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We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least common multiple of the moduli is at least 1440. Finally, we prove that if the smallest modulus of a covering system with distinct moduli is 6, then the least common multiple of the moduli is at least $5040$. The constants $108$, $1440$ and $5040$ are best possible. This resolves a conjecture of Krukenberg, a problem of Dalton and Trifonov, and a generalization thereof.
In addition to the regular Schwabe cycles of approximately 11 y, "prolonged solar activity minima" have been identified through the direct observation of sunspots and aurorae, as well as proxy data of cosmogenic isotopes. Some of these minima have been regarded as grand solar minima, which are arguably associated with the special state of the solar dynamo and have attracted significant scientific interest. In this paper, we review how these prolonged solar activity minima have been identified. In particular, we focus on the Dalton Minimum, which is named after John Dalton. We review Dalton's scientific achievements, particularly in geophysics. Special emphasis is placed on his lifelong observations of auroral displays over approximately five decades in Great Britain. Dalton's observations for the auroral frequency allowed him to notice the scarcity of auroral displays in the early 19th century. We analyze temporal variations in the annual frequency of such displays from a modern perspective. The contemporary geomagnetic positions of Dalton's observational site make his dataset extremely valuable because his site is located in the sub-auroral zone and is relatively sensitive to mino
French astronomer Honoré Flaugergues compiled astronomical observations in a series of hand-written notebooks for 1782$\unicode{x2013}$1830, which are preserved at Paris Observatory. We reviewed these manuscripts and encoded the records that contain sunspot measurements into a numerical table for further analysis. All measurements are timings and we found three types of measurements allowing the reconstruction of heliographic coordinates. In the first case, the Sun and sunspots cross vertical and horizontal wires, in the second case, one vertical and two mirror-symmetric oblique wires, and in the third case, a rhombus-shaped set of wires. Additionally, timings of two solar eclipses also provided a few sunspot coordinates. As a result, we present the time--latitude (butterfly) diagram of the reconstructed sunspot coordinates, which covers the period of the Dalton Minimum and confirms consistency with those of Derfflinger and Prantner. We identify four solar cycles in this diagram and discuss the observed peculiarities as well as the data reliability.
Visible coronal structure, in particular the spatial evolution of coronal streamers, provides indirect information about solar magnetic activity and the underlying solar dynamo. Their apparent absence of structure observed during the total eclipses of throughout the Maunder Minimum has been interpreted as evidence of a significant change in the solar magnetic field from that during modern cycles. Eclipse observations available from the more recent Dalton Minimum may be able to provide further information, sunspot activity being between the levels seen during recent cycles and in the Maunder Minimum. Here, we show and examine two graphical records of the total solar eclipse on 1806 June 16, during the Dalton Minimum. These records show significant rays and streamers around an inner ring. The ring is estimated to be ~ 0.44 R_S in width and the streamers in excess of 11.88 R_S in length. In combination with records of spicules or prominences, these eclipse records visually contrast the Dalton Minimum with the Maunder Minimum in terms of their coronal structure and support the existing discussions based on the sunspot observations. These eclipse records are broadly consistent with the
We present and analyse the sunspot observations performed by Franz I. C. Hallaschka in 1814 and 1816. These solar observations were carried out during the so-called Dalton minimum, around the maximum phase of the Solar Cycle 6. These records are very valuable because they allow us to complete observational gaps in the collection of sunspot group numbers, improving its coverage for this epoch. We have analysed and compared the observations made by Hallaschka with the records made by other contemporary observers. Unfortunately, the analysis of the sunspot areas and positions showed that they are too inaccurate for scientific use. But, we conclude that sunspot counts made by Hallaschka are similar to those made by other astronomers of that time. The observations by Hallaschka confirm a low level of the solar activity during the Dalton minimum.
As we are heading towards the next solar cycle, presumably with a relatively small amplitude, it is of significant interest to reconstruct and describe the past grand minima on the basis of actual observations of the time. The Dalton Minimum is often considered one of the grand minima captured in the coverage of telescopic observations. Nevertheless, the reconstructions of the sunspot group number vary significantly, and the existing butterfly diagrams have a large data gap during the period. This is partially because most long-term observations have remained unexplored in historical archives. Therefore, to improve our understanding on the Dalton Minimum, we have located two series of Thaddäus Derfflinger's observational records (a summary manuscript and logbooks) as well as his Brander's 5.5-feet azimuthal-quadrant preserved in the Kremsmünster Observatory. We have revised the existing Derfflinger's sunspot group number with Waldmeier classification and eliminated all the existing 'spotless days' to remove contaminations from solar meridian observations. We have reconstructed the butterfly diagram on the basis of his observations and illustrated sunspot distributions in both solar
The unexpected development of cycle 24 emphasizes the need for a better way to model future solar activity. In this article, we analyze the accumulation of spotless days during individual cycles from 1798-2010. The analysis shows that spotless days do not disappear abruptly in the transition towards an active sun. A comparison with past cycles indicates that the ongoing accumulation of spotless days is comparable to that of cycle 5 near the Dalton minimum and to that of cycles 12, 14 and 15. It also suggests that the ongoing cycle has as much as 20 \pm 8 spotless days left, from July 2010, before it reaches the next solar maximum. The last spotless day is predicted to be in December 2012, with an uncertainty of 11 months. This trend may serve as input to the solar dynamo theories.
We propose a mass sensor using optically trapped and cooled dielectric microdisks with "measuring after cooling" scheme. The center-of-mass motion of a trapped particle in vacuum can experience extremely low dissipation resulting in robust decoupling from the heat bath. Thus the measurement can be accomplished in a short time before a jump in the phonon number after the optomechanical cooling is finished. This method can lead to 10^(-6) dalton mass sensitivity which is a thousand times smaller than the mass of one electron.
Antonio Colla was a meteorologist and astronomer who made sunspot observations at the Meteorological Observatory of the Parma University (Italy). He carried out his sunspot records from 1830 to 1843, just after the Dalton Minimum. We have recovered 71 observation days for this observer. Unfortunately, many of these records are qualitative and we could only obtain the number of sunspot groups and/or single sunspots from 25 observations. However, we highlight the importance of these records because Colla is not included in the sunspot group database as an observer and, therefore, neither his sunspot observations. According to the number of groups, the sunspot observations made by Colla are similar as several observers of his time. For common observation day, only Stark significantly recorded more groups than Colla. Moreover, we have calculated the sunspot area and positions from Colla's sunspot drawings concluding that both areas and positions recorded by this observer seem unreal. Therefore, Colla's drawings can be interpreted such as sketches including reliable information on the number of groups but the information on sunspot areas and positions should not be used for scientific p
In addition to regular Schwabe cycles (~ 11 years), solar activity also shows longer periods of enhanced or reduced activity. Of these, reconstructions of the Dalton Minimum provide controversial sunspot group numbers and limited sunspot positions, partially due to limited source record accessibility. We analysed Stephan Prantner's sunspot observations from 1804--1844, the values of which had only been known through estimates despite their notable chronological coverage during the Dalton Minimum. We identified his original manuscript in Stiftsarchiv Wilten, near Innsbruck, Austria. We reviewed his biography (1782--1873) and located his observational sites at Wilten and Waidring, which housed the principal telescopes for his early and late observations: a 3.5-inch astronomical telescope and a Reichenbach 4-feet achromatic erecting telescope, respectively. We identified 215 days of datable sunspot observations, which are twice as much data as his estimated data in the existing database (= 115 days). Prantner counted up to 7--9 sunspot groups per day and measured sunspot positions, which show their distributions in both solar hemispheres. These results strikingly emphasise the differe
In a research seminar in $2006$, M. Filaseta, O. Trifonov, and G. Yu showed for each integer $n\geq3$ there is no distinct covering with all moduli in the interval $[n, 6n]$. In $2022$, this interval was subsequently improved to $[n, 8n]$ by the first author and O. Trifonov. The first author then improved this bound to $[n, 9n]$ in $2023$. Building off their method, we show that for each integer $n\geq 3$, there does not exist a distinct covering system with all moduli in the interval $[n, 10n]$.
In this paper we consider the description by a general Bell-type non-local hidden variable theory (NLHVT) of bipartite quantum states with two observables per sub-system. We derive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu (CGLMP) type which involve combinations of the probabilities of related outcomes for measurements for the four pairs of sub-system observables. It is shown that the corresponding quantum theory expressions violate the Bell inequalities in the case of the maximally entangled state of the bipartitite system. The CHSH Bell inequality is also derived from this general CGLMP Bell-type non-local hidden variable theory. This shows that quantum theory can not be underpinned by a Bell-type non-local hidden variable theory. So as a general Bell-type local hidden variable theory has already been shown to conflict with quantum theory, it follows that quantum theory can not be understood in terms of any CGLMP Bell-type hidden variable theory - local or non-local.
In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational evidence that Riemann--Roch holds for the toric rank function.
We use the Pareto Joint Inversion, together with the Particle Swarm Optimization, to invert the Love and quasi-Rayleigh surface-wave speeds, obtained from dispersion curves, in order to infer the elasticity parameters, mass densities and layer thickness of the model for which these curves are generated. For both waves, we use the dispersion relations derived by Dalton et al. (2017). Numerical results are presented for three angular frequencies, 15 Hz, 60 Hz and 100 Hz, and for two, five and seven modes, respectively. Comparisons of the model parameters with the values inverted with error-free input indicate an accurate process. If, however, we introduce a 5% error to the input, the results become significantly less accurate, which indicates that the inverse operation, even though stable, is error-sensitive. Correlations between the inverted elasticity parameters indicate that the layer parameters are more sensitive to input errors than the halfspace parameters. In agreement with Dalton et al. (2017), the fundamental mode is mainly sensitive to the layer parameters whereas higher modes are sensitive to both the layer and halfspace properties; for the second mode, the results are mor
We continue our series of papers on intergalactic medium (IGM) tracers using quasi-stellar objects (QSOs), having examined gamma-ray bursts (GRBs) and blazars in earlier studies. We have estimated the IGM properties of hydrogen column density (Nhxigm), temperature and metallicity using XMM-Newton QSO spectra over a large redshift range, with a collisional ionisation equilibrium (CIE) model for the ionised plasma. The Nhxigm parameter results were robust with respect to intrinsic power laws, spectral counts, reflection hump and soft excess features. There is scope for a luminosity bias given both luminosity and Nhxigm scale with redshift, but we find this unlikely given the consistent IGM parameter results across the other tracer types reviewed. The impact of intervening high column density absorbers was found to be minimal. The Nhxigm from the QSO sample scales as (1 + z)^1.5+/-0.2. The mean hydrogen density at z = 0 is n0 = (2.8 +/- 0.3) x 10^-7 cm^-3, the mean IGM temperature over the full redshift range is log(T/K) = 6.5+/-0.1, and the mean metallicity is [X/H] = -1.3+/-0.1(Z 0.05). Aggregating with our previous GRB and blazar tracers, we conclude that we have provided evidence
Dalton and Slawinski (2016) show that, in general, the Backus (1962) average and the Gazis et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativity. We illustrate numerically that the extent of noncommutativity is a function of the strength of anisotropy. The averages nearly commute in the case of weak anisotropy.
Researchers have created cosmic dust from scratch by recreating space-like conditions inside glass tubes。 The dust contains complex carbon-rich molecules built from elements essential to life and produces infrared signals similar to real material found in space。 By studying these laboratory samples, scientists can explore how organic chemistry unfo
Primordial black holes may occasionally pass through white dwarf stars and trigger enormous Type Ia supernova explosions。 Researchers found that these events could explain chemical patterns seen in supernova remnants, nearby explosions, and stars across the Milky Way