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The Quick Ultra-Violet Kilonova surveyor (QUVIK), a two-band UV space telescope approved for funding as a Czech national science and technology mission, will focus on detecting early UV light of kilonovae (Werner et al., 2024). In addition, it will study the UV emission of stars and stellar systems (Krti\v{c}ka et al., 2024) as well as the intense and variable emission of active galactic nuclei (AGN) or galactic nuclei activated by tidal disruption events (Zaja\v{c}ek et al., 2024). In this contribution, we describe the role of this small ($\sim 30$-cm diameter) UV telescope for studying bright, nearby AGN. With its NUV and FUV bands, the telescope will perform high-cadence ($\sim$ 0.1-1 day) two-band photometric monitoring of nearby AGN ($z<1$), which will allow us to probe accretion disk sizes/temperature profiles via photometric reverberation mapping. Thanks to its versatility, QUVIK will be able to perform a moderately fast repointing ($<20$ min) to target candidates for tidal disruption events (TDEs). Early detection of the UV emission following a TDE optical flare, in combination with the subsequent two-band UV monitoring performed simultaneously with other observatories, will enable us to infer the time delay (or its lack of) between the optical, UV, and X-ray emission. In combination with theoretical models, it will be possible to shed more light on the origin of the UV/optical emission of TDEs. Furthermore, the two-band monitoring of nuclear transients will be beneficial in distinguishing between TDEs (nearly constant blue colour) and supernovae (progressive reddening)., Comment: 28 pages, 9 figures, submitted to the JATIS special issue "Ultraviolet Science & Instrumentation: On the Way to Habitable Worlds Observatory and Beyond"

It is well known that silicon photomultipliers (SiPMs) are prone to radiation damage. With the increasing popularity of SiPMs among new spaceborne missions, especially on CubeSats, it is of paramount importance to characterize their performance in space environment. In this work, we report the in-orbit ageing of SiPM arrays, so-called multi-pixel photon counters (MPPCs), using measurements acquired by the GRBAlpha and VZLUSAT-2 CubeSats at low Earth orbit (LEO) spanning over three years, which in duration is unique. GRBAlpha is a 1U CubeSat launched on March 22, 2021, to a 550 km altitude sun-synchronous polar orbit (SSO) carrying on board a gamma-ray detector based on CsI(Tl) scintillator readout by eight MPPCs and regularly detecting gamma-ray transients such as gamma-ray bursts and solar flares in the energy range of ~30-900 keV. VZLUSAT-2 is a 3U CubeSat launched on January 13, 2022 also to a 550 km altitude SSO carrying on board, among other payloads, two gamma-ray detectors similar to the one on GRBAlpha. We have flight-proven the Hamamatsu MPPCs S13360-3050 PE and demonstrated that MPPCs, shielded by 2.5 mm of PbSb alloy, can be used in an LEO environment on a scientific mission lasting beyond three years. This manifests the potential of MPPCs being employed in future satellites., Comment: Submitted to Nuclear Instruments and Methods in Physics Research Section A, 13 pages, 14 figures

In the decimal numeral system, we prove that the well-known Graham's number, $G := \! ^{n}3$ (i.e., $3^{3^{\cdot^{\cdot^{\cdot^{3}}}}} n$-times), and any base $3$ tetration whose hyperexponent is larger than $n$ share the same $slog_3(G) - 1$ rightmost digits (where $slog$ indicates the integer super-logarithm). This is an exact result since the $slog_3(G)$-th rightmost digit of $G$ is different from the $slog_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $slog_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent is beyond $n$ is $4$., Comment: 9 pages, 1 figure

Measurements of the speed of sound in gaseous cis-1,3,3,3-tetrafluoroprop-1-ene, (R1234ze(Z)), are presented. The measurements were performed using a quasi-spherical acoustic resonator at temperatures between 307 K and 420 K and pressures up to 1.8 MPa. Ideal-gas heat capacities and acoustic virial coefficients over the same temperature range were directly calculated from the results. The relative accuracy of our determinations of the speed of sound $w$($p$,$T$) of R1234ze(Z) was approximately $\pm$ 0.02%. The accuracy of the determination of the ideal gas heat capacity ratio ${\gamma}^{0}$($T$) was approximately $\pm$ 0.25%. These data were found to be mostly consistent with the predictions of a fundamental equation of state of R1234ze(Z).

For each pair of positive integers $(a,b)$ such that $a \geq 0$ and $b > 1$, the present paper provides a necessary and sufficient condition for the existence of Hamiltonian cycles visiting all the vertices of any $k$-dimensional grid $\{0,1\}^k \subset \mathbb{R}^k$ and whose associated Euclidean distance is equal to $\sqrt{a^2+b^2}$. Our solution extends previously stated results in fairy chess on the existence of closed Euclidean $(a,b)$-leapers tours for $2 \times 2 \times \cdots \times 2$ chessboards, where the (Euclidean) knight identifies the $(1,2)$-leaper., Comment: 6 pages. Names of Lemma 2.3 and Corollary 2.5 fixed

HERMES (High Energy Rapid Modular Ensemble of Satellites) Pathfinder is a space-borne mission based on a constellation of six nano-satellites flying in a low-Earth orbit (LEO). The 3U CubeSats, to be launched in early 2025, host miniaturized instruments with a hybrid Silicon Drift Detector/GAGG:Ce scintillator photodetector system, sensitive to X-rays and gamma-rays in a large energy band. HERMES will operate in conjunction with Australian Space Industry Responsive Intelligent Thermal (SpIRIT) 6U CubeSat, launched in December 2023. HERMES will probe the temporal emission of bright high-energy transients such as Gamma-Ray Bursts (GRBs), ensuring a fast transient localization in a field of view of several steradians exploiting the triangulation technique. HERMES intrinsically modular transient monitoring experiment represents a keystone capability to complement the next generation of gravitational wave experiments. In this paper we outline the scientific case, development and programmatic status of the mission, Comment: 13 pages, 16 figures. Proceedings of SPIE "Astronomical Telescopes and Instrumentation" 2024. arXiv admin note: text overlap with arXiv:2210.13866

A generalization of Rip\`a's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$ Points Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we give a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen., Comment: 9 pages, 3 figures. This is a short version of the original paper, published in NNTDM, for the reasons explained at page 9

We introduce NNsight and NDIF, technologies that work in tandem to enable scientific study of very large neural networks. NNsight is an open-source system that extends PyTorch to introduce deferred remote execution. NDIF is a scalable inference service that executes NNsight requests, allowing users to share GPU resources and pretrained models. These technologies are enabled by the intervention graph, an architecture developed to decouple experiment design from model runtime. Together, this framework provides transparent and efficient access to the internals of deep neural networks such as very large language models (LLMs) without imposing the cost or complexity of hosting customized models individually. We conduct a quantitative survey of the machine learning literature that reveals a growing gap in the study of the internals of large-scale AI. We demonstrate the design and use of our framework to address this gap by enabling a range of research methods on huge models. Finally, we conduct benchmarks to compare performance with previous approaches. Code documentation, and materials are available at https://nnsight.net/., Comment: Code at https://nnsight.net

GRBAlpha is a 1U CubeSat launched in March 2021 to a sun-synchronous LEO at an altitude of 550 km to perform an in-orbit demonstration of a novel gamma-ray burst detector developed for CubeSats. VZLUSAT-2 followed ten months later in a similar orbit carrying as a secondary payload a pair of identical detectors as used on the first mission. These instruments detecting gamma-rays in the range of 30-900 keV consist of a 56 cm2 5 mm thin CsI(Tl) scintillator read-out by a row of multi-pixel photon counters (MPPC or SiPM). The scientific motivation is to detect gamma-ray bursts and other HE transient events and serve as a pathfinder for a larger constellation of nanosatellites that could localize these events via triangulation. At the beginning of July 2024, GRBAlpha detected 140 such transients, while VZLUSAT-2 had 83 positive detections, confirmed by larger GRB missions. Almost a hundred of them are identified as gamma-ray bursts, including extremely bright GRB 221009A and GRB 230307A, detected by both satellites. We were able to characterize the degradation of SiPMs in polar orbit and optimize the duty cycle of the detector system also by using SatNOGS radio network for downlink., Comment: Presented at SPIE.Astronomical Telescopes and Instrumentations,2024

The present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form $\{0,1\}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 \times 2 \times \cdots \times 2$ ($k$ times) chessboards concerning the wazir, the threeleaper, and the zebra, for all $k \geq 15$. Our result considers the three above-mentioned leapers and replicates for each of them the recent discovery of Euclidean knight's tours for the same set of $2 \times 2 \times \cdots \times 2$ grids, opening a new research path on the topic by studying different fairy chess leapers that perform jumps of fixed Euclidean length on given regular grids, visiting all their vertices exactly once before coming back to the starting one., Comment: 9 pages

The development of algorithms for secure state estimation in vulnerable cyber-physical systems has been gaining attention in the last years. A consolidated assumption is that an adversary can tamper a relatively small number of sensors. In the literature, block-sparsity methods exploit this prior information to recover the attack locations and the state of the system. In this paper, we propose an alternative, Lasso-based approach and we analyse its effectiveness. In particular, we theoretically derive conditions that guarantee successful attack/state recovery, independently of established time sparsity patterns. Furthermore, we develop a sparse state observer, by starting from the iterative soft thresholding algorithm for Lasso, to perform online estimation. Through several numerical experiments, we compare the proposed methods to the state-of-the-art algorithms., Comment: \textcopyright 2024 the authors. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-ND

Gamma Ray Bursts (GRBs) bridge relativistic astrophysics and multi-messenger astronomy. Space-based gamma/X-ray wide field detectors have proven essential to detect and localize the highly variable GRB prompt emission, which is also a counterpart of gravitational wave events. We study the capabilities to detect long and short GRBs by the High Energy Rapid Modular Ensemble of Satellites (HERMES) Pathfinder (HP) and SpIRIT, namely a swarm of six 3U CubeSats to be launched in early 2025, and a 6U CubeSat launched on December 1st 2023. We also study the capabilities of two advanced configurations of swarms of >8 satellites with improved detector performances (HERMES Constellations). The HERMES detectors, sensitive down to ~2-3 keV, will be able to detect faint/soft GRBs which comprise X-ray flashes and high redshift bursts. By combining state-of-the-art long and short GRB population models with a description of the single module performance, we estimate that HP will detect ~195^{+22}_{-21} long GRBs (3.4^{+0.3}_{-0.8} at redshift z>6) and ~19^{+5}_{-3} short GRBs per year. The larger HERMES Constellations under study can detect between ~1300 and ~3000 long GRBs per year and between ~160 and ~400 short GRBs per year, depending on the chosen configuration, with a rate of long GRBs above z>6 between 30 and 75 per year. Finally, we explore the capabilities of HERMES to detect short GRBs as electromagnetic counterparts of binary neutron star (BNS) mergers detected as gravitational signals by current and future ground-based interferometers. Under the assumption that the GRB jets are structured, we estimate that HP can provide up to 1 (14) yr^{-1} joint detections during the fifth LIGO-Virgo-KAGRA observing run (Einstein Telescope single triangle 10 km arm configuration). These numbers become 4 (100) yr^{-1}, respectively, for the HERMES Constellation configuration., Comment: 13 pages, 6 figures, 4 tabels. Accepted for publication by Astronomy & Astrophysics

For every non-negative integer $a$ and positive integer $b$, the congruence speed of the tetration $^{b}a$ is the difference between the number of the rightmost digits of $^{b}a$ that are the same as those of $^{b+1}a$ and the number of the rightmost digits of $^{b-1}a$ that are the same as those of $^{b}a$. In the decimal numeral system, if the given base $a$ is not a multiple of $10$, as $b:=b(a)$ becomes sufficiently large, we know that the value of the congruence speed does not depend on $b$ anymore, otherwise the number of the new rightmost zeros of $^{b}a$ drastically increases for any unit increment of $b$ and, for this reason, we have not previously described the congruence speed of $a$ when it is a multiple of $10$. This short note fills the gap by giving the formula for the congruence speed of the mentioned values of $a$ at any given height of the hyperexponent., Comment: 3 pages

Given the finite set of $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ points $G_{n_1,n_2,\ldots,n_k} \subset \mathbb{R}^k$ such that $n_k \geq \cdots \geq n_2 \geq n_1 \in \mathbb{Z}^+$, we introduce a new algorithm, called M$\Lambda$I, which returns an uncrossing covering path inside the minimum axis-aligned bounding box $[0,n_1-1] \times [0,n_2-1] \times \cdots \times [0,n_k-1]$, consisting of $3 \cdot \prod_{i=1}^{k-1} n_i-2$ links of prescribed length $n_k-1$ units. Thus, for any $n_k \geq 3$, the link length of the covering path provided by our M$\Lambda$I-algorithm is smaller than the cardinality of the set $G_{n_1,n_2,\ldots,n_k}$. Furthermore, assuming $k>2$, we present an uncrossing covering path for $G_{3,3,\ldots,3}$, consisting of $20 \cdot 3^{k-3}-2$ straight-line edges that are $2$ units long each, which is constrained by the axis-aligned bounding box $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$., Comment: 14 pages, 9 figures

This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they originate as many interesting metric spaces for any proper pair $(n,k)$. For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate $k$-dimensional environment, finding (for any $k \in \mathbb{Z}^+$) the exact values of the graph radii and diameters of the $k$-rook, $k$-king, $k$-bishop, and the corresponding values for the $3$-queen, $3$-knight, and $3$-pawn. We also provide tight bounds for the graph radii and diameters of the $k$-queen, $k$-knight, and $k$-pawn, holding for any $k \geq 4$., Comment: 35 pages, 34 figures. References and a figure added; some typos corrected; improvements on grammar and style

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present paper, we provide a $5$-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k) := \{0,1,2\}^k$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game, when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a $2$D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem~3 in [Erde, J., Gol{\'e}nia, B., \& Gol{\'e}nia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), \#P9] does not hold anymore for such a Euclidean knight, as long as a $2 \times 2 \times \cdots \times 2$ chessboard with at least $2^6$ cells is given. Moreover, we show a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of $2$ from $(1,1,1,1)$, and we finally construct closed Euclidean knight's tours on $\{0,1\}^k$ for each integer $k \geq 6$., Comment: 14 pages, 4 figures. Version 2: Theorem 4.1 has been improved by including the k = 6 case, Reference added, and other minor modifications

In this review (the third in the series focused on a small two-band UV-photometry mission), we assess possibilities for a small UV two-band photometry mission in studying accreting supermassive black holes (SMBHs; mass range $\sim 10^6$-$10^{10}\,M_{\odot}$). We focus on the following observational concepts: (i) dedicated monitoring of selected type-I Active Galactic Nuclei (AGN) in order to measure the time delay between the far-UV, the near-UV, and other wavebands (X-ray and optical), (ii) nuclear transients including (partial) tidal disruption events and repetitive nuclear transients, and (iii) the study of peculiar sources, such as changing-look AGN, hollows and gaps in accretion disks, low-luminosity AGN, and candidates for Intermediate-Mass Black Holes (IMBHs; mass range $\sim 10^2$-$10^5\,M_{\odot}$) in galactic nuclei. For tidal disruption events (TDEs), high-cadence UV monitoring is crucial for distinguishing among different scenarios for the origin of the UV emission. The small two-band UV space telescope will also provide the information about the near- and far-UV continuum variability for rare transients, such as repetitive partial TDEs and jetted TDEs. We also discuss the possibilities to study and analyze sources with non-standard accretion flows, such as AGN with gappy disks, low-luminosity active galactic nuclei with intermittent accretion, and SMBH binaries potentially involving intermediate-mass black holes., Comment: 58 pages, 23 figures, 8 tables; accepted by Space Science Reviews

We outline the impact of a small two-band UV-photometry satellite mission on the field of stellar physics, magnetospheres of stars, binaries, stellar clusters, interstellar matter, and exoplanets. On specific examples of different types of stars and stellar systems, we discuss particular requirements for such satellite missions in terms of specific mission parameters such as bandpass, precision, cadence, and mission duration. We show that such a mission may provide crucial data not only for hot stars that emit most of their light in UV, but also for cool stars, where UV traces their activity. This is important, for instance, for exoplanetary studies, because the level of stellar activity influences habitability. While the main asset of the two-band UV mission rests in time-domain astronomy, an example of open clusters proves that such a mission would be important also for the study of stellar populations. Properties of the interstellar dust are best explored when combining optical and IR information with observations in UV. It is well known that dust absorbs UV radiation efficiently. Consequently, we outline how such a UV mission can be used to detect eclipses of sufficiently hot stars by various dusty objects and study disks, rings, clouds, disintegrating exoplanets or exoasteroids. Furthermore, UV radiation can be used to study the cooling of neutron stars providing information about the extreme states of matter in the interiors of neutron stars and used for mapping heated spots on their surfaces., Comment: Accepted for publication in Space Science Reviews; corrected version including comments of the referee

This is the first in a collection of three papers introducing the science with an ultra-violet (UV) space telescope on an approximately 130~kg small satellite with a moderately fast re-pointing capability and a real-time alert communication system approved for a Czech national space mission. The mission, called Quick Ultra-Violet Kilonova surveyor - QUVIK, will provide key follow-up capabilities to increase the discovery potential of gravitational wave observatories and future wide-field multi-wavelength surveys. The primary objective of the mission is the measurement of the UV brightness evolution of kilonovae, resulting from mergers of neutron stars, to distinguish between different explosion scenarios. The mission, which is designed to be complementary to the Ultraviolet Transient Astronomy Satellite - ULTRASAT, will also provide unique follow-up capabilities for other transients both in the near- and far-UV bands. Between the observations of transients, the satellite will target other objects described in this collection of papers, which demonstrates that a small and relatively affordable dedicated UV-space telescope can be transformative for many fields of astrophysics., Comment: Accepted for publication in Space Science Reviews

Aims. Since launched on 2021 March 22, the 1U-sized CubeSat GRBAlpha operates and collects scientific data on high-energy transients, making it the smallest astrophysical space observatory to date. GRBAlpha is an in-obit demonstration of a gamma-ray burst (GRB) detector concept suitably small to fit into a standard 1U volume. As it was demonstrated in a companion paper, GRBAlpha adds significant value to the scientific community with accurate characterization of bright GRBs, including the recent outstanding event of GRB 221009A. Methods. The GRB detector is a 75x75x5 mm CsI(Tl) scintillator wrapped in a reflective foil (ESR) read out by an array of SiPM detectors, multi-pixel photon counters by Hamamatsu, driven by two separate, redundant units. To further protect the scintillator block from sunlight and protect the SiPM detectors from particle radiation, we apply a multi-layer structure of Tedlar wrapping, anodized aluminium casing and a lead-alloy shielding on one edge of the assembly. The setup allows observations of gamma radiation within the energy range of 70-890 keV with an energy resolution of ~30%. Results. Here, we summarize the system design of the GRBAlpha mission, including the electronics and software components of the detector, some aspects of the platform as well as the current way of semi-autonomous operations. In addition, details are given about the raw data products and telemetry in order to encourage the community for expansion of the receiver network for our initiatives with GRBAlpha and related experiments., Comment: Accepted for publication in Astronomy & Astrophysics, 9 pages, 10 figures

The brightest gamma-ray burst ever observed, long-duration GRB 221009A, was detected by GRBAlpha nano-satellite without saturation. We present light curves of the prompt emission in 13 energy bands, from 80 keV to 950 keV, and perform a spectral analysis to calculate the peak flux and peak isotropic-equivalent luminosity. Since the satellite's attitude information is not available for the time of this GRB, more than 200 incident directions were probed in order to find the median luminosity and its systematic uncertainty. We found that the peak flux in the $80-800$ keV range (observer frame) was $F_{\rm{ph}}^{\rm{p}}=1300_{-200}^{+1200}$ ph cm$^{-2}$s$^{-1}$ or $F_{\rm{erg}}^{\rm{p}}=5.7_{-0.7}^{+3.7}\times10^{-4}$ erg cm$^{-2}$s$^{-1}$ and the fluence in the same energy range of the first GRB episode lasting 300 s, which was observable by GRBAlpha, was $S=2.2_{-0.3}^{+1.4}\times10^{-2}$ erg cm$^{-2}$ or $S^{\rm{bol}}=4.9_{-0.5}^{+0.8}\times10^{-2}$ erg cm$^{-2}$ for the extrapolated range of $0.9-8,690$ keV. We infer the isotropic-equivalent released energy of the first GRB episode to be $E_{\rm{iso}}^{\rm{bol}}=2.8_{-0.5}^{+0.8}\times10^{54}$ erg in the $1-10,000$ keV band (rest frame at $z=0.15$). The peak isotropic-equivalent luminosity in the $92-920$ keV range (rest frame) was $L_{\rm{iso}}^{\rm{p}}=3.7_{-0.5}^{+2.5}\times10^{52}$ erg s$^{-1}$ and the bolometric peak isotropic-equivalent luminosity was $L_{\rm{iso}}^{\rm{p,bol}}=8.4_{-1.5}^{+2.5}\times10^{52}$ erg s$^{-1}$ (4 s scale) in the $1-10,000$ keV range (rest frame). The peak emitted energy is $E_p^\ast=E_p(1+z)=1120\pm470$ keV. Our measurement of $L_{\rm{iso}}^{\rm{p,bol}}$ is consistent with the Yonetoku relation. It is possible that, due to the spectral evolution of this GRB and orientation of GRBAlpha at the peak time, the true values of peak flux, fluence, $L_{\rm{iso}}$, and $E_{\rm{iso}}$ are even higher. [abridged], Comment: 7 pages, 7 figures, 1 table, accepted for publication in Astronomy & Astrophysics

Multi-messenger observations of the transient sky to detect cosmic explosions and counterparts of gravitational wave mergers critically rely on orbiting wide-FoV telescopes to cover the wide range of wavelengths where atmospheric absorption and emission limit the use of ground facilities. Thanks to continuing technological improvements, miniaturised space instruments operating as distributed-aperture constellations are offering new capabilities for the study of high energy transients to complement ageing existing satellites. In this paper we characterise the performance of the upcoming joint SpIRIT + HERMES-TP/SP nano-satellite constellation for the localisation of high-energy transients through triangulation of signal arrival times. SpIRIT is an Australian technology and science demonstrator satellite designed to operate in a low-Earth Sun-synchronous Polar orbit that will augment the science operations for the equatorial HERMES-TP/SP. In this work we simulate the improvement to the localisation capabilities of the HERMES-TP/SP when SpIRIT is included in an orbital plane nearly perpendicular (inclination = 97.6$^\circ$) to the HERMES orbits. For the fraction of GRBs detected by three of the HERMES satellites plus SpIRIT, the combined constellation is capable of localising 60% of long GRBs to within ~ 30 deg$^2$ on the sky, and 60% of short GRBs within ~ 1850 deg$^2$. Based purely on statistical GRB localisation capabilities (i.e., excluding systematic uncertainties and sky coverage), these figures for long GRBs are comparable to those reported by the Fermi GBM. Further improvements by a factor of 2 (or 4) can be achieved by launching an additional 4 (or 6) SpIRIT-like satellites into a Polar orbit, which would both increase the fraction of sky covered by multiple satellite elements, and enable $\geq$ 60% of long GRBs to be localised within a radius of ~ 1.5$^\circ$ on the sky., Comment: 17 pages, 10 figures, 1 table. Accepted for publication in PASA

We solve the general problem of visiting all the $2^k$ nodes of a $k$-dimensional hypercube by using a polygonal chain that has minimum link-length, and we show that this optimal value is given by $h(2,k):=3 \cdot 2^{k-2}$ if and only if $k \in \mathbb{N}-\{0,1\}$. Furthermore, for any $k$ above one, we constructively prove that it is possible to visit once and only once all the aforementioned nodes, $H(2,k):=\{\{0,1\} \times \{0,1\} \times \dots \times \{0,1\}\} \subset \mathbb{R}^k$, with a cycle (i.e., a closed path) having only $3 \cdot 2^{k-2}$ links., Comment: 13 pages, 5 figures. Typos corrected; misprint in Section 3 fixed

Spatiotemporal sculpturing of light pulse with ultimately sophisticated structures represents the holy grail of the human everlasting pursue of ultrafast information transmission and processing as well as ultra-intense energy concentration and extraction. It also holds the key to unlock new extraordinary fundamental physical effects. Traditionally, spatiotemporal light pulses are always treated as spatiotemporally separable wave packet as solution of the Maxwell's equations. In the past decade, however, more generalized forms of spatiotemporally nonseparable solution started to emerge with growing importance for their striking physical effects. This roadmap intends to highlight the recent advances in the creation and control of increasingly complex spatiotemporally sculptured pulses, from spatiotemporally separable to complex nonseparable states, with diverse geometric and topological structures, presenting a bird's eye viewpoint on the zoology of spatiotemporal light fields and the outlook of future trends and open challenges., Comment: This is the version of the article before peer review or editing, as submitted by an author to Journal of Optics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base $a\in\mathbb{N}_0$. The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for every tetration base which is not coprime to $10$, although a maximum gap equal to $V(a)+1$ digits (where $V(a)$ denotes the constant congruence speed of $a$) can occur, in the worst-case scenario, between the upper and lower bound. In addition, for every $a>1$ which is not a multiple of $10$, we show that $V(a)$ corresponds to the $2$-adic or $5$-adic valuation of $a-1$ or $a+1$, or even to the $5$-adic order of $a^{2}+1$, depending on the congruence class of $a$ modulo $20$., Comment: 18 pages

We solve a few open problems related to a peculiar property of the integer tetration ${^{b}a}$, which is the constancy of its congruence speed for any sufficiently large $b=b(a)$. Assuming radix-$10$ (the well-known decimal numeral system), we provide an explicit formula for the congruence speed $V(a) \in \mathbb{N}_0$ of any $a \in \mathbb{N}-\{0\}$ that is not a multiple of $10$. In particular, for any given $n \in \mathbb{N}$, we prove to be true Rip\`a's conjecture on the smallest $a$ such that $V(a)=n$. Moreover, for any $a \neq 1 : a \not\equiv 0 \pmod {10}$, we show the existence of infinitely many prime numbers $p_j:=p_j(V(a))$ such that $V(p_j)=V(a)$., Comment: 19 pages, 2 tables

We present a near-UV space telescope on a ~70kg micro-satellite with a moderately fast repointing capability and a near real-time alert communication system that has been proposed in response to a call for an ambitious Czech national mission. The mission, which has recently been approved for Phase 0, A, and B1 study shall measure the brightness evolution of kilonovae, resulting from mergers of neutron stars in the near-UV band and thus it shall distinguish between different explosion scenarios. Between the observations of transient sources, the satellite shall perform observations of other targets of interest, a large part of which will be chosen in open competition., Comment: SPIE Astronomical Telescopes and Instrumentation

We present the detector performance and early science results from GRBAlpha, a 1U CubeSat mission, which is a technological pathfinder to a future constellation of nanosatellites monitoring gamma-ray bursts (GRBs). GRBAlpha was launched in March 2021 and operates on a 550 km altitude sun-synchronous orbit. The gamma-ray burst detector onboard GRBAlpha consists of a 75x75x5 mm CsI(Tl) scintillator, read out by a dual-channel multi-pixel photon counter (MPPC) setup. It is sensitive in the ~30-900 keV range. The main goal of GRBAlpha is the in-orbit demonstration of the detector concept, verification of the detector's lifetime, and measurement of the background level on low-Earth orbit, including regions inside the outer Van Allen radiation belt and in the South Atlantic Anomaly. GRBAlpha has already detected five, both long and short, GRBs and two bursts were detected within a time-span of only 8 hours, proving that nanosatellites can be used for routine detection of gamma-ray transients. For one GRB, we were able to obtain a high resolution spectrum and compare it with measurements from the Swift satellite. We find that, due to the variable background, the time fraction of about 67 % of the low-Earth polar orbit is suitable for gamma-ray burst detection. One year after launch, the detector performance is good and the degradation of the MPPC photon counters remains at an acceptable level. The same detector system, but double in size, was launched in January 2022 on VZLUSAT-2 (3U CubeSat). It performs well and already detected three GRBs and two solar flares. Here, we present early results from this mission as well., Comment: 11 pages, 9 figures, submitted to the proceedings book of the conference: SPIE Astronomical Telescopes + Instrumentation 2022

Given any $n \in \mathbb{Z}^{+}$, we constructively prove the existence of covering paths and circuits in the plane which are characterized by the same link length of the minimum-link covering trails for the two-dimensional grid $G_n^2 := \{0,1, \ldots, n-1\} \times \{0, 1, \ldots, n-1\}$. Furthermore, we introduce a general algorithm that returns a covering cycle of analogous link length for any even value of $n$. Finally, we provide the tight upper bound $n^2 - 3 + 5 \cdot \sqrt{2}$ units for the minimum total distance travelled to visit all the nodes of $G_n^2$ with a minimum-link trail (i.e., a trail with $2 \cdot n - 2$ edges if $n$ is above two)., Comment: 18 pages, 13 figures; introduction improved, Kato's conjecture reformulated, and appendix moved after references

In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}$. In this paper we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be axis-parallel, showing that the original Theorem 1 by Kranakis et al. no longer holds when the aforementioned constraint is disregarded. Furthermore, for any given $n$ above two, as $k$ approaches infinity, the link-length of any minimal (non-rectilinear) polygonal chain does not exceed Kranakis' conjectured value of $\frac{k}{k-1} \cdot n^{k-1}+O(n^{k-2})$ only if we introduce a multiplicative constant $c \geq 1.5$ for the lower order terms (e.g., if we select $n=3$ and assume that $c<1.5$, starting from a sufficiently large $k$, it is not possible to visit all the nodes of $P(n,k)$ with a trail of link-length $\frac{k}{k-1} \cdot n^{k-1}+c \cdot n^{k-2}$)., Comment: 5 pages, extending and substantially improving some previously published results in Int. Journal of Math. Arch., Vol. 10(8), 36-38 (2019); minor stylistic corrections, reference [8] fixed; rephrasing Definitions 1.2 and 1.4, revising notations, and improving the general lower bound provided by Theorem 2.1

The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a general theorem that rules out $4/9$ of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence $A352991$ (which includes all the terms of $A001292$). Our risky conjecture states that all the perfect powers belonging to the sequence $A352991$ are perfect squares and they cannot be written as higher order perfect powers if the given term of $A352991$ is not equal to one. This challenging conjecture has been checked for any integer smaller than $10111121314151617181920212223456789$ and no counterexample has been found so far., Comment: 8 pages; typos fixed, minor style improvements, and Cross-referencing added