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This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian $p$-group $G$ such that $p^{\omega}G$ is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if $G$ is a reduced Abelian $p$-group such that $p^{\omega}G$ is bounded and $G/p^{\omega}G$ is Hopfian, then $G$ is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^{\omega}G$ an infinite elementary group such that $G$ is {\bf not} Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated., Comment: 26 pages

As a special case of perspective R-modules, an Abelian goup is called perspective if isomorphic summands have a common complement. In this paper we describe many classes of such groups.

Trying to finalize in some way the present subject, this paper targets to generalize substantially the notions of Bassian and co-Bassian groups by introducing the so-called finitely (co-)Bassian groups, semi (co-)Bassian groups, fully generalized (co-)Bassian groups, absolutely generalized (co-)Bassian groups and establishing their crucial properties and characterizations. In fact, some of the concepts give nothing new by coinciding in the reduced case with the well-known (co-)Bassian property. However, in some of the definitions, the situation is slightly more complicated and we obtain a few new and interesting things by showing that the extensions of the Bassian and co-Bassian properties are totally distinct each other., Comment: 11 pages

As a common non-trivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of p-torsion groups and groups of finite torsion-free rank by showing that these groups can be completely determined in terms of generalized finite p-ranks and also depends on their quotients modulo the maximal torsion subgroup. Surprisingly, for p-primary groups, the concept of a semi-generalized co-Bassian group is closely related to that of a generalized co-Bassian group., Comment: 11 pages

As a common non-trivial generalization of the concept of a proper generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of p-torsion groups, torsion-free groups and splitting mixed groups their complete description. Moreover, we classify the groups with the clearly related property that every subgroup is essential in a direct summand of the whole group., Comment: 14 pages

We consider two variants of those Abelian groups with all proper strongly invariant subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant (respectively, characteristic) subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015) and in J. Commut. Algebra (2023). In addition, we also explore those Abelian groups having at least one proper strongly invariant subgroup isomorphic to the whole group., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2301.08924

A famous conjecture attributed to Dardano-Dikranjan-Rinauro-Salce states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see, respectively, [5] and [6]). In this short note, we completely settle this problem in the affirmative for an arbitrary Abelian group., Comment: We give a new very short proof of the crucial Proposition 2.2 and also provide more details in the proof of Proposition 2.3

We consider two variants of those Abelian groups with all proper characteristic subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015). In addition, we also examine those Abelian groups having at least one proper characteristic subgroup isomorphic to the whole group. The established by us results somewhat extend those obtained by Grinshpon-Nikolskaya in Tomsk State Univ. J. Math. & Mech. (2011, 2012) and in Commun. Algebra (2011), respectively., Comment: 18 pages

Numerical simulation results of basic exactly solvable fluid flows using the previously proposed Lattice Boltzmann Method (LBM) formulated on a general curvilinear coordinate system are presented. As was noted in the theoretical work of H. Chen, such curvilinear Lattice Boltzmann Method preserves a fundamental one-to-one exact advection feature in producing minimal numerical diffusion, as the Cartesian lattice Boltzmann model. As we numerically show, the new model converges to exact solutions of basic fluid flows with the increase of grid resolution in the presence of both natural curvilinear geometry and/or grid non-uniform contraction, both for near equilibrium and non-equilibrium LBM parameter choices., Comment: accepted for publication in: Front. Appl. Math. Stat. - Mathematical Physics, 12/2022

We describe a fast data-driven optical camera, Tpx3Cam, with nanosecond scale timing resolution and 80 Mpixel/sec throughput. After the addition of intensifier, the camera is single photon sensitive with quantum efficiency determined primarily by the intensifier photocathode. The single photon performance of the camera was characterized with results on the gain, timing resolution and afterpulsing reported here. The intensified camera was successfully used for measurements in a variety of applications including quantum applications. As an example of such application, which requires simultaneous detection of multiple photons, we describe registration of photon pairs from the spontaneous parametric down-conversion source in a spectrometer. We measured the photon wavelength and timing with respective precisions of 0.15~nm and 3~ns, and also demonstrated that the two photons are anti-correlated in energy., Comment: prepared for submission to JINST for IWORID2022 proceedings

We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some oldstanding problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17]. Specifically, we answer Problem 44 from there in the affirmative by constructing a Krylov transitive torsion-free Abelian group which is neither fully transitive nor transitive. This extends to the torsion-free case certain similar results in the p-torsion case, obtained by Braun et al. in J. Algebra (2019). We, alternatively, also expand to the torsion-free version some of the results concerning transitivity, full transitivity and Krylov transitivity in the p-primary case due Files-Goldsmith from Proc. Amer. Math. Soc. (1998) and Danchev-Goldsmith from J. Comm. Algebra (2011)., Comment: 20 pages

If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called \textsl{strictly invariant}. For the case of $% \mathbb{Z}$-modules (i.e. Abelian groups) we prove that in many situations these submodules are invariant with respect to all homomorphisms $% K\rightarrow M$, submodules which were called \textsl{strongly invariant}., Comment: preprint version; the final version will be published by Mediterranean J. of Math

We describe how active feedback routines can be applied at limited repetition rate (5 Hz) to optimize high-power $>10$ TW) laser interactions with clustered gases. Optimization of x-ray production from an argon cluster jet, using a genetic algorithm, approximately doubled the measured energy through temporal modification of the 150 mJ driving laser pulse. This approach achieved an increased radiation yield through exploration of a multi-dimensional parameter space, without requiring detailed a priori knowledge of the complex cluster dynamics. The optimized laser pulses exhibited a slow rising edge to the intensity profile, which enhanced the laser energy coupling into the cluster medium, compared to the optimally compressed FWHM pulse (40 fs). Our work suggests that this technique can be more widely utilized for control of intense pulsed secondary radiation from petawatt-class laser systems., Comment: 5 pages, 3 figures, Published in APL

We report on the depletion and power amplification of the driving laser pulse in a strongly-driven laser wakefield accelerator. Simultaneous measurement of the transmitted pulse energy and temporal shape indicate an increase in peak power from $187 \pm 11$ TW to a maximum of $318 \pm 12$ TW after 13 mm of propagation in plasma density of $0.9 \times 10^{18}$ cm$^{-3}$. The power amplification is correlated with the injection and acceleration of electrons in the nonlinear wakefield. This process is modeled by including localized redshift and subsequent group delay dispersion at the laser pulse front., Comment: 6 pages, 4 figures. Published in PRL

Large eddy simulation (LES) of forced, homogeneous, isotropic, two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlov et al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplified, Comment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex file

The effect of small scale forcing on large scale structures in $\beta$-plane two-dimensional (2D) turbulence is studied using long-term direct numerical simulations (DNS). We find that nonlinear effects remain strong at all times and for all scales and establish an inverse energy cascade that extends to the largest scales available in the system. The large scale flow develops strong spectral anisotropy: $k^{-5/3}$ Kolmogorov scaling holds for almost all $\phi$, $\phi=\arctan (k_y/k_x)$, except in the small vicinity of $k_x=0$, where Rhines's $k^{-5}$ scaling prevails. Due to the $k^{-5}$ scaling, the spectral evolution of $\beta$-plane turbulence becomes extremely slow which, perhaps, explains why this scaling law has never before been observed in DNS. Simulations with different values of $\beta$ indicate that the $\beta$-effect diminishes at small scales where the flow is nearly isotropic. Thus, for simulations of $\b$-plane turbulence forced at small scales sufficiently removed from the scales where $\b$-effect is strong, large eddy simulation (LES) can be used. A subgrid scale (SGS) parameterization for such LES must account for the small scale forcing that is not explicitly resolved and correctly accommodate two inviscid conservation laws, viz. energy and enstrophy. This requirement gives rise to a new anisotropic stabilized negative viscosity (SNV) SGS representation which is discussed in the context of LES of isotropic 2D turbulence., Comment: 25 pages, psfig and elsart macros; tex and final postscript versions

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-in-time random noise with the spectrum $\overline{|f(k)|^2}\propto k^{-1}$ exhibit a biscaling behavior: All moments of velocity differences $S_{n\le 3}(r)=\overline{|u(x+r)-u(x)|^n}\equiv\overline{|\Delta u|^n}\propto r^{n/3}$, while $S_{n>3}\propto r^{\zeta_n}$ with $\zeta_n\approx 1$ for real $n>0$ (Chekhlov and Yakhot, Phys. Rev. E {\bf 51}, R2739, 1995). The probability density function, which is dominated by coherent shocks in the interval $\Delta u<0$, is ${\cal P}(\Delta u,r)\propto (\Delta u)^{-q}$ with $q\approx 4$., Comment: 12 pages, psfig macro, 4 figs in Postscript, accepted to Phys. Rev. E as a Brief Communication

The dynamics of velocity fluctuations, governed by the one-dimensional Burgers equation, driven by a white-in-time random force with the spatial spectrum $\overline{|f(k)|^2}\proptok^{-1}$, is considered. High-resolution numerical experiments conducted in this work give the energy spectrum $E(k)\propto k^{-\beta}$ with $\beta =5/3\pm 0.02$. The observed two-point correlation function $C(k,\omega)$ reveals $\omega\propto k^z$ with the "dynamical exponent" $z\approx 2/3$. High-order moments of velocity differences show strong intermittency and are dominated by powerful large-scale shocks. The results are compared with predictions of the one-loop renormalized perturbation expansion., Comment: 13 LaTeX pages, psfig.sty macros, Phys. Rev. E 51, R2739 (1995).

Two-parametric eddy viscosity (TPEV) and other spectral characteristics of two-dimensional (2D) turbulence in the energy transfer sub-range are calculated from direct numerical simulation (DNS) with 512$^2$ resolution. The DNS-based TPEV is compared with those calculated from the test field model (TFM) and from the renormalization group (RG) theory. Very good agreement between all three results is observed., Comment: 9 pages (RevTeX) and 5 figures, published in Phys. Fluids 6, 2548 (1994)