Let $qge3$ be an integer, $chi$ be a Dirichlet character modulo $q$, and $L(s,chi)$ denote the Dirichlet $L$-functions corresponding to $chi$. In this paper, we show some special function series, and give some new identities for the Dirichlet $L$-fun
ctions involving Gauss sums. Specially, we give specific identities for $L(2,chi)$.
The synchrotron mechanism has the radiation limit of about 160 MeV, and it is not possible to explain the very high energy (VHE) photons that are emitted by high-energy objects. Inverse Compton scattering as a traditional process is applied for the e
xplanation of the VHE emission. In this paper, jitter radiation, the relativistic electron radiation in the random and small-scale magnetic field, is proposed to be a possible mechanism to produce VHE photons. The jitter radiation frequency is associated with the perturbation field. The spectral index of the jitter radiation is dominated by the kinetic turbulence. We utilize the jitter radiation to explain the gamma-ray burst (GRB 190114C and GRB 180720B) VHE emissions that were recently detected by the Imaging Atmospheric Cherenkov Telescopes. We suggest that this mechanism can be applied to other kinds of VHE sources.
Spectropolarimetric measurements of gamma-ray burst (GRB) optical afterglows contain polarization information for both continuum and absorption lines. Based on the Zeeman effect, an absorption line in a strong magnetic field is polarized and split in
to a triplet. In this paper, we solve the polarization radiative transfer equations of the absorption lines, and obtain the degree of linear polarization of the absorption lines as a function of the optical depth. In order to effectively measure the degree of linear polarization for the absorption lines, a magnetic field strength of at least $10^3$ G is required. The metal elements that produce the polarized absorption lines should be sufficiently abundant and have large oscillation strengths or Einstein absorption coefficients. We encourage both polarization measurements and high-dispersion observations of the absorption lines in order to detect the triplet structure in early GRB optical afterglows.
This study investigates the theoretical foundations of t-distributed stochastic neighbor embedding (t-SNE), a popular nonlinear dimension reduction and data visualization method. A novel theoretical framework for the analysis of t-SNE based on the gr
adient descent approach is presented. For the early exaggeration stage of t-SNE, we show its asymptotic equivalence to a power iteration based on the underlying graph Laplacian, characterize its limiting behavior, and uncover its deep connection to Laplacian spectral clustering, and fundamental principles including early stopping as implicit regularization. The results explain the intrinsic mechanism and the empirical benefits of such a computational strategy. For the embedding stage of t-SNE, we characterize the kinematics of the low-dimensional map throughout the iterations, and identify an amplification phase, featuring the intercluster repulsion and the expansive behavior of the low-dimensional map. The general theory explains the fast convergence rate and the exceptional empirical performance of t-SNE for visualizing clustered data, brings forth the interpretations of the t-SNE output, and provides theoretical guidance for selecting tuning parameters in various applications.
Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $bar{x}$ with $0<bar{x}<q$ such that $xbar{x}equiv1(bmod q)$. A Lehmer number is defined to be any integer $a$ with $2dagger(a+b
ar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=displaystylemathop {displaystylemathop{sum{}}_{a=1}^{q} displaystylemathop{sum{}}_{bleq xq}}_{mbox{$tinybegin{array}{c} 2|a+b+1 abequiv1(bmod q)end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymptotic formula, by using estimates of Kloostermans sums and properties of trigonometric sums.
We numerically study the three-dimensional (3D) quantum Hall effect (QHE) and magnetothermoelectric transport of Weyl semimetals in the presence of disorder. We obtain a bulk picture that the exotic 3D QHE emerges in a finite range of Fermi energy ne
ar the Weyl points determined by the gap between the $n=-1$ and $n=1$ Landau levels (LLs). The quantized Hall conductivity is attributable to the chiral zeroth LLs traversing the gap, and is robust against disorder scattering for an intermediate number of layers in the direction of the magnetic field. Moreover, we predict several interesting characteristic features of the thermoelectric transport coefficients in the 3D QHE regime, which can be probed experimentally. This may open an avenue for exploring Weyl physics in topological materials.
The detection of the gamma-ray burst (GRB) X-ray emission line is important for studying the GRB physics and constraining the GRB redshift. Since the line-like feature in the GRB X-ray spectrum was first reported in 1999, several works on line search
ing have been published over the past two decades. Even though some observations on the X-ray line-like feature were performed, the significance remains controversial to date. In this paper, we utilize the down-Comptonization mechanism and present the time evolution of the Fe K$alpha$ line emitted near the GRB central engine. The line intensity decreases with the evolution time, and the time evolution depends on the the electron density and the electron temperature. In addition, the initial line with a larger broadening decreases less over time. For instance, when the emission line penetrates material with the an electron density above $10^{12}$ cm$^{-3}$ at 1 keV, it generally becomes insignificant enough after 100 s for it not to be detected. The line-like profile deviates from the Gaussian form, and it finally changes to be similar to a blackbody shape at the time of the thermal equilibrium between the line photons and the surrounding material.
We statistically examine the gamma-ray burst (GRB) photon indices obtained by the Fermi-GBM and Fermi-LAT observations and compare the LAT GRB photon indices to the GBM GRB photon indices. We apply the jitter radiation to explain the GRB spectral div
ersities in the high-energy bands. In our model, the jitter radiative spectral index is determined by the spectral index of the turbulence. We classify GRBs into three classes depending on the shape of the GRB high-energy spectrum when we compare the GBM and LAT detections: the GRB spectrum is concave (GRBs turn out to be softer and are labeled as S-GRBs), the GRB spectrum is convex (GRBs turn out to be harder and are labeled as H-GRBs), and the GRBs have no strong spectral changes (labeled as N-GRBs). A universal Kolmogorov index 7/3 in the turbulent cascade is consistent with the photon index of the N-GRBs. The S-GRB spectra can be explained by the turbulent cascade due to the kinetic magnetic reconnection with the spectral index range of the turbulence from 8/3 to 3.0. The H-GRB spectra originate from the inverse turbulent cascade with the spectral index range of the turbulence from 2.0 to 3.5 that occurred during the large lengthscale magnetic reconnection. Thus, the GRB radiative spectra are diversified because the turbulent cascade modifies the turbulent energy spectrum. More observational samples are expected in the future to further identify our suggestions.
In this paper, we propose a deep convolutional neural network-based acoustic word embedding system on code-switching query by example spoken term detection. Different from previous configurations, we combine audio data in two languages for training i
nstead of only using one single language. We transform the acoustic features of keyword templates and searching content to fixed-dimensional vectors and calculate the distances between keyword segments and searching content segments obtained in a sliding manner. An auxiliary variability-invariant loss is also applied to training data within the same word but different speakers. This strategy is used to prevent the extractor from encoding undesired speaker- or accent-related information into the acoustic word embeddings. Experimental results show that our proposed system produces promising searching results in the code-switching test scenario. With the increased number of templates and the employment of variability-invariant loss, the searching performance is further enhanced.
Modularity is a popular metric for quantifying the degree of community structure within a network. The distribution of the largest eigenvalue of a networks edge weight or adjacency matrix is well studied and is frequently used as a substitute for mod
ularity when performing statistical inference. However, we show that the largest eigenvalue and modularity are asymptotically uncorrelated, which suggests the need for inference directly on modularity itself when the network size is large. To this end, we derive the asymptotic distributions of modularity in the case where the networks edge weight matrix belongs to the Gaussian Orthogonal Ensemble, and study the statistical power of the corresponding test for community structure under some alternative model. We empirically explore universality extensions of the limiting distribution and demonstrate the accuracy of these asymptotic distributions through type I error simulations. We also compare the empirical powers of the modularity based tests with some existing methods. Our method is then used to test for the presence of community structure in two real data applications.
