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We study a flaring activity of the HSP Mrk421 that was characterized from radio to very-high-energy (VHE; E $>0.1$TeV) gamma rays with MAGIC, Fermi-LAT, Swift, XMM-Newton and several optical and radio telescopes. These observations included, for the first time for a gamma-ray flare of a blazar, simultaneous X-ray polarization measurements with IXPE. We find substantial variability in both X-rays and VHE gamma rays throughout the campaign, with the highest VHE flux above 0.2 TeV occurring during the IXPE observing window, and exceeding twice the flux of the Crab Nebula. However, the VHE and X-ray spectra are on average softer, and the correlation between these two bands weaker that those reported in previous flares of Mrk421. IXPE reveals an X-ray polarization degree significantly higher than that at radio and optical frequencies. The X-ray polarization angle varies by $\sim$100$^\circ$ on timescales of days, and the polarization degree changes by more than a factor 4. The highest X-ray polarization degree reaches 26%, around which a X-ray counter-clockwise hysteresis loop is measured with XMM-Newton. It suggests that the X-ray emission comes from particles close to the high-energy cutoff, hence possibly probing an extreme case of the Turbulent Extreme Multi-Zone model. We model the broadband emission with a simplified stratified jet model throughout the flare. The polarization measurements imply an electron distribution in the X-ray emitting region with a very high minimum Lorentz factor, which is expected in electron-ion plasma, as well as a variation of the emitting region size up to a factor of three during the flaring activity. We find no correlation between the fluxes and the evolution of the model parameters, which indicates a stochastic nature of the underlying physical mechanism. Such behaviour would be expected in a highly turbulent electron-ion plasma crossing a shock front., Comment: Submitted to Astronomy and Astrophysics. Corresponding authors: Axel Arbet-Engels, Lea Heckmann, David Paneque

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

Ferroelectric domain walls (DWs) are promising structures for assembling future nano-electronic circuit elements on a larger scale, since reporting domain wall currents of up to 1 mA per single DW. One key requirement hereto is their reproducible manufacturing by gaining preparative control over domain size and domain wall conductivity (DWC). To date, most works on DWC have focused on exploring the fundamental electrical properties of individual DWs within single shot experiments, with emphasis on quantifying the origins for DWC. Very few reports exist when it comes to compare the DWC properties between two separate DWs, and literally nothing exists where issues of reproducibility in DWC devices have been addressed. To fill this gap while facing the challenge of finding guidelines achieving predictable DWC performance, we report on a procedure that allows us to reproducibly prepare single hexagonal domains of a predefined diameter into uniaxial ferroelectric (FE) lithium niobate (LN) single crystals of 200 and 300 micrometers thickness, respectively. We show that the domain diameter can be controlled with an error of a few percent. As-grown DWs are then subjected to a standard procedure of current-controlled high-voltage DWC enhancement, repetitively reaching a DWC increase of 6 orders of magnitude. While all resulting DWs show significantly enhanced DWC values, subtle features in their individual current-voltage (I-V) characteristics hint towards different 3D shapes into the bulk, with variations probably reflecting local heterogeneities by defects, DW pinning, and surface-near DW inclination, which seem to have a larger impact than expected., Comment: 12 pages, 7 figures, with appended supplement (9 pages, 7 figures)

In the present paper, a submonoid of the well studied monoid $POI_n$ of all order-preserving partial injections on an $n$-element chain is studied. The set $IOF_n^{par}$ of all partial transformations in $POI_n$ which are fence-preserving as well as parity-preserving form a submonoid of $POI_n$. We describe the Green's relations and ideals of $IOF_n^{par}$. For each ideal of $IOF_n^{par}$, we characterize the maximal subsemigroups. We will observe that there are three different types of maximal subsemigroups., Comment: 12 pages

Lithium niobate and lithium tantalate are among the most widespread materials for nonlinear, integrated photonics. Mixed crystals with arbitrary Nb-Ta ratios provide a new degree of freedom to tune materials properties, such as the birefringence, but also leverage the advantages of the singular compounds, for example, by combining the thermal stability of lithium tantalate with the larger nonlinear or piezoelectric constants of lithium niobate. Periodic poling is the prerequisite for any nonlinear optical application. For mixed crystals this has been challenging so far due to the lack of homogeneous, mono-domain crystals, which severely inhibit domain growth and nucleation. In this work we demonstrate that surface-near ($< 1$~$\mu$m depth) periodic poling on x-cut lithium niobate tantalate mixed crystals can be achieved via electric field poling and lithographically structured electrodes. We find that naturally occurring head-to-head or tail-to-tail domain walls in the as-grown crystal inhibit domain inversion at a larger scale. However, periodic poling is possible, if the gap size between the poling electrodes is of the same order of magnitude or smaller than the average size of naturally occurring domains. This work provides the basis for the nonlinear optical application of lithium niobate tantalate mixed crystals.

Ferroelectric domain wall conductivity (DWC) is an intriguing functional property, that can be controlled through external stimuli such as electric and mechanical fields. Optical-field control, as a non-invasive flexible handle, has rarely been applied so far, but significantly expands the possibility for both tuning and probing DWC. On the one hand, as known from Second-Harmonic, Raman, and CARS micro-spectroscopy, the optical in-and-out approach delivers parameters on the DW distribution, the DW inclination, and probes the DW vibrational modes; on the other hand, photons might be applied also to directly generate charge carriers within the DW, hence acting as a functional and spectrally tunable probe to deduce the integral or local absorption properties and bandgaps of conductive DWs. Here, we report on such an optoelectronic approach by investigating the photo-induced DWC (PI-DWC) in DWs of the model system lithium niobate, a material that is well known for hosting conductive DWs. We compare three different crystals containing different numbers of domain walls: (A) none, (B) one, and (C) many conductive DWs. All samples are inspected for their current-voltage (I-V) behavior (i) in darkness, and (ii) for different illumination wavelengths swept from 500 nm down to 310 nm. All samples show their maximum PI-DWC at 310 nm, i.e., at the optical bandgap of lithium niobate; moreover, sample (C) reaches PI-DWCs of several $\mu$A. Interestingly, a noticeable PI-DWC is also observed for sub-bandgap illumination, i.e., wavelengths as high as 500 nm, hinting towards the existence and decisive role of electronic in-gap states that contribute to the electronic transport along DWs. Finally, conductive atomic force microscopy (c-AFM) investigations under illumination proved that the PI-DWC is confined to the DW area, and does not originate from photo-induced bulk conductivity., Comment: 18 pages, 3 figures, and Supplementary Material (with 10 pages, 4 figures)

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

A fully invarient congruence relations on the free algebra on a given type induces a variety of the given type. In contrast, a congruence relation of the free algebra provides algebra of that type. This algebra is given by a so-called presentation. In the present paper, we deal with an important class of algebras of type $(2)$, namely with semigroups of transformations on a finite set. Here, we are particularly interested in a presentation of a submonoid of the symmetric inverse monoid $I_n$. Our main result is a presentations for $IOF_n^{par}$, the monoid of all order-preserving, fence-preserving, and parity-preserving transformations on an $n$-element set., Comment: 19 pages

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.

In this paper we consider the submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of the dihedral inverse monoid $\mathcal{DI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids., Comment: arXiv admin note: text overlap with arXiv:2205.02196

In this paper we consider three submonoids of the dihedral inverse monoid $\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank., Comment: arXiv admin note: text overlap with arXiv:2205.02196

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner-Preston Theorem. In this article, we step forward the study of a submonoid of the symmetric inverse semigroup. We explore the monoid of all order-, fence-, and parity-preserving transformations on an $n$-element chain. We also characterize the transformations in that monoid and show that it has a rank $3n-6$. In particular, we provide a generating set $A_n$ of minimal size and exhibit concrete normal forms for the transformations generated by $A_n$.

The RTS smoother is widely used for state estimation and it is utilized here to increase the data quality with respect to physical coherence and to increase resolution. The purpose of this paper is to enhance the performance of the RTS smoother to reconstruct an aircraft landing using on board recorded data only. Thereby, errors and uncertainties of operational flight data (e.g. altitude, attitude, position, speed) recorded during flights of civil aircraft are minimized. These data can be used for subsequent analyses in terms of flight safety or efficiency, which is commonly referred to as Flight Data Monitoring (FDM). Statistical assumptions of the smoother theory are not always verified during application but (consciously or not) assumed to be fulfilled. These assumptions can hardly be verified prior to the smoother application, however, they can be verified using the results of an initial smoother iteration and modifications of specific smoother characteristics can be suggested. This project specifically verifies assumptions on the measurement noise characteristics. Variance and covariance of the measurement noise can be checked after the initial smoother application. It is discovered that these characteristics change over time and should be accounted for with a time varying covariance matrix. This sequence of matrices is estimated by kernel smoothing and replaces an initially assumed fixed and diagonal covariance matrix used for the first smoother run. The results of this second smoother iteration are mostly improved compared to the initial iteration, i.e. the errors are significantly reduced. Subsequently, the remaining dependence structures of the residuals of the second smoother iteration can be captured by copula models. Their interpretation is useful for a revision of the physical model utilized by the RTS smoother.

For $n \in \mathbb{N}$, let $[n] = \{1, 2, \ldots, n\}$ be an $n$ - element set. As usual, we denote by $I_n$ the symmetric inverse semigroup on $[n]$, i.e. the partial one-to-one transformation semigroup on $[n]$ under composition of mappings. The crown (cycle) $\cal{C}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ We say that a transformation $\alpha \in I_n$ is order-preserving if $x \prec y$ implies that $x\alpha \prec y\alpha$, for all $x, y $ from the domain of $\alpha$. In this paper, we study the inverse semigroup $IC_n$ of all partial automorphisms on a finite crown $\cal{C}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $IC_n$.

For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ - element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is \textit{fence-preserving} if $x <_f y$ implies that $x\alpha <_f y\alpha$, for all $x, y$ in the domain of $\alpha$. In this paper, we study the semigroup $PFI_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{\alpha \in PFI_n : \alpha^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and contains all regular elements of $PFI_n.$ We characterize the Green's relations for the semigroup $IF_n$. Further, we prove that the semigroup $IF_n$ is generated by its elements with \rank$\geq n-2$. Moreover, for $n \in 2\mathbb{N}$ we find the least generating set and calculate the rank of $IF_n$.

Recent astronomical data have provided the primordial deuterium abundance with percent precision. As a result, Big Bang nucleosynthesis may provide a constraint on the universal baryon to photon ratio that is as precise as, but independent from, analyses of the cosmic microwave background. However, such a constraint requires that the nuclear reaction rates governing the production and destruction of primordial deuterium are sufficiently well known. Here, a new measurement of the $^2$H($p,\gamma$)$^3$He cross section is reported. This nuclear reaction dominates the error on the predicted Big Bang deuterium abundance. A proton beam of 400-1650keV beam energy was incident on solid titanium deuteride targets, and the emitted $\gamma$-rays were detected in two high-purity germanium detectors at angles of 55$^\circ$ and 90$^\circ$, respectively. The deuterium content of the targets has been obtained in situ by the $^2$H($^3$He,$p$)$^4$He reaction and offline using the Elastic Recoil Detection method. The astrophysical S-factor has been determined at center of mass energies between 265 and 1094 keV, addressing the uppermost part of the relevant energy range for Big Bang nucleosynthesis and complementary to ongoing work at lower energies. The new data support a higher S-factor at Big Bang temperatures than previously assumed, reducing the predicted deuterium abundance., Comment: 13 pages, 9 figures, 4 tables; to be published in Phys. Rev. C

Let $X$ be an infinite linearly ordered set and let $Y$ be a nonempty subset of $X$. We calculate the relative rank of the semigroup $OP(X,Y)$ of all orientation-preserving transformations on $X$ with restricted range $Y$ modulo the semigroup $O(X,Y)$ of all order-preserving transformations on $X$ with restricted range $Y$. For $Y = X$, we characterize the relative generating sets of minimal size.

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}$., Comment: 17 pages

In this paper, we determine the relative rank of the semigroup $T(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y \subseteq X$ modulo the set $OP(X,Y)$ of all orientation-preserving transformation in $T(X,Y)$. Moreover, we state the relative rank of the semigroup $OP(X,Y)$ modulo the set $O(X,Y)$ of all order-preserving transformations in $OP(X,Y)$. In both cases we characterize the minimal relative generating sets.

In this paper, we determine the relative rank of the semigroup OP(X) of all orientation-preserving transformations on infinite chains modulo the semigroup O(X) of all order-preserving transformations.

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $\mathcal{FI}_{n}$ of all order-preserving partial injections on an $n$-element fence. In particular, we provide a minimal generating set for $\mathcal{FI}_{n}$. In the present paper, $n$ is odd since this problem for even $n$ was already solved by I. Dimitrova and J. Koppitz.

In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak endomorphisms and all endomorphisms of the undirected path $P_n$ with $n$ vertices. We also consider strong and strong weak endomorphisms of $P_n$.