arXiv subjects

Marek Wolf

Publications and source records attributed to Marek Wolf.

At least 19 recordsLinked to original sources

arXiv2026

K 1-6 is a photoionised ISM nebula shaped by a fast-moving hot white dwarf in a triple system

K 1-6 has long been classified as a planetary nebula (PN) hosting a binary central star, yet it has remained poorly studied due to its faintness. The central star exhibits pronounced photometric variability whose origin has so far been unclear. We aim to present a comprehensive characterisation of the K 1-6 system, including the physical properties of its stellar components and the nature of the surrounding nebulosity. We conducted a multi-wavelength analysis combining optical and UV spectroscopy obtained with the Gran Telescopio Canarias, the Telescopio Nazionale Galileo, the Nordic Optical Telescope, and the Hubble Space Telescope. We also present long-term multi-band ground- and space-based photometry, including high-cadence data from the Transiting Exoplanet Survey Satellite, narrow-band imaging, and the latest astrometric constraints from Gaia. Our results show that the nebula is not a remnant PN, but instead consists of interstellar medium photoionised by a hot white dwarf, which is relatively evolved. It has a cooling age of 1-2 Myr, implying that any original PN has long since dissipated. We further find that the central object is a hierarchical triple system, comprising an inner binary with an orbital period likely of the order of thousands of days and a distant tertiary companion on a timescale of tens of thousands of years. The optically dominant cool component of the inner binary is an inflated K-type star displaying extreme magnetic activity, including large-amplitude variability and flaring. Its properties resemble those of BY Dra-type binaries and Abell 35-type systems, and are difficult to reconcile with single-star evolution, pointing instead to a history of binary interaction.

astro-ph.SR↗

arXiv2026

TESS light curves and surface activity in two low-mass eclipsing binaries: NSVS 01031772 and 2MASS J04100497+2931023

The VRIC light curves were regularly measured for two eclipsing binaries, NSVS 01031772 and 2MASS J04100497+2931023 as part of our long-term observational project to study low-mass eclipsing binaries with a short orbital period and surface activity. The solution of the TESS light curves in PHOEBE results in a detached configuration. Absolute parameters of all components were improved: for N103: M1 = 0.5475 +/- 0.0035 M\sun, R1 = 0.5297 +/- 0.0035 R\sun, M2 = 0.5038 +/- 0.0040 M\sun, R2 = 0.5217 +/- 0.0035 R\sun, for 2M0410: M1 = 0.639 +/- 0.045 M\sun, R1 = 0.655 +/- 0.035 R\sun, M2 = 0.609 +/- 0.045 M\sun, R2 = 0.631 +/- 0.035 R\sun, where the temperatures of the primary components were adopted according to previous studies. The spectral type of the primary components was confirmed to be M4 and K5, and the mass ratio was derived as q = 0.920 +/- 0.003 or 0.952 +/- 0.010, respectively. We propose the presence of a third body in these systems: in the case of N103, a companion with a minimal mass of 50 MJup, orbiting the eclipsing pair with a period of about 19 years, and in 2M0410 a third body with a minimal mass of about 0.1 M\sun and a short orbital period of about 2.1 years. For N103, the hierarchical structure (2+1)+1 of a possible quadruple system was tested, but its stability was not proven. The characteristics and statistics of the flare events and dark regions on the surface of the components were estimated on the basis of the TESS and our own data. For N103, a mean frequency of flares of one per 40 hours was determined. In the case of 2M0410, practically no flares were detected.

astro-ph.SR↗

arXiv2026

Refined reduction and standardisation of 53 years of UBV photometry at Hvar

Emission-line stars classified as Be exhibit light and colour variability on various timescales, ranging from days to decades. Their evolution must be constrained by long-term observations that are accurately calibrated and stable. Here, we provide a new reduction of photoelectric UBV observations obtained at the Hvar observatory, spanning more than 50 years. This unique dataset is highly complementary to TESS. We used new, non-linear reduction equations, with temporally variable extinction over the course of the night, which allowed us to achieve long-term accuracy of 0.008 to 0.016 mag (1-sig uncertainty), as verified by the Johnson standards. We then classified 59 Be stars into five classes, based on their variability patterns; namely, long-term envelope (LTE), long-term cyclic (LTC), binarity (BIN), rapid low-amplitude (RLA), and long-term quiescence (LTQ). According to our observations, the percentages of stars in the individual classes are 44%, 24%, 25%, 66%, and 19%, respectively. The stars in the sample often exhibited more than one pattern. We confirm that the LTE-positive variability is more common than the inverse (20 vs 6); in addition, two stars exhibited both types (zeta Tau and V1294 Aql). According to our observations, the LTC variability and the LTE-positive variability are almost mutually exclusive. Among 26 binary systems, circular orbits are more common than eccentric ones (18 vs 8). As for the brightness variations between different quiescent phases, an increasing trend is less common than a decreasing one (4 vs 7), spanning from -6.5 to +6.0 mmag/yr. Our observations provide well-calibrated UBV light curves spanning several decades, offering a valuable dataset for investigations of Be-star variability and tests of various models, including the viscous decretion disc model. Continuous monitoring is important for beta Lyr, EW Lac, delta Sco, gamma Cas, and V1294 Aql.

astro-ph.SR↗

arXiv2026

New Online Database of Symbiotic Variables: Catalog and Statistical Overview of Symbiotic Binaries

We present the New Online Database of Symbiotic Variables (NODSV), a comprehensive and publicly accessible catalog of known and candidate symbiotic stars in the Milky Way and nearby galaxies. The database provides an up-to-date census of confirmed symbiotic binaries and systematically compiles information previously scattered across the literature, including photometric and spectroscopic properties, orbital parameters, and characteristics of both their cool and hot stellar components. It further records auxiliary diagnostics such as detected emission lines, flickering, X-ray emission, jets, or information about outburst activity. In its current release, NODSV contains nearly 1 400 objects, classified into confirmed symbiotic stars, three categories of candidates, and misidentified sources. Based on the collected data, though originating from heterogeneous studies, we present a statistical overview of the confirmed symbiotic population, highlighting the distributions of orbital parameters and the properties of the cool giants and their hot companions. Designed as a dynamic and evolving resource, NODSV provides a foundation for future observational campaigns and theoretical investigations of symbiotic binaries.

astro-ph.SR↗

arXiv2026

The first hours and days of the 2021 explosion of the recurrent symbiotic nova RS Ophiuchii

The accretion of matter on a massive white dwarf (WD) can lead to repeated nuclear explosions on its surface over a timescale of years to decades. The seventh explosion of the recurrent symbiotic nova RS Ophiuchi (RS Oph) was recorded on August 8, 2021. In this paper, we examine its early evolution, from 9 hours before its optical maximum until day 42. We achieved our goal by modeling the spectral energy distribution (SED) using optical spectroscopy and simultaneous $BVR_{\rm C}I_{\rm C}$ photometry, supplemented by $JHKL$ photometry and ultraviolet spectroscopy from previous explosions in 2006 and 1985. Our SED models revealed an early stage of development of the ejecta bipolar structure, consisting of a flared, density-enhanced equatorial disk and low-density regions in bipolar directions. The comparability of the internal shocks' luminosity in the equatorial outflow, inferred from our model parameters, with the luminosity of the warm WD pseudophotosphere during its presence in the spectrum (until $\sim$day 42) confirmed that a significant part of its radiation originates from reprocessed shock emission. We explain the formation and evolution of the bipolar ejecta structure during RS Oph explosions by the rotation of the accreting WD. Such an ejecta structure provides a natural framework for the generation of strong internal shocks and thus $\gamma$-ray emission inside the ejecta.

astro-ph.SR↗

arXiv2026

A Detailed Study of the Physical and Orbital Characteristics, and Eclipse Timing Variations of the Post Common Envelope Binary DD CrB

We present an in-depth analysis of the eclipsing binary DD CrB, composed of a B-type subdwarf primary and an M-type main-sequence secondary, with the main goal of investigating its eclipse timing variations (ETVs). Our new multi-color photometric observations, radial velocity measurements, and precise eclipse timings from TESS allow us to constrain the system parameters. The Romer delay between primary and secondary minima yields a mass ratio of $q = 0.299 \pm 0.009$, enabling robust simultaneous modeling of the light and radial velocity curves with {\sc phoebe} 2.17. By fixing the albedo of the secondary to its maximum physically plausible value (A$_2 = 1.0$), despite the degeneracy between albedo, surface temperature, and radius, we obtained a satisfactory fit, resulting in a significantly lower temperature ($T_2 \sim 2360$ K) and a radius ($R_2 \sim 0.16$ R$_\odot$) in agreement with literature values. Using the total mass of the components and the orbital size derived from this modeling, we interpret the ETVs and find them best explained by a Jupiter-mass tertiary companion on a $\sim13$-year orbit in all competing models, while the eccentric (e $\sim0.46$) models perform better in terms of fit statistics.

astro-ph.SR↗

arXiv2026

Apsidal motion and TESS light curves of two southern eclipsing binaries with high eccentricity: V1647 Sgr and V2283 Sgr

The study of apsidal motion rates in eccentric eclipsing binaries provides an important observational test of theoretical models of stellar structure and evolution. Precise physical parameters of the stellar components together with systematic measurements of the periastron advance are needed. We present new results of our long-term observational project to analyze the apsidal motion in early-type eccentric eclipsing binaries. New ground and space-based photometric data were obtained, and archival spectroscopic measurements were used in this study of two detached southern-hemisphere eclipsing binaries: V1647 Sgr (P=3.28 d, e=0.41), and V2283 Sgr (3.47, 0.49). Their TESS observations in four sectors have also been included and the corresponding light curves were solved using the Phoebe code. The newly completed O-C diagrams were analyzed using all reliable timings found in the literature and calculated using the TESS light curves. New or improved values were obtained for the elements of apsidal motion. Using archival spectroscopy for V1647 Sgr, the precise absolute parameters were improved: M1 = 2.184(0.035) M$_\odot$, M2 = 1.957(0.035) M$_\odot$, and R1 = 1.839(0.015) R$_\odot$, R2 = 1.716(0.015) R$_\odot$. For V2283 Sgr the absolute dimensions were newly estimated: M1 = 2.178(0.10) M$_\odot$, M2 = 1.547(0.10) M$_\odot$, and R1 = 1.796(0.01) R$_\odot$, R2 = 1.544(0.01) R$_\odot$. We improved relatively long periods of apsidal motion of about 580 and 530 years, together with the corresponding internal structure constants, log k2, -2.394, and -2.418, for V1647 Sgr and V2283 Sgr, respectively. The relativistic contribution to apsidal motion is not negligible, making about 12 resp. 14% of the total rate of apsidal motion. No signs of the presence of an additional body were revealed in the light curves or in the O-C diagrams of both eccentric systems.

astro-ph.SR↗

arXiv2025

TESS light curves and period changes in low-mass eclipsing binary BB Persei

We present a detailed analysis of the low-mass detached eclipsing binary system BB Persei, which contains two K-type stars in a circular orbit with a short period of 0.4856 d. We used light curves from the Transiting Exoplanet Survey Satellite, which observed BB Per in five sectors, to determine its photometric properties and a precise orbital ephemeris. The solution of the TESS light curve in Phoebe results in a detached configuration, where the temperature of the primary component was fixed to $T_1 = 5~300$ K according to Lamost, which gives us $T_2 = 5~050 \pm 50$ K for the secondary. The spectral type of the primary component was derived as K0 and the photometric mass ratio was estimated $q = 0.90$. Slow period changes on the current O-C diagram spanning the past 25 years indicate the presence of a third body orbiting the eclipsing pair with an orbital period of about 22 years. The companion could be a red dwarf of spectral type M6 - M7 with a minimal mass of about 0.1 M$_{\odot}$. The characteristics and temporal variation of the dark region on the surface of the secondary component were estimated.

astro-ph.SR↗

arXiv2024

WASP 0346-21: An EL CVn-Type Eclipsing Binary with Multiperiodic Pulsations in a Triple System

VLT/UVES spectroscopic and TESS photometric observations for WASP 0346-21 allow the direct determination of its physical properties, along with the detection of a circumbinary object and oscillating signals. The high-resolution spectra yielded the radial velocities of all three stars and the atmospheric parameters of $T_{\rm eff,A}$ = 7225$\pm42$ K, [M/H] = 0.30$\pm$0.03 dex, and $v_{\rm A}$$\sin i$ = 78$\pm$5 km s$^{-1}$ of the primary component. The combined analysis of these observations resulted in the fundamental parameters of the eclipsing components and the third light of $l_3$ = 0.043$\pm$0.004, which is consistent with the light contribution of the tertiary star observed in the echelle spectra. WASP 0346-21 A resides within the overlapping main-sequence domain of $δ$ Sct and $γ$ Dor variables, while the secondary component of $M_{\rm B}$ = 0.185$\pm$0.013 M$_\odot$, $R_{\rm B}$ = 0.308$\pm$0.023 R$_\odot$, $T_{\rm eff,B}$ = 10,655$\pm$146 K, and $L_{\rm B}$ = 1.09$\pm$0.17 L$_\odot$ matches well with the low-mass white dwarf (WD) model for $Z$ = 0.01, corresponding to the thick-disk population classified by the Galactic kinematics. Multifrequency analyses were performed on the residual TESS data after removing the binarity effects. The low frequencies around 26.348 day$^{-1}$ and 17.683 day$^{-1}$ are $δ$ Sct pulsations originating from WASP 0346-21 A, and the high frequencies of 97.996 day$^{-1}$ and 90.460 day$^{-1}$ are considered to be extremely low-mass WD oscillations. These results demonstrate that WASP 0346-21 is a hierarchical triple system, consisting of an EL CVn binary with multiperiodic pulsations in each component and a distant outer tertiary.

astro-ph.SR↗

arXiv2024

Exploring the MMRD Relation for Novae in M31

The results of a two decade long $R$-band photometric survey of novae in M31 are presented. From these data, $R$-band light curves have been determined for 180 novae with data sufficient for estimating peak brightness and subsequent rate of decline. The data show a weak correlation of peak brightness with fade rate consistent with the well-known Maximum Magnitude versus Rate of Decline (MMRD) relation. As generally appreciated for Galactic novae, the large scatter in the MMRD relation precludes its use in determining distances to individual novae. The novae at maximum light are distributed with standard deviation $σ=0.89$ mag about a mean $R$-band absolute magnitude given by $\langle M_R \rangle=-7.57\pm0.07$. The overall M31 luminosity distribution is in excellent agreement with that found for Galactic novae suggesting that the nova populations in M31 and the Galaxy are quite similar. The notion that all novae can be characterized by a standard luminosity 15 d after maximum light ($M_{15}$) is also explored. Surprisingly, the distribution of $M_{15}$ values is characterized by a standard deviation only slightly smaller than that for novae at maximum light and thus offers little promise for precise extragalactic distance determinations. A dozen faint and fast novae that are likely to be previously unidentified recurrent novae have been identified from their position in the MMRD plot and in the $M_{15}$ distribution.

astro-ph.SR↗

arXiv2023

Absolute Properties of the Oscillating Eclipsing Algol XZ Ursae Majoris

It is known from archival TESS data that the semi-detached Algol system XZ UMa is one of the candidate binary stars exhibiting short-period oscillations. We secured new high-resolution spectroscopic observations for the program target to better understand its binary and pulsation properties. From the echelle spectra, the radial velocities (RVs) of the eclipsing pair were derived, and the atmosphere parameters of the primary component were measured to be $v_{\rm A}\sin$$i = 80\pm7$ km s$^{-1}$, $T_{\rm eff,A}$ = $7940\pm120$ K, and [M/H] = $-0.15\pm0.20$. The combined solution of our double-lined RVs and the TESS data provides robust physical parameters for XZ UMa with mass and radius measurement precision of better than 2 \%. The outside-eclipse residuals from a mean light curve in the 0.002 phase bin were used for multifrequency analyses, and we extracted 32 significant frequencies (22 in $<$ 5.0 day$^{-1}$ and 10 in 39$-$52 day$^{-1}$). The low frequencies may be mostly aliasing sidelobes, while six of the high frequencies may be pulsation signals arising from the detached primary located inside the $δ$ Sct domain. Their periods, pulsation constants, and pulsational-orbital-period ratios indicate that the mass-accretion primary star is a $δ$ Sct pulsator and, hence, XZ UMa is an oscillating eclipsing Algol.

astro-ph.SR↗

arXiv2023

The Z Camelopardalis-type star AY Piscium: stellar and accretion disk parameters

We present a new study of the Z~Cam-type eclipsing cataclysmic variable AY~Piscium with the aim of determining the fundamental parameters of the system and the structure of the accretion flow therein. We use time-resolved photometric observations supplemented by spectroscopy in the standstill, to which we applied our light-curve modeling techniques and the Doppler tomography method, to update system parameters. We found that the system has a massive white dwarf $M_{\rm WD}=0.90(4)$ \ms, a mass ratio $q=0.50(3)$, and the effective temperature of a secondary $T_2 = 4100(50)$~K. The system inclination is $i=74.^{\circ}8(7)$. The orbital period of the system $P_{\mathrm{orb}}=0.217320523(8)\;\mathrm{d}$ is continuously increasing with the rate of $\dot{P}_{\mathrm{orb}} = +7.6(5)\times10^{-9}$ d year$^{-1}$. The mass transfer rate varies between 2.4$\times$10$^{-10}$ M$_\odot$ year$^{-1}$ in quiescence up to 1.36$\times$10$^{-8}$ M$_\odot$ year$^{-1}$ in outburst. The accretion disk transitions from the cooler, flared, steady-state disk to a warmer state with a practically constant and relatively high disk height. The mass transfer rate is about 1.6$\times$10$^{-9}$ M$_\odot$ year$^{-1}$ in the standstill. The Balmer emission lines show a multi-component structure similar to that observed in long-orbital-period nova-like systems. Out of standstill, the system exhibits outburst bimodality, with long outbursts being more prominent. We conclude that the Balmer emission lines in AY~Psc are formed by the combination of radiation from the irradiated surface of the secondary, from the outflow zone, and from winds originating in the bright spot and the disk's inner part.

astro-ph.SR↗

arXiv2023

Absolute Properties of the Oscillating Eclipsing Algol X Trianguli

We report results from the TESS photometric data and new high-resolution spectra of the Algol system X Tri showing short-period pulsations. From the echelle spectra, the radial velocities of the eclipsing pair were measured, and the rotational rate and effective temperature of the primary star were obtained to be $v_1$$\sin$$i=84\pm6$ km s$^{-1}$ and $T_{\rm eff,1}=7900 \pm 110$ K, respectively. The synthetic modeling of these observations implies that X Tri is in synchronous rotation and is physically linked to a visual companion TIC 28391715 at a separation of about 6.5 arcsec. The absolute parameters of our target star were accurately and directly determined to be $M_1 = 2.137 \pm 0.018$ $M_\odot$, $M_2 = 1.101 \pm 0.010$ $M_\odot$, $R_1 = 1.664 \pm 0.010$ $R_\odot$, $R_2 = 1.972 \pm 0.010$ $R_\odot$, $L_1 = 9.67 \pm 0.55$ $L_\odot$, and $L_2 = 2.16 \pm 0.09$ $L_\odot$. The phase-binned mean light curve was used to remove the binary effect from the observed TESS data. Multifrequency analysis of the residuals revealed 16 significant frequencies, of which the high-frequency signals between 37 day$^{-1}$ and 48 day$^{-1}$ can be considered probable pulsation modes. Their oscillation periods of 0.021$-$0.027 days and pulsation constants of 0.014$-$0.018 days are typical values of $δ$ Sct variables. The overall results demonstrate that X Tri is an oEA star system, consisting of a $δ$ Sct primary and its lobe-filling companion in the semi-detached configuration.

astro-ph.SR↗

arXiv2022

M31N 1926-07c: A Recurrent Nova in M31 with a 2.8 Year Recurrence Time

The M31 recurrent nova M31N 1926-07c has had five recorded eruptions. Well-sampled light curves of the two most recent outbursts, in January of 2020 (M31N 2020-01b) and September 2022 (M31N 2022-09a), are presented showing that the photometric evolution of the two events were quite similar, with peak magnitudes of $R=17.2\pm0.1$ and $R=17.1\pm0.1$, and $t_2$ times of $9.7\pm0.9$ and $8.1\pm0.5$ days for the 2020 and 2022 eruptions, respectively. After considering the dates of the four most recent eruptions (where the cycle count is believed to be known), a mean recurrence interval of $\langle P_\mathrm{rec}\rangle=2.78\pm0.03$ years is found, establishing that M31N 1926-07c has one of the shortest recurrence times known.

astro-ph.SR↗

arXiv2020

Predicting maximal gaps in sets of primes

Let $q>r\ge1$ be coprime integers. Let ${\mathbb P}_c={\mathbb P}_c(q,r,{\cal H})$ be an increasing sequence of primes $p$ satisfying two conditions: (i) $p\equiv r$ (mod $q$) and (ii) $p$ starts a prime $k$-tuple with a given pattern ${\cal H}$. Let $π_c(x)$ be the number of primes in ${\mathbb P}_c$ not exceeding $x$. We heuristically derive formulas predicting the growth trend of the maximal gap $G_c(x)=\max_{p'\le x}(p'-p)$ between successive primes $p,p'\in{\mathbb P}_c$. Extensive computations for primes up to $10^{14}$ show that a simple trend formula $$G_c(x) \sim {x\overπ_c(x)}\cdot(\log π_c(x) + O_k(1))$$ works well for maximal gaps between initial primes of $k$-tuples with $k\ge2$ (e.g., twin primes, prime triplets, etc.) in residue class $r$ (mod $q$). For $k=1$, however, a more sophisticated formula $$G_c(x) \sim {x\overπ_c(x)}\cdot\big(\log{π_c^2(x)\over x}+O(\log q)\big)$$ gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes ($k=1$, $q=2$, $r=1$). The distribution of appropriately rescaled maximal gaps $G_c(x)$ is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramer's conjecture. We also conjecture that the number of maximal gaps between primes in ${\mathbb P}_c$ below $x$ is $O_k(\log x)$.

math.NT↗

arXiv2020

On the first occurrences of gaps between primes in a residue class

We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O(\log^2 x)$ first-occurrence gaps between primes in (P) below $x$, while the number of maximal gaps is only $O(\log x)$. We explore the connection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun's constant. For the first occurrence of gap $d$ in (P), we expect the end-of-gap prime $p\asymp\sqrt{d}\exp(\sqrt{d/φ(q)})$ infinitely often. Finally, we study the gap size as a function of its index in the sequence of first-occurrence gaps.

math.NT↗

arXiv2020

The nature of the symbiotic candidate 2MASS J07363415+6538548 in the field of NGC 2403

New Online Database of Symbiotic Variables includes several poorly characterised objects and candidate symbiotic stars, not only in the Milky Way but also in other galaxies. The goal of the research presented in this paper was to reveal the nature of 2MASS J07363415+6538548, the object discovered as an X-ray source in the field of NGC 2403, and tentatively classified as a symbiotic candidate or a cataclysmic variable. By analysis of available photometric data from ground-based surveys, together with high precision photometry from TESS, remarkable astrometric measurements of the Gaia satellite, and observations of other surveys spanning from X-rays to infrared, we have found that the object is not a symbiotic star nor a cataclysmic variable but rather an active K-type dwarf. The star is located in the distance of 415 pc, has an effective temperature of 4 275 K, luminosity of 0.14 L$\odot$, a mass of 0.7 M$\odot$, and radius of 0.7 R$\odot$. It has a rotational period ~ 3 days and is a strong X-ray source with the X-ray luminosity of ~ $10^{30}$ erg/s. Gyrochronology and isochrone fitting confirmed that the star is young.

astro-ph.SR↗

arXiv2020

Are the Stieltjes constants irrational? Some computer experiments

Khnichin's theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss--Kuzmin theorem to several thousand high precision (up to more than 53000 significant digits) initial Stieltjes constants $γ_{n}$, $n=0,1,...,5000$ in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We study also the normality of these important constants.

math.NT↗