the cohomology and governing algebra of a weak deformation map $r$ (that unify the cohomologies of all the operators mentioned above) and also for the pair $(\mathbb{A}, r)$ that govern simultaneous deformations., Comment: 28 pages; Comments are welcome

Hybrid organic-inorganic semiconductors with strong electron-phonon interactions provide a programmable platform for developing a variety of electronic, optoelectronic, and quantum materials by controlling these interactions. However, in current hybrid semiconductors, such as halide perovskites, anharmonic vibrations with rapid dephasing hinder the ability to coherently manipulate phonons. Here, we report the observation of long-lived coherent phonons in lead organic chalcogenides (LOCs), a new family of hybrid two-dimensional semiconductors. These materials feature harmonic phonon dynamics despite distorted lattices, combining long phonon dephasing times with tunable semiconducting properties. Dephasing time as long as 75 ps at 10 K, with up to 500 cycles of phonon oscillation between scattering events, was observed, corresponding to a dimensionless harmonicity parameter more than an order of magnitude larger than that of halide perovskites. The phonon dephasing time is significantly influenced by anharmonicity and centrosymmetry, both of which can be tuned through the design of the organic ligands thanks to the direct bonding between the organic and inorganic motifs. This research opens new opportunities for the manipulation of electronic properties with coherent phonons in hybrid semiconductors.

We investigate how the signatures of the topological properties of the bandstructures for nodal-point semimetals are embedded in the response coefficients, arising in two distinct experimental set-ups, by taking the Rarita-Schwinger-Weyl semimetal as an example. The first scenario involves the computation of third-rank tensors representing second-order response coefficients, relating the charge/thermal current densities to the combined effects of the gradient of the chemical potential and an external electric field/temperature gradient. On the premises that internode scatterings can be ignored, the relaxation-time approximation leads to a quantized value for the nonvanishing components of each of these nonlinear response tensors, characterizing a single untilted RSW node. Furthermore, the final expressions turn out to be insensitive to the specific values of the chemical potential and the temperature. The second scenario involves computing the magnetoelectric conductivity under the action of collinear electric ($\mathbf E$) and magnetic ($\mathbf B$) fields, representing a planar Hall set-up. In particular, our focus is in bringing out the dependence of the linear-in-$|\mathbf B|$ parts of the conductivity tensor on the intrinsic topological properties of the bandstructure, which are nonvanishing only in the presence of a nonzero tilt in the energy spectrum., Comment: 16 pages, 1 figure; contains the review of Boltzmann equations from arXiv:2408.01422

Topological insulators (TIs) are materials with an insulating bulk characterized by a gapped band structure, along with gapless metallic surface states having a Dirac cone with a helical spin structure in momentum space. The helical spin--momentum locking of the surface states arises from intrinsic spin--orbit coupling (SOC) and provides topological protection to the surface states against scattering from external perturbations, like defects and non-magnetic impurities. Breaking the topological protection of surface states of topological insulators is an essential prerequisite for exploring their applications. Rare-earth ions typically exhibit larger magnetic moments than transition-metal ions and thus promise the opening of a wider exchange gap in the Dirac surface states of topological insulators. Bi$_2$Se$_3$ is an interesting material; on the one hand, it has semiconducting properties when it is thin sheets; on the other hand, it's a topological insulator when the structure has a minimum of six quintuple layers (QLs), with diverse applications in photothermal, thermoelectric, and optical properties. Here, we have developed a controlled colloidal synthesis with low temperature and a cost-effective process for the synthesis of undoped and Eu-doped 2D layered Bi$_2$Se$_3$ nanosheets. Scanning tunneling spectroscopy measurements demonstrated a correlation between the shift of Dirac point position and the dopant content, which has been theoretically established by the band structure calculation. At low temperatures below 10K, for the 10\% Eu-doped sample, magnetic data suggests an antiferromagnetic ordering in the sample, which may be the contribution of the mixed valence state of Europium. Our results support that antiferromagnetic exchange interaction can exist in topological surface states in rare-earth Eu-doped Bi$_2$Se$_3$, which can open a new window to novel quantum phenomena.

Understanding the loop corrections to cosmological observables is of paramount importance for having control on the quantum consistency of a theory in an expanding universe as well as for phenomenological reasons. In the present work, we begin with a systematic study of such corrections in the context scalar toy models whose perturbative Bunch-Davies wavefunction enjoys an intrinsic definition in terms of cosmological polytopes, focusing on one-loop graphs. Owing to the underlying twisted period integral representation they admit, their combinatorial structure along with their vector space structure, emerging from polynomial ideals algebra and intersection theory, are exploited to set-up and analyse the differential equations that the two- and three-site one-loop corrections have to satisfy upon variation of the external kinematic variables. We find that, while the two-site contribution can be written in terms of multiple-polylogarithms, this is no longer true for the three-site case, for which elliptic structures appear. As a non-trivial check, we consider the scattering amplitude limit, recovering the known result in terms of polylogarithms only., Comment: 16 pages, 5 figures

We explore theoretically the properties of engineered bimetallic nano-structured superlattices where an array of nano-clusters of a simple (single band) metal are embedded periodically inside another simple metal with a different work function. The exploration is done using a simplified tight-binding model with Coulomb interactions included, as well as density functional theory. Taking arrays of "Ag" clusters of fixed sizes and configurations (when unrelaxed) embedded periodically in an "Au" matrix as an example, we show that a significant enhancement of electron-phonon interactions ensues, implying possibilities for high-$T_c$ superconductivity. The enhancement stems from a strong coupling, via Coulomb interactions, between the dipolar charge distribution that forms at the Au-Ag interfaces and the breathing and other modes of vibration of the light Ag atoms caged inside the heavier Au matrix. The interface dipoles form because of the interplay between the mismatch of the local potential seen by the conduction electrons localised in Wannier orbitals at the Ag and Au sites (the Ag sites being slightly repulsive relative to the Au sites) and the (long-range) Coulomb repulsion between electrons occupying these Wannier orbitals. We also discuss the DC transport in such systems., Comment: 24 pages, 17 figures

A new database of collisional rate coefficients for transitions between the rotational states of H$_2$O collided with H$_2$ background gas is developed. The goal is to expand over the other existing databases in terms of the rotational states of water (200 states are included here) and the rotational states of hydrogen (10 states). All four symmetries of ortho and para water combined with ortho, and para hydrogen are considered.The mixed quantum/classical theory of inelastic scattering implemented in the code MQCT is employed. A detailed comparison with previous databases is conducted to ensure that this approximate method is sufficiently accurate. Integration over collision energies, summation over the final states of H$_2$ and averaging over the initial states of H$_2$ is carried out to provide state-to-state, effective, and thermal rate coefficients in a broad range of temperatures.The rate coefficients for collisions with highly excited H$_2$ molecules are presented for the first time. It is found that rate coefficients for rotational transitions in H$_2$O molecules grow with the rotational excitation of H$_2$ projectiles and exceed those of the ground state H$_2$, roughly, by a factor of 2. These data enable more accurate description of water molecules in high-temperature environments, where the hydrogen molecules of background gas are rotationally excited, and the H$_2$O + H$_2$ collision energy is high. The rate coefficients presented here are expected to be accurate up to the temperature of $\sim$ 2000 K.

We conducted a comprehensive variability analysis of the blazar PKS 2255-282 using Fermi-LAT observations spanning over four years, from MJD 57783.5 to 59358.5. Our analysis revealed a transient quasi-periodic oscillation (QPO) with a period of 93$\pm$2.6 days. We employed a variety of Fourier-based methods, including the Lomb-Scargle Periodogram (LSP) and Weighted Wavelet Z-Transform (WWZ), as well as time domain analysis techniques such as Seasonal and Non-Seasonal Autoregressive Integrated Moving Average (ARIMA) models and the Stochastic modeling with Stochastically Driven Damped Harmonic Oscillator (SHO) models. Consistently, the QPO with a period of 93 days was detected across all methods used. The observed peak in LSP and time-averaged WWZ plots has a significance level of 4.06$\sigma$ and 3.96$\sigma$, respectively. To understand the source of flux modulations in the light curve, we explored various physical models. A plausible scenario involves the precession of the jet with a high Lorentz factor or the movement of a plasma blob along a helical trajectory within the relativistic jet., Comment: 13 pages, 13 Figures, accepted for publication in the Astrophysical Journal (APJ)