Toshio Fukushima

National Astronomical Observatory of Japan | NAOJ

This is Year 2018 version of Lecture on "Introduction to Astronomy" in original Japanese text.

Part 2 of Introduction to Astronomy, a lecture to be given for college students at Hosei University and Showa Women’s University. In the winter semester, we follow the history of astronomy and physics by focusing on the challenge to the enigma of gravitation.

This is Halley's A Synopsis of the Astronomy of Comets (Halley 1705). It includes a table of comets compiled by Halley, among which the famous Halley's comet appeared as three different comets.

Presented are eight Fortran 90 programs to compute four complete elliptic integrals of first and second kind,K(m), E(m), B(m), D(m): 1) xkbed.f90, test driver of "ceik", "ceie", "ceib", "ceid", "sceik", "sceie", "sceib", and "sceid" , 2) ceik.f90 and sceik.f90, real*8 and real*4 functions to compute K(m), 3) ceie.f90 and sceie.f90, real*8 and real*...

This is Poisson's A Treatise on Mechanics, Vol.1 (Poisson, 1838) translated into English by H.H. Harte (1842)

This is Poisson's A Treatise on Mechanics, Vol.2 (Poisson, 1838) translated into English by H.H. Harte (1842)

This is Simon Newcomb's book titled "The Elements of Four Inner Planets and the Fundamental Constants for Astronomy" (Newcomb, 1895). He devoted Chapter VI to showing the discrepancy between the observed and theoretical values of orbital elements of four inner planets and proposed many hypotheses to explain the differences.

This is Routh's The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 5th ed. (Routh 1892), the second part of Routh's classic textbook on the rotational dynamics written in English

The detailed lecture schedule of "Introduction to Astronomy Part 2" is given. Also, the URL and other info of the related free movies/PDFs are provided.

The detailed lecture schedule of "Introduction to Astronomy" is given. Also, the URL and other info of the related free movies/PDFs are provided.

The computation of associated Legendre function (ALF) by the standard method, namely by using a fixed-order increasing-degree recurrence formula, suffers from the severe accumulation of round-off errors when the colatitude θ is not so large, say less than 10 degrees. When the degree of ALF increases, the manner of its relative error growth is initi...

The test Fortran program evaluates the 4pi fully normalized associated Legendre function (fnALF) and its first- through third-order derivatives for any degree and order by using the fixed-degree decreasing-order recurrence formula and a similar fixed-degree formula by employing the X-number formulation.

The test Fortran program evaluates the 4pi fully normalized associated Legendre function (fnALF) and its first- through third-order derivatives for any degree and order by using the fixed-order increasing-degree recurrence formula and similar fixed-order methods by employing the X-number formulation.

The Kepler equations are nonlinear equations consisting of the essential part of the so-called Keplerian motion, the fundamental solution of the two-body problem. It is easy to obtain the size, shape, and orientation of the orbit from the given orbital elements. However, we must solve appropriate Kepler equations to determine the position on the or...

ケプラー方程式とは、軌道力学の基礎となる二体問題の基本解、いわゆるケプラー運動の根幹をなす非線形方程式である。与えられた軌道要素から軌道の大きさや形、軌道面の向きなどは容易に計算できるが、軌道上の位置を指定するためにはケプラー方程式を解かなければいけない。軌道の種類に応じて、ケプラー方程式も楕円型、放物線型、双曲線型など多岐にわたる。放物線型を除く一般の場合に解析解は見つかっていないので数値的に解かなければならないが、その高速かつ高精度な解法はケプラー、ニュートンやハレーを悩ませた古来の難問であった。著者は２０年以上昔に全種類のケプラー方程式の新解法を考案し、従来に比して２～１３倍の高速化を達成した(Fukushima, 1996, 1997a, 1997b, 1997c, 1998, 19...

The computation of associated Legendre function (ALF) by the standard method, namely by using a fixed-order increasing-degree recurrence formula, suffers from a severe accumulation of round-off errors when the colatitude θ is not so large, say less than 60 degrees. When the degree of ALF increases, its relative error increases initially quadratic t...

This file contains the single and double precision Fortran 90 functions to compute five complete elliptic integrals: K(m), E(m), B(m) = (E(m)-(1-m)K(m))/m, D(m) = (K(m)-E(m))/m , and S(m) = (D(m)-B(m))/m: namely, 1) xcei2.f90 test driver, 2) sceik.f90: single precision function to compute K(m), 3) sceie.f90: single precision function to compute E(m...

The subroutines provide the complete elliptic integrals of the first and second kind, K(m) and E(m), and the associate complete elliptic integrals of the second kind, B(m) and D(m) in the quadruple precision. The comparison with the 40 digits computation by Mathematica shows that the maximum relative errors of K(m), E(m), B(m), and D(m) do not exce...

Abstract: As "Necessity is the mother of invention", the development of some part of mathematics has been driven by the needs in astronomy. Kepler created the Kepler equation, the first transcendental equation, to determine the position of a planet on its orbit, Newton showed his genius in regularizing a singular integral to obtain the universal at...

Frequently Asked Questions and their answers on "Introduction to Astronomy Part 2 - Enigma of Gravitation" written in Japanese

Frequently Asked Questions and their answers on "Introduction to Astronomy Part 1" written in Japanese

Frequently Asked Questions and their answers on "How to write academic article" written in Japanese

We developed new low-and high-precision procedures for computing the primary and secondary branches of the Lambert W functions, W 0 (z) and W −1 (z). The procedures for W 0 (z) are conditional switches of rational functions of transformed variables: (i) ln(z) when z is sufficiently large, and (ii) √ z + 1/e otherwise. Also, the procedures for W −1...

Fortran 90 test program package of "sw0", "swm1", "dw0c", and "dwm1c", low- and high-precision procedures computing primary and secondary branch of Lambert W function, W_0(z) or W_-1(z), by piecewise minimax rational function approximation. They are of 24- and 50-bit accuracy for the input argument expressed as arbitrary double-precision floating-p...

We developed new single-and double-precision procedures to compute the real-valued Wright ω function, ω(x), a unique solution of the nonlinear equation, ω +ln ω = x, for the real argument, x. The procedures are a conditional switch of (i) one application of the Newton correction to the asymptotic solution when x is sufficiently large, (ii) a minima...

Fortran 90 test program package of "somega.f90" and "domega.f90", the single- and double-precision functions to compute Wright's omega function, w(x), the solution of nonlinear equation, w+log(w)=x, by the piecewise minimax rational function approximation. They are of 24- and 50-bit accuracy. The functions run as fast as the exponential function pr...

Fortran 90 test program package of "womega", the double precision computation of Wright's omega function, w(x), the solution of nonlinear equation, w+log(w)=x, by a combination of the truncated Taylor series expansion, the logarithmic bisection method, and Schroeder's fourth-order method. It is numerically confirmed to be of 15 digit accuracy for a...

This is Todhunter's Spherical Trigonometry, 5th edn (Todhunter 1886), the standard text book on the spherical trigonometry.

S U M M A R Y By utilizing the addition theorems of the arctangent function and the logarithm, we developed a new expression of Bessel's exact formula to compute the prismatic gravitational field using the triple difference of certain analytic functions. The use of the new expression is fast since the number of transcendental functions required is...

By utilizing the addition theorems of the arctangent function and the logarithm, we developed a new expression of Bessel's exact formula to compute the prismatic gravitational field using the triple difference of certain analytic functions (Nagy et al. 2000). The employment of the new expression is fast since the number of transcendental functions...

By utilizing the addition theorems of the arctangent function and the logarithm, we developed a new expression of Bessel's exact formula to compute the prismatic gravitational field using the triple difference of certain analytic functions (Nagy et al. 2000). The employment of the new expression is fast since the number of transcendental functions...

We developed a double precision procedure to compute the inverse function of F −1/2 (η), the complete Fermi-Dirac integral of order −1/2, by a piecewise approximation using rational functions of type (7,7) and (3,4). The relative errors of the approximations do not exceed 6 double precision machine epsilons and the CPU time to evaluate the new form...

Fortran 90 test program package of "difd1h" and "difdm1h", the double precision computation of the inverse function of the un-normalized complete Fermi-Dirac integral of order 1/2 and -1/2 by a piecewise mini-max rational function approximation. They are numerically confirmed to be of 15 digit accuracy for arbitrary positive value of the integrals....

We developed a double precision procedure to compute the inverse function of b ≡ B −1/2 (η), the complete Bose-Einstein integral of order −1/2, by a conditional switch of the rational functions of type (7,7) with respect to the transformed integral value, u ≡ b/(1 + b). The maximum error of the new approximation is 11 double precision machine epsil...

Fortran 90 test program package of "bei1h" and "beim1h", the double precision computation of the inverse function of the complete Bose-Einstein integral of order 1/2 and -1/2 by a single or piecewise minimax rational function approximation. They are numerically confirmed to be of 15 digit accuracy. The routines run fairly fast in the sense that its...

Analytical formulae are developed to compute the complete Bose-Einstein integral of degeneracy η and of half integer orders, k = −9/2, −7/2, · · · , and 39/2, and of integer orders, k = 1, 2, · · · , and 19. If 13/2 ≤ k, the formula is a mini max rational function of z ≡ exp(η). Otherwise, the formula is a conditional switch of two mini max rationa...

Fortran 90 test program package of "bem9h", "bem7h", ..., and "be39h" and "be2h", "be4h", ..., "be38h", the double precision computation of normalized complete Bose-Einstein integrals of half integer orders, -9/2, -7/2, ..., and 39/2, and of integer orders, 1, 2, ..., 19, by rational function approximation. The maximum magnitude of their relative e...

Fortran 90 test program package of "beNh", namely "bem9h", "bem7h", ..., and "be21h", the double precision computation of normalized complete Bose-Einstein integrals of orders, -9/2, -7/2, ..., and 21/2 by rational function approximation. The maximum magnitude of their relative errors do not exceed 9 double precision machine epsilon. Their CPU time...

Fortran 90 test program package of "qbenNh", namely "qbenm9h", "qbenm7h", ..., and "qben21h", the quadruple precision computation of normalized complete Bose-Einstein integrals of orders, -9/2, -7/2, ..., and 21/2 by conditionally switching (1) Dingle's analytical series expansion around eta=0, and (2) the polylogarithm series expansion around eta=...

The single and double precision procedures are developed for the inverse Bose-Einstein integral of order 1/2. They are of the form of a logarithm of the rational function of type (3,3) and (6,5), respectively. The maximum error of the new approximations is one and 8 machine epsilons in the single and double precision computations, respectively. The...

Fortran 90 test program package of "sbe5h", "sbe3h", "sbe1h", and "sbem1h", the single precision functions to compute the complete Bose-Einstein integrals of orders, 5/2, 3/2, 1/2, and -1/2 by a piecewise mini-max rational function approximation. They are numerically confirmed to be of the 24 bit relative accuracy when -infinity < eta <= 0. The fun...

Fortran 90 test program package of "dbe5h", "dbe3h", "dbe1h", and "dbem1h", the double precision functions to compute the complete Bose-Einstein integrals of orders, 5/2, 3/2, 1/2, and -1/2 by a piecewise mini-max rational function approximation. They are numerically confirmed to be of the 15 digit relative accuracy when -infinity < eta <= 0. The f...

Fortran 90 test program package of "qbes5h", "qbes3h", "qbes1h", and "qbesm1h", the quadruple precision computation of complete Bose-Einstein integrals of orders, 5/2, 3/2, 1/2, and -1/2 by conditionally switching (1) Dingle's analytical series expansion around eta=0, and (2) the polylogarithm series expansion around eta=-infinity. They are numeric...

Fortran 90 test program package of "sbei1h" and "bei1h", the single and double precision computation of the inverse function of the complete Bose-Einstein integral of order 1/2 by a minimax rational function approximation of the polylogarithm representation. They are numerically confirmed to be of the 24 bit and 15 digit accuracy. The routines run...

Fortran 90 test program package of "bes5h", "bes3h", "bes1h", and "besm1h", double precision computation of complete Bose-Einstein integrals of orders, 5/2, 3/2, 1/2, and -1/2 by conditionally switching (1) Dingle's analytical series expansion around eta=0, and (2) the polylogarithm series expansion around eta=-infinity. They are numerically confir...

By utilizing the addition theorems of the arc tangent function and the logarithm, we developed a new expression of the analytic exact formulae of the gravitational field of a homogeneous rectangular prism. The evaluation of the expression is fast since the number of transcendental functions required is significantly reduced. The numerical experimen...

S U M M A R Y The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows cubically with respect to the distance from the prism. This phenomenon is eminent in not only the gravitational potential...

We developed analytical procedures to compute the incomplete Fermi-Dirac integral of orders ±3/2 and ±1/2. They are based on (i) the analytical integration after replacing the kernel function by its piecewise minimax polynomial approximations, (ii) the Taylor series expansion of the obtained analytical formulae when their direct evaluation suffers...

Fortran 90 test program package of "sifd3h", "sifd1h", "sifdm1h", and "sifdm3h", fast and reliable computation of incomplete Fermi-Dirac integrals of orders, 3/2, 1/2, -1/2, and -3/2 by combining (1) the analytical integration after approximating the kernel function by piecewise polynomial, (2) the Taylor series expansion of the analytical formulae...

This is a Fortran 90 test program package of "qlog1p", "dlog1p", and "slog1p", the quadruple, double, and single precision programs to compute a special logarithm function, log1p(x) = log(1+x), accurately by mini-max rational approximations. "qlog1p", "dlog1p", and "slog1p" keep 33, 16, and 9 digit accuracy for arbitrary value of x, especially even...

This is a Fortran 90 test program package of "dexpm1" and "sexpm1", the double and single precision functions to compute a special exponential function, expm1(x) = exp(x)-1, accurately for arbitrary value of argument, x. They are fast in the sense that "dexpm1" and "sexpm1" run 3.2 and 4.0 times faster than the standard approach using "sinh" provid...

This is a Fortran 90 test program package of (i) "qgami", "dgami", and "sgami", the quadruple, double, and single precision functions to compute the incomplete Gamma function, Gamma(a,x), and (ii) "qgamci", "dgamci", and "sgamci", the quadruple, double, and single precision functions to compute the complementary incomplete Gamma function, gamma(a,x...

This is a Fortran 90 test program package of "qgam", "dgam", and "sgam", the quadruple, double, and single precision functions to compute Gamma function by using the piecewise minimax polynomial approximation, respectively. Their maximum relative errors are 4.3E-35, 5.5E-17, and 3.3E-8, respectively. The functions "dgam" and "sgam" run 20% and 30%...

This is a fully augmented version of "xgfdpa.txt", a Fortran 90 test program package of "dgfdp" and "sgfdp", the double and single precision subroutines to compute the generalized Fermi-Dirac integrals of contiguous orders simultaneously.

Fortran 90 test program package of "fdi", "fdih", "fdi5h", "fdi3h", "fdi1h", "fdim1h", and "fdim3h", fast computation of incomplete Fermi-Dirac integrals of arbitrary order, arbitrary half integer order, and some specific half-integer orders, 5/2, 3/2, 1/2, -1/2, and -3/2 by analytical integration after approximating the kernel function by piecewis...

The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows cubically with respect to the distance from the prism. This phenomenon is eminent in not only the gravitational potential but also the...

Fortran 90 test program package of "fpri", "fvpri", "fgpri", "fvgpri", and "fggpri", several Fortran codes on the fast computation of gravitational potential of a uniform rectangular prism by full utilization of the general addition theorem of the arc tangent function, which is newly established, and the addition theorem of logarithm. The observed...

By utilizing the general addition theorem of the principal value of the arctangent function newly established and the existing addition theorem of the logarithm, we developed a new method to compute the analytic exact formulae of the gravitational field quickly. The method is fast since the number of transcendental functions required is significant...

Fortran 90 program package to evaluate the gravitational potential, the gravity vector, and the gravity gradient tensor of a uniform rectangular prism by using the Taylor series expansion methods of order 0, 2, 4, 6, 8, 10, 12, 14, and 16.

The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows cubicaly with respect to the distance from the prism. This phenomenon is prominent in not only the gravitational potential but also the...

The subroutines provide the associate incomplete elliptic integrals, B(phi|m), D(phi|m), and J(phi,n|m), simultaneously. As an internal device, they give also the associate complete elliptic integrals, B(m), D(m), and J(n|m), simultaneously. The standard incomplete/complete elliptic integrals F(phi|m), E(phi|m), and Pi(phi,n|m) and K(m), E(m), and...

This is the new edition of Lagrange's Analytical Mechanics (Lagrange 1811) translated in English.

This is Cline's Variational Principles in Classical Mechanics 2nd edn (Cline 2018) being an excellent textbook for beginners to study Lagrangian and Hamiltonian formalism of analytical mechanics.

The Cartesian to geodetic coordinate transformation can be characterized in terms of a solution of the so-called the latitude equation (Fukushima 1999, J. Geodesy, 73:603-610). We present an analytical method to obtain its approximate solution for the tangent of the geodetic latitude. It arises from applying Lagrange expansion formula to a suitable...

We review a recent progress in the forward modelling of the gravitational and magnetic fields of vertically inhomogeneous prism and tesseroid. First, we present a numerical procedure to evaluate the tesseroidal field by (i) the analytical integration in the vertical direction, (ii) the split quadrature method in the horizontal directions, (iii) the...

The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows in a cubic manner with respect to the distance from the prism. This phenomenon is common to the gravitational potential, the gravity ve...

Fortran 90 program package to evaluate the gravitational potential, the gravity vector, and the gravity gradient tensor a uniform rectangular prism by using the Taylor series expansion methods of order 0, 2, 4, 6, and 8.

We reviewed a recent progress in the computation of the gravitational field of a general object. Among them, the most general method is realized by the regularization of the Newton kernels of the defining integrals of the field by the integration variable transformation to the local spherical polar coordinates (Fukushima 2016d, Mon. Not. Royal Astr...

Astronomy is one of the most classic sciences. Yet, it may not be appropriate to regard it as one branch of the natural sciences. For instance, the so-called cosmological principle, which Jordano Bruno advocated during the debate of the Ptolemaic/Copernican theories, namely the assertion that "Since the Earth can not be a special place in the Unive...

“Serendipity” is a word created by Horace Walpole, the inventor of the gothic thriller, in 1754 based on a Persian fairy tale, “The Three Princes of Serendip (= Sri Lanka)”. The word means an unplanned, fortuitous discovery. Its well-quoted examples are the X-ray by W. C. Rontgen (1895) and the penicillin by A. Fleming (1928). However, there exist...

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully- and Schmidt quasi-normalized associated Legendre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two to four-term recurrence form...

“Serendipity” is a word created by Horace Walpole, the inventor of the gothic thriller, in 1754 based on a Persian fairy tale, “The Three Princes of Serendip (= Sri Lanka)”. The word means an unplanned, fortuitous discovery. Its well-quoted examples are the X-ray by W. C. Rontgen (1895) and the penicillin by A. Fleming (1928). However, there exist...

“Serendipity” is a word created by Horace Walpole, the inventor of the gothic thriller, in 1754 based on a Persian fairy tale, “The Three Princes of Serendip (= Sri Lanka)”. The word means an unplanned, fortuitous discovery. Its well-quoted examples are the X-ray by W. C. Rontgen (1895) and the penicillin by A. Fleming (1928). However, there exist...

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recur-sive method to compute the sine/cosine series coefficient of the 4π fully-and Schmidt quasi-normalized associated Leg-endre functions. The key of the method is a set of increasing-degree/order mixed-wavenumber two-to four-term recurrence for...

Astronomy is one of the most classic sciences. Yet, it may not be appropriate to regard it as one branch of the natural sciences. For instance, the so-called cosmological principle, which Jordano Bruno advocated during the debate of the Ptolemaic/Copernican theories, namely the assertion that "Since the Earth can not be a special place in the Unive...

Astronomy is one of the most classic sciences. Yet, it may not be appropriate to regard it as one branch of the natural sciences. For instance, the so-called cosmological principle, which Jordano Bruno advocated during the debate of the Ptolemaic/Copernican theories, namely the assertion that "Since the Earth can not be a special place in the Unive...

We present a recursive formulation to compute the potential, the acceleration vector and the force gradient tensor of the gravitational field of a right rectangular parallelepiped or a prism when its volume mass density varies vertically by following an arbitrary degree polynomial. First, the potential of the parallelepiped is expressed as the trip...

We reviewed a recent progress in the computation of the gravitational field of a general object. Among them, the most general method is realized by the regularization of the Newton kernels of the defining integrals of the field by the integration variable transformation to the local spherical polar coordinates (Fukushima 2016d, Mon. Not. Royal Astr...

Reviewed is the recent progress in the numerical computation of the gravitational field of a general object (Fukushima 2013, 2015, 2016a, 2016b, 2016c, 2016d, 2017, 2018a, 2018b). Among them, the most general method is realized by the regularization of the Newton kernels by the integration variable transformation to the local spherical polar coordi...

In order to construct a realistic rotation curve of Milky Way theoretically, we present a novel method to integrate the gravitational field for general three-dimensional objects (Fukushima 2016, MNRAS, 463:1500). By adopting the spherical polar coordinates centered at the evaluation point as the integration variables, we numerically compute the vol...

In order to accelerate the spherical/spheroidal harmonic synthesis of any function, we developed a new recursive method to compute the sine/cosine series coefficient of the 4π fully normalized associated Legendre function. The key of the method is a set of increasing-degree/order mixed-wavenumber two-to four-term recurrence formulas to compute the...

Triggered by another type of the negative parameter transformation, or the so-called imaginary modulus transformation, of the incomplete elliptic integral of the first kind presented in \citet[Formula 17.4.17]{AS64}, we obtained a group of similar transformation of the incomplete elliptic integrals of the second and third kind and of the complete e...

In order to obtain the gravitational field of a general finite body like Ryugu inside its Brillouin sphere, we developed a new method to compute the field accurately (Fukushima 2017, Astron. J., 154:145). First, the body is assumed to consist of some layers in a certain spherical polar coordinate system and the volume mass density of each layer is...

This is Legendre's "Excercices de Calcul Integral" (Legendre 1811), which contains a great variety of integrals reducible to the incomplete/complete elliptic integrals.

We developed a numerical method to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile (Fukushima 2016, MNRAS, 456:3702). We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating...

We developed a numerical method[-70pt] to compute the gravitational field of an infinitely-thin axisymmetric disc with an arbitrary surface mass density profile. We evaluate the gravitational potential by a split quadrature using the double exponential rule and obtain the acceleration vector by numerically differentiating the potential by Ridders’...

We present a recursive formulation to compute the potential, the acceleration vector, and the force gradient tensor of the gravitational field of a right rectangular parallelepiped or a prism when its volume mass density varies vertically by following an arbitrary degree polynomial. First, the potential of the parallelepiped is expressed as the tri...

Presented is a group of new variable transformation of the incomplete elliptic integrals of all three kinds and of the complete elliptic integral of the third kind. The new transformation is similar to but significantly different from the imaginary modulus transformation.

This is Hancock's "Elliptic Integrals" (Hancock 1917), an authoritative text book on the elliptic integrals.

Selected are 10 movies to know mathematics: The Wife Who Raised a Genius (2018), gifted (2017), The Man who Knew Infinity (2015), x+y (2014), 21 (2008), The Housekeeper and the Professor (2006), Proof (2005), Pi (1998), Cube (1997), and Good Will Hunting (1997).

Selected are 10 movies to know physics: Spider-man: Homecoming (2017), Coherence (2013), Mr. Nobody (2009), Angels and Demons (2009), Hawking (2004), Timeline (2003), October Sky (1999), Back to the Future (1985), The Girl who Leapt through Time (1983), and The China Syndrome (1979).

Selected are 10 movies to know physics: Spider-man: Homecoming (2017), Coherence (2013), Mr. Nobody (2009), Angels and Demons (2009), Hawking (2004), Timeline (2003), October Sky (1999), Back to the Future (1985), The Girl who Leapt through Time (1983), and The China Syndrome (1979).

Selected are 10 movies to know biology: Life (2017), Hachi: A Dog's Tale (2009), Resident Evil (2002), Outbreak (1995), Jurassic Park (1975), Jaws (1975), The Planet of the Apes (1968), The Birds (1963), Godzilla (1954), and Frankenstein (1931).

Selected are 10 movies to know biology: Life (2017), Hachi: A Dog's Tale (2009), Resident Evil (2002), Outbreak (1995), Jurassic Park (1975), Jaws (1975), The Planet of the Apes (1968), The Birds (1963), Godzilla (1954), and Frankenstein (1931).

We developed a new recursive method to compute the sine/cosine series coefficient of the 4p fully normalized associated Legendre function. The key of the method is a set of increasing-degree/order mixed-wavenumber two- to four-term recurrence formulas to compute the diagonal terms. They are used in preparing the seed values of the decreasing-order...

Reviewed is the recent progress in the numerical computation of the gravitational field of a general object (Fukushima 2013, 2015, 2016a, 2016b, 2016c, 2016d, 2017, 2018). Among them, the most general method is realized by the regularization of the Newton kernels by the integration variable transformation to the local spherical polar coordinates (F...

Selected are 10 movies to watch for knowing computer: Ex Machina (2014), her (2013), A.I. Artificial Intelligence (2001), The Matrix (1999), Bicentennial Man (1999), Ghost in the Shell (1995), The Terminator (1984), Blade Runner (1982), TRON (1982), and Colossus: The Forbin Project (1970).

Selected are 10 movies to watch for knowing computer: Ex Machina (2014), her (2013), A.I. Artificial Intelligence (2001), The Matrix (1999), Bicentennial Man (1999), Ghost in the Shell (1995), The Terminator (1984), Blade Runner (1982), TRON (1982), Colossus: The Forbin Project (1970).

Selected are 10 movies to watch for knowing the Earth: An Inconvenient Sequel: Truth to Power (2017), Earth: One Amazing Day (2017), The Impossible (2012), The Day After Tomorrow (2004), Dante's Peak (1997), Twister (1996), The Abyss (1989), Earthquake (1974), Japan Sinks (1973), Voyage to the Bottom of the Sea (1961).