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## Mathematics (数学)

Mount Usu / Sarobetsu post-mined peatland
From left: Crater basin in 1986 and 2006. Cottongrass / Daylily

Gr. μαθημα. Engl. Mathematics

(History of Natural sciences)

#### History

Three Greek geometric problems (ギリシア幾何学三大問題)
a set of geometric problems whose solution was sought using only compass and straightedge:
1. circle squaring: construct a square equal in area to a circle
2. cube duplication (Delian problem): give the length of an edge of a cube, that a second cube be constructed having double the volume of the first
3. angle trisection: divide an arbitrary angle into three equal angles

Fibonacci (alias), Leonardo da Pisa 1170-1250

"Liber Abaci": introduced Indian-Arabic mathematics, numeration, calculation, etc.

Descartes René 1596-1650: founder of analytic geometry
Tartaglia-Cardano: solution of third degree equation or cubic equation
Tartaglia, Niccolò Fontana (1499/1500-1557) and
Cardano, Gerolamo (1501-1576)

Cardano–Tartaglia formula (= cubic equation): ax3 + bx2 + cx + d = 0

Oresme, Nicole 1323-1382: concept and symbol of power, n on an
Stifel, Michael 1486-1567: "Arithmetica integra" - concept of logarithm
Ferrari, Lodovico 1522-1565: solution of biquadratic equation
Gunter, Edmund 1581-1626: trigonometric function, logarithmic table

invented Gunter rule (prototype of slide rule)

Barrow Isaac 1630-1677: reverse tangent (逆接線) →

find out the original curve by detecting the characteristics of tangent = the beginning of infinitesimal calculus

Newton: differentiation (called fluxion at that time) and integration

Leibniz, Gottfried Wilhelm 1646-1716, Germany: competition to the priority of differential and integral calculus

Bernoulli, Jacques 1654-1705: one of the Bernoulli Family

1/12 + 1/22 + 1/32 + … → convergence

Bernoulli, Jean 1667-1748: Farther of Daniel
Bernoulli, Daniel 1700-1812, Switzerland (born in Holland)

mathematics and physics

Euler, Leonhard 1707-1783, math and astronomy

1735 solved Basel problem
≈1735 discovered Euler-Maclaurin formula (オイラーの和公式)
1736 The Seven Bridges of Königsberg (ケーニヒスベルクの橋)

Königsberger Brückenproblem
the foundations of graph theory

1748 proved Euler's formula  eiz = cosz + isinz
1748 published Introductio in analysin infinitorum
1755 published Institutiones calculi differentialis
1760 developed Euler's equations of rigid body dynamics
1761 discovered Euler's totient function

#### Graph theory (グラフ理論)

Def. the study of graphs, which are mathematical structures used to model pairwise relations between objects
Def. Graph: an ordered pair G = (V, E) comprising,

V, a set of vertices (nodes or points, 頂点/節点)
E ⊆ {(x, y)}|x, yV and xy}, a set of edges (links or lines, 枝/辺), which are unordered pairs of vertices

Def. degree or valency (次数): the number of edges that lead to a vertex
Ex. Eulerian trail or path (オイラー路): solution of Königsberg bridges
Q. Reaching an island or mainland bank other than via one of the bridges, or accessing any bridge without crossing to its other end

Pr. Euler path: a route around a graph that visits every edge once

The number of vertices of odd degree must be either zero or two
If not then there is no Euler Path
And if there are two vertices with odd degree, then they are the starting and ending vertices
⇒ Vertices N, E and S = 3. W = 5 → no Euler Path //

[ calculus ]

Hamilton, William Rowan (1805-1865, Ireland)

quaternion (四元数)
Hamiltonian path or traceable path (ハミルトン路): a graph that visits each vertex exactly once
Hamiltonian cycle or circuit: a cycle that visits each vertex exactly once

icosian game or Hamilton puzzle

Def. complete graph (完全グラフ), Kn: a graph in which every pair of distinct vertices is connected by a unique edge
Alembert (d'), Jean Le Rond 1717-1783, France, mathematics, physics and enlightenment:

edited Encyclopedie universelle des arts et sciences (百科全書) with Diderot, Denis (1713-1784)

Laplace, Pierre Simon, Marquis de 1749-1827, France, mathematics, astronomy and physics (probability, equation and geodesy)

left his name Laplace transformation or transform

Gauss, Karl Friedrich 1777-1855, Germany: algebra, geometry, analysis

left his name gauss (G) on the electromagnetic unit (EMU), gauss
Basic theorem of algebra: every non-constant single-variable polynomial with complex coefficients has at least one complex root
1801 Disquisitiones arithmeticae (数論研究)

Fourier, Jean Baptiste Joseph 1768-1830, France: mathematics and thermodynamics

proposed the analytical theory of heat or thermodynamics (contributed the development of mathematical physics)

Poisson, Siméon Denis 1781-1840, France

Probability theory and static electromagnetics

Poncelet, Jean Victor 1788-1867, France

Study on geometry when he was in jail because he became captive during joining an army for the conquest to Russia by Napoleon I
→ founder of projective geometry
left his name a unit of power Poncelet (= 100 kg·m/sec)

Cauchy Augustine Louis 1789-1856: series (級数)

1831: complex number (複素数)

Lobachevsky NI 1793-1856, Russia / Bolyai J 1802-1860, Hungary

Lobachevskian geometry → hyperbolic geometry
Non-Euclidean axiom: denied Euclidean axiom 5
= for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part

Abel Niels Henrik 1802-1829 (only 26 years), Nowray
Galois E (ガロア) 1811-1832

Research on the bacground of theory proposed by Abel → establishing group theory (群論)

Riemann, Georg Friedrich Bolyai 1826-1866, Germany

Non-Euclidean axiom: no pararell lines exist

Poincaré, Jules Henri 1854-1912, France: mathematics, physics and philosophy

Cousin of The president Poincaré R (1860-1934)
Multidisciplinary research represented by differential equation

Hadamard, Jacques 1865-1963 & Poussin, de la Valle 1866-1962

Prime number theorem (PNT)
D ≡ prime density = (count prime number < N)/N)
N → ∞ ⇒ D = 1/lnN

1959 International Mathematical Olympiad, IMO: for pre-college students

### Elementary function (初等関数)

Def. Algebraic function (代数関数): a function defined as the root of a polynomial equation

= rational function (有理関数) + irrational function (無理関数)

#### Transcendental function (超越関数)

##### Elementary transcendental function (初等超越関数)
function that is not algebraic

Ex. trigonometric function (f), exponent f, logarithmic f

Elementary f = algebraic f + elementary transcendental f
##### Exponent function (指数関数)
Law. exponential law (指数法則): b > 0, ax > 0 ⇒

axay = ax+yax/ay = ax-y
(ax)y = axy
(ab)x = axbx → (b/a)x = bx/ax

Case. zero to the power of zero (0 to the 0th power): 00
##### Def. Power function (冪関数)
Ex. limx→∞(1 + a/x)x = ea
Pr. a/x = 1/y → (1 + a/x)x = (1+ 1/y)ay = ((1 + 1/y)y)a continuous

x → +∞, a > 0 → y → +∞, a < 0 → y → –∞
∴ (1 + 1/y)y = e //

##### Def. Hyperbolic function (双曲線関数)
x (-∞ < x < ∞) ⇒
coshx = (ex + e-x)/2 (hyperbolic cosine 双曲線余弦)
sinhx = (ex - e-x)/2 (hyperbolic sine 双曲線正弦)

⇒ graph: catenary (懸垂線)

tanhx = sinhx/coshx
cothx = coshx/sinhx
sechx = 1/coshx
coshx (or cosechx) = 1/sinhx
⇒ inverse hyperbolic function (逆双曲線関数)
Prop. (1) cosh2x - sinh2x = 1

(2) tanhx = coshx/sinhx
(3) cosh(x ± y) = coshxcoshy ± sinhxsinhy, and

sinh(x ± y) = sinhxcoshy ± coshxsinhy
(double-sign corresponds)

(4) limx→∞tanhx = 1, and limx→-∞tanhx = -1

Pr._(1) cosh2x - sinh2x = (coshx + sinhx)(coshx - sinhx) = ex·e-x = 1 //

(2) trivial //
(3) coshxcoshy + sinhxsinhy

= (ex + e-x)/2·(ey + e-y)/2 + (ex - e-x)/2·(ey - e-y)/2
= (ex+y + e-x+y + ex-y + e-x-y)/4 + (ex+y - e-x+y - ex-y + e-x-y)/4
= (ex+y + e-(x+y))/2 = cosh(x + y), and so forth //

(4) limx→∞tanhx = limx→∞((1 - e-2x)/(1 + e-2x)) = 1
__ limx→-∞tanhx = limx→-∞((e2x - 1)/(e2x + 1)) = -1

Ex. sinh(log2) = (elog2 - e-log2)/2 = (2 - 1/2)/2 = 3/4
##### Def. Trigonometric function (三角関数)
sine (正弦): sinθ = y/r
cosine (余弦): cosθ = x/r
tangent (正接): tanθ = x/y = sinθ/cosθ
cotangent (余接): cotθ = x/y, = 1/tanθ
secant (正割): secθ = r/x, = 1/cosθ
cosecant (余割): cosecθ = y/x = 1/sinθ
tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
Cycle (周期): sinθ, cosθ, secθ, cosecθ → 2π, tanθ, cotθπ

→ sin(π/2 – θ) = cosθ, cos(π/2 – θ) = sinθ, sin = 0

limθ→0(sinθ/θ) = 1, |θ| ∠ 1 → sinθθ

sin(–θ) = –sinθ, cos(–θ) = cosθ, tan(–θ) = –tanθ
sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, 1 + cot2θ = cosec2θ
Th. limx→0(sinx/x) = 1
Ex. limx→0(tanx/x) = limx→0(sinx/x)·(1/cosx) = 1
Ex. limx→0sin(sinx)/x = limx→0(sin(sinx)/sinx)·(sinx/x)

= 1·1 = 1 (∵ x → 0 ⇒ sinx → 0)

Equations of trigonometric function

ΔABC, AB = a, BC = b, CA = c, ∠A, ∠B, ∠C ⇒

Th. sine theorem (正弦定理): a/sinA = b/sinB = c/sinC

= 2R (R: radius of circumcircle)

Th. cosine theorem (余弦定理)

a2 = b2 + c2 + bccosA
b2 = a2 + c2 + accosB
c2 = a2 + b2 + abcosC

sin(x ± y) = sinxcosy ± cosxsiny
cos(x ± y) = cosxcosy –(±) sinxsiny
tan(x ± y) = (tanx ± tany)/(1 –(±) tanxtany)

Eq. Double-angle formula (2倍角公式)

sin2x = 2sinxcosx
cos2x = cos2x – sin2x = 2cos2x – 1 = 1 – 2sin2x

Eq. Triple-angle formula (3倍角公式)

sin3x =3sinx – 4sin3x
cos3x = 4cos3x – 3cosx

Eq. Half-angle formula (半角公式)

sin2x/2 = (1 – cosx/2)
cos2x/2 = (1 + cosx)/2
tan2x/2 = (1 – cosx)/(1 + cosx)

Eq. Product (積) → sum (difference) 和(差)

sinαcosβ = 1/2{sin(α + β) + sin(αβ)}
cosαsinβ = 1/2{sin(α + β) – sin(αβ)}
cosαcosβ = 1/2{cos(α + β) + cos(αβ)}
sinαsinβ = 1/2{cos(α + β) – cos(αβ)}

Eq. Sum (difference) 和(差) → product (積)

sinα + sinβ = 2sin((α + β)/2)cos((αβ)/2)
sinα – sinβ = 2cos((α + β)/2)sin((αβ)/2)
cosα + cosβ = 2cos((α + β)/2)cos((αβ)/2)
cosα – cosβ = 2sin((α + β)/2)sin((αβ)/2)

Ex. limxasinx = sina, limxacosx = cosa
Pr. ε > 0, δ > 0, |xa| < δ → |sinx – sina| < ε

|sinx – sina| = |2sin((xa)/2)·cos((x + a)/2)| < xa < δ (= ε)
∵ |cos((x + a)/2)| ≤ 1, |sin((xa)/2)| < (xa)/2___ditto cosine //

Eq. Combination of trigonometric function (三角関数の合成)

asinθ + bcosθ = √(a2 + b2)·sin(θ + α)

Here, α is obtained by: cosα = a/√(a2 + b2), sinα = b/√(a2 + b2)

Pr. asinθ + bcosθ = √(a2 + b2)·((a/√(a2 + b2))·sinθ + (b/√(a2 + b2))·cosθ)

= √(a2 + b2)·(cosαsinθ + sinαcosθ) = √(a2 + b2)·sin(θ + α) // _____________________from addition theorem

Ax. t := tan(x/2) ⇒

cosx = (1 - t2)/(1 + t2), sinx = 2t/(1 + t2), dx = 2/(1 + t2dt

Pr. cos2(x/2) = 1/(1 + tan2(x/2)) = 1/(1 + t2)

From double-angle formula,
cosx = 2cos2(x/2) - 1 = 2·1/(1 + t2) -1 = (1 - t2)/(1 + t2)
sinx = 2sin(x/2)cos(x/2) = 2cos2(x/2)tan(x/2)

= 2·1/(1 + t2t = 2t/(1 + t2)

t = tan(x/2), differentiate both sides
dt = 1/(cos2(x/2))·1/2·dxdx = 2cos2(x/2)·dt = 2/(1 + t2dt

##### Def. Inverse trigonometric function (逆三角関数)
= cyclometric function
sin-1x = arcsinx: x = siny (-1 ≤ x ≤ +1)
cos-1x = arccosx: x = cosy (-1 ≤ x ≤ +1)
tan-1x = arctanx: x = tany (ℜ)
cot-1x[-∞, +∞] = arccotx: x = coty (ℜ)
sec-1x = arcsecx: x = secy (x ≤ -1 or 1 ≤ x)
csc-1x = arccscx: x = cscy (x ≤ -1 or 1 ≤ x)
Q. Solve (1) sin-1(1/2), (2) tan-1(-√3) and (3) sin-1(sin(3/5·π))

(-π/2 ≤ θπ/2)

A._(1) θ := sin-1(1/2), sinθ = 1/2 ∴ θ = π/6 ∴ sin-1(1/2) = π/6
__ (2) θ := tan-1(-√3), tanθ = -√3 ∴ θ = π/3 ∴ tan-1(-√3) = π/3
__ (3) θ := sin-1(sin(3/5·π)), sinθ = sin(3/5·π) ∴ θ = 2/5·π

∴ sin-1(sin(3/5·π)) = θ = 2/5·π

## Set theory (集合論)

1874 Cantor Georg (1845-1918): established set theory
##### Law. Basic formulae (基本公式)
Idempotent law (冪等法則)

AA = A, AA = A

Commutative law 交換法則

AB = BA, AB = BA

Distributive law (分配法則)

A∪(BC) = (AB)∪C
A∩(BC) = (AB)∩C
A∪(BC) = (AB)∩(AC)
A∩(BC) = (AB)∪(AC)

Absorptive law (吸収法則)

A∪(AB) = A
A∩(AB) = A

Double complement law (復元法則)

(AC)C = A

De Morgan's law (ド・モルガンの法則)

(AB)C = ACBC
(AB)C = ACBC

## Number theory (整数論)

Def. Gauss notation (ガウス記号), [•]: x, aZ, a < x

⇒ max(a) ≡ [x]

Ex. [-2] = -2, [π] = 3, [1 - √5] = -2
Ax. xx - 1 < [x] or [x] ≤ x < [x] + 1
Ex. [2 + √5] + [2 - √5] = 4 + (-1)

#### Fermat's Last Theorem (フェルマーの最終定理) 1670

= Fermat's conjecture
Th. No three positive integers a, b and c satisfy the equation

an + bn = cn

for any integer value of n greater than 2
Prior to prove the theory:
n = 1: a1 + b1 = c1a + b = c ⇒ infinite solution
n = 2: a2 + b2 = c2 (≡ the Pythagorean theorem) ⇒ infinite solution
Pr.
Case. n = 3 (proven by Euler 1753)

Case. n = 4 (proven by Fermat ≈1630)

Case. n = 5 (proven by Dirichlet & Legendre 1825)

Case. n = 7 (proven by Lamé 1839)

## Geometry (幾何学)

 Concerned with shape, size, relative position of figures, and the properties of space Euclidean geometry (ユークリッド幾何学): the study of points, lines, shapes, figures and spaces based on the Euclidean five axioms Plane geometry (平面幾何学) = two-dimensional Euclidean geometry Solid geometry (立体幾何学) = three-dimensional Euclidean geometry Differential geometry (微分幾何学): using calculus and linear algebra to study problems in geometry Topology (位相幾何学, トポロジー): dealing with the properties of geometry that are unchanged by continuous function Algebraic geometry (代数幾何学): studies geometry through the use of multivariate polynomials and other algebraic techniques

### Euclidean geometry (ユークリッド幾何学)

#### Euclid's five axioms (5公理)

1. Things that are equal to the same thing are also equal to one another
2. If equals are added to equals, then the wholes are equal (Addition property of equality)
3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality)
4. Things that coincide with one another are equal to one another (Reflexive Property)
5. The whole is greater than the part.

#### Euclid's five postulates (5公準)

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

#### Trochoid (トロコイド)

a roulette formed by a circle rolling along a line (s.l.), including curve and circle

x = - bsinθ
y = a - bcosθ

L: a line (the circle center C moves parallel to L)
P (trochoid) ≡ every other point in the rotating plane attached to the circle traces
θ: the variable angle through which the circle rolls

##### Eepitrochoid (外トロコイド)/Hypotrochoid (内トロコイド)
a roulette traced by a point attached to a circle of radius r rolling around the outside/inside of a fixed circle of radius R

Cycloid (CP = b)

##### Cycloid (サイクロイド)
= common trochoid

If the cycloid has a cusp (尖点) at the origin and its humps are oriented upward, its parametric equation is:
x = a(t - sint), y = a(1 - cost)

Generation of the involute of the cycloid unwrapping a tense wire placed on half cycloid arc shown in red
Astroid (アステロイド, 星芒形) = tetracuspid (四尖点形), cubocycloid, paracycle: a hypocycloid with four cusps
If the radius of the fixed circle is a then the equation is given by:

x2/3 + y2/3 = a2/3 ⊂ superellipse
x = acos3t = a/4·(3cost + cos3t), y = asin3t = a/4·(3sint + sin3t)

Cardioid (カージオイド, 心臓形): shown by

polar coordinates: r = a(1 + cosθ)
orthogonal coordinates: (x2 + y2)(x2 + y2 + 2ax) - a2y2 = 0
parameters: x = a(1 + cosθ)cosθ, y = a(1 + cosθ)sinθ

S = 3/2·πa2, L = 8a

#### Non-Euclidean geometry (非ユークリッド幾何学)

Hyperbolic_______Euclidean_______Elliptic

### Determinant (行列式)

a scalar that is computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix (linear albegra)

Expression: det(A), det A, or |A|
det(A) = = a11a22 - a12a21

#### Simultaneous equation (連立方程式)

Q. Solve the simultaneous equations (1) and (2)
(1) 7x – 4y = 3, -6x + 5y = -1
(2) ax + by + cz = 1, bx + cy + az = 1, cx + ay + bz = 1
A.
(1) D2 = = 35 - 24 = 11,

D12 = = 15 - 4 = 11, D22 = = -7 + 18 = 11
x = D12/D = 1, y = D22/D = 1

(2) D3 = = 3abc - a3 - b3 - c3,

D13 = = D23 = = D33 =

= ab + bc + ca - a2 - b2 - c2

a) a + b + c ≠ 0, a ≠ b ≠ c →

x = D13/D = 1/(a + b + c),
y = D23/D = 1/(a + b + c),
z = D33/D = 1/(a + b + c)

b) a + b + c = 0 → incompatible (不能)
c) a + b + c ≠ 0, a = b = c → indeterminate (不定)

## Matrix (行列)

### Frame of reference (座標系)

#### Two dimensions (二次元)

1) Orthogonal coordinates (直交座標)
2) Oblique coordinate (斜交座標)
3) Polar coordinate (極座標)
Terms:
coordinates 座標
axis (pl. axes) 軸
horizontal plane 水平面
horizontal axis 水平軸
hypotenuse 斜辺
normal line 法線 (adj. normal 法線、法線の)
orthogonal 直角な
parabola 放物線(放物線の)