arXiv subjects
Publications and source records attributed to Tomohiro Yamada.
We give explicit upper bounds for coefficients of polynomials appearing in Gauss-Kraïtchik formula for cyclotomic polynomials. We use a certain relation between elementary symmetric polynomials and power sums polynomials.
We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$.
The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0, 1, \ldots$. We determine all pairs $(P_m, P_n)$ satisfying $P_m^b=2^{g_1} 3^{g_2} 5^{g_3} 7^{g_4} P_n^a$ for some integers $g_1, g_2, g_3, g_4$, $a$, and $b$. More generally, for a linear recurrence sequence $u_n$ satisfying the dominant root condition and a given set of primes $p_1, \ldots, p_k$, there exist only finitely many pairs $(u_m, u_n)$ satisfying $u_m^b=p_1^{g_1} \cdots p_k^{g_k} u_n^a$ for some integers $g_1, \ldots, g_k$, $a$, and $b$.
We shall show that $2160$ is the only biunitary triperfect number divisible by $27=3^3$.
In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.
We shall show that there exist only finitely many nondegenerate three-term Machin-type formulae and give explicit upper bounds for the sizes of variables.
We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.
A compact accelerator design for irradiation purposes is being proposed at KEK. This design targets an energy of 10 MeV and a current of 50 mA. Current design includes a 100 kV thermionic DC electron gun with an RF grid, 1-cell normal-conducting buncher cavity, and Nb$_{3}$Sn superconducting cavities to accelerate the beam to the final energy of 10 MeV. The goal of the present beam dynamics study is the beam loss suppression (to the ppm level), since it results in a thermal load on the cavity. Then the beam performance at the accelerator exit should be confirmed. The main issue was to transport the beam without loss, since the initial electron energy (100 keV) is low, and the beam parameters are intricately correlated. In addition, the space charge effect is considerable. For this reason, simultaneous optimization of multiple parameters was necessary. Here we report optimization results and their effect on the design of the machine.
We shall prove that if $N=p^αq_1^{2β_1} q_2^{2β_2} \cdots q_{r-1}^{2β_{r-1}}$ is an odd perfect number such that $p, q_1, \ldots, q_{r-1}$ are distinct primes, $p\equivα\equiv 1\mod{4}$ and $t$ divides $2β_i+1$ for all $i=1, 2, \ldots, r-1$, then $t^5$ divides $N$, improving an eighty-year old result of Kanold.
We give an explicit version of Brun-Titchmarsh theorem applicable for arbitrary moduli and arbitrary intervals. For example, we show that $π(x+y; k, a)-π(x; k, a)<2y/(φ(k)(\log (y/k)+0.8601))$ for any relatively prime positive integers $k$ and $a$ and any positive real numbers $x$ and $y$ such that $y>k$.
We show that if $N\pm 1=Mφ(N)$ with $N\neq 15, 255$ composite, then $M<15.76515\log\log\log N$ and $M<16.03235\log\logω(N)$, together with similar results for the unitary totient function, Dedekind function, and the sum of unitary divisors.
Monitoring motion originating from ultra low-temperature cooling systems like cryocoolers is important for vibration sensitive cryogenic experiments like KAGRA. Since no commercial cryogenic accelerometers are available, we developed a compact self-calibrating accelerometer with a Michelson interferometer readout for cryogenic use. Change in calibration factor and drop in interferometer output originating from temperature drop were the main concerns which were tackled. Sensitivity of $3.38\times10^{-11}$ m/$\sqrt{\mathrm{Hz}}$ at 1 Hz was achieved at 300 K. The accelerometer was tested inside the KAGRA cryostat; showed stable operation down to 12 K in 0.1-100 Hz band with only 1% visibility drop. Our accelerometer can be employed in low vibration cryogenic environment for a multitude of applications.
We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{α_1}\cdots p_s^{α_s} q_1^{β_1}\cdots q_t^{β_t}$ with $β_1, \ldots, β_t$ bounded by a given constant. We shall also give similar results for quasiperfect numbers and relatively prime amicable pairs of opposite parity.
Let $σ(n)$ and $γ(n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\neq 1, 1782$ and $σ(n)=(γ(n))^2$, then there exist odd (not necessarily distinct) primes $p, p^\prime$ and (not necessarily odd) distinct primes $q_i (i=1, 2, \ldots, k)$ such that $p, p^\prime\mid\mid n$, $q_i^2\mid\mid n (i=1, 2, \ldots, k)$ and $q_1\mid σ(p^2), q_{i+1}\midσ(q_i^2) (1\leq i\leq k-1), p^\prime \midσ(q_k^2)$.
We report the mirror suspension design for Large-scale Cryogenic Gravitational wave Telescope, KAGRA, during bKAGRA Phase 1. Mirror thermal noise is one of the fundamental noises for room-temperature gravitational-wave detectors such as Advanced LIGO and Advanced Virgo. Thus, reduction of thermal noise is required for further improvement of their sensitivity. One effective approach for reducing thermal noise is to cool the mirrors. There are many technical challenges that must be overcome to cool the mirrors, such as cryocooler induced vibrations, thermal drift in suspensions, and reduction in duty cycling due to the increased number of potential failure mechanisms. Our mirror suspension has a black coating that makes radiative cooling more efficient. For conduction cooling, we developed ultra high purity aluminum heat links, which yield high thermal conductivity while keeping the spring constant sufficiently small. A unique inclination adjustment system, called moving mass, is used for aligning the mirror orientation in pitch. Photo-reflective displacement sensors, which have a large range, are installed for damping control on marionette recoil mass and intermediate recoil mass. Samarium cobalt magnets are used for coil-magnet actuators to prevent significant change of magnetism between room temperature and cryogenic temperature. In this paper, the design of our first cryogenic payload and its performance during bKAGRA Phase 1 are discussed.
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.
Some new results concerning the equation $σ(N)=aM, σ(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
Upgrades to improve the sensitivity of gravitational wave detectors enable more frequent detections and more precise source parameter estimation. Unlike other advanced interferometric detectors such as Advanced LIGO and Advanced Virgo, KAGRA requires different approach for the upgrade since it is the only detector which employs cryogenic cooling of the test masses. In this paper, we describe possible KAGRA upgrades with technologies focusing on different detector bands, and compare the impacts on the detection of compact binary coalescences. We show that either fivefold improvement in the $100 M_{\odot}$--$100 M_{\odot}$ binary black hole range, a factor of 1.3 improvement in the binary neutron star range, or a factor of 1.7 improvement in the sky localization of binary neutron stars is well feasible with upgrades that do not require changes in the existing cryogenic or vacuum infrastructure. We also show that twofold broadband sensitivity improvement is possible by applying multiple upgrades to the detector.