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Context. Cassiopeia A occupies an important place among supernova remnants (SNRs) in low-frequency radio astronomy. The analysis of its continuum spectrum from low frequency observations reveals the evolution of the SNR absorption properties over time and suggests a method for probing unshocked ejecta and the SNR interaction with the circumstellar medium (CSM). Aims. In this paper we present low-frequency measurements of the integrated spectrum of Cassiopeia A to find the typical values of free-free absorption parameters towards this SNR in the middle of 2023. We also add new results to track its slowly evolving and decreasing integrated flux density. Methods. We used the New Extension in Nan\c{c}ay Upgrading LOFAR (NenuFAR) and the Ukrainian Radio Interferometer of NASU (URAN-2, Poltava) for measuring the continuum spectrum of Cassiopeia A within the frequency range of 8-66 MHz. The radio flux density of Cassiopeia A has been obtained on June-July, 2023 with two sub-arrays for each radio telescope, used as a two-element correlation interferometer. Results. We measured magnitudes of emission measure, electron temperature and an average number of charges of the ions for both internal and external absorbing ionized gas towards Cassiopeia A from its integrated spectrum. Generally, their values are comparable to those presented by Stanislavsky et al. (2023), but their slight changes show the evolution of free-free absorption parameters in this SNR. Based on high accuracy of the measurements, we have detected the SNR-CSM interaction. Conclusions. The integrated flux-density spectrum of Cassiopeia A obtained with the NenuFAR and URAN-2 interferometric observations opens up new possibilities for continuous monitoring the ionized gas properties in and around Cassiopeia A to observe theevolution of unshocked ejecta and the SNR-CSM interaction in future studies.

Random trajectories of single particles in living cells contain information about the interaction between particles, as well as, with the cellular environment. However, precise consideration of the underlying stochastic properties, beyond normal diffusion, remains a challenge as applied to each particle trajectory separately. In this paper, we show how positions of confined particles in living cells can obey not only the Laplace distribution, but the Linnik one. This feature is detected in experimental data for the motion of G proteins and coupled receptors in cells, and its origin is explained in terms of stochastic resetting. This resetting process generates power-law waiting times, giving rise to the Linnik statistics in confined motion, and also includes exponentially distributed times as a limit case leading to the Laplace one. The stochastic process, which is affected by the resetting, can be Brownian motion commonly found in cells. Other possible models producing similar effects are discussed., Comment: 9 pages, 3 figures, 1 table

Resetting a stochastic process is an important problem describing the evolution of physical, biological and other systems which are continually returned to their certain fixed point. We consider the motion of a subdiffusive particle with a constant drift under Poissonian resetting. In this model the stochastic process is Brownian motion subordinated by an inverse infinitely divisible process (subordinator). Although this approach includes a wide class of subdiffusive system with Poissonian resetting by using different subordinators, each of such systems has a stationary state with the asymmetric Laplace distribution in which the scale and asymmetric parameters depend on the Laplace exponent of the subordinators used. Moreover, the mean time for the particle to reach a target is finite and has a minimum, optimal with respect to the resetting rate. Features of L\'evy motion under this resetting and the effect of a linear potential are discussed., Comment: 12 pages, 5 figures

Context. Cassiopeia A is one of the most extensively studied supernova remnants (SNRs) in our Galaxy. The analysis of its spectral features with the help of low frequency observations plays an important role for understanding the evolution of the radio source through the propagation of synchrotron emission to observers through the SNR environment and the interstellar medium. Aims. In this paper we present measurements of the integrated spectrum of Cas A to characterize the properties of free-free absorption towards this SNR. We also add new measurements to track its slowly evolving and decreasing integrated flux density. Methods. We use the Giant Ukrainian radio telescope (GURT) for measuring the continuum spectrum of Cassiopeia A within the frequency range of 16-72 MHz. The radio flux density of Cassiopeia A relative to the reference source of the radio galaxy Cygnus A has been measured on May-October, 2019 with two subarrays of the GURT, used as a two-element correlation interferometer. Results. We determine magnitudes of emission measure, electron temperature and an average number of charges of the ions for both internal and external absorbing ionized gas towards in Cassiopeia A. Generally, their values are close to the ones suggested by Arias et al. (2018), although for some there are slight differences. In the absence of clumping we find the unshocked ejecta of M = 2.61 solar mass at the electron density of 15.3 cm^-3 has a gas temperature of T=100 K. If the clumping factor is 0.67, then the unshocked ejecta of 0.96 solar mass the electron density of 18.7 cm^-3. Conclusions. The integrated flux density spectrum of Cassiopeia A obtained with the GURT interferometric observations is consistent with the theoretical model within measurement errors and also reasonably consistent with other recent results in the literature., Comment: 9 pages, 9 figures, 2 tables

In this article we study the trapped motion of a molecule undergoing diffusivity fluctuations inside a harmonic potential. For the same diffusing-diffusivity process, we investigate two possible interpretations. Depending on whether diffusivity fluctuations are interpreted as temperature or friction fluctuations, we show that they display drastically different statistical properties inside the harmonic potential. We compute the characteristic function of the process under both types of interpretations and analyse their limit behavior. Based on the integral representations of the processes we compute the mean-squared displacement and the normalized excess kurtosis. In the long-time limit, we show for friction fluctuations that the probability density function (PDF) always converges to a Gaussian whereas in the case of temperature fluctuations the stationary PDF can display either Gaussian distribution or generalized Laplace (Bessel) distribution depending on the ratio between diffusivity and positional correlation times., Comment: 14 pages, 2 eps figures

The interpretation of solar radio bursts observed by Parker Solar Probe (PSP) in the encounter phase plays a key role in understanding intrinsic properties of the emission mechanism in the solar corona. Lower time-frequency resolution of the PSP receiver can be overcome by simultaneous ground-based observations using more advanced antennas and receivers. In this paper we present such observations for which the active active region 12765, begetter of type III, J, and U solar bursts, was within sight of ground-based instruments and behind the solar limb of the PSP spacecraft. We used a subarray of the Giant Ukrainian Radio Telescope (GURT) to get the spectral properties of radio bursts at the frequency range of 8-80 MHz, as well as the PSP radio instruments with a bandwidth of 10.5 kHz - 19.2 MHz, during solar observations on June 5, 2020. We directly detected the radio events initiated by the active region behind the solar limb of the PSP spacecraft, using special conditions in the solar corona, due to the absence of active regions from the PSP side. Following the generation mechanism of solar radio emission, we refined the density model for the solar corona above the active region 12765 responsible for the radio bursts. Based on the PSP spacecraft position near the Sun and delays of radio waves between space- and ground-based records, we found the corresponding radio responses on the PSP spectrogram. The absence of sunspots from the PSP side contributes to the propagation of radio waves from a dense loop of the Sun to quiet regions with low densities, through which PSP instruments can detect the radiation., Comment: 8 pages, 6 figures, 2 tables, accepted to be published in Astronomy and Astrophysics 2021

The solar radiation focusing effect is related to the specific phenomenon of propagation of the Sun-emitted HF and VHF waves through terrestrial ionosphere. This natural effect is observed with ground-based radio instruments running within 10-200 MHz range, as distinctive patterns - the Spectral Caustics (SCs) - on the solar dynamic spectra. It has been suggested that SCs are associated with medium-scale traveling ionospheric disturbances (MSTIDs). In this paper, we present the first direct observations of SCs induced by MSTIDs, using solar dynamic spectra with SCs obtained by different European radio telescopes on January 8, 2014 and simultaneous two-dimensional detrended total electron content (dTEC) maps over Europe. Spatial examination of dTEC maps as well as precise timing analysis of the maps and the dynamic spectra have been performed. First, we found several pairs of one-to-one (TID-SC) correspondences. The study provides strong observational evidence supporting the suggestion that MSTIDs are the cause of SCs., Comment: 10 pages, 5 figures

For the first time we present simulation results of the focusing effect of the ionospheric plasma density irregularities, namely, Medium Scale Traveling Ionospheric Disturbances (MSTIDs), on solar radio emission by applying a ray-tracing method to the Earth's ionosphere with MSTIDs. With this technique we investigate the focusing effect which manifests itself in the form of peculiar spectral disturbances in intensity with specific morphology, so-called Spectral Caustics (SCs), occasionally appearing in dynamic spectra of different solar radio instruments operating in the meter-decameter wavelength range. We show that the simulated spectral shapes of SCs are in good agreement with the ones detected in real solar radio spectrograms. In particular, SCs that are classified as inverted V-like, V-like, X-like, and fiber-like types have been reproduced. It is also found that the seasonal dependence in the occurrence of SCs, which has been discovered recently, can be understood through a strong relation between the focusing frequency - the most important characteristic point in most SCs patterns - and the elevation angle of the Sun. We find that under typical parameters of MSTIDs with spatial and temporal periods set to be 300 km and 40 min, respectively, the focusing frequency decreases with the growth of the elevation angle. Physical interpretations of the results and implications on the analysis of solar radio data with SCs are discussed.

The paper is devoted to recent advances in stochastic modeling of anomalous kinetic processes observed in dielectric materials which are prominent examples of disordered (complex) systems. Theoretical studies of dynamical properties of those "structures with variations" [Goldenfield and Kadanoff 1999 Science 284 87--9] require application of such mathematical tools by means of which their random nature can be analyzed and, independently of the details differing various systems (dipolar materials, glasses, semiconductors, liquid crystals, polymers, etc.), the empirical universal kinetic patterns can be derived. We begin with a brief survey of the historical background of the dielectric relaxation study. After a short outline of the theoretical ideas providing the random tools applicable to modeling of relaxation phenomena, we present probabilistic implications for the study of the relaxation-rate distribution models. In the framework of the probability distribution of relaxation rates we consider description of complex systems, in which relaxing entities form random clusters interacting with each other and single entities. Then we focus on stochastic mechanisms of the relaxation phenomenon. We discuss the diffusion approach and its usefulness for understanding of anomalous dynamics of the relaxing systems. We also discuss extensions of the diffusive approach to the systems under tempered random processes. Useful relationships among different stochastic approaches to the anomalous dynamics of complex systems allow us to get a fresh look on this subject. The paper closes with a final discussion on achievements of stochastic tools describing the anomalous time evolution of the complex systems., Comment: Extended version, 80 pages, 14 figures, 4 tables

First-of-its-kind radio imaging of decameter solar stationary type IV radio burst has been presented in this paper. On 6 September 2014 the observations of type IV burst radio emission have been carried out with the two-dimensional heliograph based on the Ukrainian T-shaped radio telescope (UTR-2) together with other telescope arrays. Starting at 09:55 UT and throughout 3 hours, the radio emission was kept within the observational session of UTR-2. The interesting observation covered the full evolution of this burst, "from birth to death". During the event lifetime, two C-class solar X-ray flares with peak times 11:29 UT and 12:24 UT took place. The time profile of this burst in radio has a double-humped shape that can be explained by injection of energetic electrons, accelerated by the two flares, into the burst source. According to the heliographic observations we suggest the burst source was confined within a high coronal loop, which was a part of a relatively slow coronal mass ejection. The latter has been developed for several hours before the onset of the event. Through analyzing about 1.5 million of heliograms (3700 temporal frames with 4096 images in each frame that correspond to the number of frequency channels) the radio burst source imaging shows a fascinating dynamical evolution. Both space-based (GOES, SDO, SOHO, STEREO) data and various ground-based instrumentation (ORFEES, NDA, RSTO, NRH) records have been used for this study.

The broadband analog-digital heliograph based on the UTR-2 radio telescope is described in detail. This device operates by employing the parallel-series principle when five equi-spaced array pattern beams which scan the given radio source (e.g. solar corona) are simultaneously shaped. As a result, the obtained image presents a frame of 5 x 8 pixels with the space resolution 25' x 25' at 25 MHz. Each pixel corresponds to the signal from the appropriate pattern beam. The most essential heliograph component is its phase shift module for fast sky scanning by pencil-shape antenna beams. Its design, as well as its switched cable lengths calculation procedure, are presented, too. Each heliogram is formed in the actual heliograph just by using this phase shifter. Every pixel of a signal received from the corresponding antenna pattern beam is the cross-correlation dynamic spectrum (time-frequency-intensity) measured in real time with the digital spectrum processor. This new generation heliograph gives the solar corona images in the frequency range 8-32 MHz with the frequency resolution 4 kHz, time resolution to 1 ms, and dynamic range about 90 dB. The heliographic observations of radio sources and solar corona made in summer of 2010 are demonstrated as examples., Comment: 48 pages, 9 figures, 2 tables

Using the method of the Laplace transform, we consider fractional oscillations. They are obtained by the time-clock randomization of ordinary harmonic vibrations. In contrast to sine and cosine, the functions describing the fractional oscillations exhibit a finite number of damped oscillations with an algebraic decay. Their fractional differential equation is derived., Comment: 12 pages, 2 figures

We consider the usual Langevin equation depending on an internal time. This parameter is substituted by a first passage time of a self-similar Markov process. Then the Gaussian process is parent, and the hitting time process is directing. The probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes. The corresponding master equation becomes the fractional Fokker-Planck equation. We show that the resulting process has non-Markovian properties, all its moments are finite, the fluctuation-dissipation relation and the H-theorem hold., Comment: 6 pages, 2 figures

In this paper we consider a new mathematical extension of the Black-Scholes model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly \alpha-stable process. The subordinated process represents the Brownian motion indexed by an independent, continuous and increasing process. This allows us to introduce the long-term memory effects in the classical Black-Scholes model., Comment: 8 pages

Decameter wavelength radio emission is finely structured in solar bursts. For their research it is very important to use a sufficient sensitivity of antenna systems. In this paper we study an influence of the radiotelescope-antenna effective area on the results of decameter solar radio observations. For this purpose we compared the solar bursts received by the array of 720 ground-based dipoles and the single dipole of the radiotelescope UTR-2. It's shown that a larger effective area of the ground-based antenna allows us to measure a weaker solar emission and to distinguish a fine structure of strong solar events. This feature has been also verified by simultaneous ground- and space-based observations in the overlapping frequency range., Comment: 7 pages, 3 figures

We consider the evolution of logistic maps under long-term memory. The memory effects are characterized by one parameter \alpha. If it equals to zero, any memory is absent. This leads to the ordinary discrete dynamical systems. For \alpha = 1 the memory becomes full, and each subsequent state of the corresponding discrete system accumulates all past states with the same weight just as the ordinary integral of first order does in the continuous space. The case with 0 < \alpha < 1 has the long-term memory effects. The characteristic features are also observed for the fractional integral depending on time, and the parameter \alpha is equivalent to the order index of fractional integral. We study the evolution of the bifurcation diagram among \alpha = 0 and \alpha = 0.15 . The main result of this work is that the long-term memory effects make difficulties for developing the chaos motion in such logistic maps. The parameter \alpha\ resembles a governing parameter for the bifurcation diagram. For \alpha\ > 0.15 the memory effects win over chaos., Comment: 10 pages, 2 figures

The stochastic scenario of relaxation in the complex systems is presented. It is based on a general probabilistic formalism of limit theorems. The nonexponential relaxation is shown to result from the asymptotic self-similar properties in the temporal behavior of such systems. This model provides a rigorous justification of the energy criterion introduced by Jonscher. The meaning of the parameters into the empirical response functions is clarified. This treatment sheds a fresh light on the nature of not only the dielectric relaxation but also mechanical, luminescent and radiochemical ones. In the case of the Cole-Cole response there exists a direct link between the notation of the fractional derivative (appearing in the fractional macroscopic equation often proposed) and the model. But the macroscopic response equations, relating to the Cole-Davidson and Havriliak-Negami relaxations, have a more general integro-differential form in comparison with the ordinary fractional one., Comment: 20 pages, 1 table

In this paper we consider a random motion of magnetic bright points (MBP) associated with magnetic fields at the solar photosphere. The MBP transport in the short time range [0-20 minutes] has a subdiffusive character as the magnetic flux tends to accumulate at sinks of the flow field. Such a behavior can be rigorously described in the framework of a continuous time random walk leading to the fractional Fokker-Planck dynamics. This formalism, applied for the analysis of the solar subdiffusion of magnetic fields, generalizes the Leighton's model., Comment: 7 pages

In this paper we consider a system of three fractional differential equations describing a nonlinear reaction. Our analysis includes both analytical technique and numerical simulation. This allows us to control the efficiency of the numerical method. We combine the numerical approximation of fractional integral with finding zeros of a function of one variable. The variable gives a desired solution. The solution has a peak. Its position in time depends on the order of fractional derivative. The two other solutions of this system have a completely monotonic character., Comment: 13 pages, 5 figures

We use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory. Our consideration touches two physical examples in detail. First we study a temporal evolution of the species concentration for the trapping reaction in which a diffusing reactant is surrounded by a sea of randomly moving traps. The analysis is based on the random-variable formalism of anomalous diffusive processes. We find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by the two-time scale subordination of random processes. Another example is connected with a state equation for continuous media with memory. If the pressure and the density of a medium are subordinated in two different random processes, then the ordinary state equation becomes fractional with two time scales. This allows one to arrive at the state equation of Bagley-Torvik type., Comment: 15 pages

In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional equations of motion are derived using the Hamiltonian formalism. The approach is illustrated with a simple-fractional oscillator in a free state and under an external force. Besides the behavior of the coupled fractional oscillators is analyzed. The natural extension of this approach to continuous systems is stated. The interpretation of the mechanics is discussed., Comment: 16 pages, 5 figures

We consider the fractional oscillator being a generalization of the conventional linear oscillator in the framework of fractional calculus. It is interpreted as an ensemble average of ordinary harmonic oscillators governed by stochastic time arrow. The intrinsic absorption of the fractional oscillator results from the full contribution of the harmonic oscillators' ensemble: these oscillators differs a little from each other in frequency so that each response is compensated by an antiphase response of another harmonic oscillator. This allows to draw a parallel in the dispersion analysis for the media described by the fractional oscillator and the ensemble of ordinary harmonic oscillators with damping. The features of this analysis are discussed., Comment: 6 pages, 3 figures

This work is motivated by the relaxation data for materials which exhibit a change of the relationship between the fractional power-law exponents when different relaxation peaks in their dielectric susceptibility are observed. Within the proposed framework we derive a frequency-domain relaxation function fitting the whole range of the two-power-law dielectric spectroscopy data with independent low- and high-frequency fractional exponents \gamma\ and -\alpha, respectively. We show that this effect results from a contribution of different processes. For high frequencies it is determined by random stops and movement of relaxing components, and the low-frequency slope is caused by clustering in their temporal changes., Comment: 13 pages, 3 figures, 2 tables

We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered \alpha-stable processes. Its most important application is to overcome the infinite-moment difficulty for the \alpha-stable random operational time \tau. The tempering results in the existence of all moments of \tau. The subordination by the inverse tempered \alpha-stable process provides diffusion(relaxation) that occupies an intermediate place between subdiffusion (Cole-Cole law) and normal diffusion (exponential law). Here we obtain explicitly the Fokker-Planck equation, the mean square displacement and the relaxation function. This model includes subdiffusion as a particular case., Comment: 4 pages, 3 figures

An integro-differential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the "3/2 law"., Comment: 4 pages

We present dielectric spectroscopy data obtained for gallium-doped Cd$_{0.99}$Mn$_{0.01}$Te:Ga mixed crystals which exhibit a very special case of the two-power-law relaxation pattern with the high-frequency power-law exponent equal to 1. We explain this behavior, which cannot be fitted by none of the well-known empirical relaxation functions, in a subordinated diffusive framework. We propose diffusion scenario based on a renormalized clustering of random number of spatio-temporal steps in the continuous time random walk. Such a construction substitutes the renewal counting process, used in the classical continuous time random walk methodology, by a compound counting one. As a result, we obtain a novel relaxation function governing the observed non-standard pattern, and we show the importance of the compound counting processes in studying fractional dynamics of complex systems., Comment: 6 pages, 5 figures; corrected version

The non-exponential relaxation is shown to result from subordination by inverse tempered \alpha-stable processes. The main feature of tempered \alpha-stable processes is a finiteness of their moments, and the class of random processes includes ordinary \alpha-stable processes as a particular case. Starting with the Cole-Cole response this subordination approach establishes its direct link with the Cole-Davidson law. We derive the relaxation function describing the tempered relaxation. The meaning of the empirical response function is clarified., Comment: 14 pages, 3 figures

In this paper we analyze a coupling between the very large jumps in physical and operational times as applied to anomalous diffusion. The approach is based on subordination of a skewed Levy-stable process by its inverse to get two types of operational time - the spent and the residual waiting time, respectively. The studied processes have different properties which display both subdiffusive and superdiffusive features of anomalous diffusion underlying the two-power-law relaxation patterns., Comment: 6 pages, 3 figures; corrected version

We derive a general pattern of the nonexponential, two-power-law relaxation from the compound subordination theory of random processes applied to anomalous diffusion. The subordination approach is based on a coupling between the very large jumps in physical and operational times. It allows one to govern a scaling for small and large times independently. Here we obtain explicitly the relaxation function, the kinetic equation and the susceptibility expression applicable to the range of experimentally observed power-law exponents which cannot be interpreted by means of the commonly known Havriliak-Negami fitting function. We present a novel two-power relaxation law for this range in a convenient frequency-domain form and show its relationship to the Havriliak-Negami one., Comment: 5 pages; 3 figures; corrected version