Conjugate - Kyoyaku (English spelling) conjugate

Japanese: 共役 - きょうやく(英語表記)conjugate
Conjugate - Kyoyaku (English spelling) conjugate
Also written as conjugate. It means to be in a pair or to join in a pair, but in mathematics, it is used with a special meaning in each case. (1) Conjugate diameters c.diameters The locus of the midpoints of chords parallel to one diameter of an ellipse or hyperbola is also the other diameter of the ellipse or hyperbola. These two diameters are said to be conjugate, and one is called the conjugate diameter of the other, or simply the conjugate diameter. (2) Conjugate axis c.axis In the hyperbola x 2 / a 2 - y 2 / b 2 = 1, the y axis does not intersect with the curve. In this way, of the two symmetric axes of a hyperbola, the axes that do not intersect with the curve are called conjugate axes. (3) Conjugate hyperbola c.hyperbola The two hyperbolae x 2 / a 2 - y 2 / b 2 = 1 and x 2 / a 2 - y 2 / b 2 = -1 are said to be conjugate hyperbolas. Their asymptote is the same. (4) Conjugate poles c.poles A quadratic curve (or quadratic surface) has the property that if the polar line (or polar plane) of point P passes through point P', then the polar line (or plane) of P' also passes through P (→ polar lines and poles). Two points P and P' that have this relationship are said to be conjugate with respect to this quadratic curve (or quadratic surface). (5) Conjugate polar lines (polar planes) c.polars A quadratic curve (or quadratic surface) has the property that if the pole P of a line (plane) p is on p ', then the pole P' of p ' is also on p . Two lines (planes) p and p ' that have this relationship are said to be conjugate with respect to this quadratic curve (quadratic surface). (6) Conjugate complex numbers For z = x + yi ( x and y are real numbers), the complex number = x - yi is called the conjugate complex number. Geometrically, on a Gaussian plane, z and are symmetric with respect to the x- axis. The only numbers for which z = 1 are real numbers. (7) Conjugate quaternion c.quateernion Two quaternions q = x0 + x1i + x2j + x3k and = x0 - x1i - x2j - x3k ( i , j , k are imaginary numbers ) are said to be conjugate to each other. (8) Conjugate roots c.roots If an equation with real coefficients has an imaginary root a + bi ( a and b are real numbers, b ≠ 0), then its conjugate complex number a - bi is also a root of the equation. Two roots that are conjugate complex numbers to each other like this are called conjugate roots. (9) Conjugate subgroup c.subgroup Let H be a subgroup of a group G , and let a be any element of G. If we consider the set a -1 Ha = { a -1 ha | hH }, then a -1 Ha is also a subgroup of G. This is called a conjugate subgroup of H. If a -1 Ha = H holds for any element a of G , then H is said to be a normal subgroup or an invariant subgroup of G ( a -1 is the inverse of a ). (10) Conjugate fields c.fields Let K 1 and K 2 be two extension fields of field K. If there exists a correspondence σ that maps K 1 onto K 2 , and σ(α) = α holds for any element α of K , then K 2 is called a conjugate field of K 1 with respect to K. (11) Conjugate elements c.elements If two elements α 1 and α 2 of an extension field K ' of field K are both algebraic elements with respect to K and are roots of the same irreducible polynomial f ( x ) in K , then α 1 and α 2 are called conjugate elements with respect to K.

Conjugation
Conjugation

In organic compounds, two or more multiple bonds, i.e., double bonds or triple bonds, exist with one single bond between them, and these bonds exhibit interaction. This interaction is based on the delocalization of electrons, and single bonds between unsaturated bonds exhibit the properties of unsaturated bonds to a certain extent. For example, the interatomic distances in butadiene CH 2 =CH-CH=CH 2 are 1.37 Å for C=C and 1.47 Å for C-C, the former being longer than the 1.34 Å for C=C in ethylene, and the latter being shorter than the 1.54 Å for C-C in ethane. This is the result of interactions due to conjugation, and can be explained by considering the spread of π electrons involved in the unsaturated bonds. Conjugation affects both physical properties such as light absorption of the substance, and chemical properties such as addition reactivity. (→Conjugated double bonds)

Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information

Japanese:
共軛とも書く。対になっていること,あるいは対になって結合することをいうが,数学では,それぞれの場合に特別の意味をもって使用される。(1) 共役直径 c.diameters 楕円あるいは双曲線の,一つの直径に平行な弦の中点の軌跡はまた,これらの楕円あるいは双曲線の,他の直径となる。この二つの直径は共役であるといい,一方を他方の共役直径,あるいは単に共役径という。(2) 共役軸 c.axis 双曲線 x2/a2y2/b2=1 において,y 軸はこの曲線と交わらない。このように双曲線の二つの対称軸のうち,曲線と交わらない軸を共役軸という。(3) 共役双曲線 c.hyperbola 二つの双曲線 x2/a2y2/b2=1 と x2/a2y2/b2=-1 とは,互いに共役な双曲線であるという。これらの漸近線は一致する。(4) 共役な極 c.poles 二次曲線(または二次曲面)には,点 Pの極線(または極平面)が点 P'を通れば,P'の極線(または平面)も Pを通るという性質がある(→極線と極点)。このような関係にある 2点 P,P'はこの二次曲線(または二次曲面)に関して共役であるという。(5) 共役な極線(極平面) c.polars 二次曲線(または二次曲面)には,直線(平面)p の極 Pが p'の上にあれば,p'の極 P'も p の上にあるという性質がある。このような関係にある 2直線(平面)pp'は,この二次曲線(二次曲面)に関して共役であるという。(6) 共役複素数 zxyixy は実数)に対して,複素数 =xyi を共役複素数という。幾何学的には,ガウス平面上で,z と とは x軸に関して,対称である。z= となる数は実数だけである。(7) 共役四元数 c.quateernion 二つの四元数 qx0x1ix2jx3k と =x0x1ix2jx3kijk は虚数)は,互いに共役であるという。(8) 共役根 c.roots 実係数の方程式が虚根 abiab は実数,b≠0)をもてば,必ずその共役複素数 abi もこの方程式の根となる。このように互いに共役複素数となっているような 2根を共役根という。(9) 共役部分群 c.subgroup 群 G の部分群を H とし,aG の任意の元とするとき,集合 a-1Ha={a-1hahH}を考えれば,a-1Ha もまた G の部分群である。これを H の共役部分群という。G の任意の元 a に対して,a-1HaH が成り立てば,HG の正規部分群,あるいは不変部分群といわれる(a-1a の逆元)。(10) 共役体 c.fields 体 K の二つの拡大体を K1K2 とするとき,K1K2 の上へ写像する対応σが存在し,K の任意の元αに対して,σ(α)=αが成り立てば,K2K に関する K1 の共役体という。(11) 共役元 c.elements 体 K の拡大体 K'の 2元α1,α2が,ともに K に関する代数的元であって,K における同一の既約多項式 fx)の根となっているとき,このα1,α2K に関する共役元という。

共役
きょうやく
conjugation

有機化合物で,2個以上の多重結合,すなわち二重結合や三重結合が1個の単結合をはさんで存在し,これらの結合が相互作用を示すこと。この相互作用は電子の非局在化に基づくもので,不飽和結合にはさまれた単結合はある程度不飽和結合の性質を示す。たとえばブタジエン CH2=CH-CH=CH2 における原子間距離はC=Cで 1.37Å,C-Cで 1.47Åであり,前者はエチレンのC=Cの 1.34Åより長く,後者はエタンのC-Cの 1.54Åより短い。これは共役による相互作用の結果であり,不飽和結合に関与するπ電子の広がりがあると考えると説明がつく。共役はその物質の光の吸収のような物理的性質にも,付加反応性のような化学的性質にも影響を及ぼす。 (→共役二重結合 )

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