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Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by characterizing the singular values and singular vectors of the corresponding incidence matrix through group algebras, we prove a bound on the number of incidences between points and varieties of a certain form over finite fields. Our result leads to a new incidence bound for points and flats in finite geometries, which improves previous results for certain parameter regimes. As another application of our point-variety incidence bound, we prove a result on pinned distance problems by Phuong et al. and Cilleruelo et al. under a weaker condition., Comment: 20 pages

Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal point of research in coding theory. In this paper, we present several new combinatorial upper and lower bounds on the maximum size of $q$-ary insdel codes. Our main upper bound is a sphere-packing bound obtained by solving a linear programming (LP) problem. It improves upon previous results for cases when the distance $d$ or the alphabet size $q$ is large. Our first lower bound is derived from a connection between insdel codes and matchings in special hypergraphs. This lower bound, together with our upper bound, shows that for fixed block length $n$ and edit distance $d$, when $q$ is sufficiently large, the maximum size of insdel codes is $ \frac{q^{n-\frac{d}{2}+1}}{{n\choose \frac{d}{2}-1}}(1 \pm o(1))$. The second lower bound refines Alon et al.'s recent logarithmic improvement on Levenshtein's GV-type bound and extends its applicability to large $q$ and $d$., Comment: 20 pages

We present a fully-kinetic simulation of the full life cycle of strong Langmuir turbulence (SLT) excited by electron beams that are accelerated under the solar corona conditions. We find that (1) most packets ($\sim$80%) are affected by their neighbors during their collapse, as a result, their spatial scale variations present non-standard evolutionary features, i.e., deviating away from what was predicted by the Zakharov model; (2) the collapsing cavity is too shallow to trap the wave packet due to the growth of the Coulomb force, as a result a majority ($\sim$70%) of the packet energy runs away and a secondary localization may occur. The study indicates that the non-standard Langmuir collapse may play an important role in coronal plasmas interacting with an intense electron beam, that may be eventually confirmed by humanity's first mission to fly through the corona.

Where and how flares efficiently accelerate charged particles remains an unresolved question. Recent studies revealed that a "magnetic bottle" structure, which forms near the bottom of a large-scale reconnection current sheet above the flare arcade, is an excellent candidate for confining and accelerating charged particles. However, further understanding its role requires linking the various observational signatures to the underlying coupled plasma and particle processes. Here we present the first study combining multiwavelength observations with data-informed macroscopic magnetohydrodynamics and particle modeling in a realistic eruptive flare geometry. The presence of an above-the-loop-top magnetic bottle structure is strongly supported by the observations, which feature not only a local minimum of magnetic field strength but also abruptly slowing down plasma downflows. It also coincides with a compact hard X-ray source and an extended microwave source that bestrides above the flare arcade. Spatially resolved spectral analysis suggests that nonthermal electrons are highly concentrated in this region. Our model returns synthetic emission signatures that are well matched to the observations. The results suggest that the energetic electrons are strongly trapped in the magnetic bottle region due to turbulence, with only a small fraction managing to escape. The electrons are primarily accelerated by plasma compression and facilitated by a fast-mode termination shock via the Fermi mechanism. Our results provide concrete support for the magnetic bottle as the primary electron acceleration site in eruptive solar flares. They also offer new insights into understanding the previously reported small population of flare-accelerated electrons entering interplanetary space., Comment: 19 pages, 13 figures (12 pages and 10 figures for main text). Accepted for publication in The Astrophysical Journal. Minor proof corrections have been incorporated

The origin of multiple peaks in lightcurves of various wavelengths remains illusive during flares. Here we discuss the flare of SOL2023-05-09T03:54M6.5 with six flux peaks as recorded by a tandem of new microwave and Hard X-ray instruments. According to its microwave spectra, the flare represents a high-turnover frequency (>15 GHz) event. The rather-complete microwave and HXR spectral coverage provides a rare opportunity to uncover the origin of such event together with simultaneous EUV images. We concluded that (1) the microwave sources originates around the top section of the flaring loops with a trend of source spatial dispersion with frequency;(2) the visible movement of the microwave source from peak to peak originates from the process of new flaring loops appearing sequentially along the magnetic neutral line; 3) the optically-thin microwave spectra are hard with the indices varying from -1.2 to -0.4, and the turnover frequency always exceeds 15 GHz; 4) higher turnover/peak frequency corresponds to stronger peak intensity and harder optically-thin spectra. Using the Fokker-Planck and GX simulator codes we obtained a good fit to the observed microwave spectra and spatial distribution of the sources at all peaks, if assuming the radiating energetic electrons have the same spatial distribution and single-power-law spectra but with the number density varying in a range of 30%. We conclude that the particle acceleration in this flare happens in a compact region nearing the looptop. These results provide new constraints on the acceleration of energetic electrons and the underlying flare intermittent reconnection process., Comment: 23 pages, 11 figures

Batch codes are a type of codes specifically designed for coded distributed storage systems and private information retrieval protocols. These codes have got much attention in recent years due to their ability to enable efficient and secure storage in distributed systems. In this paper, we study an array code version of the batch codes, which is called the \emph{batch array code} (BAC). Under the setting of BAC, each node stores a bucket containing multiple code symbols and responds with a locally computed linear combination of the symbols in its bucket during the recovery of a requested symbol. We demonstrate that BACs can support the same type of requests as the original batch codes but with reduced redundancy. Specifically, we establish information theoretic lower bounds on the code lengths and provide several code constructions that confirm the tightness of the lower bounds for certain parameter regimes., Comment: 24 pages

Private information retrieval (PIR) codes and batch codes are two important types of codes that are designed for coded distributed storage systems and private information retrieval protocols. These codes have been the focus of much attention in recent years, as they enable efficient and secure storage and retrieval of data in distributed systems. In this paper, we introduce a new class of codes called \emph{$(s,t)$-batch codes}. These codes are a type of storage codes that can handle any multi-set of $t$ requests, comprised of $s$ distinct information symbols. Importantly, PIR codes and batch codes are special cases of $(s,t)$-batch codes. The main goal of this paper is to explore the relationship between the number of redundancy symbols and the $(s,t)$-batch code property. Specifically, we establish a lower bound on the number of redundancy symbols required and present several constructions of $(s,t)$-batch codes. Furthermore, we extend this property to the case where each request is a linear combination of information symbols, which we refer to as \emph{functional $(s,t)$-batch codes}. Specifically, we demonstrate that simplex codes are asymptotically optimal functional $(s,t)$-batch codes, in terms of the number of redundancy symbols required, under certain parameter regime., Comment: 25 pages

Modern large-scale distributed storage systems use erasure codes to protect against node failures with low storage overhead. In practice, the failure rate and other factors of storage devices in the system may vary significantly over time, and leads to changes of the ideal code parameters. To maintain the storage efficiency, this requires the system to adjust parameters of the currently used codes. The changing process of code parameters on encoded data is called code conversion. As an important class of storage codes, locally repairable codes (LRCs) can repair any codeword symbol using a small number of other symbols. This feature makes LRCs highly efficient for addressing single node failures in the storage systems. In this paper, we investigate the code conversions for locally repairable codes in the merge regime. We establish a lower bound on the access cost of code conversion for general LRCs and propose a general construction of LRCs that can perform code conversions with access cost matching this bound. This construction provides a family of LRCs together with optimal conversion process over the field of size linear in the code length., Comment: 25 pages

Ground level enhancements (GLEs) are extreme solar energetic particle (SEP) events that are of particular importance in space weather. In solar cycle 24, two GLEs were recorded on 2012 May 17 (GLE 71) and 2017 September 10 (GLE 72), respectively, by a range of advanced modern instruments. Here we conduct a comparative analysis of the two events by focusing on the effects of large-scale magnetic field configuration near active regions on particle acceleration and release. Although the active regions both located near the western limb, temporal variations of SEP intensities and energy spectra measured in-situ display different behaviors at early stages. By combining a potential field model, we find the CME in GLE 71 originated below the streamer belt, while in GLE 72 near the edge of the streamer belt. We reconstruct the CME shock fronts with an ellipsoid model based on nearly simultaneous coronagraph images from multi-viewpoints, and further derive the 3D shock geometry at the GLE onset. The highest-energy particles are primarily accelerated in the shock-streamer interaction regions, i.e., likely at the nose of the shock in GLE 71 and the eastern flank in GLE 72, due to quasi-perpendicular shock geometry and confinement of closed fields. Subsequently, they are released to the field lines connecting to near-Earth spacecraft when the shocks move through the streamer cusp region. This suggests that magnetic structures in the corona, especially shock-streamer interactions, may have played an important role in the acceleration and release of the highest-energy particles in the two events., Comment: Accepted for publication in ApJ

Solar coronal waves frequently appear as bright disturbances that propagate globally from the eruption center in the solar atmosphere, just like the tsunamis in the ocean on Earth. Theoretically, coronal waves can sweep over the underlying chromosphere and leave an imprint in the form of Moreton wave, due to the enhanced pressure beneath their coronal wavefront. Despite the frequent observations of coronal waves, their counterparts in the chromosphere are rarely detected. Why the chromosphere rarely bears the imprints of solar tsunamis remained a mystery since their discovery three decades ago. To resolve this question, all coronal waves and associated Moreton waves in the last decade have been initially surveyed, though the detection of Moreton waves could be hampered by utilising the low-quality H$\alpha$ data from Global Oscillations Network Group. Here, we present 8 cases (including 5 in Appendix) of the coexistence of coronal and Moreton waves in inclined eruptions where it is argued that the extreme inclination is key to providing an answer to address the question. For all these events, the lowest part of the coronal wavefront near the solar surface appears very bright, and the simultaneous disturbances in the solar transition region and the chromosphere predominantly occur beneath the bright segment. Therefore, evidenced by observations, we propose a scenario for the excitation mechanism of the coronal-Moreton waves in highly inclined eruptions, in which the lowest part of a coronal wave can effectively disturb the chromosphere even for a weak (e.g., B-class) solar flare., Comment: 24 pages, 14 figures

Preflare activities contain critical information about the pre-cursors and causes of solar eruptions. Here we investigate the characteristics and origin of a group of broadband pulsations (BBPs) in the decimetric-metric wavelengths, taking place during the preflare stage of the M7.1 flare dated on 2011 September 24. The event was recorded by multiple solar instruments including the Nan\c{c}ay Radioheliograh that measure the properties of the radio source. The BBPs start $\sim$24 min before the flare onset, extending from $<$ 360 to above 800 MHz with no discernible spectral drift. The BBPs consist of two stages, during the first stage the main source remains stationary, during the second stage it moves outward along with a steepening extreme-ultraviolet (EUV) wave driven by the eruption of a high-temperature structure. In both stages, we observe frequent EUV brightenings and jets originating from the flare region. During the second stage, the BBPs become denser in number and stronger in general, with the level of the polarization increasing gradually from $<$ 20% to $>$ 60% in the right-handed sense. These observations indicate the steepening EUV wave is important to the BBPs during the second stage, while the preflare reconnections causing the jets and EUV brightenings are important in both stages. This is the first time such a strong association of an EUV wave with BBPs is reported. We suggest a reconnection plus shock-sweeping-across-loop scenario for the cause of the BBPs., Comment: 17 pages, 7 figures

The acceleration and transport of energetic electrons during solar flares is one of the outstanding topics in solar physics. Recent X-ray and radio imaging and spectroscopy observations have provided diagnostics of the distribution of nonthermal electrons and suggested that, in certain flare events, electrons are primarily accelerated in the loop-top and likely experience trapping and/or scattering effects. By combining the focused particle transport equation with magnetohydrodynamic (MHD) simulations of solar flares, we present a macroscopic particle model that naturally incorporates electron acceleration and transport. Our simulation results indicate that the physical processes such as turbulent pitch-angle scattering can have important impacts on both electron acceleration in the loop-top and transport in the flare loop, and their influences are highly energy dependent. A spatial-dependent turbulent scattering with enhancement in the loop-top can enable both efficient electron acceleration to high energies and transport of abundant electrons to the footpoints. We further generate spatially resolved synthetic hard X-ray (HXR) emission images and spectra, revealing both the loop-top and footpoint HXR sources. Similar to the observations, we show that the footpoint HXR sources are brighter and harder than the loop-top HXR source. We suggest that the macroscopic particle model provides new insights into understanding the connection between the observed loop-top and footpoint nonthermal emission sources by combining the particle model with dynamically evolving MHD simulations of solar flares., Comment: Accepted for publication in ApJL

The standard theory of plasma emission is based on kinetic couplings between a single beam of energetic electrons and unmagnetized thermal plasmas, involving multi-step nonlinear wave-particle and wave-wave interactions. The theory has not yet been completely verified with fully-kinetic electromagnetic particle-in-cell (PIC) simulations. Earlier studies, greatly limited by available computational resources, are controversial regarding whether the fundamental emission can be generated according to the standard theory. To resolve the controversy, we conducted PIC simulations with a large domain of simulation and a large number of macroparticles, among the largest ones of similar studies. We found significant fundamental emission if the relative beam density is small enough (say, $\le$ 0.01), in line with earlier study with a much-smaller domain; the relative intensity (normalized by the total initial beam energy) of all modes, except the mode associated with the beam-electromagnetic Weibel instability, decreases with increasing relative density of the beam. We also found significant transverse magnetic component associated with the superluminal Langmuir turbulence, which has been mistakenly regarded as evidence of the F emission in earlier study. Further investigations are required to reveal their origin.

Recent years, several new types of codes were introduced to provide fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) have played an important role. A linear code is said to have locality $r$ if each of its code symbols can be repaired by accessing at most $r$ other code symbols. For an LRC with length $n$, dimension $k$ and locality $r$, its minimum distance $d$ was proved to satisfy the Singleton-like bound $d\leq n-k-\lceil k/r\rceil+2$. Since then, many works have been done for constructing LRCs meeting the Singleton-like bound over small fields. In this paper, we study quaternary LRCs meeting Singleton-like bound through a parity-check matrix approach. Using tools from finite geometry, we provide some new necessary conditions for LRCs being optimal. From this, we prove that there are $27$ different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented., Comment: 23 pages, the Chinese version of this paper will appear in SCIENTIA SINICA Mathematica (DOI: 10.1360/SSM-2022-0041)

Recent observations have shown that in many large solar energetic particle (SEP) events the event-integrated differential spectra resemble double power laws. We perform numerical modeling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation, including protons and heavier ions. We find that for all ion species the energy spectra integrated over the simulation domain can be described by a double power law, and the break energy depends on the ion charge-to-mass ratio as $E_B \sim (Q/A)^\alpha$, with $\alpha$ varying from 0.16 to 1.2 by considering different turbulence spectral indices. We suggest that the double power law distribution may emerge as a result of the superposition of energetic particles from different source regions where the acceleration rates differ significantly due to particle diffusion. The diffusion and mixing of energetic particles could also provide an explanation for the increase of Fe/O at high energies as observed in some SEP events. Although further mixing processes may occur, our simulations indicate that either power-law break or rollover can occur near the Sun and predict that the spectral forms vary significantly along the shock front, which may be examined by upcoming near-Sun SEP measurements from Parker Solar Probe and Solar Orbiter., Comment: Accepted for publication in The Astrophysical Journal Letters

A number of double coronal X-ray sources have been observed during solar flares by RHESSI, where the two sources reside at different sides of the inferred reconnection site. However, where and how are these X-ray-emitting electrons accelerated remains unclear. Here we present the first model of the double coronal hard X-ray (HXR) sources, where electrons are accelerated by a pair of termination shocks driven by bi-directional fast reconnection outflows. We model the acceleration and transport of electrons in the flare region by numerically solving the Parker transport equation using velocity and magnetic fields from the macroscopic magnetohydrodynamic simulation of a flux rope eruption. We show that electrons can be efficiently accelerated by the termination shocks and high-energy electrons mainly concentrate around the two shocks. The synthetic HXR emission images display two distinct sources extending to $>$100 keV below and above the reconnection region, with the upper source much fainter than the lower one. The HXR energy spectra of the two coronal sources show similar spectral slopes, consistent with the observations. Our simulation results suggest that the flare termination shock can be a promising particle acceleration mechanism in explaining the double-source nonthermal emissions in solar flares., Comment: Accepted for publication in ApJ

Previous studies on the beam-driven plasma emission process were done mainly for unmagnetized plasmas. Here we present fully-kinetic electromagnetic particle-in-cell simulations to investigate such process in weakly-magnetized plasmas of the solar corona conditions. The primary mode excited is the beam-Langmuir (BL) mode via the classical bump-on-tail instability. Other modes include the whistler (W) mode excited by the electron cyclotron resonance instability, the generalized Langmuir (GL) waves that include a superluminal Z-mode component with smaller wave number $k$ and a thermal Langmuir component with larger $k$, and the fundamental (F) and harmonic (H) branches of plasma emission. Further simulations of different mass and temperature ratios of electrons and protons indicate that the GL mode and the two escaping modes (F and H) correlate positively with the BL mode in intensity, supporting that they are excited through nonlinear wave-wave coupling processes involving the BL mode. We suggest that the dominant process is the decay of the primary BL mode. This is consistent with the standard theory of plasma emission. Yet, the other possibility of the Z+W$\rightarrow$O--F coalescing process for the F emission cannot be ruled out completely.

Compound eruptions represent that multiple closely spaced magnetic structures erupt consecutively within a short interval, and then lead to a single flare and a single CME. However, it is still subtle for the links between multiple eruptions and the associated single flare or/and single CME. In this Letter, we report the compound eruptions of twin close flux ropes (FR1 and FR2) within a few minutes that resulted in a flare with a single soft X-ray peak and a CME with two cores. The successive groups of expanding loops and double peaks of intensity flux in AIA cool wavelengths indicate two episodes of internal magnetic reconnections during the compound eruptions. Following the eruption of FR2, the erupting FR1 was accelerated, and then the expanding loops overlying FR2 were deflected. Moreover, the eruption of FR2 likely involved the external magnetic reconnection between the bottom of the overlying stretching field lines and the rebounding loops that were previously pushed by the eruption of FR1, which was evidenced by a pair of groups of newly-formed loops. All results suggest that the compound eruptions involved both internal and external magnetic reconnections, and two erupting structures of twin FRs interacted at the initial stage. We propose that two episodes of internal magnetic reconnections were likely united within a few minutes to form the continuous impulsive phase of the single peaked flare, and two separated cores of the CME was possibly because that the latter core was too slow to merge with the former one.

Electron cyclotron maser emission (ECME) is regarded as a plausible source for the coherent radio radiations from solar active regions (e.g., solar radio spikes). In this Letter, we present a 2D3V fully kinetic electromagnetic particle-in-cell (PIC) simulation to investigate the wave excitations and subsequent nonlinear processes induced by the energetic electrons in the loss-cone distribution. The ratio of the plasma frequency to the electron gyrofrequency ${\omega}_{pe}/{\Omega}_{ce}$ is set to 0.25, adequate for solar active region conditions. As a main result, we obtain strong emissions at the second-harmonic X mode (X2). While the fundamental X mode (X1) and the Z mode are amplified directly via the electron cyclotron maser instability, the X2 emissions can be produced by the nonlinear coalescence between two Z modes and between Z and X1 modes. This represents a novel generation mechanism for the harmonic emissions in plasmas with a low value of ${\omega}_{pe}/{\Omega}_{ce}$, which may resolve the escaping difficulty of explaining solar radio emissions with the ECME mechanism., Comment: 17 pages, 4 figures

Content. Electron-cyclotron maser emission (ECME) is the favored mechanism for solar radio spikes and has been investigated extensively since the 1980s. Most studies relevant to solar spikes employ a loss-cone-type distribution of energetic electrons, generating waves mainly in the fundamental X/O mode (X1/O1), with a ratio of plasma oscillation frequency to electron gyrofrequency (${\omega}_ {pe}/{\Omega}_{ce}$) lower than 1. Despite the great progress made in this theory, one major problem is how the fundamental emissions pass through the second-harmonic absorption layer in the corona and escape. This is generally known as the escaping difficulty of the theory. Aims. We study the harmonic emissions generated by ECME driven by energetic electrons with the horseshoe distribution to solve the escaping difficulty of ECME for solar spikes. Methods. We performed a fully kinetic electromagnetic PIC simulation with ${\omega}_ {pe}/{\Omega}_{ce}$ = 0.1, corresponding to the strongly magnetized plasma conditions in the flare region, with energetic electrons characterized by the horseshoe distribution. We also varied the density ratio of energetic electrons to total electrons ($n_e/n_0$) in the simulation. Results. We obtain efficient amplification of waves in Z and X2 modes, with a relatively weak growth of O1 and X3. With a higher-density ratio, the X2 emission becomes more intense, and the rate of energy conversion from energetic electrons into X2 modes can reach $\sim$0.06% and 0.17%, with $n_e/n_0$= 5% and 10%, respectively. Conclusions. We find that the horseshoe-driven ECME can lead to an efficient excitation of X2 and X3 with a low value of ${\omega}_ {pe}/{\Omega}_{ce}$, providing novel means for resolving the escaping difficulty of ECME when applied to solar radio spikes. The simultaneous growth of X2 and X3 can be used to explain some harmonic structures observed in solar spikes., Comment: 10 pages, 9 figures

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,\delta)$-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. \cite{Prakash2012Optimal} are considered. Through parity check matrix approach, constructions of both optimal $(r,\delta)$-LRCs with all symbol locality ($(r,\delta)_a$-LRCs) and optimal $(r,\delta)$-LRCs with information locality ($(r,\delta)_i$-LRCs) are provided. As a generalization of a work of Xing and Yuan \cite{XY19}, these constructions are built on a connection between sparse hypergraphs and optimal $(r,\delta)$-LRCs. With the help of constructions of large sparse hypergraphs, the length of codes constructed can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3\delta+1$. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed., Comment: 21 pages

A fast-mode shock can form in the front of reconnection outflows and has been suggested as a promising site for particle acceleration in solar flares. Recent development of magnetic reconnection has shown that numerous plasmoids can be produced in a large-scale current layer. Here we investigate the dynamical modulation of electron acceleration in the looptop region when plasmoids intermittently arrive at the shock by combining magnetohydrodynamics simulations with a particle kinetic model. As plasmoids interact with the shock, the looptop region exhibits various compressible structures that modulate the production of energetic electrons. The energetic electron population varies rapidly in both time and space. The number of 5$-$10 keV electrons correlates well with the area with compression, while that of $>$50 keV electrons shows good correlation with strong compression area but only moderate correlation with shock parameters. We further examine the impacts of the first plasmoid, which marks the transition from a quasi-steady shock front to a distorted and dynamical shock. The number of energetic electrons is reduced by $\sim 20\%$ at 15$-$25 keV and nearly 40\% for 25$-$50 keV, while the number of 5$-$10 keV electrons increases. In addition, the electron energy spectrum above 10 keV evolves softer with time. We also find double or even multiple distinct sources can develop in the looptop region when the plasmoids move across the shock. Our simulations have strong implications to the interpretation of nonthermal looptop sources, as well as the commonly observed fast temporal variations in flare emissions, including the quasi-periodic pulsations., Comment: accepted for publication in ApJL

Ever since the famous Erd\H{o}s-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, the authors proposed a new quantitative intersection problem for families of subsets: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection number} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection number among all families in ${[n]\choose k}$ with the same family size? In \cite{KG2020}, the authors studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $|\mathcal{F}|$ and characterize the relation between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes., Comment: 33 pages, to appear in SCIENCE CHINA Mathematics

In the standard model of solar flares, a large-scale reconnection current sheet is postulated as the central engine for powering the flare energy release and accelerating particles. However, where and how the energy release and particle acceleration occur remain unclear due to the lack of measurements for the magnetic properties of the current sheet. Here we report the measurement of spatially-resolved magnetic field and flare-accelerated relativistic electrons along a current-sheet feature in a solar flare. The measured magnetic field profile shows a local maximum where the reconnecting field lines of opposite polarities closely approach each other, known as the reconnection X point. The measurements also reveal a local minimum near the bottom of the current sheet above the flare loop-top, referred to as a "magnetic bottle". This spatial structure agrees with theoretical predictions and numerical modeling results. A strong reconnection electric field of ~4000 V/m is inferred near the X point. This location, however, shows a local depletion of microwave-emitting relativistic electrons. These electrons concentrate instead at or near the magnetic bottle structure, where more than 99% of them reside at each instant. Our observations suggest that the loop-top magnetic bottle is likely the primary site for accelerating and/or confining the relativistic electrons., Comment: 16 pages, 8 figures

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is to characterize its extremal structure and its corresponding stability. The famous Erd\H{o}s-Ko-Rado theorem answered both direct and inverse problems and led the era of studying intersection problems for finite sets. In this paper, we consider the following quantitative intersection problem which can be viewed an inverse problem for Erd\H{o}s-Ko-Rado type theorems: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection among all families in ${[n]\choose k}$ with the same family size? Using a pure combinatorial approach, we provide two structural characterizations of the optimal family of given size that maximizes the total intersection. As a consequence, for $n$ large enough and $\mathcal{F}$ of proper size, these characterizations show that the optimal family $\mathcal{F}$ is indeed $t$-intersecting ($t\geq 1$). To a certain extent, this reveals the relationship between properties of being intersecting and maximizing the total intersection. Also, we provide an upper bound on $\mathcal{I}(\mathcal{F})$ for several ranges of $|\mathcal{F}|$ and determine the unique optimal structure for families with sizes of certain values., Comment: 36 pages

Nonthermal loop-top sources in solar flares are the most prominent observational signature that suggests energy release and particle acceleration in the solar corona. Although several scenarios for particle acceleration have been proposed, the origin of the loop-top sources remains unclear. Here we present a model that combines a large-scale magnetohydrodynamic simulation of a two-ribbon flare with a particle acceleration and transport model for investigating electron acceleration by a fast-mode termination shock at the looptop. Our model provides spatially resolved electron distribution that evolves in response to the dynamic flare geometry. We find a concave-downward magnetic structure located below the flare termination shock, induced by the fast reconnection downflows. It acts as a magnetic trap to confine the electrons at the looptop for an extended period of time. The electrons are energized significantly as they cross the shock front, and eventually build up a power-law energy spectrum extending to hundreds of keV. We suggest that this particle acceleration and transport scenario driven by a flare termination shock is a viable interpretation for the observed nonthermal loop-top sources., Comment: submitted to ApJL

Let $\{(A_i,B_i)\}_{i=1}^{m}$ be a collection of pairs of sets with $|A_i|=a$ and $|B_i|=b$ for $1\leq i\leq m$. Suppose that $A_i\cap B_j=\emptyset$ if and only if $i=j$, then by the famous Bollob\'{a}s theorem, we have the size of this collection $m\leq {a+b\choose a}$. In this paper, we consider a variant of this problem by setting $\{A_i\}_{i=1}^{m}$ to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollob\'{a}s type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we have a similar theorem for finite sets, which settles a recent conjecture of Gerbner et. al \cite{GKMNPTX2019}. Moreover, we also determine the unique extremal structure of $\{(A_i,B_i)\}_{i=1}^{m}$ for the primary case of the theorem for finite sets.

We present numerical modelling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation with spatial diffusion both along and across the magnetic field. We show that the location on the shock where the high-energy particle intensity is the largest, depends on the energy of the particles and on time. The acceleration of particles to more than 100 MeV mainly occurs in the shock-streamer interaction region, due to perpendicular shock geometry and the trapping effect of closed magnetic fields. A comparison of the particle spectra to that in a radial magnetic field shows that the intensity at 100 MeV (200 MeV) is enhanced by more than one order (two orders) of magnitude. This indicates that the streamer-like magnetic field can be an important factor in producing large solar energetic particle events. We also show that the energy spectrum integrated over the simulation domain consists of two different power laws. Further analysis suggests that it may be a mixture of two distinct populations accelerated in the streamer and open field regions, where the acceleration rate differs substantially. Our calculations also show that the particle spectra are affected considerably by a number of parameters, such as the streamer tilt angle, particle spatial diffusion coefficient, and shock compression ratio. While the low-energy spectra agree well with standard diffusive shock acceleration theory, the break energy ranges from $\sim$1 MeV to $\sim$90 MeV and the high-energy spectra can extend to $\sim$1 GeV with a slope of $\sim$2-3.

It has been suggested that the Z-mode instability driven by energetic electrons with a loss-cone type velocity distribution is one candidate process behind the continuum and zebra pattern of solar type-IV radio bursts. Both the temperature of background plasma ($T_0$) and the energy of energetic electrons ($v_e$) are considered to be important to the variation of the maximum growth rate ($\gamma_{max}$). Here we present a detailed parameter study on the effect of $T_0$ and $v_e$, within a regime of the frequency ratio ($10 \leq \frac{\omega_{pe}}{\Omega_{ce}} \leq 30$). In addition to $\gamma_{max}$, we also analyze the effect on the corresponding wave frequency ($\omega^r_{max}$) and propagation angle ($\theta_{max}$). We find that (1) $\gamma_{max}$ in-general decreases with increasing $v_e$, while its variation with $T_0$ is more complex depending on the exact value of $v_e$; (2) with increasing $T_0$ and $v_e$, $\omega^r_{max}$ presents step-wise profiles with jumps separated by gradual or very-weak variations, and due to the warm-plasma effect on the wave dispersion relation $\omega^r_{max}$ can vary within the hybrid band (the harmonic band containing the upper hybrid frequency) and the band higher; (3) the propagation is either perpendicular or quasi-perpendicular, and $\theta_{max}$ presents variations in line with those of $\omega^r_{max}$, as constrained by the resonance condition. We also examine the profiles of $\gamma_{max}$ with $\frac{\omega_{pe}}{\Omega_{ce}}$ for different combinations of $T_0$ and $v_e$ to clarify some earlier calculations which show inconsistent results.

As a crucial technique for integrated circuits (IC) test response compaction, $X$-compact employs a special kind of codes called $X$-codes for reliable compressions of the test response in the presence of unknown logic values ($X$s). From a combinatorial view point, Fujiwara and Colbourn \cite{FC2010} introduced an equivalent definition of $X$-codes and studied $X$-codes of small weights that have good detectability and $X$-tolerance. An $(m,n,d,x)$ $X$-code is an $m\times n$ binary matrix with column vectors as its codewords. The parameters $d,x$ correspond to the test quality of the code. In this paper, bounds and constructions for constant weighted $X$-codes are investigated. First, we obtain a general result on the maximum number of codewords $n$ for an $(m,n,d,x)$ $X$-code of weight $w$, and we further improve this lower bound for the case with $x=2$ and $w=3$ through the probabilistic method. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted $X$-codes with $d=3,7$ and $x=2$, which are optimal for the case when $d=3, w=4$ and nearly optimal for the case when $d=3,w=3$. We also consider a special class of $X$-codes introduced in \cite{FC2010} and improve the best known lower bound on the maximum number of codewords for this kind of $X$-codes., Comment: 14 pages, 2 figures, 1 table; accept by IEEE Transactions on Information Theory

In recent years, the rapidly increasing amounts of data created and processed through the internet resulted in distributed storage systems employing erasure coding based schemes. Aiming to balance the tradeoff between data recovery for correlated failures and efficient encoding and decoding, distributed storage systems employing maximally recoverable codes came up. Unifying a number of topologies considered both in theory and practice, Gopalan et al. \cite{Gopalan2017} initiated the study of maximally recoverable codes for grid-like topologies. In this paper, we focus on the maximally recoverable codes that instantiate grid-like topologies $T_{m\times n}(1,b,0)$. To characterize the property of codes for these topologies, we introduce the notion of \emph{pseudo-parity check matrix}. Then, using the Combinatorial Nullstellensatz, we establish the first polynomial upper bound on the field size needed for achieving the maximal recoverability in topologies $T_{m\times n}(1,b,0)$. And using hypergraph independent set approach, we further improve this general upper bound for topologies $T_{4\times n}(1,2,0)$ and $T_{3\times n}(1,3,0)$. By relating the problem to generalized \emph{Sidon sets} in $\mathbb{F}_q$, we also obtain non-trivial lower bounds on the field size for maximally recoverable codes that instantiate topologies $T_{4\times n}(1,2,0)$ and $T_{3\times n}(1,3,0)$., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1605.05412 by other authors

In eruptive solar flares, termination shocks (TSs), formed when high-speed reconnection outflows collide with closed dense flaring loops, are believed to be one of the possible candidates for plasma heating and particle acceleration. In this work, we perform resistive magnetohydrodynamic simulations in a classic Kopp-Pneuman flare configuration to study the formation and evolution of TSs, and analyze in detail the dynamic features of TSs and variations of the shock strength in space and time. This research focuses on the fast reconnection phase when plasmoids form and produce small-scale structures inside the flare current sheet. It is found that the TS emerges once the downward outflow colliding with closed magnetic loops becomes super-magnetosonic, and immediately becomes highly dynamical. The morphology of a TS can be flat, oblique, or curved depending on the detailed interactions between the outflows/plasmoids and the highly dynamic plasma in the looptop region. The TS becomes weaker when a plasmoid is crossing through, or may even be destroyed by well developed plasmoids and then re-constructed above the plasmoids. We also perform detailed statistical analysis on important physical quantities along and across the shock front. The density and temperature ratios range from 1 to 3 across the TS front, and the pressure ratio typically has larger values up to 10. We show that weak guide fields do not strongly affect the Mach number and compression ratios, and the TS length becomes slightly larger in the case with thermal conduction., Comment: accepted for publication in ApJ

Latest observational reports of solar flares reveal some uncommon features of microwave spectra, such as unusually hard (or even positive) spectra, and/or a super-high peak frequency. For a better understanding of these features, we conduct a parameter study to investigate the effect of broken-power-law spectra of energetic electrons on microwave emission on the basis of gyrosynchrotron mechanism. The electron broken-power-law energy distribution is characterized by three parameters, the break energy ($E_B$), the power-law indices below ($\delta_1$) and above ($\delta_2$) the break energy. We find that with the addition of the $\delta_2$ component of the electron spectra, the total flux density can increase by several times in the optically-thick regime, and by orders of magnitude in the optically-thin regime; the peak frequency ($\nu_p$) also increases and can reach up to tens of GHz; and the degree of polarization ($r_c$) decreases in general. We also find that (1) the variation of the flux density is much larger in the optically-thin regime, and the microwave spectra around the peak frequency manifest various profiles with the softening or soft-hard pattern; (2) the parameters $\delta_1$ and $E_B$ affect the microwave spectral index ($\alpha$) and the degree of polarization ($r_c$) mainly in the optically-thick regime, while $\delta_2$ mainly affects the optically-thin regime. The results are helpful in understanding the lately-reported microwave bursts with unusual spectral features and point out the demands for a more-complete spectral coverage of microwave bursts, especially, in the high-frequency regime, say, $>10-20$ GHz.

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$, $B\in\mathcal{B}$. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton's and Frankl--Tohushige's on cross-intersecting families in the single-part setting., Comment: 17 pages

This paper presents the latest observations from the newly-built solar radio spectrograph at the \emph{Chashan Solar Observatory}. On July 18 2016, the spectrograph records a solar spike burst event, which has several episodes showing harmonic structures, with the second, third, and fourth harmonics. The lower harmonic radio spike emissions are observed later than the higher harmonic bands, and the temporal delay of the second (third) harmonic relative to the fourth harmonic is about 30\ --\ 40 (10) ms. Based on the electron cyclotron maser emission mechanism, we analyze possible causes of the temporal delay and further infer relevant coronal parameters, such as the magnetic field strength and the electron density at the radio source., Comment: 17 pages,6 figures

Recent observations have shown that coronal shocks driven by coronal mass ejections can develop and accelerate particles within several solar radii in large solar energetic particle (SEP) events. Motivated by this, we present an SEP acceleration study including the process that a fast shock propagates through a streamer-like magnetic field with both closed and open field lines in the low corona region. The acceleration of protons is modeled by numerically solving the Parker transport equation with spatial diffusion both along and across the magnetic field. We show that particles can be sufficiently accelerated to up to several hundred MeV within 2-3 solar radii. When the shock propagates through a streamer-like magnetic field, particles are more efficiently accelerated compared to the case with a simple radial magnetic field, mainly due to perpendicular shock geometry and the natural trapping effect of closed magnetic fields. Our results suggest that the coronal magnetic field configuration is an important factor for producing large SEP events. We further show that the coronal magnetic field configuration strongly influences the distribution of energetic particles, leading to different locations of source regions along the shock front where most of high energy particles are concentrated. This work may have strong implications to SEP observations. The upcoming Parker Solar Probe will provide in situ observations for the distribution of energetic particles in the coronal shock region, and test the results of the study., Comment: accepted for publication in ApJ

Type-I bursts (i.e. noise storms) are the earliest-known type of solar radio emission at the metre wavelength. They are believed to be excited by non-thermal energetic electrons accelerated in the corona. The underlying dynamic process and exact emission mechanism still remain unresolved. Here, with a combined analysis of extreme ultraviolet (EUV), radio and photospheric magnetic field data of unprecedented quality recorded during a type-I storm on 30 July 2011, we identify a good correlation between the radio bursts and the co-spatial EUV and magnetic activities. The EUV activities manifest themselves as three major brightening stripes above a region adjacent to a compact sunspot, while the magnetic field there presents multiple moving magnetic features (MMFs) with persistent coalescence or cancelation and a morphologically similar three-part distribution. We find that the type-I intensities are correlated with those of the EUV emissions at various wavelengths with a correlation coefficient of 0.7-0.8. In addition, in the region between the brightening EUV stripes and the radio sources there appear consistent dynamic motions with a series of bi-directional flows, suggesting ongoing small-scale reconnection there. Mainly based on the induced connection between the magnetic motion at the photosphere and the EUV and radio activities in the corona, we suggest that the observed type-I noise storms and the EUV brightening activities are the consequence of small-scale magnetic reconnection driven by MMFs. This is in support of the original proposal made by Bentely et al. (Solar Phys. 193, 227, 2000).

Hot channel (HC) structure, observed in the high-temperature passbands of the AIA/SDO, is regarded as one candidate of coronal flux rope which is an essential element of solar eruptions. Here we present the first radio imaging study of an HC structure in the metric wavelength. The associated radio emission manifests as a moving type-IV (t-IVm) burst. We show that the radio sources co-move outwards with the HC, indicating that the t-IV emitting energetic electrons are efficiently trapped within the structure. The t-IV sources at different frequencies present no considerable spatial dispersion during the early stage of the event, while the sources spread gradually along the eruptive HC structure at later stage with significant spatial dispersion. The t-IV bursts are characterized by a relatively-high brightness temperature ($\sim$ 10$^{7}$ $-$ 10$^{9}$ K), a moderate polarization, and a spectral shape that evolves considerably with time. This study demonstrates the possibility of imaging the eruptive HC structure at the metric wavelength and provides strong constraints on the t-IV emision mechanism, which, if understood, can be used to diagnose the essential parameters of the eruptive structure., Comment: 16 pages, 5 figures, Accepted for Publication in APJ Letters

Type III and type-III-like radio bursts are produced by energetic electron beams guided along coronal magnetic fields. As a variant of type III bursts, Type N bursts appear as the letter "N" in the radio dynamic spectrum and reveal a magnetic mirror effect in coronal loops. Here, we report a well-observed N-shaped burst consisting of three successive branches at metric wavelength with both fundamental and harmonic components and a high brightness temperature ($>$10$^9$ K). We verify the burst as a true type N burst generated by the same electron beam from three aspects of the data. First, durations of the three branches at a given frequency increase gradually, may due to the dispersion of the beam along its path. Second, the flare site, as the only possible source of non-thermal electrons, is near the western feet of large-scale closed loops. Third, the first branch and the following two branches are localized at different legs of the loops with opposite sense of polarization. We also find that the sense of polarization of the radio burst is in contradiction to the O-mode and there exists a fairly large time delay ($\sim$3-5 s) between the fundamental and harmonic components. Possible explanations accounting for these observations are presented. Assuming the classical plasma emission mechanism, we can infer coronal parameters such as electron density and magnetic field near the radio source and make diagnostics on the magnetic mirror process., Comment: Accepted by ApJ

With a test-particle simulation, we investigate the effect of large-scale coronal magnetic fields on electron acceleration at an outward-propagating coronal shock with a circular front. The coronal field is approximated by an analytical solution with a streamer-like magnetic field featured by partially open magnetic field and a current sheet at the equator atop the closed region. We show that the large-scale shock-field configuration, especially the relative curvature of the shock and the magnetic field line across which the shock is sweeping, plays an important role in the efficiency of electron acceleration. At low shock altitudes, when the shock curvature is larger than that of magnetic field lines, the electrons are mainly accelerated at the shock flanks; at higher altitudes, when the shock curvature is smaller, the electrons are mainly accelerated at the shock nose around the top of closed field lines. The above process reveals the shift of efficient electron acceleration region along the shock front during its propagation. It is also found that in general the electron acceleration at the shock flank is not so efficient as that at the top of closed field since at the top a collapsing magnetic trap can be formed. In addition, we find that the energy spectra of electrons is power-law like, first hardening then softening with the spectral index varying in a range of -3 to -6. Physical interpretations of the results and implications on the study of solar radio bursts are discussed., Comment: accepted by ApJ

Band split of solar type II radio bursts, discovered several decades ago, is a fascinating phenomenon with the type-II lanes exhibiting two almost-parallel sub-bands with similar morphology. The underlying split mechanism remains elusive. One popular interpretation is that the splitting bands are emitted from the shock upstream and downstream, respectively, with their frequency ratio ({\gamma}) determined by the shock compression ratio. This interpretation has been taken as the physical basis for many published references. Here we report an observational analysis of type II events with nice split selected from the ground-based RSTN data from 2001 to 2014, in the metric-decametric wavelength. We investigate the temporal variation and distribution of {\gamma}, and conduct correlation analyses on the deduced spectral values. It is found that {\gamma} varies in a very narrow range with >80% of {\gamma} (one-minute averaged data) being between 1.15 to 1.25. For some well-observed and long-lasting events, {\gamma} does not show a systematic variation trend within observational uncertainties, from the onset to the termination of the splits. In addition, the parameters representing the propagation speed of the radio source (presumably the coronal shock) show a very weak or basically no correlation with {\gamma}. We suggest that these results do not favor the upstreamdownstream scenario of band splits.

Two solar type II radio bursts, separated by ~24 hours in time, are examined together. Both events are associated with coronal mass ejections (CMEs) erupting from the same active region (NOAA 11176) beneath a well-observed helmet streamer. We find that the type II emissions in both events ended once the CME/shock fronts passed the white-light streamer tip, which is presumably the magnetic cusp of the streamer. This leads us to conjecture that the closed magnetic arcades of the streamer may play a role in electron acceleration and type II excitation at coronal shocks. To examine such a conjecture, we conduct a test-particle simulation for electron dynamics within a large-scale partially-closed streamer magnetic configuration swept by a coronal shock. We find that the closed field lines play the role of an electron trap, via which the electrons are sent back to the shock front for multiple times, and therefore accelerated to high energies by the shock. Electrons with an initial energy of 300eV can be accelerated to tens of keV concentrating at the loop apex close to the shock front with a counter-streaming distribution at most locations. These electrons are energetic enough to excite Langmuir waves and radio bursts. Considering the fact that most solar eruptions originate from closed field regions, we suggest that the scenario may be important to the generation of more metric type IIs. This study also provides an explanation to the general ending frequencies of metric type IIs at or above 20-30 MHz and the disconnection issue between metric and interplanetary type IIs., Comment: 27 pages, 10 figures, accepted by ApJ

Simultaneous radio and extreme ultraviolet (EUV)/white-light imaging data are examined for a solar type II radio burst occurring on 2010 March 18 to deduce its source location. Using a bow-shock model, we reconstruct the 3-dimensional EUV wave front (presumably the type-II emitting shock) based on the imaging data of the two STEREO spacecraft. It is then combined with the Nan\c{c}ay radio imaging data to infer the 3-dimensional position of the type II source. It is found that the type II source coincides with the interface between the CME EUV wave front and a nearby coronal ray structure, providing evidence that the type II emission is physically related to the CME-ray interaction. This result, consistent with those of previous studies, is based on simultaneous radio and EUV imaging data for the first time., Comment: 5 figures