ShantanuT01/DACTYL-Pretraining huggingface dataset:
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that \vspace{1mm} \noindent {\bf Conjecture (Chowla).} {\em \begin{equation} \label{a.1} \sum_{n\le x} \lambda (f(n)) =o(x) \end{equation} for any polynomial $f(x)$ with integer coefficients which is not of form $bg(x)^2$. } \vspace{1mm} \noindent The prime number theorem is equivalent to \eqref{a.1} when $f(x)=x$. Chowla's conjecture is proved for linear functions but for the degree greater than 1, the conjecture seems to be extremely hard and still remains wide open. One can consider a weaker form of Chowla's conjecture, namely, \vspace{1mm} \noindent {\bf Conjecture 1 (Cassaigne, et al).} {\em If $f(x) \in \Z [x]$ and is not in the form of $bg^2(x)$ for some $g(x)\in \Z[x]$, then $\lambda (f(n))$ changes sign infinitely often.}   Clearly, Chowla's conjecture implies Conjecture 1. Although it is weaker, Conjecture 1 is still wide open for polynomials of degree $>1$. In this article, we study Conjecture 1 for the quadratic polynomials. One of our main theorems is   {\bf Theorem 1.} {\em Let $f(x) = ax^2+bx +c $ with $a>0$ and $l$ be a positive integer such that $al$ is not a perfect square. Then if the equation $f(n)=lm^2 $ has one solution $(n_0,m_0) \in \Z^2$, then it has infinitely many positive solutions $(n,m) \in \N^2$.}   As a direct consequence of Theorem 1, we prove some partial results of Conjecture 1 for quadratic polynomials are also proved by using Theorem 1.
We investigate the equilibrium and real-time properties of the spin correlation function $\langle \vec{S}_1\vec{S}_2 \rangle$ in the two-impurity Kondo model for different distances $R$ between the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of $\langle \vec{S}_1\vec{S}_2 \rangle$. For distances $R$ exceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings $J$ between the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain length scale introduced by finite temperature. The real-time dynamics after a sudden quench of the system reveals that correlations propagate through the conduction band with Fermi velocity. We identify two distinct timescales for the long time behavior which reflects that for small $J$ the system is driven by the RKKY interaction while for large $J$ the Kondo effect dominates. Interestingly, we find that at certain distances a one-dimensional dispersion obeying $\epsilon(k)=\epsilon(-k)$ may lead to a local parity conservation of the impurities such that $\langle \vec{S}_1\vec{S}_2 \rangle$ becomes a conserved quantity for long times and does not decay to its equilibrium value.
We consider an infinite graphene geometry where bonds and sites have been removed selectively to map it onto an effective Sierpinski gasket comprising of hexagons. We show that such a structure is capable of sustaining an infinite number of extended single particle states inspite of the absence of any translational order. When each basic hexagonal plaquette in the Sierpinski geometry is threaded by a magnetic flux, the spectrum exhibits bands of extended eigenstates. The bands persist for any arbitrary value of the flux but disappear again as the flux becomes equal to half the fundamental flux quantum. The localization - de-localization issues are discussed thoroughly along with the computation of two terminal magneto-transport of finite versions of the lattice. The numerical results corroborate our analytical findings.
Traffic Management Centers (TMCs) routinely use traffic cameras to provide situational awareness regarding traffic, road, and weather conditions. Camera footage is quite useful for a variety of diagnostic purposes; yet, most footage is kept for only a few days, if at all. This is largely due to the fact that currently, identification of notable footage is done via manual review by human operators---a laborious and inefficient process. In this article, we propose a semantics-oriented approach to analyzing sequential image data, and demonstrate its application for automatic detection of real-world, anomalous events in weather and traffic conditions. Our approach constructs semantic vector representations of image contents from textual labels which can be easily obtained from off-the-shelf, pretrained image labeling software. These semantic label vectors are used to construct semantic topic signals---time series representations of physical processes---using the Latent Dirichlet Allocation (LDA) topic model. By detecting anomalies in the topic signals, we identify notable footage corresponding to winter storms and anomalous traffic congestion. In validation against real-world events, anomaly detection using semantic topic signals significantly outperforms detection using any individual label signal.
We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully-flat covers, extending some results of [4]. As applications we will construct semi-orthogonal decompositions for root stacks of log pairs (X,D) where D is a (not necessarily simple) normal crossing divisors, generalizing results from [17] and [3]. Further we will compute the Kummer flat K-theory of general log pairs (X,D), generalizing earlier results of Hagihara and Nizio{\l} in the simple normal crossing case [15], [23].
We list all perfect colorings of $Z^2$ by 9 or less colors. Keywords: perfect colorings, equitable partitions
We shortly review the bi-local, collective dipole mechanism to a construction of Higher Spin AdS Gravity from the O(N) vector field theory. We also make some comments regarding an exact nonlinear comparison and in this regard we present a bi-local (symmetric) gauge description of Vasiliev's theory.
The detection of hazardous conditions near public transit stations is necessary for ensuring the safety and accessibility of public transit. Smart city infrastructures aim to facilitate this task among many others through the use of computer vision. However, most state-of-the-art computer vision models require thousands of images in order to perform accurate detection, and there exist few images of hazardous conditions as they are generally rare. In this paper, we examine the detection of snow-covered sidewalks along bus routes. Previous work has focused on detecting other vehicles in heavy snowfall or simply detecting the presence of snow. However, our application has an added complication of determining if the snow covers areas of importance and can cause falls or other accidents (e.g. snow covering a sidewalk) or simply covers some background area (e.g. snow on a neighboring field). This problem involves localizing the positions of the areas of importance when they are not necessarily visible.   We introduce a method that utilizes Structure from Motion (SfM) rather than additional annotated data to address this issue. Specifically, our method learns the locations of sidewalks in a given scene by applying a segmentation model and SfM to images from bus cameras during clear weather. Then, we use the learned locations to detect if and where the sidewalks become obscured with snow. After evaluating across various threshold parameters, we identify an optimal range at which our method consistently classifies different categories of sidewalk images correctly. Although we demonstrate an application for snow coverage along bus routes, this method can extend to other hazardous conditions as well. Code for this project is available at https://github.com/ind1010/SfM_for_BusEdge.
We present evidence for an exotic gapless insulating phase of hard-core bosons on multi-leg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at $\nu=1/3$ filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other $N$-leg systems and to $N$-layer two-dimensional structures.
We attract attention to the fact that the ultimate (shot-noise-limited) polarimetric sensitivity can be enhanced by orders of magnitude leaving the photon flux incident onto the photodetector on the same low level. This opportunity is of crucial importance for present-day spin noise spectroscopy, where a direct increase of sensitivity by increasing the probe beam power is strongly restricted by the admissible input power of the broadband photodetectors. The gain in sensitivity is achieved by replacing the 45-deg polarization geometry commonly used in conventional schemes with balanced detectors by geometries with stronger polarization extinction. The efficiency of these high-extinction polarization geometries with enhancement of the detected signal by more than an order of magnitude is demonstrated by measurements of the spin noise spectra of bulk n:GaAs in the spectral range 835-918 nm. It is shown that the inevitable growth of the probe beam power with the sensitivity gain makes spin noise spectroscopy much more perturbative, but, at the same time, opens up fresh opportunities for studying nonlinear interactions of strong light fields with spin ensembles.
We show that the energy gap induced by ferromagnetically aligned magnetic impurities on the surface of a topological insulator can be filled, due to scattering off the non-magnetic potential of the impurities. In both a continuum surface model and a three-dimensional tight-binding lattice model, we find that the energy gap disappears already at weak potential scattering as impurity resonances add spectral weight at the Dirac point. This can help explain seemingly contradictory experimental results as to the existence of a gap.
We have realized a tunable coupling over a large frequency range between an asymmetric Cooper pair transistor (charge qubit) and a dc SQUID (phase qubit). Our circuit enables the independent manipulation of the quantum states of each qubit as well as their entanglement. The measurements of the charge qubit's quantum states is performed by resonant read-out via the measurement of the quantum states of the SQUID. The measured coupling strength is in agreement with an analytic theory including a capacitive and a tunable Josephson coupling between the two qubits.
This paper proposes a deep neural network structure that exploits edge information in addressing representative low-level vision tasks such as layer separation and image filtering. Unlike most other deep learning strategies applied in this context, our approach tackles these challenging problems by estimating edges and reconstructing images using only cascaded convolutional layers arranged such that no handcrafted or application-specific image-processing components are required. We apply the resulting transferrable pipeline to two different problem domains that are both sensitive to edges, namely, single image reflection removal and image smoothing. For the former, using a mild reflection smoothness assumption and a novel synthetic data generation method that acts as a type of weak supervision, our network is able to solve much more difficult reflection cases that cannot be handled by previous methods. For the latter, we also exceed the state-of-the-art quantitative and qualitative results by wide margins. In all cases, the proposed framework is simple, fast, and easy to transfer across disparate domains.
This paper derives the asymptotic behavior of realized power variation of pure-jump It\^{o} semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled It\^{o} semimartingale over a fixed interval.
We demonstrate phase super-resolution in the absence of entangled states. The key insight is to use the inherent time-reversal symmetry of quantum mechanics: our theory shows that it is possible to \emph{measure}, as opposed to prepare, entangled states. Our approach is robust, requiring only photons that exhibit classical interference: we experimentally demonstrate high-visibility phase super-resolution with three, four, and six photons using a standard laser and photon counters. Our six-photon experiment demonstrates the best phase super-resolution yet reported with high visibility and resolution.
We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.
I provide a calculation at full two-loop order of the complex pole squared mass of the W boson in the Standard Model in the pure MS-bar renormalization scheme, with Goldstone boson mass effects resummed. This approach is an alternative to earlier ones that use on-shell or hybrid renormalization schemes. The renormalization scale dependence of the real and imaginary parts of the resulting pole mass are studied. Both deviate by about $\pm 4$ MeV from their median values as the renormalization scale is varied from 50 GeV to 200 GeV, but the theory error is likely larger. A surprising feature of this scheme is that the 2-loop QCD correction has a larger scale-dependence, but a smaller magnitude, than the 2-loop non-QCD correction, unless the renormalization scale is chosen very far from the top-quark mass.
For the task of semantic segmentation, high-resolution (pixel-level) ground truth is very expensive to collect, especially for high resolution images such as gigapixel pathology images. On the other hand, collecting low resolution labels (labels for a block of pixels) for these high resolution images is much more cost efficient. Conventional methods trained on these low-resolution labels are only capable of giving low-resolution predictions. The existing state-of-the-art label super resolution (LSR) method is capable of predicting high resolution labels, using only low-resolution supervision, given the joint distribution between low resolution and high resolution labels. However, it does not consider the inter-instance variance which is crucial in the ideal mathematical formulation. In this work, we propose a novel loss function modeling the inter-instance variance. We test our method on a real world application: infiltrating breast cancer region segmentation in histopathology slides. Experimental results show the effectiveness of our method.
We present newly discovered radio emission in the galaxy cluster Abell 2443 which is (1) diffuse, (2) extremely steep spectrum, (3) offset from the cluster center, (4) of irregular morphology and (5) not clearly associated with any of the galaxies within the cluster. The most likely explanation is that this emission is a cluster radio relic, associated with a cluster merger. We present deep observations of Abell 2443 at multiple low frequencies (1425, 325 and 74 MHz) which help characterize the spectrum and morphology of this relic. Based on the curved spectral shape of the relic emission and the presence of small scale structure, we suggest that this new source is likely a member of the radio phoenix class of radio relics.
We present the absolute properties of the eclipsing binary AO Ser with a pulsating component from our $BV$ photometric and high-resolution spectroscopic observations, which were performed between April and May 2017. The radial velocities (RVs) for both components were measured, and the effective temperature and projected rotational velocity of the primary star were determined to be $T_{\rm eff,1}$ = 8,820 $\pm$ 62 K and $v_1 \sin i_{1}$ = 90 $\pm$ 18 km s$^{-1}$, respectively, by comparing the observed spectrum with the Kurucz models. The accurate fundamental parameters of AO Ser were determined by a simultaneous analysis of the light and RV curves. The masses and radii of the primary and secondary components are $M_1$ = 2.55 $\pm$ 0.09 M$_\odot$ and $R_1$ = 1.64 $\pm$ 0.02 R$_\odot$ and $M_2$ = 0.49 $\pm$ 0.02 M$_\odot$ and $R_2$ = 1.38 $\pm$ 0.02 R$_\odot$, respectively. Multiple frequency analyses for the eclipse-subtracted light residuals were conducted. As a result, we detected two frequencies of $f_1$ = 21.852 days$^{-1}$ and $f_2$ = 23.484 days$^{-1}$. The evolutionary position on the HR diagram and the pulsational characteristics indicate that the primary star is a $\delta$ Sct pulsator with a radial fundamental mode. On the other hand, the relatively evolved secondary is oversized for its own mass.
Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and $\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal basin of attraction of stable distribution of index $\alpha\in(0,2]$ and $\beta\in(0,2]$. When $d=1$ and $\alpha\ne 1$, a functional limit theorem has been established in \cite{KestenSpitzer} and a local limit theorem in \cite{BFFN}. In this paper, we establish the convergence of the finite-dimensional distributions and a local limit theorem when $\alpha=d$ (i.e. $\alpha = d=1$ or $\alpha=d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in \cite{bolthausen} and recently in \cite{DU} in the particular case where $\beta=2$ (respectively for $\alpha=d=2$ and $\alpha=d=1$).
We determine the asymptotic behavior of the Arakelov metric, the Arakelov-Green's function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang's theory of admissible Green's functions on polarized metrized graphs.
A graph is Hamiltonian if it contains a cycle which visits every vertex of the graph exactly once. In this paper, we consider the problem of Hamiltonicity of a graph $G_n$, which will be called the prime difference graph of order $n$, with vertex set $\{1,2,\cdots, n\}$ and edge set $\{uv: |u-v|$ is a prime number$\}$. A recent result, conjectured by Sun and later proved by Chen, asserts that $G_n$ is Hamiltonian for $n\geq 5$. This paper extends their result in three directions. First, we prove that for any two integers $a$ and $b$ with $1\leq a<b\leq n$, there is a Hamilton path in $G_n$ from $a$ to $b$ except some cases of small $n$. This result implies robustness of the Hamiltonicity property of the prime difference graph in a sense that for any edge $e$ in $G_n$ there exists a Hamilton cycle containing $e$. Second, we show that the prime difference graph contains considerably more about the cycle structure than Hamiltonicity; precisely, for any integer $n\geq 7$, the prime difference graph $G_n$ contains any 2-factor of the complete graph of order $n$ as a subgraph. Finally, we find that $G_n$ may contain more edge-disjoint Hamilton cycles. In particular, these Hamilton cycles are generated by two prime differences.
We study the properties of the p-wave holographic superconductor for the scalar hairy black holes in the probe limit. The black hole solutions in question possess planar topology, which are derived from the Einstein gravity theory minimally coupled to a scalar field with a generic scalar potential. These solutions can be viewed as characterized by two independent parameters, namely, $\alpha$ and $k_0$, where AdS vacuum is manifestly restored when $\alpha\to \infty$. Consequently, the p-wave holographic superconductor is investigated by employing the above static planar black hole spacetime as the background metric, where a Maxwell field is introduced to the model by nonminimally coupling it to a complex vector field. The latter is shown to condensate and furnish the superconducting phase when the temperature is below a critical value. By numerical calculations, we examine in detail how the scalar field in the background affects the properties of the superconductivity. It is found that the critical temperature depends crucially on the parameters $\alpha$ and $k_0$, which subsequently affects the condensation process. By employing the Kubo formula, the real, as well as imaginary parts of the conductivity, are calculated and presented as functions of frequency. The results are discussed regarding the poles of the Green function, and the typical values of the BCS theory.
The well known black hole candidate H~1743-322 exhibited temporal and spectral variabilities during several outbursts. Daily variation of the accretion rates and the flow geometry change on a daily basis during each of the outbursts could be understood very well using the recent implementation of two component advective flow (TCAF) solution of the viscous transonic flow equations as an additive table model in XSPEC. This has dramatically improved our understanding about the accretion flow dynamics. Most interestingly, the solution allows to treat mass of the black hole candidate as a free parameter and there mass could be estimated from spectral fits. In this paper, we fit the data of two successive outbursts of H~1743-322 in 2010 and 2011 and studied evolutions of accretion flow parameters, such as, two component (Keplerian and sub-Keplerian) accretion rates, shock location (i.e., size of the Compton cloud), etc. We assume that the model Normalization remains the same accross the states in both these outbursts. We use this to estimate mass of the black hole and found that it comes out in the range of $9.25-12.86 M_\odot$. For the sake of comparison, we also estimated mass using Photon index vs. QPO frequency correlation method which turns out to be $11.65 \pm 0.67 M_\odot$ using GRO J1655-40 as reference source. Combining these two estimates, the most probable mass of the compact object becomes $11.21^{+1.65}_{-1.96} M_\odot$.
Some short-distance properties of the fourth-rank hadronic vacuum polarization tensor are re-examined.Their consequences are critically discussed in the context of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon.