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We study adiabatic oscillations of rotating self-gravitating gaseous stars in mathematically rigorous manner. The internal motion of the star is supposed to be governed by the Euler-Poisson equations with rotation of constant angular velocity under the equation of state of the ideal gas. The motion is supposed to be adiabatic, but not to be barotropic in general. This causes a free boundary problem to gas-vacuum interface. Existence of solutions to the linearized equation in the Lagrange coordinates of the perturbations around a fixed stationary solution, the eigenvalue problem with concept of quadratic pencil of operators, and the stability problem with a new concept of stability introduced in this article are discussed., Comment: Discussion on spectrum of the generator in a dense subspace added, and presentation in whole ameliorated

The cryogenic sapphire oscillator (CSO) is a highly specialized machine, which delivers a reference signal exhibiting the lowest frequency fluctuations. For the best units, the Allan deviation (ADEV) is <1e-15 for integration time between 1 and 10,000 s, with a drift <1e-14 in one day.The oscillator is based on a sapphire monocrystal resonating at 10 GHz in a whispering-gallery mode, cooled at 6 K for highest Q-factor and zero thermal coefficient. We report on the progress accomplished implementing eleven CSOs in about 10 years since the first sample delivered to the ESA station in Argentina. Short-term stability is improved by a factor of 3-10, depending on the integration time, and the refrigerator's electric power is reduced to 3 kW. Frequency stability and overall performances are reproducible, with unattended operation between scheduled maintenance every two years. The CSO is suitable to scientific applications requiring extreme frequency stability with reliable long-term operation. For example, the flywheel for primary frequency standards, the ground segment of GNSS, astrometry, VLBI, and radio astronomy stations., Comment: 7 pages, 12 figures

In meteorology the analysis of motions of the atmosphere on the Earth has been done using various mathematical models and using various approximations. In this article as the simplest model the compressible Euler equations with barotropic equation of state of the ideal gas is analyzed under the co-ordinate system which rotates with constant angular velocity. Mathematically rigorous inquiry is tried. Although problems remain to be open, some fundamental results are exhibited., Comment: Section 3 revised. Confusions of symbols corrected

We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid. Although the reduced system of equations for the potentials in the co-rotating co-ordinate system is known, we derive the system of equations for potentials in the so called zero angular momentum observer co-ordinate system. We newly give a proof of the equivalence between the reduced system and the full system of Einstein equations. It is done under the assumption that the angular velocity is constant on the support of the density. Also the consistency of the equations of the system is analyzed. On this basic theory we construct on the whole space the stationary asymptotically flat metric generated by a slowly rotating compactly supported perfect fluid with vacuum boundary., Comment: Revision enlarged by adding an explanation of the assumption on EOS, comments to compare the present research with preceding achievements, and a glossary of variables and parameters

We consider the oscillations of the atmosphere around a stratified back ground density and entropy distribution under the gravitation on the flat Earth. The atmosphere is supposed to be an ideal gas and the motion is supposed to be governed by the compressible Euler equations. The density distribution of the back ground equilibrium is supposed to touch the vacuum at the finite height of the stratosphere. Considering the linearized approximation for small perturbations, we show that time periodic oscillations with a sequence of time periods which accumulate to infinity, say, slow and slow oscillations, so called `g-modes', can appear when the square of the Brunt-Vaisala frequency is positive everywhere for the considered back ground equilibrium.

We consider the vibrations and waves in the atmosphere under the gravitation on the flat earth. The stratified density distribution of the back ground equilibrium is supposed to touch the vacuum at the finite height of the stratosphere. We show that incompressible motions are possible to create vibrations or progressive waves with countably many time frequencies of the vibrations or speeds of the wave propagation which accumulate to 0, say, of countably infinitely many slow and slow modes.

We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid. Although the derived equations are not already known except for the case of the constant angular velocity described in the corotating coordinate system, the main content of this article is not to derive the equations, but to prove the equivalence of the derived equations with the full set of the Einstein equations, and to prove the consistency of the derived equations. These affairs have not yet been discussed except for the vacuum case, and are far from being self-evident, requiring tedious careful calculations and some tricks. The proof is done under the assumption that the angular velocity is constant on a neighborhood of the support of the density. The conclusions seem to be doubtful if this assumption does not hold., Comment: Abstract rewritten. No changes in the main body of the article. arXiv admin note: text overlap with arXiv:1705.07392

Axially symmetric stationary metrics governed by the Einstein-Euler equations for slowly rotating perfect fluids have been constructed in an arbitrarily large bounded domain containing the support of the mass density. However the problem of global prolongation of the metric is still open. On the other hand the so called matter-vacuum matching problem, particularly as the source problem for the Kerr metric, has been discussed by several authors. This can be regarded as the approach to the same open problem in the opposite direction. We give a remark on this open problem.

Lyra2REv2 is a hashing algorithm that consists of a chain of individual hashing algorithms, and it is used as a proof-of-work function in several cryptocurrencies. The most crucial and exotic hashing algorithm in the Lyra2REv2 chain is a specific instance of the general Lyra2 algorithm. This work presents the first hardware implementation of the specific instance of Lyra2 that is used in Lyra2REv2. Several properties of the aforementioned algorithm are exploited in order to optimize the design. In addition, an FPGA-based hardware implementation of a standalone miner for Lyra2REv2 on a Xilinx Multi-Processor System on Chip is presented. The proposed Lyra2REv2 miner is shown to be significantly more energy efficient than both a GPU and a commercially available FPGA-based miner. Finally, we also explain how the simplified Lyra2 and Lyra2REv2 architectures can be modified with minimal effort to also support the recent Lyra2REv3 chained hashing algorithm., Comment: 13 pages, accepted for publication in IEEE Trans. Circuits Syst. I. arXiv admin note: substantial text overlap with arXiv:1807.05764

The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gaseous stars is analyzed in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler-Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed. The existence of eigenvalues which accumulate to 0 is proved in a mathematically rigorous fashion.The absence of continuous spectra and the completeness of eigenfunctions for the operators reduced by spherical harmonics is discussed.

The hydrodynamic evolution of self-gravitating gaseous stars is governed by the Euler-Poisson equations. We study the structure of the linear approximation of barotropic perturbations around spherically symmetric equilibria based on functional analytic tools. In contrast to folklore, we show that the spectrum of the linearized operator for general perturbations is not of the Sturm-Liouville type unless the perturbations are restricted. In particular, we prove that it is of the Sturm-Liouville type for irrotational perturbations., Comment: Final revision accepted by Journal of Mathematical Physics

We consider stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by $6/5<\gamma <2$ and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implicit function theorem. Oblateness of star surface is shown and also relationship between the central density and the total mass is given., Comment: Typos have been corrected

In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which give mathematical models of slowly uniformly rotating gaseous stars. We try to extend this result to the study of solutions of the Einstein-Euler equations in the framework of the general theory of relativity. Although many interesting studies have been done about axisymmetric metric in the general theory of relativity, they are restricted to the region of the vacuum. Mathematically rigorous existence theorem of the axisymmetric interior solutions of the stationary metric corresponding to the energy-momentum tensor of the perfect fluid with non-zero pressure may be not yet established until now except only one found in the pioneering work by U. Heilig done in 1993. In this article, along a different approach to that of Heilig's work, axisymmetric stationary solutions of the Einstein-Euler equations are constructed near those of the Euler-Poisson equations when the speed of light is sufficiently large in the considered system of units, or, when the gravitational field is sufficiently weak.

In this note we derive the descriptions of the system of Euler-Poisson equations which governs the hydrodynamic evolution of gaseous stars in various co-ordinate systems. This note does not contain essentially new results for astrophysicists, but mathematically rigorous derivations cannot be found in the literatures written by physicists so that it will be useful to prepare details of rather stupidly honest derivations of the equations in various flames when we are going to push forward the mathematical research of the problem., Comment: Few typos corrected

We construct stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent $\gamma$ belongs to $(\frac65,\frac32]$. The problem is formulated as a nonlinear integral equation, and is solved by iteration technique. By this method, not only we get the existence, but also we clarify properties of the solutions such as the physical vacuum condition and oblateness of the star surface.

We have been studying spherically symmetric motions of gaseous stars with physical vacuum boundary governed either by the Euler-Poisson equations in the non-relativistic theory or by the Einstein-Euler equations in the relativistic theory. The problems are to construct solutions whose first approximations are small time-periodic solutions to the linearized problem at an equilibrium and to construct solutions to the Cauchy problem near an equilibrium. These problems can be solved when $1/(\gamma-1)$ is an integer, where $\gamma$ is the adiabatic exponent of the gas near the vacuum, by the formulation by R. Hamilton of the Nash-Moser theorem. We discuss on an application of the formulation by J. T. Schwartz of the Nash-Moser theorem to the case in which $1/(\gamma-1)$ is not an integer but sufficiently large., Comment: Typos corrected, and References are up-dated. No substantial changes

Investigations of spherically symmetric motions of self-gravitating gaseous stars governed by the non-relativistic Newtonian gravitation theory or by the general relativistic theory lead us to a certain type of non-linear hyperbolic equations defined on a finite interval of the space variable. The linearized principal part has regular singularities at the both ends of the interval of the space variable. But the regularity loss caused by the singularities at the boundaries requires application of the Nash-Moser technique. An abstract unified treatment of the problem is presented.

We investigate the time evolution of a homogeneous isotropic universe which consists of a perfect gas governed by the Einstein-Euler-de Sitter equations. Under suitable assumptions on the equation of state and assumptions on the present states, we give a mathematically rigorous proof of the existence of the Big Bang at the finite past and expanding of the universe in the course to the infinite future., Comment: Few words have been added in Appendix with related references

We construct spherically symmetric solutions to the Einstein-Euler equations, which contains a positive cosmological constant, say, the Einstein-Euler-de Sitter equations. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler-de Sitter equations are given by the Tolman-Oppenheimer-Volkoff-de Sitter equation. We can construct solutions near time periodic linearized solutions around the equilibrium. The Cauchy problem around the equilibrium can be solved. This work can be considered as a trial of the generalization of the previous work on the problem without cosmological constants.

Spherically symmetric static solutions of the Einstein equations with a positive cosmological constant for the energy-momentum tensor of a barotropic perfect fluid are governed by the Tolman-Oppenheimer-Volkoff-de Sitter equation. Existence and regularity at the vacuum boundary of the solutions with finite radii are investigated. The interior metric of the solution is connected with the Schwarzschild-de Sitter metric on the vacuum region., Comment: Typos corrected

We construct spherically symmetric solutions to the Einstein-Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying so called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash-Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the non-relativistic Euler-Poisson equations.

The `catuskoti' or tetralemma in Buddhist logic is a problematic subject from the modern logical point of view. Recently a many-valued paraconsistent logic was proposed in order to formalize catuskoti adequately by G. Priest. On the other hand a slight modification of the formalization of catuskoti seems to allow an appropriate interpretation in the framework of the classical propositional calculus in the mathematical logic developed by Russell-Whitehead and Hilbert-Ackermann., Comment: Appendix has been slightly corrected. No substantial changes

We study spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations. Equilibria are given as solutions of the Lane-Emden equations, and the linearized equation around one of these equilibria admits time-periodic solutions. To justify the linearization, we should construct true solutions of the nonlinear evolution equation for which this time-periodic solutions plus the equilibrium is the first approximation. To do so, we apply the Nash-Moser theory. We presents a partial result under some assumptions on the adiabatic exponent of the gas, and propose an open problem to be studied in the future., Comment: Typos corrected and rude expressions revised thanks to suggestions by an anonymous referee. arXiv admin note: text overlap with arXiv:1210.3670

We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem., Comment: Finally revised

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is not easy to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Interestingly, the equation can be transformed to a nonlinear wave equation on a higher dimensional space, for which the space dimension, being larger than 4, is related to the adiabatic exponent of the original one-dimensional problem. We try to find a family of solutions expanded by a small parameter. Applying the Nash-Moser theory, we justify this expansion.The application of the Nash-Moser theory is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

We report the operation of a $^{87}$Rb maser in the self-oscillating mode, using a laser diode as the optical pumping source. The maser uses a TE$_{021}$ cavity surrounding a cell containing the atoms and 11 Torr of N$_{2}$ as the buffer gas. The optical pumping is accomplished by using a commercial laser diode frequency-locked on the linear absorption line of an external rubidium cell. The maser output power is maximized when the laser spectrum is spread by modulating its frequency through the variation of its injection current. The maser output power is also presented for various modulation waveforms and frequencies. We find that the spurious sidebands induced by the modulation can be minimized. Finally, the linear dependence between the laser and maser frequencies is shown. We find that the relative maser frequency pulling from the laser is about $4.6 \times 10^{-3}$.