A new proof of compactness in G(S)BD

We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.

Limits on long-timescale radio transients at 150 MHz using the TGSS ADR1 and LoTSS DR2 catalogues

We present a search for transient radio sources on timescales of 2-9 years at 150 MHz. This search is conducted by comparing the first Alternative Data Release of the TIFR GMRT Sky Survey (TGSS ADR1) and the second data release of the LOFAR Two-metre Sky Survey (LoTSS DR2). The overlapping survey area covers 5570 $\rm {deg}^2$ on the sky, or 14 per cent of the total sky. We introduce a method to compare the source catalogues that involves a pair match of sources, a flux density cutoff to meet the survey completeness limit and a newly developed compactness criterion. This method is used to identify both transient candidates in the TGSS source catalogue that have no counterpart in the LoTSS catalogue and transient candidates in LoTSS without a counterpart in TGSS. We find that imaging artefacts and uncertainties and variations in the flux density scales complicate the transient search. Our method to search for transients by comparing two different surveys, while taking into account imaging artefacts around bright sources and misaligned flux scales between surveys, is universally applicable to future radio transient searches. No transient sources were identified, but we are able to place an upper limit on the transient surface density of <5.4 · 10−4 deg−2 at 150 MHz for compact sources with an integrated flux density over 100 mJy. Here we define a transient as a compact source with flux density greater than 100 mJy that appears in the catalogue of one survey without a counterpart in the other survey.

Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order $1<\alpha <2$ 1 < α < 2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval $J=[0,\infty )$ J = [ 0 , ∞ ) . We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

Arithmeticity of discrete subgroups

The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

Results of Positive Solutions for the Fractional Differential System on an Infinite Interval

The chief topic of this paper is to investigate the fractional differential system on an infinite interval. By introducing an appropriate compactness criterion in a special function space and applying the Schauder fixed-point theorem and the Banach contraction mapping principle, we established the results for the existence and uniqueness of positive solutions. An example is then given to show the utilization of the main results.

A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

We present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an $H^1$-conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.

STUDY OF THE THERMAL EMISSION IN THE RADIO CONTINUUM AT 1.4 GHZ IN SELECTED COMPACT GALAXIES WITH ACTIVE STAR FORMATION

The fraction of thermal (free-free) emission in the radio continuum at the frequency of 1.4 GHz is derived in subsample of 92 compact star-forming galaxies (CSFGs). These galaxies with detected radio emission and selected with a more strong compactness criterion represent the subsample of a larger sample of 193 galaxies (Parnovsky, S., & Izotova, I.: 2018, BTSNU, 57(1), 41). We use the 1.4 GHz fluxes from the FIRST (Becker R.H.,White R.L., & Helfand D.J.: 1995, ApJ, 450, 559). The fluxes of the thermal component at 1.4 GHz are derived from the extinction- and aperture-corrected fluxes of the Hβ emission line in the SDSS spectra following to (Caplan, J., & Deharveng, L.: 1986, A&A, 155, 297) and are compared with the total fluxes in radio continuum. As well as for a complete sample of 193 galaxies, the distribution of the fraction of thermal emission A in the radio continuum at 1.4 GHz in subsample of 92 CSFGs is similar to the log-normal one with a median value of about 15%. We find that the fraction of thermal emission at 1.4 GHz is lower for the older starbursts. However, the results of the statistical analysis of the dependence of A on various parameters have been changed. For the complete sample, correlations of A with the equivalent width EW(Hβ) of the Hβ emission line and the g – r colour index are revealed. For the subsample of 92 galaxies, the statistical significance of the correlation A with the colour index becomes negligible, and it corresponds to a confidence level of 92% for dependence of A on the equivalent width EW(Hβ). The reasons for the differences in the properties of these samples are discussed.