wpmath0000001_18

Proper_motion.html

  1. μ α = α 1 - α \ \mu_{\alpha}=\alpha_{1}-\alpha
  2. μ δ = δ 1 - δ . \ \mu_{\delta}=\delta_{1}-\delta\ .
  3. μ 2 \ \mu^{2}
  4. = μ δ 2 + μ α 2 cos 2 δ ={\mu_{\delta}}^{2}+{\mu_{\alpha}}^{2}\cdot\cos^{2}\delta
  5. μ 2 \ \mu^{2}
  6. = μ δ 2 + μ < m t p l > α 2 ={\mu_{\delta}}^{2}+{\mu_{<}mtpl>{{\alpha\ast}}}^{2}
  7. μ δ \ \mu_{\delta}
  8. = μ cos θ =\mu\cos\theta
  9. μ α cos δ \ \mu_{\alpha}\cos\delta
  10. = μ sin θ =\mu\sin\theta
  11. μ < m t p l > α \ \mu_{<}mtpl>{{\alpha\ast}}
  12. = μ sin θ =\mu\sin\theta

Propositional_calculus.html

  1. P Q P\to Q
  2. P P
  3. Q Q
  4. P Q , P Q P\to Q,P\vdash Q
  5. P P
  6. Q Q
  7. A A
  8. B B
  9. C C
  10. P P
  11. Q Q
  12. R R
  13. φ φ
  14. ψ ψ
  15. χ χ
  16. a = 5 a=5
  17. P P
  18. P P
  19. ¬ P ¬P
  20. ¬ P ¬P
  21. P P
  22. P P
  23. ¬ P ¬P
  24. P P
  25. ¬ P ¬P
  26. P P
  27. ¬ P ¬P
  28. ¬ ¬ P ¬¬P
  29. P P
  30. P P
  31. Q Q
  32. P P
  33. Q Q
  34. P Q P∧Q
  35. P Q P∧Q
  36. P P
  37. Q Q
  38. P P
  39. Q Q
  40. P P
  41. Q Q
  42. P P
  43. Q Q
  44. P P
  45. Q Q
  46. P P
  47. Q Q
  48. P P
  49. Q Q
  50. P Q P∧Q
  51. P Q P∧Q
  52. P Q P∧Q
  53. P Q P∨Q
  54. P P
  55. Q Q
  56. P P
  57. Q Q
  58. P P
  59. Q Q
  60. P P
  61. Q Q
  62. P Q P→Q
  63. P P
  64. Q Q
  65. Q Q
  66. P P
  67. P P
  68. Q Q
  69. P P
  70. Q Q
  71. P Q P↔Q
  72. P P
  73. Q Q
  74. P P
  75. Q Q
  76. P P
  77. Q Q
  78. φ φ
  79. ¬ φ ¬φ
  80. φ φ
  81. ψ ψ
  82. φ ψ φ∧ψ
  83. φ ψ φ∧ψ
  84. P Q R P∧Q∧R
  85. P Q P∧Q
  86. R R
  87. P P
  88. Q R Q∧R
  89. ( P Q ) R (P∧Q)∧R
  90. P ( Q R ) P∧(Q∧R)
  91. P ( Q R ) P∧(Q∨R)
  92. ( P Q ) R (P∧Q)∨R
  93. P P
  94. Q Q
  95. Z Z
  96. k k
  97. k k
  98. P P
  99. Q Q
  100. 1. P Q 2. P Q \begin{array}[]{rl}1.&P\to Q\\ 2.&P\\ \hline\therefore&Q\end{array}
  101. C C
  102. ( P 1 , , P n ) (P_{1},...,P_{n})
  103. C C
  104. ( P 1 , , P n ) (P_{1},...,P_{n})
  105. P P
  106. Q Q
  107. P Q P→Q
  108. P P
  109. Q Q
  110. P P
  111. P Q P→Q
  112. Q Q
  113. Q Q
  114. φ φ
  115. ψ ψ
  116. 1. φ ψ 2. φ ψ \begin{array}[]{rl}1.&\varphi\to\psi\\ 2.&\varphi\\ \hline\therefore&\psi\end{array}
  117. Q Q
  118. Q Q
  119. A = { P Q , ¬ Q and R , ( P Q ) R } A=\{PQ,\neg Q\and R,(PQ)\to R\}
  120. Γ Γ
  121. A A
  122. P Q , ¬ Q and R , ( P Q ) R Γ PQ,\neg Q\and R,(PQ)\to R\in\Gamma
  123. A A
  124. R R
  125. R Γ R\in\Gamma
  126. ( P Q ) ( ¬ P Q ) (PQ)\leftrightarrow(\neg P\to Q)
  127. = ( \Alpha , Ω , \Zeta , \Iota ) \mathcal{L}=\mathcal{L}\left(\Alpha,\ \Omega,\ \Zeta,\ \Iota\right)
  128. \Alpha \Alpha
  129. \mathcal{L}
  130. \Alpha \Alpha
  131. p p
  132. q q
  133. r r
  134. Ω Ω
  135. Ω Ω
  136. Ω = Ω 0 Ω 1 Ω j Ω m . \Omega=\Omega_{0}\cup\Omega_{1}\cup\ldots\cup\Omega_{j}\cup\ldots\cup\Omega_{m}.
  137. Ω j \Omega_{j}
  138. j j
  139. Ω Ω
  140. Ω 1 = { ¬ } , \Omega_{1}=\{\lnot\},
  141. Ω 2 { , , , } . \Omega_{2}\subseteq\{\land,\lor,\to,\leftrightarrow\}.
  142. Ω 0 = { 0 , 1 } . \Omega_{0}=\{0,1\}.
  143. ¬ ¬
  144. \cdot
  145. \wedge
  146. { , } \{\bot,\top\}
  147. \Zeta \Zeta
  148. \Iota \Iota
  149. \mathcal{L}
  150. \Alpha \Alpha
  151. \mathcal{L}
  152. p 1 , p 2 , , p j p_{1},p_{2},\ldots,p_{j}
  153. f f
  154. Ω j \Omega_{j}
  155. ( f ( p 1 , p 2 , , p j ) ) \left(f(p_{1},p_{2},\ldots,p_{j})\right)
  156. \mathcal{L}
  157. p p
  158. ¬ p \neg p
  159. q q
  160. ( ¬ p q ) (\neg p\lor q)
  161. 1 = ( \Alpha , Ω , \Zeta , \Iota ) \mathcal{L}_{1}=\mathcal{L}(\Alpha,\Omega,\Zeta,\Iota)
  162. \Alpha \Alpha
  163. Ω \Omega
  164. \Zeta \Zeta
  165. \Iota \Iota
  166. \Alpha \Alpha
  167. \Alpha = { p , q , r , s , t , u } . \Alpha=\{p,q,r,s,t,u\}.
  168. , \wedge,\lor
  169. ¬ ¬
  170. a b a\leftrightarrow b
  171. ( a b ) ( b a ) (a\to b)\land(b\to a)
  172. Ω = Ω 1 Ω 2 \Omega=\Omega_{1}\cup\Omega_{2}
  173. Ω 1 = { ¬ } , \Omega_{1}=\{\lnot\},
  174. Ω 2 = { } . \Omega_{2}=\{\to\}.
  175. ( p ( q p ) ) (p\to(q\to p))
  176. ( ( p ( q r ) ) ( ( p q ) ( p r ) ) ) ((p\to(q\to r))\to((p\to q)\to(p\to r)))
  177. ( ( ¬ p ¬ q ) ( q p ) ) ((\neg p\to\neg q)\to(q\to p))
  178. p p
  179. ( p q ) (p\to q)
  180. q q
  181. a b a\lor b
  182. ¬ a b \neg a\to b
  183. a b a\land b
  184. ¬ ( a ¬ b ) \neg(a\to\neg b)
  185. 2 = ( \Alpha , Ω , \Zeta , \Iota ) \mathcal{L}_{2}=\mathcal{L}(\Alpha,\Omega,\Zeta,\Iota)
  186. \Alpha \Alpha
  187. Ω \Omega
  188. \Zeta \Zeta
  189. \Iota \Iota
  190. \Alpha \Alpha
  191. \Alpha = { p , q , r , s , t , u } . \Alpha=\{p,q,r,s,t,u\}.
  192. Ω = Ω 1 Ω 2 \Omega=\Omega_{1}\cup\Omega_{2}
  193. Ω 1 = { ¬ } , \Omega_{1}=\{\lnot\},
  194. Ω 2 = { , , , } . \Omega_{2}=\{\land,\lor,\to,\leftrightarrow\}.
  195. \Iota = \Iota=\varnothing
  196. \Zeta \Zeta
  197. \vdash
  198. Γ ψ \Gamma\vdash\psi
  199. Γ Γ
  200. ψ ψ
  201. Γ ψ \Gamma\vdash\psi
  202. Γ Γ
  203. ψ ψ
  204. Γ Γ
  205. Γ Γ
  206. Γ Γ
  207. Γ Γ
  208. ( p q ) (p\to q)
  209. ( p ¬ q ) (p\to\neg q)
  210. ¬ p \neg p
  211. { ( p q ) , ( p ¬ q ) } ¬ p \{(p\to q),(p\to\neg q)\}\vdash\neg p
  212. ¬ p \neg p
  213. ( p r ) (p\to r)
  214. { ¬ p } ( p r ) \{\neg p\}\vdash(p\to r)
  215. ¬ ¬ p \neg\neg p
  216. p p
  217. ¬ ¬ p p \neg\neg p\vdash p
  218. p p
  219. q q
  220. ( p q ) (p\land q)
  221. { p , q } ( p q ) \{p,q\}\vdash(p\land q)
  222. ( p q ) (p\land q)
  223. p p
  224. ( p q ) (p\land q)
  225. q q
  226. ( p q ) p (p\land q)\vdash p
  227. ( p q ) q (p\land q)\vdash q
  228. p p
  229. ( p q ) (p\lor q)
  230. q q
  231. ( p q ) (p\lor q)
  232. p ( p q ) p\vdash(p\lor q)
  233. q ( p q ) q\vdash(p\lor q)
  234. ( p q ) (p\lor q)
  235. ( p r ) (p\to r)
  236. ( q r ) (q\to r)
  237. r r
  238. { p q , p r , q r } r \{p\lor q,p\to r,q\to r\}\vdash r
  239. ( p q ) (p\to q)
  240. ( q p ) (q\to p)
  241. ( p q ) (p\leftrightarrow q)
  242. { p q , q p } ( p q ) \{p\to q,q\to p\}\vdash(p\leftrightarrow q)
  243. ( p q ) (p\leftrightarrow q)
  244. ( p q ) (p\to q)
  245. ( p q ) (p\leftrightarrow q)
  246. ( q p ) (q\to p)
  247. ( p q ) ( p q ) (p\leftrightarrow q)\vdash(p\to q)
  248. ( p q ) ( q p ) (p\leftrightarrow q)\vdash(q\to p)
  249. p p
  250. ( p q ) (p\to q)
  251. q q
  252. { p , p q } q \{p,p\to q\}\vdash q
  253. p p
  254. q q
  255. ( p q ) (p\to q)
  256. ( p q ) ( p q ) (p\vdash q)\vdash(p\to q)
  257. ( ( p q ) p ) q ((p\to q)\land p)\vdash q
  258. p p
  259. q q
  260. p p
  261. q q
  262. ( ( p q ) ¬ q ) ¬ p ((p\to q)\land\neg q)\vdash\neg p
  263. p p
  264. q q
  265. q q
  266. p p
  267. ( ( p q ) ( q r ) ) ( p r ) ((p\to q)\land(q\to r))\vdash(p\to r)
  268. p p
  269. q q
  270. q q
  271. r r
  272. p p
  273. r r
  274. ( ( p q ) ¬ p ) q ((p\lor q)\land\neg p)\vdash q
  275. p p
  276. q q
  277. p p
  278. q q
  279. ( ( p q ) ( r s ) ( p r ) ) ( q s ) ((p\to q)\land(r\to s)\land(p\lor r))\vdash(q\lor s)
  280. p p
  281. q q
  282. r r
  283. s s
  284. p p
  285. r r
  286. q q
  287. s s
  288. ( ( p q ) ( r s ) ( ¬ q ¬ s ) ) ( ¬ p ¬ r ) ((p\to q)\land(r\to s)\land(\neg q\lor\neg s))\vdash(\neg p\lor\neg r)
  289. p p
  290. q q
  291. r r
  292. s s
  293. q q
  294. s s
  295. p p
  296. r r
  297. ( ( p q ) ( r s ) ( p ¬ s ) ) ( q ¬ r ) ((p\to q)\land(r\to s)\land(p\lor\neg s))\vdash(q\lor\neg r)
  298. p p
  299. q q
  300. r r
  301. s s
  302. p p
  303. s s
  304. q q
  305. r r
  306. ( p q ) p (p\land q)\vdash p
  307. p p
  308. q q
  309. p p
  310. p , q ( p q ) p,q\vdash(p\land q)
  311. p p
  312. q q
  313. p ( p q ) p\vdash(p\lor q)
  314. p p
  315. p p
  316. q q
  317. ( ( p q ) ( p r ) ) ( p ( q r ) ) ((p\to q)\land(p\to r))\vdash(p\to(q\land r))
  318. p p
  319. q q
  320. p p
  321. r r
  322. p p
  323. q q
  324. r r
  325. ¬ ( p q ) ( ¬ p ¬ q ) \neg(p\land q)\vdash(\neg p\lor\neg q)
  326. p p
  327. q q
  328. p p
  329. q q
  330. ¬ ( p q ) ( ¬ p ¬ q ) \neg(p\lor q)\vdash(\neg p\land\neg q)
  331. p p
  332. q q
  333. p p
  334. q q
  335. ( p q ) ( q p ) (p\lor q)\vdash(q\lor p)
  336. p p
  337. q q
  338. q q
  339. p p
  340. ( p q ) ( q p ) (p\land q)\vdash(q\land p)
  341. p p
  342. q q
  343. q q
  344. p p
  345. ( p q ) ( q p ) (p\leftrightarrow q)\vdash(q\leftrightarrow p)
  346. p p
  347. q q
  348. q q
  349. p p
  350. ( p ( q r ) ) ( ( p q ) r ) (p\lor(q\lor r))\vdash((p\lor q)\lor r)
  351. p p
  352. q q
  353. r r
  354. p p
  355. q q
  356. r r
  357. ( p ( q r ) ) ( ( p q ) r ) (p\land(q\land r))\vdash((p\land q)\land r)
  358. p p
  359. q q
  360. r r
  361. p p
  362. q q
  363. r r
  364. ( p ( q r ) ) ( ( p q ) ( p r ) ) (p\land(q\lor r))\vdash((p\land q)\lor(p\land r))
  365. p p
  366. q q
  367. r r
  368. p p
  369. q q
  370. p p
  371. r r
  372. ( p ( q r ) ) ( ( p q ) ( p r ) ) (p\lor(q\land r))\vdash((p\lor q)\land(p\lor r))
  373. p p
  374. q q
  375. r r
  376. p p
  377. q q
  378. p p
  379. r r
  380. p ¬ ¬ p p\vdash\neg\neg p
  381. p p
  382. p p
  383. ( p q ) ( ¬ q ¬ p ) (p\to q)\vdash(\neg q\to\neg p)
  384. p p
  385. q q
  386. q q
  387. p p
  388. ( p q ) ( ¬ p q ) (p\to q)\vdash(\neg p\lor q)
  389. p p
  390. q q
  391. p p
  392. q q
  393. ( p q ) ( ( p q ) ( q p ) ) (p\leftrightarrow q)\vdash((p\to q)\land(q\to p))
  394. p p
  395. q q
  396. p p
  397. q q
  398. q q
  399. p p
  400. ( p q ) ( ( p q ) ( ¬ p ¬ q ) ) (p\leftrightarrow q)\vdash((p\land q)\lor(\neg p\land\neg q))
  401. p p
  402. q q
  403. p p
  404. q q
  405. p p
  406. q q
  407. ( p q ) ( ( p ¬ q ) ( ¬ p q ) ) (p\leftrightarrow q)\vdash((p\lor\neg q)\land(\neg p\lor q))
  408. p p
  409. q q
  410. p p
  411. q q
  412. p p
  413. q q
  414. ( ( p q ) r ) ( p ( q r ) ) ((p\land q)\to r)\vdash(p\to(q\to r))
  415. p p
  416. q q
  417. r r
  418. q q
  419. r r
  420. p p
  421. ( p ( q r ) ) ( ( p q ) r ) (p\to(q\to r))\vdash((p\land q)\to r)
  422. p p
  423. q q
  424. r r
  425. p p
  426. q q
  427. r r
  428. p ( p p ) p\vdash(p\lor p)
  429. p p
  430. p p
  431. p p
  432. p ( p p ) p\vdash(p\land p)
  433. p p
  434. p p
  435. p p
  436. ( p ¬ p ) \vdash(p\lor\neg p)
  437. p p
  438. p p
  439. ¬ ( p ¬ p ) \vdash\neg(p\land\neg p)
  440. p p
  441. p p
  442. A A A→A
  443. A A
  444. A A A\lor A
  445. ( A A ) A (A\lor A)\land A
  446. A A
  447. A A A\vdash A
  448. A A \vdash A\to A
  449. A A A\vdash A
  450. A A
  451. A A
  452. A A \vdash A\to A
  453. A A
  454. A A
  455. A A
  456. A A
  457. A A
  458. A A
  459. A A
  460. A A
  461. P P
  462. A ( P ) = t r u e A(P)=true
  463. A A
  464. ¬ φ ¬φ
  465. A A
  466. φ φ
  467. A A
  468. ( φ ψ ) (φ∧ψ)
  469. A A
  470. φ φ
  471. ψ ψ
  472. A A
  473. ( φ ψ ) (φ∨ψ)
  474. A A
  475. φ φ
  476. ψ ψ
  477. A A
  478. ( φ ψ ) (φ→ψ)
  479. A A
  480. φ φ
  481. ψ ψ
  482. A A
  483. ( φ ψ ) (φ↔ψ)
  484. A A
  485. φ φ
  486. ψ ψ
  487. φ φ
  488. S S
  489. S S
  490. φ φ
  491. S S
  492. φ φ
  493. S S
  494. φ φ
  495. φ φ
  496. S S
  497. S S
  498. φ φ
  499. S S
  500. φ φ
  501. S S
  502. φ φ
  503. S S
  504. φ φ
  505. G G
  506. A , B A,B
  507. C C
  508. G G
  509. A A
  510. G G
  511. A A
  512. G G
  513. A A
  514. G G
  515. A A
  516. ( A ) ( G ) (A)(G)
  517. G G
  518. A A
  519. G G
  520. A A
  521. G G
  522. A A
  523. G G
  524. A A
  525. G G
  526. G G
  527. G G
  528. A A
  529. A A
  530. B B
  531. A A
  532. A A
  533. A A
  534. B B
  535. A A
  536. B B
  537. A A
  538. G G
  539. G G
  540. G G
  541. A A
  542. A A
  543. A A
  544. B B
  545. G G
  546. A A
  547. B B
  548. A A
  549. B B
  550. G G
  551. G G
  552. A A
  553. G G
  554. A A
  555. G G
  556. A A
  557. G G
  558. A A
  559. P P
  560. S S
  561. P P
  562. S S
  563. S S
  564. 𝒫 \mathcal{P}
  565. 𝒫 \mathcal{P}
  566. 𝒫 \mathcal{P}
  567. n n
  568. 2 n 2^{n}
  569. a a
  570. 2 1 = 2 2^{1}=2
  571. a a
  572. a a
  573. a a
  574. b b
  575. 2 2 = 4 2^{2}=4
  576. a a
  577. b b
  578. a a
  579. b b
  580. 𝒫 \mathcal{P}
  581. 0 \aleph_{0}
  582. 2 0 = 𝔠 2^{\aleph_{0}}=\mathfrak{c}
  583. 𝒫 \mathcal{P}
  584. φ φ
  585. ψ ψ
  586. 𝒫 \mathcal{P}
  587. \mathcal{I}
  588. 𝒫 \mathcal{P}
  589. \mathcal{I}
  590. \mathcal{I}
  591. φ φ
  592. \mathcal{I}
  593. φ φ
  594. \mathcal{I}
  595. ϕ \models\phi
  596. φ φ
  597. ψ ψ
  598. φ φ
  599. φ φ
  600. ψ ψ
  601. φ φ
  602. ¬ ϕ \neg\phi
  603. φ φ
  604. ¬ ϕ \neg\phi
  605. φ φ
  606. ( ϕ ψ ) (\phi\to\psi)
  607. ψ ψ
  608. P ϕ \models_{\mathrm{P}}\phi
  609. P ( ϕ ψ ) \models_{\mathrm{P}}(\phi\to\psi)
  610. P ψ \models_{\mathrm{P}}\psi
  611. ¬ ϕ \neg\phi
  612. \mathcal{I}
  613. φ φ
  614. \mathcal{I}
  615. ( ϕ ψ ) (\phi\to\psi)
  616. \mathcal{I}
  617. φ φ
  618. \mathcal{I}
  619. ψ ψ
  620. \mathcal{I}
  621. ψ ψ
  622. φ φ
  623. ( ϕ ψ ) (\phi\to\psi)
  624. ϕ P ψ \phi\models_{\mathrm{P}}\psi
  625. P ( ϕ ψ ) \models_{\mathrm{P}}(\phi\to\psi)
  626. φ φ
  627. χ χ
  628. ψ ψ
  629. ϕ ( χ ϕ ) \phi\to(\chi\to\phi)
  630. χ χ
  631. ( ϕ ( χ ψ ) ) ( ( ϕ χ ) ( ϕ ψ ) ) (\phi\to(\chi\to\psi))\to((\phi\to\chi)\to(\phi\to\psi))
  632. φ φ
  633. ϕ χ ϕ \phi\land\chi\to\phi
  634. ϕ χ χ \phi\land\chi\to\chi
  635. ϕ ( χ ( ϕ χ ) ) \phi\to(\chi\to(\phi\land\chi))
  636. ϕ ϕ χ \phi\to\phi\lor\chi
  637. χ ϕ χ \chi\to\phi\lor\chi
  638. ( ϕ ψ ) ( ( χ ψ ) ( ϕ χ ψ ) ) (\phi\to\psi)\to((\chi\to\psi)\to(\phi\lor\chi\to\psi))
  639. ( ϕ χ ) ( ( ϕ ¬ χ ) ¬ ϕ ) (\phi\to\chi)\to((\phi\to\neg\chi)\to\neg\phi)
  640. ϕ ( ¬ ϕ χ ) \phi\to(\neg\phi\to\chi)
  641. ϕ ¬ ϕ \phi\lor\neg\phi
  642. ( ϕ χ ) ( ϕ χ ) (\phi\leftrightarrow\chi)\to(\phi\to\chi)
  643. ( ϕ χ ) ( χ ϕ ) (\phi\leftrightarrow\chi)\to(\chi\to\phi)
  644. ( ϕ χ ) ( ( χ ϕ ) ( ϕ χ ) ) (\phi\to\chi)\to((\chi\to\phi)\to(\phi\leftrightarrow\chi))
  645. ϕ , ϕ χ χ \phi,\ \phi\to\chi\vdash\chi
  646. ϕ 1 , ϕ 2 , , ϕ n , χ ψ \phi_{1},\ \phi_{2},\ ...,\ \phi_{n},\ \chi\vdash\psi
  647. ϕ 1 , ϕ 2 , , ϕ n χ ψ \phi_{1},\ \phi_{2},\ ...,\ \phi_{n}\vdash\chi\to\psi
  648. ϕ 1 , ϕ 2 , , ϕ n χ ψ \phi_{1},\ \phi_{2},\ ...,\ \phi_{n}\vdash\chi\to\psi
  649. ϕ 1 , ϕ 2 , , ϕ n , χ ψ \phi_{1},\ \phi_{2},\ ...,\ \phi_{n},\ \chi\vdash\psi
  650. ϕ 1 , , ϕ n χ ψ \phi_{1},\ ...,\ \phi_{n}\vdash\chi\to\psi
  651. ϕ 1 , , ϕ n , χ χ ψ \phi_{1},\ ...,\ \phi_{n},\ \chi\vdash\chi\to\psi
  652. ϕ 1 , , ϕ n , χ χ \phi_{1},\ ...,\ \phi_{n},\ \chi\vdash\chi
  653. ϕ 1 , , ϕ n , χ ψ \phi_{1},\ ...,\ \phi_{n},\ \chi\vdash\psi
  654. ϕ χ ϕ \vdash\phi\wedge\chi\to\phi
  655. ϕ χ ϕ \phi\wedge\chi\vdash\phi
  656. A A A\to A
  657. ( A ( ( B A ) A ) ) ( ( A ( B A ) ) ( A A ) ) (A\to((B\to A)\to A))\to((A\to(B\to A))\to(A\to A))
  658. ϕ = A , χ = B A , ψ = A \phi=A,\chi=B\to A,\psi=A
  659. A ( ( B A ) A ) A\to((B\to A)\to A)
  660. ϕ = A , χ = B A \phi=A,\chi=B\to A
  661. ( A ( B A ) ) ( A A ) (A\to(B\to A))\to(A\to A)
  662. A ( B A ) A\to(B\to A)
  663. ϕ = A , χ = B \phi=A,\chi=B
  664. A A A\to A
  665. ϕ \phi
  666. ϕ = 1 \phi=1
  667. x = y x=y
  668. ( x y ) ( y x ) (x\to y)\land(y\to x)
  669. x y x\equiv y
  670. x = y x=y
  671. ( x y ) ( ¬ x ¬ y ) (x\land y)\lor(\neg x\land\neg y)
  672. x y x\leq y
  673. x = y x=y
  674. x y x\leq y
  675. y x y\leq x
  676. x y x\leq y
  677. x y = x x\land y=x
  678. x y = y x\lor y=y
  679. \vdash
  680. ϕ 1 , ϕ 2 , , ϕ n ψ \phi_{1},\ \phi_{2},\ \dots,\ \phi_{n}\vdash\psi
  681. ϕ 1 ϕ 2 ϕ n ψ \phi_{1}\ \land\ \phi_{2}\ \land\ \dots\ \land\ \phi_{n}\ \ \leq\ \ \psi
  682. x y x\leq y
  683. x y x\ \vdash\ y
  684. x y x\to y
  685. x y x\leq y
  686. x y x\ \vdash\ y
  687. p p
  688. p p
  689. p p
  690. p p

Prosecutor's_fallacy.html

  1. 1 - ( 1 - 1 10000 ) 20000 86 % 1-\left(1-\frac{1}{10000}\right)^{20000}\approx 86\%
  2. P ( I | E ) = P ( E | I ) P ( I ) P ( E ) P(I|E)=P(E|I)\cdot\frac{P(I)}{P(E)}
  3. P ( E ) = P ( E | I ) P ( I ) + P ( E | I ) [ 1 - P ( I ) ] P(E)=P(E|I)\cdot P(I)+P(E|\sim I)\cdot[1-P(I)]
  4. Odds ( I | E ) Odds ( I ) P ( E | I ) \operatorname{Odds}(I|E)\geq\operatorname{Odds}(I)\cdot P(E|I)
  5. P ( I | E ) P ( E | I ) Odds ( I ) P(I|E)\approx P(E|I)\cdot\operatorname{Odds}(I)

Protactinium.html

  1. Th 90 232 + 0 1 n 90 233 Th β - 22.3 min 91 233 Pa β - 26.967 d 92 233 U \mathrm{{}^{232}_{\ 90}Th\ +\ ^{1}_{0}n\ \longrightarrow\ ^{233}_{\ 90}Th\ % \xrightarrow[22.3\ min]{\beta^{-}}\ ^{233}_{\ 91}Pa\ \xrightarrow[26.967\ d]{% \beta^{-}}\ ^{233}_{\ 92}U}
  2. × 10 - 6 \times 10^{-}6
  3. 3 ¯ \overline{3}
  4. 3 ¯ \overline{3}
  5. 3 ¯ \overline{3}
  6. 3 ¯ \overline{3}
  7. × 10 4 \times 10^{4}
  8. × 10 4 \times 10^{4}
  9. × 10 1 0 \times 10^{1}0
  10. × 10 1 3 \times 10^{1}3
  11. × 10 7 \times 10^{7}
  12. × 10 8 \times 10^{8}
  13. × 10 8 \times 10^{8}
  14. × 10 8 \times 10^{8}

Protein_primary_structure.html

  1. C α \mathrm{C^{\alpha}}
  2. - C ( = O ) - CH 3 \mathrm{-C(=O)-CH_{3}}
  3. - C ( = O ) H \mathrm{-C(=O)H}
  4. - C ( = O ) - ( CH 2 ) 12 - CH 3 \mathrm{-C(=O)-\left(CH_{2}\right)_{12}-CH_{3}}
  5. O η \mathrm{O^{\eta}}
  6. - C ( = O ) - ( CH 2 ) 14 - CH 3 \mathrm{-C(=O)-\left(CH_{2}\right)_{14}-CH_{3}}
  7. S γ \mathrm{S^{\gamma}}
  8. \rightarrow
  9. \rightarrow

Protein_secondary_structure.html

  1. ± q 1 0.42 e \pm q_{1}\equiv 0.42e
  2. ± q 2 0.20 e \pm q_{2}\equiv 0.20e
  3. E = q 1 q 2 [ 1 r O N + 1 r C H - 1 r O H - 1 r C N ] 332 kcal / mol . E=q_{1}q_{2}\left[\frac{1}{r_{ON}}+\frac{1}{r_{CH}}-\frac{1}{r_{OH}}-\frac{1}{% r_{CN}}\right]\cdot 332\ \mathrm{kcal/mol}.
  4. E E
  5. 3 10 3_{10}

Proton_decay.html

  1. q q q l Λ 2 \frac{qqql}{\Lambda^{2}}
  2. d c u c u c e c Λ 2 \frac{d^{c}u^{c}u^{c}e^{c}}{\Lambda^{2}}
  3. e c ¯ u c ¯ q q Λ 2 \frac{\overline{e^{c}}\overline{u^{c}}qq}{\Lambda^{2}}
  4. d c ¯ u c ¯ q l Λ 2 \frac{\overline{d^{c}}\overline{u^{c}}ql}{\Lambda^{2}}
  5. Λ \Lambda
  6. 1 Λ G U T 2 \frac{1}{\Lambda_{GUT}^{2}}
  7. T ¯ \overline{T}
  8. 3 ¯ \overline{3}
  9. 1 M M S U S Y \frac{1}{MM_{SUSY}}
  10. 1 M S U S Y 2 \frac{1}{M_{SUSY}^{2}}

Prudence.html

  1. u ( x ) u(x)
  2. u ( x ) u(x)
  3. u ′′′ ( x ) > 0 u^{{}^{\prime\prime\prime}}\left(x\right)>0
  4. - u ′′′ ( x ) u ′′ ( x ) -\frac{u^{{}^{\prime\prime\prime}}\left(x\right)}{u^{{}^{\prime\prime}}\left(x% \right)}

Pseudocode.html

  1. k S x k \sum_{k\in S}x_{k}

Pseudometric_space.html

  1. ( X , d ) (X,d)
  2. X X
  3. d : X × X 0 d\colon X\times X\longrightarrow\mathbb{R}_{\geq 0}
  4. x , y , z X x,y,z\in X
  5. d ( x , x ) = 0 d(x,x)=0
  6. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)
  7. d ( x , z ) d ( x , y ) + d ( y , z ) d(x,z)\leq d(x,y)+d(y,z)
  8. d ( x , y ) = 0 d(x,y)=0
  9. x y x\neq y
  10. ( X ) \mathcal{F}(X)
  11. f : X f\colon X\to\mathbb{R}
  12. x 0 X x_{0}\in X
  13. d ( f , g ) = | f ( x 0 ) - g ( x 0 ) | d(f,g)=|f(x_{0})-g(x_{0})|
  14. f , g ( X ) f,g\in\mathcal{F}(X)
  15. V V
  16. p p
  17. V V
  18. d ( x , y ) = p ( x - y ) . d(x,y)=p(x-y).
  19. ( Ω , 𝒜 , μ ) (\Omega,\mathcal{A},\mu)
  20. d ( A , B ) := μ ( A Δ B ) d(A,B):=\mu(A\Delta B)
  21. A , B 𝒜 A,B\in\mathcal{A}
  22. f : X 1 X 2 f:X_{1}\rightarrow X_{2}
  23. d 1 ( x , y ) := d 2 ( f ( x ) , f ( y ) ) d_{1}(x,y):=d_{2}(f(x),f(y))
  24. B r ( p ) = { x X d ( p , x ) < r } , B_{r}(p)=\{x\in X\mid d(p,x)<r\},
  25. x y x\sim y
  26. d ( x , y ) = 0 d(x,y)=0
  27. X * = X / X^{*}=X/{\sim}
  28. d * ( [ x ] , [ y ] ) = d ( x , y ) d^{*}([x],[y])=d(x,y)
  29. d * d^{*}
  30. X * X^{*}
  31. ( X * , d * ) (X^{*},d^{*})
  32. A X A\subset X
  33. ( X , d ) (X,d)
  34. π ( A ) = [ A ] \pi(A)=[A]
  35. ( X * , d * ) (X^{*},d^{*})

Pseudorandom_number_generator.html

  1. × 10 6 001 \times 10^{6}001
  2. P P
  3. ( , 𝔅 ) \left(\mathbb{R},\mathfrak{B}\right)
  4. 𝔅 \mathfrak{B}
  5. 𝔉 \mathfrak{F}
  6. 𝔉 𝔅 \mathfrak{F}\subseteq\mathfrak{B}
  7. 𝔉 = { ( - , t ] : t } \mathfrak{F}=\left\{\left(-\infty,t\right]:t\in\mathbb{R}\right\}
  8. 𝔉 \mathfrak{F}
  9. 𝔅 \mathfrak{B}
  10. { ( - , t ] : t } \left\{\left(-\infty,t\right]:t\in\mathbb{R}\right\}
  11. A A\subseteq\mathbb{R}
  12. A A
  13. P P
  14. P P
  15. ( 0 , 1 ] \left(0,1\right]
  16. A A
  17. ( 0 , 1 ] \left(0,1\right]
  18. A A
  19. P P
  20. f : 1 f:\mathbb{N}_{1}\rightarrow\mathbb{R}
  21. 1 = { 1 , 2 , 3 , } \mathbb{N}_{1}=\left\{1,2,3,\dots\right\}
  22. P P
  23. 𝔉 \mathfrak{F}
  24. A A
  25. f ( 1 ) A f\left(\mathbb{N}_{1}\right)\subseteq A
  26. E 𝔉 0 < ε N 1 N n 1 , | # { i { 1 , 2 , , n } : f ( i ) E } n - P ( E ) | < ε \forall E\in\mathfrak{F}\quad\forall 0<\varepsilon\in\mathbb{R}\quad\exists N% \in\mathbb{N}_{1}\quad\forall N\leq n\in\mathbb{N}_{1},\quad\left|\frac{\#% \left\{i\in\left\{1,2,\dots,n\right\}:f(i)\in E\right\}}{n}-P(E)\right|<\varepsilon
  27. # S \#S
  28. S S
  29. f f
  30. ( 0 , 1 ) \left(0,1\right)
  31. F F
  32. P P
  33. F * f F^{*}\circ f
  34. P P
  35. F * : ( 0 , 1 ) F^{*}:\left(0,1\right)\rightarrow\mathbb{R}
  36. P P
  37. F * ( x ) := inf { t : x F ( t ) } F^{*}(x):=\inf\left\{t\in\mathbb{R}:x\leq F(t)\right\}
  38. F ( b ) F(b)
  39. f ( b ) f(b)
  40. F ( b ) = - b f ( b ) d b F(b)=\int_{-\infty}^{b}f(b^{\prime})db^{\prime}
  41. 0 = F ( - ) F ( b ) F ( ) = 1 0=F(-\infty)\leq F(b)\leq F(\infty)=1
  42. F ( b ) = c F(b)=c
  43. b = F - 1 ( c ) b=F^{-1}(c)
  44. f ( b ) f(b)
  45. erf - 1 ( x ) \operatorname{erf}^{-1}(x)
  46. x x
  47. erf - 1 ( x ) \operatorname{erf}^{-1}(x)

Pseudosphere.html

  1. t ( t - tanh t , sech t ) , 0 t < . t\mapsto\left(t-\tanh{t},\operatorname{sech}\,{t}\right),\quad\quad 0\leq t<\infty.
  2. ( x , y ) ( v ( arcosh y ) cos x , v ( arcosh y ) sin x , u ( arcosh y ) ) (x,y)\mapsto(v(\operatorname{arcosh}y)\cos x,v(\operatorname{arcosh}y)\sin x,u% (\operatorname{arcosh}y))
  3. t ( u ( t ) , v ( t ) ) t\mapsto(u(t),v(t))

PSPACE-complete.html

  1. x 1 x 2 x 3 x 4 : ( x 1 ¬ x 3 x 4 ) and ( ¬ x 2 x 3 ¬ x 4 ) \exists x_{1}\,\exists x_{2}\,\exists x_{3}\,\exists x_{4}:(x_{1}\neg x_{3}x_{% 4})\and(\neg x_{2}x_{3}\neg x_{4})
  2. x 1 x 2 x 3 x 4 : ( x 1 ¬ x 3 x 4 ) and ( ¬ x 2 x 3 ¬ x 4 ) \exists x_{1}\,\forall x_{2}\,\exists x_{3}\,\forall x_{4}:(x_{1}\neg x_{3}x_{% 4})\and(\neg x_{2}x_{3}\neg x_{4})

PSPACE.html

  1. 𝐏𝐒𝐏𝐀𝐂𝐄 = k 𝐒𝐏𝐀𝐂𝐄 ( n k ) . \mathbf{PSPACE}=\bigcup_{k\in\mathbb{N}}\mathbf{SPACE}(n^{k}).
  2. 𝐍𝐋 𝐏 𝐍𝐏 𝐏𝐇 𝐏𝐒𝐏𝐀𝐂𝐄 \mathbf{NL}\subseteq\mathbf{P}\subseteq\mathbf{NP}\subseteq\mathbf{PH}% \subseteq\mathbf{PSPACE}
  3. 𝐏𝐒𝐏𝐀𝐂𝐄 𝐄𝐗𝐏𝐓𝐈𝐌𝐄 𝐄𝐗𝐏𝐒𝐏𝐀𝐂𝐄 \mathbf{PSPACE}\subseteq\mathbf{EXPTIME}\subseteq\mathbf{EXPSPACE}
  4. 𝐍𝐋 𝐏𝐒𝐏𝐀𝐂𝐄 𝐄𝐗𝐏𝐒𝐏𝐀𝐂𝐄 \mathbf{NL}\subsetneq\mathbf{PSPACE}\subsetneq\mathbf{EXPSPACE}
  5. 𝐏 𝐄𝐗𝐏𝐓𝐈𝐌𝐄 \mathbf{P}\subsetneq\mathbf{EXPTIME}
  6. p \leq_{p}
  7. p \leq_{p}

Public_capital.html

  1. Y t = A t * ( N t , K t , G t ) \qquad\qquad Y_{t}=A_{t}*(N_{t},K_{t},G_{t})

Puff_pastry.html

  1. l = ( f + 1 ) n l=(f+1)^{n}
  2. l l
  3. f f
  4. n n
  5. ( 2 + 1 ) 4 = 81 (2+1)^{4}=81

Pulley.html

  1. n T - W = 0. nT-W=0.
  2. M A = W T = n . MA=\frac{W}{T}=n.

Pulse-amplitude_modulation.html

  1. 2 2 2^{2}
  2. 2 3 2^{3}
  3. 2 4 2^{4}

Pulsed_plasma_thruster.html

  1. Δ v = v e ln m 0 m 1 \Delta v=v\text{e}\ln\frac{m_{0}}{m_{1}}
  2. Δ v \Delta v
  3. v e v\text{e}
  4. v e = I sp g 0 v\text{e}=I\text{sp}\cdot g_{0}
  5. I sp I\text{sp}
  6. g 0 g_{0}
  7. ln \ln
  8. m 0 m_{0}
  9. m 1 m_{1}

Pump.html

  1. P = Δ p Q η P=\frac{\Delta pQ}{\eta}
  2. Δ P = ( v 2 2 - v 1 2 ) 2 + Δ z g + Δ p static ρ \Delta P={(v_{2}^{2}-v_{1}^{2})\over 2}+\Delta zg+{\Delta p_{\mathrm{static}}% \over\rho}

Purchasing_power_parity.html

  1. PPPrate X , i = PPPrate X , b GDPdef X , i GDPdef X , b PPPrate U , b GDPdef U , i GDPdef U , b \textrm{PPPrate}_{X,i}=\frac{\textrm{PPPrate}_{X,b}\cdot\frac{\textrm{GDPdef}_% {X,i}}{\textrm{GDPdef}_{X,b}}}{\textrm{PPPrate}_{U,b}\cdot\frac{\textrm{GDPdef% }_{U,i}}{\textrm{GDPdef}_{U,b}}}

Pushdown_automaton.html

  1. Γ * \Gamma^{*}
  2. Γ \Gamma
  3. ε \varepsilon
  4. M = ( Q , Σ , Γ , δ , q 0 , Z , F ) M=(Q,\ \Sigma,\ \Gamma,\ \delta,\ q_{0},\ Z,\ F)
  5. Q \,Q
  6. Σ \,\Sigma
  7. Γ \,\Gamma
  8. δ \,\delta
  9. Q × ( Σ { ε } ) × Γ × Q × Γ * Q\times(\Sigma\cup\{\varepsilon\})\times\Gamma\times Q\times\Gamma^{*}
  10. q 0 Q \,q_{0}\in\,Q
  11. Z Γ \ Z\in\,\Gamma
  12. F Q F\subseteq Q
  13. ( p , a , A , q , α ) δ (p,a,A,q,\alpha)\in\delta
  14. M M
  15. M M
  16. p Q p\in Q
  17. a Σ { ε } a\in\Sigma\cup\{\varepsilon\}
  18. A Γ A\in\Gamma
  19. a a
  20. q q
  21. A A
  22. α Γ * \alpha\in\Gamma^{*}
  23. ( Σ { ε } ) (\Sigma\cup\{\varepsilon\})
  24. δ \,\delta
  25. Q × ( Σ { ε } ) × Γ Q\times(\Sigma\cup\{\varepsilon\})\times\Gamma
  26. Q × Γ * Q\times\Gamma^{*}
  27. δ ( p , a , A ) \delta(p,a,A)
  28. p p
  29. A A
  30. a a
  31. ( q , α ) δ ( p , a , A ) (q,\alpha)\in\delta(p,a,A)
  32. ( p , a , A , q , α ) δ (p,a,A,q,\alpha)\in\delta
  33. ( p , w , β ) Q × Σ * × Γ * (p,w,\beta)\in Q\times\Sigma^{*}\times\Gamma^{*}
  34. M M
  35. δ \delta
  36. M \vdash_{M}
  37. M M
  38. ( p , a , A , q , α ) δ (p,a,A,q,\alpha)\in\delta
  39. ( p , a x , A γ ) M ( q , x , α γ ) (p,ax,A\gamma)\vdash_{M}(q,x,\alpha\gamma)
  40. x Σ * x\in\Sigma^{*}
  41. γ Γ * \gamma\in\Gamma^{*}
  42. ( p , w , β ) (p,w,\beta)
  43. q 0 q_{0}
  44. Z Z
  45. w w
  46. ( q 0 , w , Z ) (q_{0},w,Z)
  47. F F
  48. ε \varepsilon
  49. L ( M ) = { w Σ * | ( q 0 , w , Z ) M * ( f , ε , γ ) L(M)=\{w\in\Sigma^{*}|(q_{0},w,Z)\vdash_{M}^{*}(f,\varepsilon,\gamma)
  50. f F f\in F
  51. γ Γ * } \gamma\in\Gamma^{*}\}
  52. N ( M ) = { w Σ * | ( q 0 , w , Z ) M * ( q , ε , ε ) N(M)=\{w\in\Sigma^{*}|(q_{0},w,Z)\vdash_{M}^{*}(q,\varepsilon,\varepsilon)
  53. q Q } q\in Q\}
  54. M * \vdash_{M}^{*}
  55. M \vdash_{M}
  56. M M
  57. M M^{\prime}
  58. L ( M ) = N ( M ) L(M)=N(M^{\prime})
  59. M M
  60. M M^{\prime}
  61. N ( M ) = L ( M ) N(M)=L(M^{\prime})
  62. { 0 n 1 n n 0 } \{0^{n}1^{n}\mid n\geq 0\}
  63. M = ( Q , Σ , Γ , δ , p , Z , F ) M=(Q,\ \Sigma,\ \Gamma,\ \delta,\ p,\ Z,\ F)
  64. Q = { p , q , r } Q=\{p,q,r\}
  65. Σ = { 0 , 1 } \Sigma=\{0,1\}
  66. Γ = { A , Z } \Gamma=\{A,Z\}
  67. q 0 = p q_{0}=p
  68. Z Z
  69. F = { r } F=\{r\}
  70. δ \delta
  71. ( p , 0 , Z , p , A Z ) (p,0,Z,p,AZ)
  72. ( p , 0 , A , p , A A ) (p,0,A,p,AA)
  73. ( p , ϵ , Z , q , Z ) (p,\epsilon,Z,q,Z)
  74. ( p , ϵ , A , q , A ) (p,\epsilon,A,q,A)
  75. ( q , 1 , A , q , ϵ ) (q,1,A,q,\epsilon)
  76. ( q , ϵ , Z , r , Z ) (q,\epsilon,Z,r,Z)
  77. p p
  78. 0
  79. A A
  80. A A
  81. A A
  82. A A
  83. A A AA
  84. A A
  85. Z Z
  86. p p
  87. q q
  88. q q
  89. 1 1
  90. A A
  91. q q
  92. r r
  93. Z Z
  94. ( p , a , A , q , α ) (p,a,A,q,\alpha)
  95. p p
  96. q q
  97. a ; A / α a;A/\alpha
  98. a a
  99. A A
  100. α \alpha
  101. M M
  102. \vdash
  103. p p
  104. q q
  105. ( p , 0011 , Z ) ( q , 0011 , Z ) ( r , 0011 , Z ) (p,0011,Z)\vdash(q,0011,Z)\vdash(r,0011,Z)
  106. ( p , 0011 , Z ) ( p , 011 , A Z ) ( q , 011 , A Z ) (p,0011,Z)\vdash(p,011,AZ)\vdash(q,011,AZ)
  107. ( p , 0011 , Z ) ( p , 011 , A Z ) ( p , 11 , A A Z ) ( q , 11 , A A Z ) (p,0011,Z)\vdash(p,011,AZ)\vdash(p,11,AAZ)\vdash(q,11,AAZ)
  108. ( q , 1 , A Z ) ( q , ϵ , Z ) \vdash(q,1,AZ)\vdash(q,\epsilon,Z)
  109. ( r , ϵ , Z ) \vdash(r,\epsilon,Z)
  110. ( p , 00111 , Z ) ( q , 00111 , Z ) ( r , 00111 , Z ) (p,00111,Z)\vdash(q,00111,Z)\vdash(r,00111,Z)
  111. ( p , 00111 , Z ) ( p , 0111 , A Z ) ( q , 0111 , A Z ) (p,00111,Z)\vdash(p,0111,AZ)\vdash(q,0111,AZ)
  112. ( p , 00111 , Z ) ( p , 0111 , A Z ) ( p , 111 , A A Z ) ( q , 111 , A A Z ) (p,00111,Z)\vdash(p,0111,AZ)\vdash(p,111,AAZ)\vdash(q,111,AAZ)
  113. ( q , 11 , A Z ) ( q , 1 , Z ) \vdash(q,11,AZ)\vdash(q,1,Z)
  114. ( r , 1 , Z ) \vdash(r,1,Z)
  115. ( 1 , ε , A , 1 , α ) (1,\varepsilon,A,1,\alpha)
  116. A α A\to\alpha
  117. ( 1 , a , a , 1 , ε ) (1,a,a,1,\varepsilon)
  118. a a
  119. 1 1
  120. M M
  121. G G
  122. N ( M ) = L ( G ) N(M)=L(G)
  123. M = ( Q , Σ , Γ , δ , q 0 , F ) M=(Q,\ \Sigma,\ \Gamma,\ \delta,\ q_{0},\ F)
  124. Σ \Sigma\,
  125. Γ \Gamma\,
  126. δ \,\delta
  127. Q × Σ ϵ × Γ * P ( Q × Γ * ) Q\times\Sigma_{\epsilon}\times\Gamma^{*}\longrightarrow P(Q\times\Gamma^{*})
  128. δ \delta
  129. \longrightarrow
  130. q 1 , q 2 Q q_{1},q_{2}\in Q
  131. w Σ ϵ w\in\Sigma_{\epsilon}
  132. x 1 , x 2 , , x m Γ * x_{1},x_{2},\ldots,x_{m}\in\Gamma^{*}
  133. m 0 m\geq 0
  134. y 1 , y 2 , , y n Γ * y_{1},y_{2},\ldots,y_{n}\in\Gamma^{*}
  135. n 0 n\geq 0
  136. δ \delta^{{}^{\prime}}
  137. \longrightarrow
  138. ϵ \epsilon
  139. δ \delta^{{}^{\prime}}
  140. ϵ \epsilon
  141. \longrightarrow
  142. ϵ \epsilon
  143. \vdots
  144. δ \delta^{{}^{\prime}}
  145. ϵ \epsilon
  146. \longrightarrow
  147. ϵ \epsilon
  148. δ \delta^{{}^{\prime}}
  149. ϵ \epsilon
  150. ϵ \epsilon
  151. \longrightarrow
  152. δ \delta^{{}^{\prime}}
  153. ϵ \epsilon
  154. ϵ \epsilon
  155. \longrightarrow
  156. \vdots
  157. δ \delta^{{}^{\prime}}
  158. ϵ \epsilon
  159. ϵ \epsilon
  160. \longrightarrow

Pyridine.html

  1. log 10 p = A - B C + T \log_{10}p=A-\frac{B}{C+T}

Pyrite.html

  1. 3 ¯ \overline{3}
  2. a a

Pythagoras.html

  1. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}

Pythagorean_triple.html

  1. a = m 2 - n 2 , b = 2 m n , c = m 2 + n 2 a=m^{2}-n^{2},\ \,b=2mn,\ \,c=m^{2}+n^{2}
  2. a = k ( m 2 - n 2 ) , b = k ( 2 m n ) , c = k ( m 2 + n 2 ) a=k\cdot(m^{2}-n^{2}),\ \,b=k\cdot(2mn),\ \,c=k\cdot(m^{2}+n^{2})
  3. a 2 + b 2 = ( m 2 - n 2 ) 2 + ( 2 m n ) 2 = ( m 2 + n 2 ) 2 = c 2 . a^{2}+b^{2}=(m^{2}-n^{2})^{2}+(2mn)^{2}=(m^{2}+n^{2})^{2}=c^{2}.
  4. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  5. c 2 - a 2 = b 2 c^{2}-a^{2}=b^{2}
  6. ( c - a ) ( c + a ) = b 2 (c-a)(c+a)=b^{2}
  7. ( c + a ) b = b ( c - a ) \tfrac{(c+a)}{b}=\tfrac{b}{(c-a)}
  8. ( c + a ) b \tfrac{(c+a)}{b}
  9. m n \tfrac{m}{n}
  10. ( c - a ) b \tfrac{(c-a)}{b}
  11. b ( c - a ) \tfrac{b}{(c-a)}
  12. ( c + a ) b \tfrac{(c+a)}{b}
  13. n m \tfrac{n}{m}
  14. c b + a b = m n , c b - a b = n m \frac{c}{b}+\frac{a}{b}=\frac{m}{n},\quad\quad\frac{c}{b}-\frac{a}{b}=\frac{n}% {m}
  15. c b \tfrac{c}{b}
  16. a b \tfrac{a}{b}
  17. c b = 1 2 ( m n + n m ) = m 2 + n 2 2 m n , a b = 1 2 ( m n - n m ) = m 2 - n 2 2 m n . \frac{c}{b}=\frac{1}{2}\left(\frac{m}{n}+\frac{n}{m}\right)=\frac{m^{2}+n^{2}}% {2mn},\quad\quad\frac{a}{b}=\frac{1}{2}\left(\frac{m}{n}-\frac{n}{m}\right)=% \frac{m^{2}-n^{2}}{2mn}.
  18. c b \tfrac{c}{b}
  19. a b \tfrac{a}{b}
  20. m n \tfrac{m}{n}
  21. m 2 + n 2 2 m n \tfrac{m^{2}+n^{2}}{2mn}
  22. c b \tfrac{c}{b}
  23. a = m 2 - n 2 , b = 2 m n , c = m 2 + n 2 a=m^{2}-n^{2},\ \,b=2mn,\ \,c=m^{2}+n^{2}
  24. m 2 - n 2 , 2 m n m^{2}-n^{2},\,2mn
  25. m 2 + n 2 m^{2}+n^{2}
  26. m 2 - n 2 m^{2}-n^{2}
  27. m 2 + n 2 m^{2}+n^{2}
  28. θ \theta
  29. tan θ = 2 m n m 2 - n 2 \tan{\theta}=\tfrac{2mn}{m^{2}-n^{2}}
  30. tan θ 2 = n m \tan{\tfrac{\theta}{2}}=\tfrac{n}{m}
  31. K s 2 = n ( m - n ) m ( m + n ) = 1 - c s . \tfrac{K}{s^{2}}=\tfrac{n(m-n)}{m(m+n)}=1-\tfrac{c}{s}.
  32. a = 2 m n , b = m 2 - n 2 , c = m 2 + n 2 a=2mn,\quad b=m^{2}-n^{2},\quad c=m^{2}+n^{2}
  33. ( a c ) 2 + ( b c ) 2 = 1. \left(\frac{a}{c}\right)^{2}+\left(\frac{b}{c}\right)^{2}=1.
  34. x = a c , y = b c x=\frac{a}{c},\quad y=\frac{b}{c}
  35. x = a c , y = b c x=\frac{a}{c},\quad y=\frac{b}{c}
  36. ( a c ) 2 + ( b c ) 2 = 1 a 2 + b 2 = c 2 , \left(\frac{a}{c}\right)^{2}+\left(\frac{b}{c}\right)^{2}=1\implies a^{2}+b^{2% }=c^{2},
  37. P = ( m n , 0 ) . P^{\prime}=\left(\frac{m}{n},0\right).
  38. P = ( 2 ( m n ) ( m n ) 2 + 1 , ( m n ) 2 - 1 ( m n ) 2 + 1 ) = ( 2 m n m 2 + n 2 , m 2 - n 2 m 2 + n 2 ) . P=\left(\frac{2\left(\frac{m}{n}\right)}{\left(\frac{m}{n}\right)^{2}+1},\frac% {\left(\frac{m}{n}\right)^{2}-1}{\left(\frac{m}{n}\right)^{2}+1}\right)=\left(% \frac{2mn}{m^{2}+n^{2}},\frac{m^{2}-n^{2}}{m^{2}+n^{2}}\right).
  39. ( x 1 - y , 0 ) \left(\frac{x}{1-y},0\right)
  40. ( a - 1 ) ( b - 1 ) - gcd ( a , b ) + 1 2 ; \tfrac{(a-1)(b-1)-\gcd{(a,b)}+1}{2};
  41. ( a - 1 ) ( b - 1 ) 2 . \tfrac{(a-1)(b-1)}{2}.
  42. a b 2 \tfrac{ab}{2}
  43. X = [ c + b a a c - b ] . X=\begin{bmatrix}c+b&a\\ a&c-b\end{bmatrix}.
  44. det X = c 2 - a 2 - b 2 \det X=c^{2}-a^{2}-b^{2}\,
  45. X = 2 [ m n ] [ m n ] = 2 ξ ξ T X=2\begin{bmatrix}m\\ n\end{bmatrix}[m\ n]=2\xi\xi^{T}\,
  46. A = [ α β γ δ ] A=\begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}
  47. [ m - v n u ] [ 1 0 ] = [ m n ] \begin{bmatrix}m&-v\\ n&u\end{bmatrix}\begin{bmatrix}1\\ 0\end{bmatrix}=\begin{bmatrix}m\\ n\end{bmatrix}
  48. Γ = SL ( 2 , 𝐙 ) SL ( 2 , 𝐙 2 ) \Gamma=\mathrm{SL}(2,\mathbf{Z})\to\mathrm{SL}(2,\mathbf{Z}_{2})
  49. U = [ 1 2 0 1 ] , L = [ 1 0 2 1 ] . U=\begin{bmatrix}1&2\\ 0&1\end{bmatrix},\qquad L=\begin{bmatrix}1&0\\ 2&1\end{bmatrix}.
  50. c 2 = a 2 + b 2 = ( a + b i ) ( a + b i ) ¯ = ( a + b i ) ( a - b i ) . c^{2}=a^{2}+b^{2}=(a+bi)\overline{(a+bi)}=(a+bi)(a-bi).
  51. a + b i = ε ( m + n i ) 2 , ε { ± 1 , ± i } . a+bi=\varepsilon\left(m+ni\right)^{2},\quad\varepsilon\in\{\pm 1,\pm i\}.
  52. { ε = + 1 , a = + ( m 2 - n 2 ) , b = + 2 m n ; ε = - 1 , a = - ( m 2 - n 2 ) , b = - 2 m n ; ε = + i , a = - 2 m n , b = + ( m 2 - n 2 ) ; ε = - i , a = + 2 m n , b = - ( m 2 - n 2 ) . \begin{cases}\varepsilon=+1,&\quad a=+\left(m^{2}-n^{2}\right),\quad b=+2mn;\\ \varepsilon=-1,&\quad a=-\left(m^{2}-n^{2}\right),\quad b=-2mn;\\ \varepsilon=+i,&\quad a=-2mn,\quad b=+\left(m^{2}-n^{2}\right);\\ \varepsilon=-i,&\quad a=+2mn,\quad b=-\left(m^{2}-n^{2}\right).\end{cases}
  53. ( m + n i ) 2 = ( m 2 - n 2 ) + 2 m n i . (m+ni)^{2}=(m^{2}-n^{2})+2mni.
  54. | p | 2 |p|^{2}
  55. | p | 2 |p|^{2}
  56. | q | 2 |q|^{2}
  57. | p | | q | |p||q|
  58. a 2 / 4 n a^{2}/4n
  59. | n - a 2 / 4 n | |n-a^{2}/4n|
  60. n + a 2 / 4 n n+a^{2}/4n
  61. n = ( b + c ) / 2 n=(b+c)/2
  62. b = | n - a 2 / 4 n | b=|n-a^{2}/4n|
  63. side a : side b = a 2 - 1 2 : side c = a 2 + 1 2 . \,\text{side }a:\,\text{side }b={a^{2}-1\over 2}:\,\text{side }c={a^{2}+1\over 2}.
  64. side a : side b = ( a 2 ) 2 - 1 : side c = ( a 2 ) 2 + 1 \,\text{side }a:\,\text{side }b=\left({a\over 2}\right)^{2}-1:\,\text{side }c=% \left({a\over 2}\right)^{2}+1
  65. a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}
  66. ( a 2 + a b + b 2 ) 2 + ( c 2 + c d + d 2 ) 2 = ( ( a + b ) 2 + ( a + b ) ( c + d ) + ( c + d ) 2 ) 2 (a^{2}+ab+b^{2})^{2}+(c^{2}+cd+d^{2})^{2}=((a+b)^{2}+(a+b)(c+d)+(c+d)^{2})^{2}
  67. a , b , c , d = - 2634 , 955 , 1770 , 5400 a,b,c,d=-2634,955,1770,5400
  68. a , b , c , d = - 31764 , 7590 , 27385 , 48150 a,b,c,d=-31764,7590,27385,48150
  69. a 2 + b 2 = c 2 + d 2 a^{2}+b^{2}=c^{2}+d^{2}
  70. ( m 2 + n 2 ) ( p 2 + q 2 ) = ( m p - n q ) 2 + ( n p + m q ) 2 = ( m p + n q ) 2 + ( n p - m q ) 2 . (m^{2}+n^{2})(p^{2}+q^{2})=(mp-nq)^{2}+(np+mq)^{2}=(mp+nq)^{2}+(np-mq)^{2}.
  71. ( a 2 - b 2 ) 2 + ( 2 a b ) 2 = ( a 2 + b 2 ) 2 (a^{2}-b^{2})^{2}+(2ab)^{2}=(a^{2}+b^{2})^{2}
  72. ( c 2 - d 2 ) 2 + ( 2 c d ) 2 = ( c 2 + d 2 ) 2 (c^{2}-d^{2})^{2}+(2cd)^{2}=(c^{2}+d^{2})^{2}
  73. ( a 2 - b 2 ) ( a 2 + b 2 ) = ( c 2 - d 2 ) ( c 2 + d 2 ) (a^{2}-b^{2})(a^{2}+b^{2})=(c^{2}-d^{2})(c^{2}+d^{2})
  74. a 4 - b 4 = c 4 - d 4 a^{4}-b^{4}=c^{4}-d^{4}
  75. a , b , c , d = 133 , 59 , 158 , 134 a,b,c,d=133,59,158,134
  76. 2 ( a 4 + b 4 + c 4 + d 4 ) = ( a 2 + b 2 + c 2 + d 2 ) 2 2(a^{4}+b^{4}+c^{4}+d^{4})=(a^{2}+b^{2}+c^{2}+d^{2})^{2}
  77. ( 2 a b ) 2 + ( 2 c d ) 2 = ( a 2 + b 2 - c 2 - d 2 ) 2 (2ab)^{2}+(2cd)^{2}=(a^{2}+b^{2}-c^{2}-d^{2})^{2}
  78. ( 2 a c ) 2 + ( 2 b d ) 2 = ( a 2 - b 2 + c 2 - d 2 ) 2 (2ac)^{2}+(2bd)^{2}=(a^{2}-b^{2}+c^{2}-d^{2})^{2}
  79. ( 2 a d ) 2 + ( 2 b c ) 2 = ( a 2 - b 2 - c 2 + d 2 ) 2 (2ad)^{2}+(2bc)^{2}=(a^{2}-b^{2}-c^{2}+d^{2})^{2}
  80. a + b = c a+b=c
  81. 4 ( a 2 + a b + b 2 ) = d 2 4(a^{2}+ab+b^{2})=d^{2}
  82. a , b , c , d = 3 , 5 , 8 , 14 a,b,c,d=3,5,8,14
  83. 3 2 + 4 2 = 5 2 3^{2}+4^{2}=5^{2}
  84. 20 2 + 21 2 = 29 2 20^{2}+21^{2}=29^{2}
  85. ( x - 1 2 ) 2 + ( x + 1 2 ) 2 = y 2 (\tfrac{x-1}{2})^{2}+(\tfrac{x+1}{2})^{2}=y^{2}
  86. x 2 - 2 y 2 = - 1 x^{2}-2y^{2}=-1
  87. 5 2 + 12 2 = 13 2 5^{2}+12^{2}=13^{2}
  88. 7 2 + 24 2 = 25 2 7^{2}+24^{2}=25^{2}
  89. ( 2 m + 1 ) 2 + ( 2 m 2 + 2 m ) 2 = ( 2 m 2 + 2 m + 1 ) 2 (2m+1)^{2}+(2m^{2}+2m)^{2}=(2m^{2}+2m+1)^{2}
  90. ( m 2 + n 2 - p 2 - q 2 ) 2 + ( 2 m q + 2 n p ) 2 + ( 2 n q - 2 m p ) 2 = ( m 2 + n 2 + p 2 + q 2 ) 2 . (m^{2}+n^{2}-p^{2}-q^{2})^{2}+(2mq+2np)^{2}+(2nq-2mp)^{2}=(m^{2}+n^{2}+p^{2}+q% ^{2})^{2}.
  91. ( x 1 2 - x 0 ) 2 + ( 2 x 1 ) 2 x 0 = ( x 1 2 + x 0 ) 2 (x_{1}^{2}-x_{0})^{2}+(2x_{1})^{2}x_{0}=(x_{1}^{2}+x_{0})^{2}
  92. ( a 2 - b 2 - c 2 - d 2 ) 2 + ( 2 a b ) 2 + ( 2 a c ) 2 + ( 2 a d ) 2 = ( a 2 + b 2 + c 2 + d 2 ) 2 . (a^{2}-b^{2}-c^{2}-d^{2})^{2}+(2ab)^{2}+(2ac)^{2}+(2ad)^{2}=(a^{2}+b^{2}+c^{2}% +d^{2})^{2}.
  93. F ( k , m ) = k m ( k - 1 + m ) + k ( k - 1 ) ( 2 k - 1 ) 6 F(k,m)=km(k-1+m)+\frac{k(k-1)(2k-1)}{6}
  94. m = v 4 - 24 v 2 - 25 48 , k = v 2 , F ( m , k ) = v 5 + 47 v 48 m=\tfrac{v^{4}-24v^{2}-25}{48},\;k=v^{2},\;F(m,k)=\tfrac{v^{5}+47v}{48}
  95. 0 2 + 1 2 + 2 2 + + 24 2 = 70 2 0^{2}+1^{2}+2^{2}+\dots+24^{2}=70^{2}
  96. F ( k , m ) + p 2 = ( p + 1 ) 2 F(k,m)+p^{2}=(p+1)^{2}
  97. p = F ( k , m ) - 1 2 p=\tfrac{F(k,m)-1}{2}
  98. 1 2 + 2 2 + 3 2 + 4 2 + 5 2 + 27 2 = 28 2 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+27^{2}=28^{2}
  99. x 2 + ( x + 1 ) 2 + + ( x + q ) 2 + p 2 = ( p + 1 ) 2 , x^{2}+(x+1)^{2}+\cdots+(x+q)^{2}+p^{2}=(p+1)^{2},
  100. p = ( q + 1 ) x 2 + q ( q + 1 ) x + q ( q + 1 ) ( 2 q + 1 ) 6 - 1 2 . p=\frac{(q+1)x^{2}+q(q+1)x+\frac{q(q+1)(2q+1)}{6}-1}{2}.
  101. x 3 + y 3 + z 3 = w 3 , x^{3}+y^{3}+z^{3}=w^{3},
  102. - 1 , \sqrt{-1},

Pythagorean_tuning.html

  1. ( 2 3 ) 6 × 2 4 \left(\frac{2}{3}\right)^{6}\times 2^{4}
  2. 1024 729 \frac{1024}{729}
  3. ( 2 3 ) 5 × 2 3 \left(\frac{2}{3}\right)^{5}\times 2^{3}
  4. 256 243 \frac{256}{243}
  5. ( 2 3 ) 4 × 2 3 \left(\frac{2}{3}\right)^{4}\times 2^{3}
  6. 128 81 \frac{128}{81}
  7. ( 2 3 ) 3 × 2 2 \left(\frac{2}{3}\right)^{3}\times 2^{2}
  8. 32 27 \frac{32}{27}
  9. ( 2 3 ) 2 × 2 2 \left(\frac{2}{3}\right)^{2}\times 2^{2}
  10. 16 9 \frac{16}{9}
  11. 2 3 × 2 \frac{2}{3}\times 2
  12. 4 3 \frac{4}{3}
  13. 1 1 \frac{1}{1}
  14. 1 1 \frac{1}{1}
  15. 3 2 \frac{3}{2}
  16. 3 2 \frac{3}{2}
  17. ( 3 2 ) 2 × 1 2 \left(\frac{3}{2}\right)^{2}\times\frac{1}{2}
  18. 9 8 \frac{9}{8}
  19. ( 3 2 ) 3 × 1 2 \left(\frac{3}{2}\right)^{3}\times\frac{1}{2}
  20. 27 16 \frac{27}{16}
  21. ( 3 2 ) 4 × ( 1 2 ) 2 \left(\frac{3}{2}\right)^{4}\times\left(\frac{1}{2}\right)^{2}
  22. 81 64 \frac{81}{64}
  23. ( 3 2 ) 5 × ( 1 2 ) 2 \left(\frac{3}{2}\right)^{5}\times\left(\frac{1}{2}\right)^{2}
  24. 243 128 \frac{243}{128}
  25. ( 3 2 ) 6 × ( 1 2 ) 3 \left(\frac{3}{2}\right)^{6}\times\left(\frac{1}{2}\right)^{3}
  26. 729 512 \frac{729}{512}
  27. S 1 = 256 243 90.225 cents S_{1}={256\over 243}\approx 90.225\ \hbox{cents}
  28. S 2 = 3 7 2 11 = 2187 2048 113.685 cents S_{2}={3^{7}\over 2^{11}}={2187\over 2048}\approx 113.685\ \hbox{cents}
  29. S E = 2 12 = 100.000 cents . S_{E}=\sqrt[12]{2}=100.000\ \hbox{cents}.

Quadratic_equation.html

  1. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  2. x x
  3. a a
  4. b b
  5. c c
  6. a a
  7. 0
  8. a = 0 a=0
  9. a a
  10. b b
  11. c c
  12. x x
  13. x 2 - x - 1 = 0. x^{2}-x-1=0.
  14. ( p x + q ) ( r x + s ) = 0 (px+q)(rx+s)=0
  15. p x + q = 0 px+q=0
  16. r x + s = 0 rx+s=0
  17. ( x + q ) ( x + s ) (x+q)(x+s)
  18. q q
  19. s s
  20. b b
  21. c c
  22. ( x + 3 ) ( x + 2 ) (x+3)(x+2)
  23. a a
  24. 1 1
  25. b = 0 b=0
  26. c = 0 c=0
  27. x 2 + 2 h x + h 2 = ( x + h ) 2 , x^{2}+2hx+h^{2}=(x+h)^{2},
  28. a a
  29. c / a c/a
  30. b / a b/a
  31. x x
  32. 1 ) x 2 + 2 x - 2 = 0 1)\ x^{2}+2x-2=0
  33. 2 ) x 2 + 2 x = 2 2)\ x^{2}+2x=2
  34. 3 ) x 2 + 2 x + 1 = 2 + 1 3)\ x^{2}+2x+1=2+1
  35. 4 ) ( x + 1 ) 2 = 3 4)\ \left(x+1\right)^{2}=3
  36. 5 ) x + 1 = ± 3 5)\ x+1=\pm\sqrt{3}
  37. 6 ) x = - 1 ± 3 6)\ x=-1\pm\sqrt{3}
  38. x = 1 + 3 x=−1+√3
  39. x = 1 3 x=−1−√3
  40. ( x + b 2 a ) 2 = b 2 - 4 a c 4 a 2 . \left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}-4ac}{4a^{2}}.
  41. x x
  42. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}.
  43. b b
  44. x 2 + p x + q = 0 , x^{2}+px+q=0,
  45. x = 1 2 ( - p ± p 2 - 4 q ) . x=\frac{1}{2}\left(-p\pm\sqrt{p^{2}-4q}\right).
  46. D D
  47. Δ = b 2 - 4 a c . \Delta=b^{2}-4ac.
  48. - b + Δ 2 a and - b - Δ 2 a , \frac{-b+\sqrt{\Delta}}{2a}\quad\,\text{and}\quad\frac{-b-\sqrt{\Delta}}{2a},
  49. - b 2 a , -\frac{b}{2a},
  50. - b 2 a + i - Δ 2 a and - b 2 a - i - Δ 2 a , \frac{-b}{2a}+i\frac{\sqrt{-\Delta}}{2a}\quad\,\text{and}\quad\frac{-b}{2a}-i% \frac{\sqrt{-\Delta}}{2a},
  51. i i
  52. a a
  53. b b
  54. c c
  55. a > 0 a>0
  56. a Align l t ; 0 a&lt;0
  57. x x
  58. x = - b 2 a \scriptstyle x=\tfrac{-b}{2a}
  59. y y
  60. x x
  61. y y
  62. ( 0 , c ) (0,c)
  63. x x
  64. f ( x ) = 0 f(x)=0
  65. a a
  66. b b
  67. c c
  68. f f
  69. f f
  70. x x
  71. x x
  72. x x
  73. x x
  74. x - r x-r
  75. a x 2 + b x + c ax^{2}+bx+c
  76. r r
  77. a x 2 + b x + c = 0. ax^{2}+bx+c=0.
  78. a x 2 + b x + c = a ( x - - b + b 2 - 4 a c 2 a ) ( x - - b - b 2 - 4 a c 2 a ) . ax^{2}+bx+c=a\left(x-\frac{-b+\sqrt{b^{2}-4ac}}{2a}\right)\left(x-\frac{-b-% \sqrt{b^{2}-4ac}}{2a}\right).
  79. a x 2 + b x + c = a ( x + b 2 a ) 2 . ax^{2}+bx+c=a\left(x+\frac{b}{2a}\right)^{2}.
  80. y = f ( x ) y=f(x)
  81. f ( x ) = 0 f(x)=0
  82. x x
  83. x x
  84. b b
  85. b 2 - 4 a c \sqrt{b^{2}-4ac}
  86. 4 a c −4ac
  87. x + y = p , x y = q x+y=p,\ \ xy=q
  88. x 2 + q = p x x^{2}+q=px
  89. x x
  90. x = p 2 + ( p 2 ) 2 - q x=\frac{p}{2}+\sqrt{\left(\frac{p}{2}\right)^{2}-q}
  91. x = 4 a c + b 2 - b 2 a . x=\frac{\sqrt{4ac+b^{2}}-b}{2a}.
  92. a x / c = y ax/c=y
  93. b b
  94. b b
  95. b b
  96. b b
  97. x 1 + x 2 = - b a x_{1}+x_{2}=-\frac{b}{a}
  98. x 1 x 2 = c a . x_{1}\ x_{2}=\frac{c}{a}.
  99. ( x - x 1 ) ( x - x 2 ) = x 2 - ( x 1 + x 2 ) x + x 1 x 2 = 0 , \left(x-x_{1}\right)\ \left(x-x_{2}\right)=x^{2}\ -\left(x_{1}+x_{2}\right)x+x% _{1}x_{2}=0,
  100. x 2 + ( b / a ) x + c / a = 0. x^{2}+(b/a)x+c/a=0.
  101. x x
  102. x x
  103. x V = x 1 + x 2 2 = - b 2 a . x_{V}=\frac{x_{1}+x_{2}}{2}=-\frac{b}{2a}.
  104. y y
  105. y V = - b 2 4 a + c = - b 2 - 4 a c 4 a . y_{V}=-\frac{b^{2}}{4a}+c=-\frac{b^{2}-4ac}{4a}.
  106. x 1 - b a . x_{1}\approx-\frac{b}{a}.
  107. x 2 = c a x 1 - c b . x_{2}=\frac{c}{a\ x_{1}}\approx-\frac{c}{b}.
  108. b b
  109. b b
  110. b b
  111. a x 2 + b x ± c = 0 , ax^{2}+bx\pm c=0,
  112. a a
  113. c c
  114. x = c / a tan θ x=\sqrt{c/a}\tan\theta
  115. sin 2 θ + b a c sin θ cos θ ± cos 2 θ = 0. \sin^{2}\theta+\frac{b}{\sqrt{ac}}\sin\theta\cos\theta\pm\cos^{2}\theta=0.
  116. 2 θ
  117. tan 2 θ n = + 2 a c b , \tan 2\theta_{n}=+2\frac{\sqrt{ac}}{b},
  118. sin 2 θ p = - 2 a c b , \sin 2\theta_{p}=-2\frac{\sqrt{ac}}{b},
  119. n n
  120. p p
  121. 4.16130 x 2 + 9.15933 x - 11.4207 = 0 4.16130x^{2}+9.15933x-11.4207=0
  122. log a = 0.6192290 , log b = 0.9618637 , log c = 1.0576927 \log a=0.6192290,\log b=0.9618637,\log c=1.0576927
  123. 2 a c / b = 2 × 10 ( 0.6192290 + 1.0576927 ) / 2 - 0.9618637 = 1.505314 2\sqrt{ac}/b=2\times 10^{(0.6192290+1.0576927)/2-0.9618637}=1.505314
  124. θ = ( tan - 1 1.505314 ) / 2 = 28.20169 or - 61.79831 \theta=(\tan^{-1}1.505314)/2=28.20169^{\circ}\,\text{ or }-61.79831^{\circ}
  125. log | tan θ | = - 0.2706462 or 0.2706462 \log\left|\tan\theta\right|=-0.2706462\,\text{ or }0.2706462
  126. log c / a = ( 1.0576927 - 0.6192290 ) / 2 = 0.2192318 \log\sqrt{c/a}=(1.0576927-0.6192290)/2=0.2192318
  127. x 1 = 10 0.2192318 - 0.2706462 = 0.888353 x_{1}=10^{0.2192318-0.2706462}=0.888353
  128. x 2 = - 10 0.2192318 + 0.2706462 = - 3.08943 x_{2}=-10^{0.2192318+0.2706462}=-3.08943
  129. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  130. b 2 - 4 a c < 0 , b^{2}-4ac<0,
  131. x 1 , x 2 = r ( cos θ ± i sin θ ) , x_{1},\,x_{2}=r(\cos\theta\pm i\sin\theta),
  132. r = c a r=\sqrt{\tfrac{c}{a}}
  133. θ = cos - 1 ( - b 2 a c ) . \theta=\cos^{-1}\left(\tfrac{-b}{2\sqrt{ac}}\right).
  134. a a
  135. b b
  136. c c
  137. a a
  138. a a
  139. 1 1
  140. a a
  141. b b
  142. c c
  143. 2 2
  144. 2 a 2a
  145. ± b 2 - 4 a c \pm\sqrt{b^{2}-4ac}
  146. 2 2
  147. 2 2
  148. 2 2
  149. x 2 + b x + c x^{2}+bx+c
  150. 2 2
  151. b = 0 b=0
  152. x = c x=\sqrt{c}
  153. - c = - c + 2 c = c . -\sqrt{c}=-\sqrt{c}+2\sqrt{c}=\sqrt{c}.
  154. x 2 + c = ( x + c ) 2 . \displaystyle x^{2}+c=(x+\sqrt{c})^{2}.
  155. b 0 b≠0
  156. R ( c ) R(c)
  157. c c
  158. R ( c ) + 1 R(c)+1
  159. b a R ( a c b 2 ) \frac{b}{a}R\left(\frac{ac}{b^{2}}\right)
  160. b a ( R ( a c b 2 ) + 1 ) . \frac{b}{a}\left(R\left(\frac{ac}{b^{2}}\right)+1\right).
  161. a a
  162. F < s u b > 4 F<sub>4

Quadratic_formula.html

  1. a x 2 + b x + c = 0. ax^{2}+bx+c=0.
  2. x x
  3. a a
  4. b b
  5. c c
  6. a a
  7. x x
  8. y = a x 2 + b x + c , y=ax^{2}+bx+c,
  9. x x
  10. a a
  11. a a
  12. x 2 + b a x + c a = 0. x^{2}+\frac{b}{a}x+\frac{c}{a}=0.
  13. c c
  14. x 2 + b a x = - c a . x^{2}+\frac{b}{a}x=-\frac{c}{a}.
  15. x 2 + b a x + ( b 2 a ) 2 = - c a + ( b 2 a ) 2 , x^{2}+\frac{b}{a}x+\left(\frac{b}{2a}\right)^{2}=-\frac{c}{a}+\left(\frac{b}{2% a}\right)^{2},
  16. ( x + b 2 a ) 2 = - c a + b 2 4 a 2 \left(x+\frac{b}{2a}\right)^{2}=-\frac{c}{a}+\frac{b^{2}}{4a^{2}}
  17. ( x + b 2 a ) 2 = b 2 - 4 a c 4 a 2 . \left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}-4ac}{4a^{2}}.
  18. x + b 2 a = ± b 2 - 4 a c 2 a . x+\frac{b}{2a}=\pm\frac{\sqrt{b^{2}-4ac\ }}{2a}.
  19. x x
  20. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}.
  21. x = - b + b 2 - 4 a c 2 a and x = - b - b 2 - 4 a c 2 a x=\frac{-b+\sqrt{b^{2}-4ac}}{2a}\quad\,\text{and}\quad x=\frac{-b-\sqrt{b^{2}-% 4ac}}{2a}
  22. a a
  23. b b
  24. p 2 ( x ) = a 2 x 2 + a 1 x + a 0 p_{2}\left(x\right)=a_{2}x^{2}+a_{1}x+a_{0}
  25. x x
  26. x = - b ± b 2 - 4 a c 2 a = - b 2 a ± b 2 - 4 a c 2 a x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}=-\frac{b}{2a}\pm\frac{\sqrt{b^{2}-4ac\ }}% {2a}
  27. - b 2 a \;-\frac{b}{2a}\;
  28. b 2 - 4 a c 2 a \frac{\sqrt{b^{2}-4ac\ }}{2a}
  29. x x
  30. b 2 - 4 a c = 0 , \sqrt{b^{2}-4ac\ }=0,
  31. b 2 - 4 a c = 0 , b^{2}-4ac=0,\;
  32. i i
  33. i = - 1 i=\sqrt{-1\ }
  34. x x
  35. x x
  36. x = 4 a c + b 2 - b 2 a . x=\frac{\sqrt{4ac+b^{2}}-b}{2a}.
  37. a a
  38. b / a b/a
  39. 4 a 4a
  40. a x 2 + b x + c \displaystyle ax^{2}+bx+c
  41. b b
  42. x x
  43. x = y + m x=y+m
  44. a ( y + m ) 2 + b ( y + m ) + c = 0. a(y+m)^{2}+b(y+m)+c=0.
  45. y y
  46. a y 2 + y ( 2 a m + b ) + ( a m 2 + b m + c ) = 0. ay^{2}+y(2am+b)+(am^{2}+bm+c)=0.
  47. y y
  48. m m
  49. m m
  50. 2 a m + b = 0 2am+b=0
  51. m = - b 2 a m=\frac{-b}{2a}
  52. a a
  53. y 2 = - ( a m 2 + b m + c ) a . y^{2}=\frac{-(am^{2}+bm+c)}{a}.
  54. m m
  55. y 2 = - ( b 2 4 a + - b 2 2 a + c ) a = b 2 - 4 a c 4 a 2 . y^{2}=\frac{-(\frac{b^{2}}{4a}+\frac{-b^{2}}{2a}+c)}{a}=\frac{b^{2}-4ac}{4a^{2% }}.
  56. y = ± b 2 - 4 a c 2 a y=\pm\frac{\sqrt{b^{2}-4ac}}{2a}
  57. x = y + m = y - b 2 a x=y+m=y-\frac{b}{2a}
  58. ( r 1 - r 2 ) 2 = ( r 1 + r 2 ) 2 - 4 r 1 r 2 . (r_{1}-r_{2})^{2}=(r_{1}+r_{2})^{2}-4r_{1}r_{2}.
  59. r 1 - r 2 = ± ( r 1 + r 2 ) 2 - 4 r 1 r 2 . r_{1}-r_{2}=\pm\sqrt{(r_{1}+r_{2})^{2}-4r_{1}r_{2}}.
  60. a 0 a≠0
  61. a a
  62. x 2 + b a x + c a = ( x - r 1 ) ( x - r 2 ) = x 2 - ( r 1 + r 2 ) x + r 1 r 2 . x^{2}+\frac{b}{a}x+\frac{c}{a}=(x-r_{1})(x-r_{2})=x^{2}-(r_{1}+r_{2})x+r_{1}r_% {2}.
  63. - b a -\frac{b}{a}
  64. c a . \frac{c}{a}.
  65. r 1 - r 2 = ± ( - b a ) 2 - 4 c a = ± b 2 a 2 - 4 a c a 2 = ± b 2 - 4 a c a . r_{1}-r_{2}=\pm\sqrt{(-\frac{b}{a})^{2}-4\frac{c}{a}}=\pm\sqrt{\frac{b^{2}}{a^% {2}}-\frac{4ac}{a^{2}}}=\pm\frac{\sqrt{b^{2}-4ac}}{a}.
  66. r 1 = ( r 1 + r 2 ) + ( r 1 - r 2 ) 2 = - b a ± b 2 - 4 a c a 2 = - b ± b 2 - 4 a c 2 a . r_{1}=\frac{(r_{1}+r_{2})+(r_{1}-r_{2})}{2}=\frac{-\frac{b}{a}\pm\frac{\sqrt{b% ^{2}-4ac}}{a}}{2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  67. r 2 = - r 1 - b a r_{2}=-r_{1}-\frac{b}{a}
  68. r 1 = - b + b 2 - 4 a c 2 a r_{1}=\frac{-b+\sqrt{b^{2}-4ac}}{2a}
  69. r 2 = - b - b 2 - 4 a c 2 a r_{2}=\frac{-b-\sqrt{b^{2}-4ac}}{2a}
  70. r 1 = - b - b 2 - 4 a c 2 a r_{1}=\frac{-b-\sqrt{b^{2}-4ac}}{2a}
  71. r 2 = - b + b 2 - 4 a c 2 a . r_{2}=\frac{-b+\sqrt{b^{2}-4ac}}{2a}.
  72. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  73. x 2 + p x + q , x^{2}+px+q,
  74. x 2 + p x + q = ( x - α ) ( x - β ) , x^{2}+px+q=(x-\alpha)(x-\beta),
  75. x 2 + p x + q = x 2 - ( α + β ) x + α β , x^{2}+px+q=x^{2}-(\alpha+\beta)x+\alpha\beta,
  76. p = - ( α + β ) p=-(\alpha+\beta)
  77. q = α β q=\alpha\beta
  78. α \alpha
  79. β \beta
  80. α \alpha
  81. β \beta
  82. α \alpha
  83. β \beta
  84. α + β \alpha+\beta
  85. α β . \alpha\beta.
  86. S n . S_{n}.
  87. α \alpha
  88. β , \beta,
  89. r 1 \displaystyle r_{1}
  90. α \displaystyle\alpha
  91. r 1 = α + β r_{1}=\alpha+\beta
  92. α \alpha
  93. β , \beta,
  94. r 1 = - p , r_{1}=-p,
  95. r 2 = α - β r_{2}=\alpha-\beta
  96. α \alpha
  97. β \beta
  98. - r 2 = β - α -r_{2}=\beta-\alpha
  99. r 2 r_{2}
  100. r 2 r_{2}
  101. - 1 , -1,
  102. r 2 2 = ( α - β ) 2 \scriptstyle r_{2}^{2}=(\alpha-\beta)^{2}
  103. ( α - β ) 2 = ( α + β ) 2 - 4 α β (\alpha-\beta)^{2}=(\alpha+\beta)^{2}-4\alpha\beta\!
  104. r 2 2 = p 2 - 4 q r_{2}^{2}=p^{2}-4q\!
  105. r 2 = ± p 2 - 4 q . r_{2}=\pm\sqrt{p^{2}-4q}.\!
  106. r 1 \displaystyle r_{1}
  107. α = 1 2 ( - p + p 2 - 4 q ) β = 1 2 ( - p - p 2 - 4 q ) \begin{aligned}\displaystyle\alpha&\displaystyle=\textstyle{\frac{1}{2}}\left(% -p+\sqrt{p^{2}-4q}\right)\\ \displaystyle\beta&\displaystyle=\textstyle{\frac{1}{2}}\left(-p-\sqrt{p^{2}-4% q}\right)\end{aligned}
  108. 1 2 ( - p ± p 2 - 4 q ) \textstyle{\frac{1}{2}}\left(-p\pm\sqrt{p^{2}-4q}\right)
  109. p = b a , q = c a \scriptstyle p=\tfrac{b}{a},q=\tfrac{c}{a}\!
  110. r 1 2 = - p 2 = - b 2 a \scriptstyle\frac{r_{1}}{2}=\frac{-p}{2}=\frac{-b}{2a}\!
  111. r 2 2 = p 2 - 4 q \scriptstyle r_{2}^{2}=p^{2}-4q\!
  112. r 2 r_{2}
  113. r 3 , r_{3},

Quadratic_programming.html

  1. n n
  2. m m
  3. 𝐜 \mathbf{c}
  4. n n
  5. Q Q
  6. n × n n\times n
  7. A A
  8. m × n m\times n
  9. b b
  10. m m
  11. 1 2 𝐱 T Q 𝐱 + 𝐜 T 𝐱 . \tfrac{1}{2}\mathbf{x}^{T}Q\mathbf{x}+\mathbf{c}^{T}\mathbf{x}.
  12. A 𝐱 𝐛 A\mathbf{x}\leq\mathbf{b}
  13. 𝐱 T \mathbf{x}^{T}
  14. 𝐱 \mathbf{x}
  15. A 𝐱 𝐛 A\mathbf{x}\leq\mathbf{b}
  16. A 𝐱 A\mathbf{x}
  17. 𝐛 \mathbf{b}
  18. [ Q E T E 0 ] [ 𝐱 λ ] = [ - 𝐜 𝐝 ] \begin{bmatrix}Q&E^{T}\\ E&0\end{bmatrix}\begin{bmatrix}\mathbf{x}\\ \lambda\end{bmatrix}=\begin{bmatrix}-\mathbf{c}\\ \mathbf{d}\end{bmatrix}
  19. λ \lambda
  20. 𝐱 \mathbf{x}
  21. Q Q
  22. 𝐝 = 0 \mathbf{d}=0
  23. E 𝐱 = 0 E\mathbf{x}=0
  24. 𝐲 \mathbf{y}
  25. Z 𝐲 = 𝐱 Z\mathbf{y}=\mathbf{x}
  26. 𝐲 \mathbf{y}
  27. 𝐱 \mathbf{x}
  28. E Z 𝐲 = 0 EZ\mathbf{y}=0
  29. Z Z
  30. E Z = 0 EZ=0
  31. Z Z
  32. E E
  33. E E
  34. 1 2 𝐱 T Q 𝐱 + 𝐜 T 𝐱 1 2 𝐲 T Z T Q Z 𝐲 + ( Z T 𝐜 ) T 𝐲 \tfrac{1}{2}\mathbf{x}^{T}Q\mathbf{x}+\mathbf{c}^{T}\mathbf{x}\quad\Rightarrow% \quad\tfrac{1}{2}\mathbf{y}^{T}Z^{T}QZ\mathbf{y}+(Z^{T}\mathbf{c})^{T}\mathbf{y}
  35. Z T Q Z 𝐲 = - Z T 𝐜 Z^{T}QZ\mathbf{y}=-Z^{T}\mathbf{c}
  36. Q Q
  37. Z T Q Z Z^{T}QZ
  38. Z Z
  39. c = 0 c=0
  40. L ( x , λ ) = 1 2 x T Q x + λ T ( A x - b ) . L(x,\lambda)=\tfrac{1}{2}x^{T}Qx+\lambda^{T}(Ax-b).
  41. g ( λ ) g(\lambda)
  42. g ( λ ) = inf x L ( x , λ ) g(\lambda)=\inf_{x}L(x,\lambda)
  43. L L
  44. x L ( x , λ ) = 0 \nabla_{x}L(x,\lambda)=0
  45. x * = - Q - 1 A T λ . x^{*}=-Q^{-1}A^{T}\lambda.
  46. g ( λ ) = - 1 2 λ T A Q - 1 A T λ - λ T b g(\lambda)=-\tfrac{1}{2}\lambda^{T}AQ^{-1}A^{T}\lambda-\lambda^{T}b
  47. - 1 2 λ T A Q - 1 A T λ - λ T b -\tfrac{1}{2}\lambda^{T}AQ^{-1}A^{T}\lambda-\lambda^{T}b
  48. λ 0 \lambda\geqslant 0

Quadratic_reciprocity.html

  1. ( p q ) ( q p ) = ( - 1 ) p - 1 2 q - 1 2 \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}% {2}}
  2. ( p q ) \left(\frac{p}{q}\right)
  3. ( a / p ) (a/p)
  4. x 2 a ( mod p ) x^{2}\equiv a\;\;(\mathop{{\rm mod}}p)
  5. The congruence x 2 - 1 ( mod p ) is solvable if and only if p 1 ( mod 4 ) . \,\text{The congruence }x^{2}\equiv-1\;\;(\mathop{{\rm mod}}p)\,\text{ is % solvable if and only if }p\equiv 1\;\;(\mathop{{\rm mod}}4).
  6. The congruence x 2 2 ( mod p ) is solvable if and only if p ± 1 ( mod 8 ) . \,\text{The congruence }x^{2}\equiv 2\;\;(\mathop{{\rm mod}}p)\,\text{ is % solvable if and only if }p\equiv\pm 1\;\;(\mathop{{\rm mod}}8).
  7. If q 1 ( mod 4 ) then \,\text{If }q\equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{ then}
  8. the congruence x 2 p ( mod q ) is solvable if and only if x 2 q ( mod p ) is, but \,\text{the congruence }x^{2}\equiv p\;\;(\mathop{{\rm mod}}q)\,\text{ is % solvable if and only if }x^{2}\equiv q\;\;(\mathop{{\rm mod}}p)\,\text{ is, but}
  9. If q 3 ( mod 4 ) then \,\text{If }q\equiv 3\;\;(\mathop{{\rm mod}}4)\,\text{ then}
  10. the congruence x 2 p ( mod q ) is solvable if and only if x 2 - q ( mod p ) is. \,\text{the congruence }x^{2}\equiv p\;\;(\mathop{{\rm mod}}q)\,\text{ is % solvable if and only if }x^{2}\equiv-q\;\;(\mathop{{\rm mod}}p)\,\text{ is.}
  11. If p 1 ( mod 4 ) or q 1 ( mod 4 ) (or both), then \,\text{If }p\equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{ or }q\equiv 1\;\;(% \mathop{{\rm mod}}4)\,\text{ (or both), then}
  12. x 2 q ( mod p ) is solvable if and only if x 2 p ( mod q ) is solvable. x^{2}\equiv q\;\;(\mathop{{\rm mod}}p)\,\text{ is solvable if and only if }x^{% 2}\equiv p\;\;(\mathop{{\rm mod}}q)\,\text{ is solvable.}
  13. If p q 3 ( mod 4 ) , then \,\text{If }p\equiv q\equiv 3\;\;(\mathop{{\rm mod}}4),\,\text{ then}
  14. x 2 q ( mod p ) is solvable if and only if x 2 p ( mod q ) is not solvable. x^{2}\equiv q\;\;(\mathop{{\rm mod}}p)\,\text{ is solvable if and only if }x^{% 2}\equiv p\;\;(\mathop{{\rm mod}}q)\,\text{ is not solvable.}
  15. p = x 2 + y 2 if and only if p = 2 or p 1 ( mod 4 ) , p=x^{2}+\;\,y^{2}\,\text{ if and only if }p=2\,\text{ or }p\equiv 1\;\;(% \mathop{{\rm mod}}4),
  16. p = x 2 + 2 y 2 if and only if p = 2 or p 1 , 3 ( mod 8 ) , \!\,p=x^{2}+2y^{2}\,\text{ if and only if }p=2\,\text{ or }p\equiv 1,3\;\;(% \mathop{{\rm mod}}8),
  17. p = x 2 + 3 y 2 if and only if p = 3 or p 1 ( mod 3 ) . \!\,p=x^{2}+3y^{2}\,\text{ if and only if }p=3\,\text{ or }p\equiv 1\;\;(% \mathop{{\rm mod}}3).
  18. p q = x 2 + 5 y 2 . pq=x^{2}+5y^{2}.
  19. If p 1 , 9 ( mod 20 ) then p = x 2 + 5 y 2 , \,\text{If }\;\,p\equiv 1,9\;\;(\mathop{{\rm mod}}20)\,\text{ then }\;\,p=x^{2% }+5y^{2},
  20. if p , q 3 , 7 ( mod 20 ) then p q = x 2 + 5 y 2 . \!\,\,\text{if }p,q\equiv 3,7\;\;(\mathop{{\rm mod}}20)\,\text{ then }pq=x^{2}% +5y^{2}.
  21. a p a ( mod p ) . a^{p}\equiv a\;\;(\mathop{{\rm mod}}p).
  22. a ( p - 1 ) / 2 ± 1 ( mod p ) . a^{(p-1)/2}\equiv\pm 1\;\;(\mathop{{\rm mod}}p).
  23. b ( a - 1 ) / 2 + 1 ( mod a ) b^{(a-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}a)\;
  24. a ( b - 1 ) / 2 + 1 ( mod b ) a^{(b-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}b)\;
  25. a ( b - 1 ) / 2 - 1 ( mod b ) a^{(b-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}b)\;
  26. b ( a - 1 ) / 2 - 1 ( mod a ) b^{(a-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}a)\;
  27. a ( A - 1 ) / 2 + 1 ( mod A ) a^{(A-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}A)\;
  28. A ( a - 1 ) / 2 + 1 ( mod a ) A^{(a-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}a)\;
  29. a ( A - 1 ) / 2 - 1 ( mod A ) a^{(A-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}A)\;
  30. A ( a - 1 ) / 2 - 1 ( mod a ) A^{(a-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}a)\;
  31. a ( b - 1 ) / 2 + 1 ( mod b ) a^{(b-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}b)\;
  32. b ( a - 1 ) / 2 + 1 ( mod a ) b^{(a-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}a)\;
  33. b ( a - 1 ) / 2 - 1 ( mod a ) b^{(a-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}a)\;
  34. a ( b - 1 ) / 2 - 1 ( mod b ) a^{(b-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}b)\;
  35. b ( B - 1 ) / 2 + 1 ( mod B ) b^{(B-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}B)\;
  36. B ( b - 1 ) / 2 - 1 ( mod b ) B^{(b-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}b)\;
  37. b ( B - 1 ) / 2 - 1 ( mod B ) b^{(B-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}B)\;
  38. B ( b - 1 ) / 2 + 1 ( mod b ) B^{(b-1)/2}\equiv+1\;\;(\mathop{{\rm mod}}b)\;
  39. N ( c - 1 ) / 2 ( mod c ) N^{(c-1)/2}\;\;(\mathop{{\rm mod}}c)
  40. ( N c ) N ( c - 1 ) / 2 ( mod c ) = ± 1. \left(\frac{N}{c}\right)\equiv N^{(c-1)/2}\;\;(\mathop{{\rm mod}}c)=\pm 1.
  41. ( a p ) = { 0 if a 0 ( mod p ) + 1 if a 0 ( mod p ) and for some integer x , a x 2 ( mod p ) - 1 if there is no such x . \left(\frac{a}{p}\right)=\begin{cases}\;\;\,0\,\text{ if }a\equiv 0\;\;(% \mathop{{\rm mod}}p)\\ +1\,\text{ if }a\not\equiv 0\;\;(\mathop{{\rm mod}}p)\,\text{ and for some % integer }x,\;a\equiv x^{2}\;\;(\mathop{{\rm mod}}p)\\ -1\,\text{ if there is no such }x.\end{cases}
  42. ( p q ) = { + ( q p ) if p 1 ( mod 4 ) or q 1 ( mod 4 ) - ( q p ) if p q 3 ( mod 4 ) \left(\frac{p}{q}\right)=\begin{cases}+\left(\frac{q}{p}\right)\,\text{ if }p% \equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{ or }q\equiv 1\;\;(\mathop{{\rm mod}}% 4)\\ -\left(\frac{q}{p}\right)\,\text{ if }p\equiv q\equiv 3\;\;(\mathop{{\rm mod}}% 4)\end{cases}
  43. ( p q ) ( q p ) = ( - 1 ) p - 1 2 q - 1 2 . \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}% {2}}.
  44. ( p q ) = sgn i = 1 q - 1 2 k = 1 p - 1 2 ( k p - i q ) \left(\frac{p}{q}\right)=\operatorname{sgn}\prod_{i=1}^{\frac{q-1}{2}}\prod_{k% =1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right)
  45. ( - 1 p ) = ( - 1 ) p - 1 2 = { + 1 if p 1 ( mod 4 ) - 1 if p 3 ( mod 4 ) \left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}=\left\{\begin{array}[]{cl}+1&\,% \text{if}\;p\equiv 1\;\;(\mathop{{\rm mod}}4)\\ -1&\,\text{if}\;p\equiv 3\;\;(\mathop{{\rm mod}}4)\end{array}\right.
  46. ( 2 p ) = ( - 1 ) p 2 - 1 8 = { + 1 if p 1 or 7 ( mod 8 ) - 1 if p 3 or 5 ( mod 8 ) {\left(\frac{2}{p}\right)=(-1)^{\frac{p^{2}-1}{8}}=\left\{\begin{array}[]{cl}+% 1&\,\text{if}\;p\equiv 1\;\,\text{ or }\;7\;\;(\mathop{{\rm mod}}8)\\ -1&\,\text{if}\;p\equiv 3\;\,\text{ or }\;5\;\;(\mathop{{\rm mod}}8)\end{array% }\right.}
  47. Let a , b , and c be integers that satisfy \,\text{Let }a,b,\,\text{ and }c\,\text{ be integers that satisfy}
  48. gcd ( a , b ) = gcd ( b , c ) = gcd ( c , a ) = 1. \gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1.\;
  49. At least one of a b , b c , c a < 0. \,\text{At least one of }ab,\;bc,\;ca<0.
  50. u 2 - b c ( mod a ) , v 2 - c a ( mod b ) , and w 2 - a b ( mod c ) are solvable. u^{2}\equiv-bc\;\;(\mathop{{\rm mod}}a),\;v^{2}\equiv-ca\;\;(\mathop{{\rm mod}% }b),\,\text{ and }w^{2}\equiv-ab\;\;(\mathop{{\rm mod}}c)\,\text{ are solvable.}
  51. Then the equation a x 2 + b y 2 + c z 2 = 0 has a nontrivial solution in integers. \,\text{Then the equation }ax^{2}+by^{2}+cz^{2}=0\,\text{ has a nontrivial % solution in integers. }
  52. ( b a ) = 1 (\tfrac{b}{a})=1
  53. ( a b ) = - 1. (\tfrac{a}{b})=-1.
  54. x 2 + a y 2 - b z 2 = 0 x^{2}+ay^{2}-bz^{2}=0
  55. ( B b ) = ( b B ) = - 1. (\tfrac{B}{b})=(\tfrac{b}{B})=-1.
  56. ( p b ) = ( p B ) = - 1 , (\tfrac{p}{b})=(\tfrac{p}{B})=-1,
  57. B x 2 + b y 2 - p z 2 = 0 Bx^{2}+by^{2}-pz^{2}=0
  58. If a 1 ( mod 4 ) is prime there exists a prime β such that ( a β ) = - 1 , \,\text{If }a\equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{ is prime there exists a% prime }\beta\,\text{ such that }\left(\frac{a}{\beta}\right)=-1,\,
  59. a x 2 + b y 2 + c z 2 = 0 ax^{2}+by^{2}+cz^{2}=0
  60. If p 1 ( mod 8 ) is prime, then there exists an odd prime q < 2 p + 1 such that ( p q ) = - 1. \,\text{If }p\equiv 1\;\;(\mathop{{\rm mod}}8)\,\text{ is prime, then there % exists an odd prime }q<2\sqrt{p}+1\,\text{ such that }\left(\frac{p}{q}\right)% =-1.
  61. ( p q ) = { ( q p ) if q 1 ( mod 4 ) ( - q p ) if q 3 ( mod 4 ) \left(\frac{p}{q}\right)=\begin{cases}\left(\frac{q}{p}\right)\;\;\,\text{ if % }q\equiv 1\;\;(\mathop{{\rm mod}}4)\\ \left(\frac{-q}{p}\right)\,\text{ if }q\equiv 3\;\;(\mathop{{\rm mod}}4)\end{cases}
  62. Let q * = ( - 1 ) q - 1 2 q (in other words | q * | = | q | and q * 1 ( mod 4 ) ). \,\text{Let }q^{*}=(-1)^{\frac{q-1}{2}}q\;\;\,\text{ (in other words }|q^{*}|=% |q|\,\text{ and }q^{*}\equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{).}\;
  63. Then ( p q ) = ( q * p ) . \,\text{ Then }\left(\frac{p}{q}\right)=\left(\frac{q^{*}}{p}\right).
  64. If p ± q ( mod 4 a ) then ( a p ) = ( a q ) . \,\text{If }p\equiv\pm q\;\;(\mathop{{\rm mod}}4a)\,\text{ then }\left(\frac{a% }{p}\right)=\left(\frac{a}{q}\right).
  65. If p q , p q , p p ( mod 4 ) , and q q ( mod 4 ) then ( p q ) ( q p ) = ( p q ) ( q p ) . \,\text{If }p\neq q,p^{\prime}\neq q^{\prime},p\equiv p^{\prime}\;\;(\mathop{{% \rm mod}}4),\,\text{ and }q\equiv q^{\prime}\;\;(\mathop{{\rm mod}}4)\,\text{ % then }\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=\left(\frac{p^{\prime}}% {q^{\prime}}\right)\left(\frac{q^{\prime}}{p^{\prime}}\right).
  66. Let a , b , and c be integers. Then for every prime p that divides a b c , \,\text{Let }a,b,\,\text{ and }c\,\text{ be integers. Then for every prime }p% \,\text{ that divides }abc,
  67. if a x 2 + b y 2 + c z 2 0 ( mod 4 a b c / p ) has a nontrivial solution \,\text{if }ax^{2}+by^{2}+cz^{2}\equiv 0\;\;(\mathop{{\rm mod}}4abc/p)\,\text{% has a nontrivial solution }
  68. so does a x 2 + b y 2 + c z 2 0 ( mod 4 a b c ) . \,\text{so does }ax^{2}+by^{2}+cz^{2}\equiv 0\;\;(\mathop{{\rm mod}}4abc).
  69. ( - 1 n ) = ( - 1 ) ( n - 1 ) / 2 = { 1 if n 1 ( mod 4 ) - 1 if n 3 ( mod 4 ) \left(\frac{-1}{n}\right)=(-1)^{(n-1)/2}=\left\{\begin{array}[]{cl}1&\,\text{% if}\;n\equiv 1\;\;(\mathop{{\rm mod}}4)\\ -1&\,\text{if}\;n\equiv 3\;\;(\mathop{{\rm mod}}4)\end{array}\right.
  70. ( 2 n ) = ( - 1 ) ( n 2 - 1 ) / 8 = { 1 if n 1 or 7 ( mod 8 ) - 1 if n 3 or 5 ( mod 8 ) {\left(\frac{2}{n}\right)=(-1)^{(n^{2}-1)/8}=\left\{\begin{array}[]{cl}1&\,% \text{if}\;n\equiv 1\;\,\text{ or }\;7\;\;(\mathop{{\rm mod}}8)\\ -1&\,\text{if}\;n\equiv 3\;\,\text{ or }\;5\;\;(\mathop{{\rm mod}}8)\end{array% }\right.}
  71. ( m n ) = ( - 1 ) ( m - 1 ) ( n - 1 ) / 4 ( n m ) . \left(\frac{m}{n}\right)=(-1)^{(m-1)(n-1)/4}\left(\frac{n}{m}\right).
  72. ( M p ) = ( - 1 ) ( p - 1 ) ( M - 1 ) / 4 ( p M ) , \left(\frac{M}{p}\right)=(-1)^{(p-1)(M-1)/4}\Bigg(\frac{p}{M}\Bigg),
  73. ( a m ) = ( a m ± 4 a n ) where n is an integer and m ± 4 a n > 0. \left(\frac{a}{m}\right)=\left(\frac{a}{m\pm 4an}\right)\,\text{ where }n\,% \text{ is an integer and }m\pm 4an>0.
  74. ( 2 7 ) = ( 2 15 ) = ( 2 23 ) = ( 2 31 ) = 1 , (\tfrac{2}{7})=(\tfrac{2}{15})=(\tfrac{2}{23})=(\tfrac{2}{31})\dots=1,
  75. ( a m ) = - 1 (\tfrac{a}{m})=-1
  76. If a , b , a and b are positive and odd and \,\text{If }a,b,a^{\prime}\,\text{ and }b^{\prime}\,\text{ are positive and % odd and}
  77. gcd ( a , b ) = gcd ( a , b ) = 1 , then \gcd(a,b)=\gcd(a^{\prime},b^{\prime})=1,\,\text{ then}
  78. if a a ( mod 4 ) and b b ( mod 4 ) , ( a b ) ( b a ) = ( a b ) ( b a ) . \,\text{if }a\equiv a^{\prime}\;\;(\mathop{{\rm mod}}4)\,\text{ and }b\equiv b% ^{\prime}\;\;(\mathop{{\rm mod}}4),\;\bigg(\frac{a}{b}\bigg)\left(\frac{b}{a}% \right)=\left(\frac{a^{\prime}}{b^{\prime}}\right)\left(\frac{b^{\prime}}{a^{% \prime}}\right).
  79. ( a , b ) v (a,b)_{v}
  80. ( a , b ) v (a,b)_{v}
  81. a x 2 + b y 2 = z 2 ax^{2}+by^{2}=z^{2}
  82. x = y = z = 0 x=y=z=0
  83. ( a , b ) v (a,b)_{v}
  84. ( a , b ) v (a,b)_{v}
  85. [ α π ] 2 α N π - 1 2 ( mod π ) = { + 1 if gcd ( α , π ) = 1 and there is a Gaussian integer η such that α η 2 ( mod π ) - 1 if gcd ( α , π ) = 1 and there is no such η . \begin{aligned}\displaystyle\left[\frac{\alpha}{\pi}\right]_{2}&\displaystyle% \equiv\alpha^{\frac{\mathrm{N}\pi-1}{2}}\;\;(\mathop{{\rm mod}}\pi)\\ &\displaystyle=\begin{cases}+1\,\text{ if }\gcd(\alpha,\pi)=1\,\text{ and % there is a Gaussian integer }\eta\,\text{ such that }\alpha\equiv\eta^{2}\;\;(% \mathop{{\rm mod}}\pi)\\ -1\,\text{ if }\gcd(\alpha,\pi)=1\,\text{ and there is no such }\eta.\end{% cases}\end{aligned}
  86. [ λ μ ] 2 = [ μ λ ] 2 , [ i λ ] 2 = ( - 1 ) b 2 , and [ 1 + i λ ] 2 = ( 2 a + b ) , \Bigg[\frac{\lambda}{\mu}\Bigg]_{2}=\Bigg[\frac{\mu}{\lambda}\Bigg]_{2},\;\;\;% \;\Bigg[\frac{i}{\lambda}\Bigg]_{2}=(-1)^{\frac{b}{2}},\;\;\,\text{ and }\;\;% \Bigg[\frac{1+i}{\lambda}\Bigg]_{2}=\Bigg(\frac{2}{a+b}\Bigg),
  87. ( a b ) (\tfrac{a}{b})
  88. ω = - 1 + - 3 2 = e 2 π i 3 \omega=\frac{-1+\sqrt{-3}}{2}=e^{\frac{2\pi i}{3}}
  89. [ α π ] 2 = ± 1 α N π - 1 2 ( mod π ) = { + 1 if gcd ( α , π ) = 1 and there is an Eisenstein integer η such that α η 2 ( mod π ) - 1 if gcd ( α , π ) = 1 and there is no such η . \begin{aligned}\displaystyle\left[\frac{\alpha}{\pi}\right]_{2}&\displaystyle=% \pm 1\equiv\alpha^{\frac{\mathrm{N}\pi-1}{2}}\;\;(\mathop{{\rm mod}}\pi)\\ &\displaystyle=\begin{cases}+1\,\text{ if }\gcd(\alpha,\pi)=1\,\text{ and % there is an Eisenstein integer }\eta\,\text{ such that }\alpha\equiv\eta^{2}\;% \;(\mathop{{\rm mod}}\pi)\\ -1\,\text{ if }\gcd(\alpha,\pi)=1\,\text{ and there is no such }\eta.\end{% cases}\end{aligned}
  90. [ λ μ ] 2 [ μ λ ] 2 = ( - 1 ) N λ - 1 2 N μ - 1 2 , [ 1 - ω λ ] 2 = ( a 3 ) , and [ 2 λ ] 2 = ( 2 N λ ) , \left[\frac{\lambda}{\mu}\right]_{2}\bigg[\frac{\mu}{\lambda}\bigg]_{2}=(-1)^{% \frac{\mathrm{N}\lambda-1}{2}\frac{\mathrm{N}\mu-1}{2}},\;\;\;\;\bigg[\frac{1-% \omega}{\lambda}\bigg]_{2}=\bigg(\frac{a}{3}\bigg),\;\;\,\text{ and }\;\;\bigg% [\frac{2}{\lambda}\bigg]_{2}=\bigg(\frac{2}{\mathrm{N}\lambda}\bigg),
  91. ( a b ) (\tfrac{a}{b})
  92. 𝒪 k . \mathcal{O}_{k}.
  93. 𝔭 𝒪 k \mathfrak{p}\subset\mathcal{O}_{k}
  94. N 𝔭 \mathrm{N}\mathfrak{p}
  95. α 𝒪 k , \alpha\in\mathcal{O}_{k},\;\;
  96. 𝒪 k \mathcal{O}_{k}
  97. [ α 𝔭 ] 2 α N 𝔭 - 1 2 ( mod 𝔭 ) = { + 1 if α 𝔭 and there is an η 𝒪 k such that α - η 2 𝔭 - 1 if α 𝔭 and there is no such η 0 if α 𝔭 , \begin{aligned}\displaystyle\left[\frac{\alpha}{\mathfrak{p}}\right]_{2}&% \displaystyle\equiv\alpha^{\frac{\mathrm{N}\mathfrak{p}-1}{2}}\;\;(\mathop{{% \rm mod}}\mathfrak{p})\\ &\displaystyle=\begin{cases}+1\,\text{ if }\alpha\not\in\mathfrak{p}\,\text{ % and there is an }\eta\in\mathcal{O}_{k}\,\text{ such that }\alpha-\eta^{2}\in% \mathfrak{p}\\ -1\,\text{ if }\alpha\not\in\mathfrak{p}\,\text{ and there is no such }\eta\\ \;\;\;0\,\text{ if }\alpha\in\mathfrak{p},\end{cases}\end{aligned}
  98. 𝔞 𝒪 k \mathfrak{a}\subset\mathcal{O}_{k}
  99. 𝔞 = 𝔭 1 𝔭 2 𝔭 n \mathfrak{a}=\mathfrak{p}_{1}\mathfrak{p}_{2}\dots\mathfrak{p}_{n}
  100. [ α 𝔞 ] 2 = [ α 𝔭 1 ] 2 [ α 𝔭 2 ] 2 [ α 𝔭 n ] 2 , \bigg[\frac{\alpha}{\mathfrak{a}}\bigg]_{2}=\left[\frac{\alpha}{\mathfrak{p}_{% 1}}\right]_{2}\left[\frac{\alpha}{\mathfrak{p}_{2}}\right]_{2}\dots\left[\frac% {\alpha}{\mathfrak{p}_{n}}\right]_{2},
  101. β 𝒪 k \beta\in\mathcal{O}_{k}
  102. [ α β ] 2 = [ α β 𝒪 k ] 2 . \bigg[\frac{\alpha}{\beta}\bigg]_{2}=\bigg[\frac{\alpha}{\beta\mathcal{O}_{k}}% \bigg]_{2}.
  103. { ω 1 , ω 2 } \left\{\omega_{1},\omega_{2}\right\}
  104. 𝒪 k = ω 1 ω 2 . \mathcal{O}_{k}=\mathbb{Z}\omega_{1}\oplus\mathbb{Z}\omega_{2}.
  105. ν 𝒪 k \nu\in\mathcal{O}_{k}
  106. ν ω 1 = a ω 1 + b ω 2 ν ω 2 = c ω 1 + d ω 2 \begin{aligned}\displaystyle\nu\omega_{1}&\displaystyle=a\omega_{1}+b\omega_{2% }\\ \displaystyle\nu\omega_{2}&\displaystyle=c\omega_{1}+d\omega_{2}\end{aligned}
  107. χ ( ν ) = i ( b 2 - a + 2 ) c + ( a 2 - b + 2 ) d + a d . \chi(\nu)=i^{(b^{2}-a+2)c+(a^{2}-b+2)d+ad}.
  108. [ μ ν ] 2 [ ν μ ] 2 = ( - 1 ) m - 1 2 n - 1 2 χ ( μ ) m n - 1 2 χ ( ν ) - n m - 1 2 . \Bigg[\frac{\mu}{\nu}\Bigg]_{2}\left[\frac{\nu}{\mu}\right]_{2}=(-1)^{\frac{m-% 1}{2}\frac{n-1}{2}}\chi(\mu)^{m\frac{n-1}{2}}\chi(\nu)^{-n\frac{m-1}{2}}.
  109. μ μ ( mod 4 ) and ν ν ( mod 4 ) \mu\equiv\mu^{\prime}\;\;(\mathop{{\rm mod}}4)\,\text{ and }\nu\equiv\nu^{% \prime}\;\;(\mathop{{\rm mod}}4)
  110. [ μ ν ] 2 [ ν μ ] 2 = [ μ ν ] 2 [ ν μ ] 2 . \Bigg[\frac{\mu}{\nu}\Bigg]_{2}\left[\frac{\nu}{\mu}\right]_{2}=\Bigg[\frac{% \mu^{\prime}}{\nu^{\prime}}\Bigg]_{2}\left[\frac{\nu^{\prime}}{\mu^{\prime}}% \right]_{2}.
  111. f , g F [ x ] f,g\in\mathrm{F}[x]
  112. ( g f ) (\tfrac{g}{f})
  113. ( g f ) = { + 1 if gcd ( f , g ) = 1 and there are h , k F [ x ] such that g - h 2 = k f - 1 if gcd ( f , g ) = 1 and g is not a square ( mod f ) 0 if gcd ( f , g ) 1. \left(\frac{g}{f}\right)=\begin{cases}+1\,\text{ if }\gcd(f,g)=1\,\text{ and % there are }h,k\in\mathrm{F}[x]\,\text{ such that }g-h^{2}=kf\\ -1\,\text{ if }\gcd(f,g)=1\,\text{ and }g\,\text{ is not a square }\;\;(% \mathop{{\rm mod}}f)\\ \;\;\;0\,\text{ if }\gcd(f,g)\neq 1.\end{cases}
  114. f = f 1 f 2 f n f=f_{1}f_{2}\dots f_{n}
  115. ( g f ) = ( g f 1 ) ( g f 2 ) ( g f n ) . \left(\frac{g}{f}\right)=\left(\frac{g}{f_{1}}\right)\left(\frac{g}{f_{2}}% \right)\dots\left(\frac{g}{f_{n}}\right).
  116. f , g F [ x ] f,g\in\mathrm{F}[x]
  117. ( g f ) ( f g ) = ( - 1 ) q - 1 2 ( deg f ) ( deg g ) . \left(\frac{g}{f}\right)\left(\frac{f}{g}\right)=(-1)^{\frac{q-1}{2}(\deg f)(% \deg g)}.
  118. q * . \sqrt{q^{*}}.
  119. Q ( q * ) Q(\sqrt{q^{*}})
  120. Q ( e 2 π i / q ) Q(e^{2\pi i/q})

Quadrature_amplitude_modulation.html

  1. s ( t ) \displaystyle s(t)
  2. i 2 = - 1 \scriptstyle i^{2}\;=\;-1
  3. I ( t ) \scriptstyle I(t)
  4. Q ( t ) \scriptstyle Q(t)
  5. f 0 \scriptstyle f_{0}
  6. { } \Re\{\}
  7. I ( t ) \scriptstyle I(t)
  8. Q ( t ) \scriptstyle Q(t)
  9. I ( t ) \scriptstyle I(t)
  10. r ( t ) \displaystyle r(t)
  11. r ( t ) \displaystyle r(t)
  12. r ( t ) \scriptstyle r(t)
  13. 4 π f 0 t \scriptstyle 4\pi f_{0}t
  14. I ( t ) \scriptstyle I(t)
  15. Q ( t ) \scriptstyle Q(t)
  16. s ( t ) \scriptstyle s(t)
  17. Q ( t ) \scriptstyle Q(t)
  18. S ( f ) = 1 2 [ M I ( f - f 0 ) + M I ( f + f 0 ) ] + i 2 [ M Q ( f - f 0 ) - M Q ( f + f 0 ) ] S(f)=\frac{1}{2}\left[M_{I}(f-f_{0})+M_{I}(f+f_{0})\right]+\frac{i}{2}\left[M_% {Q}(f-f_{0})-M_{Q}(f+f_{0})\right]
  19. f 0 \scriptstyle f_{0}
  20. H t \scriptstyle H_{t}
  21. s ( t ) = n = - [ v c [ n ] h t ( t - n T s ) cos ( 2 π f 0 t ) - v s [ n ] h t ( t - n T s ) sin ( 2 π f 0 t ) ] s(t)=\sum_{n=-\infty}^{\infty}\left[v_{c}[n]\cdot h_{t}(t-nT_{s})\cos(2\pi f_{% 0}t)-v_{s}[n]\cdot h_{t}(t-nT_{s})\sin(2\pi f_{0}t)\right]
  22. v c [ n ] \scriptstyle v_{c}[n]
  23. v s [ n ] \scriptstyle v_{s}[n]
  24. n \scriptstyle n
  25. H r \scriptstyle H_{r}
  26. H r \scriptstyle H_{r}
  27. M \scriptstyle M
  28. E b \scriptstyle E_{b}
  29. E s \scriptstyle E_{s}
  30. k E b \scriptstyle kE_{b}
  31. N 0 \scriptstyle N_{0}
  32. P b \scriptstyle P_{b}
  33. P b c \scriptstyle P_{bc}
  34. P s \scriptstyle P_{s}
  35. P s c \scriptstyle P_{sc}
  36. Q ( x ) = 1 2 π x e - 1 2 t 2 d t , x 0 \scriptstyle Q(x)\;=\;\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{1}{2}t^{% 2}}dt,\ x\geq 0
  37. Q ( x ) \scriptstyle Q(x)
  38. Q ( x ) = 1 2 erfc ( 1 2 x ) \scriptstyle Q(x)\;=\;\frac{1}{2}\operatorname{erfc}\left(\frac{1}{\sqrt{2}}x\right)
  39. k \scriptstyle k
  40. P s c = 2 ( 1 - 1 M ) Q ( 3 M - 1 E s N 0 ) P_{sc}=2\left(1-\frac{1}{\sqrt{M}}\right)Q\left(\sqrt{\frac{3}{M-1}\frac{E_{s}% }{N_{0}}}\right)
  41. P s = 1 - ( 1 - P s c ) 2 \,P_{s}=1-\left(1-P_{sc}\right)^{2}
  42. E b / N 0 1 \scriptstyle E_{b}/N_{0}\gg 1
  43. P b c P s c 1 2 k = 4 k ( 1 - 1 M ) Q ( 3 k M - 1 E b N 0 ) P_{bc}\approx\frac{P_{sc}}{\frac{1}{2}k}=\frac{4}{k}\left(1-\frac{1}{\sqrt{M}}% \right)Q\left(\sqrt{\frac{3k}{M-1}\frac{E_{b}}{N_{0}}}\right)
  44. P b = P b c \,P_{b}=P_{bc}
  45. ρ \rho
  46. P b ( E | ρ ) = 1 log 2 ( L M ) ( i = 1 log 2 L P b ( E i L | ρ ) + i = 1 log 2 M P b ( E i M | ρ ) ) P_{b}(E|\rho)=\frac{1}{\log_{2}\left(L\cdot M\right)}\left(\sum_{i=1}^{\log_{2% }L}P_{b}(E^{L}_{i}|\rho)+\sum_{i=1}^{\log_{2}M}P_{b}(E^{M}_{i}|\rho)\right)
  47. P b ( E i P | ρ ) = 2 P j = 0 ( 1 - 2 - i ) P - 1 ( - 1 ) j 2 i - 1 P ( 2 i - 1 - j 2 i - 1 P + 1 2 ) \Q [ ( 2 j + 1 ) 6 ρ ( L 2 + M 2 - 2 ) ] P_{b}(E^{P}_{i}|\rho)=\frac{2}{P}\sum_{j=0}^{(1-2^{-i})P-1}\left(-1\right)^{% \left\lfloor\frac{j\cdot 2^{i-1}}{P}\right\rfloor}\cdot\left(2^{i-1}-\left% \lfloor\frac{j\cdot 2^{i-1}}{P}+\frac{1}{2}\right\rfloor\right)\cdot\Q\left[% \left(2j+1\right)\sqrt{\frac{6\rho}{(L^{2}+M^{2}-2)}}\right]
  48. k \scriptstyle k
  49. k = 3 \scriptstyle k\;=\;3
  50. P s 4 Q ( 3 k E b ( M - 1 ) N 0 ) P_{s}\leq{}4Q\left(\sqrt{\frac{3kE_{b}}{(M-1)N_{0}}}\;\right)
  51. P b \scriptstyle P_{b}
  52. P s < ( M - 1 ) Q ( d m i n 2 2 N 0 ) P_{s}<(M-1)Q\left(\sqrt{\frac{d_{min}^{2}}{2N_{0}}}\right)
  53. M \scriptstyle M

Quadrilateral.html

  1. A + B + C + D = 360 . \angle A+\angle B+\angle C+\angle D=360^{\circ}.
  2. K = 1 2 p q sin θ , K=\tfrac{1}{2}pq\cdot\sin\theta,
  3. K = 1 2 p q K=\tfrac{1}{2}pq
  4. K = m n sin φ , K=mn\cdot\sin\varphi,
  5. K \displaystyle K
  6. K = 1 2 a d sin A + 1 2 b c sin C . K=\tfrac{1}{2}ad\cdot\sin{A}+\tfrac{1}{2}bc\cdot\sin{C}.
  7. K = 1 2 ( a d + b c ) sin A . K=\tfrac{1}{2}(ad+bc)\sin{A}.
  8. K = a b sin A . K=ab\cdot\sin{A}.
  9. K = | tan θ | 4 | a 2 + c 2 - b 2 - d 2 | . K=\frac{|\tan\theta|}{4}\cdot\left|a^{2}+c^{2}-b^{2}-d^{2}\right|.
  10. K = 1 2 | tan θ | | a 2 - b 2 | . K=\tfrac{1}{2}|\tan\theta|\cdot\left|a^{2}-b^{2}\right|.
  11. K = 1 4 ( 2 ( a 2 + c 2 ) - 4 x 2 ) ( 2 ( b 2 + d 2 ) - 4 x 2 ) sin φ K=\tfrac{1}{4}\sqrt{(2(a^{2}+c^{2})-4x^{2})(2(b^{2}+d^{2})-4x^{2})}\sin{\varphi}
  12. K = 1 2 a b sin α + 1 4 4 c 2 d 2 - ( c 2 + d 2 - a 2 - b 2 + 2 a b cos α ) 2 , K=\tfrac{1}{2}ab\cdot\sin{\alpha}+\tfrac{1}{4}\sqrt{4c^{2}d^{2}-(c^{2}+d^{2}-a% ^{2}-b^{2}+2ab\cdot\cos{\alpha})^{2}},
  13. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - 1 4 ( a c + b d + p q ) ( a c + b d - p q ) , K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},
  14. K = 1 4 4 p 2 q 2 - ( a 2 + c 2 - b 2 - d 2 ) 2 . K=\tfrac{1}{4}\sqrt{4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}.
  15. K = 1 2 ( m + n + p ) ( m + n - p ) ( m + n + q ) ( m + n - q ) , K=\tfrac{1}{2}\sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},
  16. K = 1 2 p 2 q 2 - ( m 2 - n 2 ) 2 . K=\tfrac{1}{2}\sqrt{p^{2}q^{2}-(m^{2}-n^{2})^{2}}.
  17. p 2 + q 2 = 2 ( m 2 + n 2 ) . p^{2}+q^{2}=2(m^{2}+n^{2}).
  18. K = 1 2 [ ( m + n ) 2 - p 2 ] [ p 2 - ( m - n ) 2 ] , K=\tfrac{1}{2}\sqrt{[(m+n)^{2}-p^{2}]\cdot[p^{2}-(m-n)^{2}]},
  19. K = 1 4 [ ( p + q ) 2 - 4 m 2 ] [ 4 m 2 - ( p - q ) 2 ] , K=\tfrac{1}{4}\sqrt{[(p+q)^{2}-4m^{2}]\cdot[4m^{2}-(p-q)^{2}]},
  20. K = 1 2 | 𝐀𝐂 × 𝐁𝐃 | , K=\tfrac{1}{2}|\mathbf{AC}\times\mathbf{BD}|,
  21. K = 1 2 | x 1 y 2 - x 2 y 1 | . K=\tfrac{1}{2}|x_{1}y_{2}-x_{2}y_{1}|.
  22. p = a 2 + b 2 - 2 a b cos B = c 2 + d 2 - 2 c d cos D p=\sqrt{a^{2}+b^{2}-2ab\cos{B}}=\sqrt{c^{2}+d^{2}-2cd\cos{D}}
  23. q = a 2 + d 2 - 2 a d cos A = b 2 + c 2 - 2 b c cos C . q=\sqrt{a^{2}+d^{2}-2ad\cos{A}}=\sqrt{b^{2}+c^{2}-2bc\cos{C}}.
  24. p = ( a c + b d ) ( a d + b c ) - 2 a b c d ( cos B + cos D ) a b + c d p=\sqrt{\frac{(ac+bd)(ad+bc)-2abcd(\cos{B}+\cos{D})}{ab+cd}}
  25. q = ( a b + c d ) ( a c + b d ) - 2 a b c d ( cos A + cos C ) a d + b c . q=\sqrt{\frac{(ab+cd)(ac+bd)-2abcd(\cos{A}+\cos{C})}{ad+bc}}.
  26. a 2 + b 2 + c 2 + d 2 = p 2 + q 2 + 4 x 2 a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}
  27. p 2 q 2 = a 2 c 2 + b 2 d 2 - 2 a b c d cos ( A + C ) . p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos{(A+C)}.
  28. X Y = | a 2 + c 2 - b 2 - d 2 | 2 p . XY=\frac{|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}.
  29. e f g h ( a + c + b + d ) ( a + c - b - d ) = ( a g h + c e f + b e h + d f g ) ( a g h + c e f - b e h - d f g ) efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)
  30. det [ 0 a 2 p 2 d 2 1 a 2 0 b 2 q 2 1 p 2 b 2 0 c 2 1 d 2 q 2 c 2 0 1 1 1 1 1 0 ] = 0. \det\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\ a^{2}&0&b^{2}&q^{2}&1\\ p^{2}&b^{2}&0&c^{2}&1\\ d^{2}&q^{2}&c^{2}&0&1\\ 1&1&1&1&0\end{bmatrix}=0.
  31. m = 1 2 - a 2 + b 2 - c 2 + d 2 + p 2 + q 2 m=\tfrac{1}{2}\sqrt{-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}
  32. n = 1 2 a 2 - b 2 + c 2 - d 2 + p 2 + q 2 . n=\tfrac{1}{2}\sqrt{a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}.
  33. p 2 + q 2 = 2 ( m 2 + n 2 ) . \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).
  34. m = 1 2 2 ( b 2 + d 2 ) - 4 x 2 m=\tfrac{1}{2}\sqrt{2(b^{2}+d^{2})-4x^{2}}
  35. n = 1 2 2 ( a 2 + c 2 ) - 4 x 2 . n=\tfrac{1}{2}\sqrt{2(a^{2}+c^{2})-4x^{2}}.
  36. sin A + sin B + sin C + sin D = 4 sin A + B 2 sin A + C 2 sin A + D 2 \sin{A}+\sin{B}+\sin{C}+\sin{D}=4\sin{\frac{A+B}{2}}\sin{\frac{A+C}{2}}\sin{% \frac{A+D}{2}}
  37. tan A tan B - tan C tan D tan A tan C - tan B tan D = tan ( A + C ) tan ( A + B ) . \frac{\tan{A}\tan{B}-\tan{C}\tan{D}}{\tan{A}\tan{C}-\tan{B}\tan{D}}=\frac{\tan% {(A+C)}}{\tan{(A+B)}}.
  38. tan A + tan B + tan C + tan D cot A + cot B + cot C + cot D = tan A tan B tan C tan D . \frac{\tan{A}+\tan{B}+\tan{C}+\tan{D}}{\cot{A}+\cot{B}+\cot{C}+\cot{D}}=\tan{A% }\tan{B}\tan{C}\tan{D}.
  39. K 1 4 ( a + c ) ( b + d ) K\leq\tfrac{1}{4}(a+c)(b+d)
  40. K 1 4 ( a 2 + b 2 + c 2 + d 2 ) K\leq\tfrac{1}{4}(a^{2}+b^{2}+c^{2}+d^{2})
  41. K 1 4 ( p 2 + q 2 ) K\leq\tfrac{1}{4}(p^{2}+q^{2})
  42. K 1 2 ( a 2 + c 2 ) ( b 2 + d 2 ) K\leq\tfrac{1}{2}\sqrt{(a^{2}+c^{2})(b^{2}+d^{2})}
  43. K ( s - a ) ( s - b ) ( s - c ) ( s - d ) K\leq\sqrt{(s-a)(s-b)(s-c)(s-d)}
  44. K 1 2 ( a b + c d ) ( a c + b d ) ( a d + b c ) 3 . \displaystyle K\leq\tfrac{1}{2}\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.
  45. K 1 16 L 2 , K\leq\tfrac{1}{16}L^{2},
  46. K 1 2 p q K\leq\tfrac{1}{2}pq
  47. a 2 + b 2 + c 2 + d 2 p 2 + q 2 a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}
  48. p q a c + b d pq\leq ac+bd
  49. p q m 2 + n 2 , pq\leq m^{2}+n^{2},
  50. m 2 + n 2 = 1 2 ( p 2 + q 2 ) . m^{2}+n^{2}=\frac{1}{2}(p^{2}+q^{2}).
  51. a 2 + b 2 + c 2 > d 2 3 a^{2}+b^{2}+c^{2}>\frac{d^{2}}{3}
  52. a 4 + b 4 + c 4 d 4 27 . a^{4}+b^{4}+c^{4}\geq\frac{d^{4}}{27}.
  53. K 1 16 L 2 K\leq\tfrac{1}{16}L^{2}
  54. K = 1 2 p q sin θ 1 2 p q , K=\tfrac{1}{2}pq\sin{\theta}\leq\tfrac{1}{2}pq,
  55. A P + B P + C P + D P A C + B D . AP+BP+CP+DP\geq AC+BD.

Quantile.html

  1. q q
  2. q q
  3. k k
  4. q q
  5. x x
  6. x x
  7. k / q k/q
  8. x x
  9. ( q k ) / q = 1 ( k / q ) (q−k)/q=1−(k/q)
  10. q 1 q−1
  11. q q
  12. k k
  13. k k
  14. q q
  15. k k
  16. q q
  17. p p
  18. p p
  19. k / q k/q
  20. m m
  21. X X
  22. k k
  23. q q
  24. p = k / q p=k/q
  25. N N
  26. h h
  27. h h
  28. h h
  29. N N
  30. h h
  31. N p + 1 / 2 Np+1/2\,
  32. x h - 1 / 2 x_{\lceil h\,-\,1/2\rceil}
  33. N p + 1 / 2 Np+1/2\,
  34. x h - 1 / 2 + x h + 1 / 2 2 \frac{x_{\lceil h\,-\,1/2\rceil}+x_{\lfloor h\,+\,1/2\rfloor}}{2}
  35. N p Np\,
  36. x h x_{\lfloor h\rceil}\,
  37. N p Np\,
  38. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  39. N p + 1 / 2 Np+1/2\,
  40. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  41. ( N + 1 ) p (N+1)p\,
  42. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  43. ( N - 1 ) p + 1 (N-1)p+1\,
  44. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  45. ( N + 1 / 3 ) p + 1 / 3 (N+1/3)p+1/3\,
  46. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  47. ( N + 1 / 4 ) p + 3 / 8 (N+1/4)p+3/8\,
  48. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  49. ( N + 2 ) p - 1 / 2 (N+2)p-1/2\,
  50. x h + ( h - h ) ( x h + 1 - x h ) x_{\lfloor h\rfloor}+(h-\lfloor h\rfloor)(x_{\lfloor h\rfloor+1}-x_{\lfloor h% \rfloor})
  51. h h
  52. h = ( N + 1 ) / 2 h=(N+1)/2
  53. p = 1 / 2 p=1/2

Quantization_(physics).html

  1. \hbar

Quantum_chemistry.html

  1. n 3 n^{3}
  2. n n

Quantum_chromodynamics.html

  1. ψ i ( x ) \psi_{i}(x)\,
  2. i , j , i,\,j,\,\ldots
  3. 𝒜 μ a ( x ) \mathcal{A}^{a}_{\mu}(x)\,
  4. G μ ν a G^{a}_{\mu\nu}\,
  5. G μ ν a = μ 𝒜 ν a - ν 𝒜 μ a + g f a b c 𝒜 μ b 𝒜 ν c , G^{a}_{\mu\nu}=\partial_{\mu}\mathcal{A}^{a}_{\nu}-\partial_{\nu}\mathcal{A}^{% a}_{\mu}+gf^{abc}\mathcal{A}^{b}_{\mu}\mathcal{A}^{c}_{\nu}\,,
  6. r \propto r
  7. P W \,\langle P_{W}\rangle
  8. s i = ± 1 s_{i}=\pm 1\,
  9. J i , k = ϵ i J 0 ϵ k . J_{i,k}=\epsilon_{i}\,J_{0}\,\epsilon_{k}\,.
  10. ( s i s i ϵ i J i , k ϵ i J i , k ϵ k s k s k ϵ k ) . (\,s_{i}\to s_{i}\cdot\epsilon_{i}\quad\,J_{i,k}\to\epsilon_{i}J_{i,k}\epsilon% _{k}\,\quad s_{k}\to s_{k}\cdot\epsilon_{k}\,)\,.
  11. := - s i J i , k s k , {\mathcal{H}}:=-\sum s_{i}\,J_{i,k}\,s_{k}\,,
  12. J i , k J_{i,k}
  13. P W : = J i , k J k , l J n , m J m , i P_{W}:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}
  14. 0 λ w R c 0\leftarrow\lambda_{w}\ll R_{c}
  15. g G μ a ψ ¯ i γ μ T i j a ψ j , \propto gG^{a}_{\mu}\bar{\psi}_{i}\gamma^{\mu}T^{a}_{ij}\psi_{j}\,,
  16. T 5 10 12 T\cong 5\cdot 10^{12}

Quantum_computing.html

  1. n n
  2. 2 n 2^{n}
  3. 2 n 2^{n}
  4. 2 n 2^{n}
  5. n n
  6. | |{\downarrow}\rangle
  7. | |{\uparrow}\rangle
  8. | 0 |0{\rangle}
  9. | 1 |1{\rangle}
  10. 2 3 = 8 2^{3}=8
  11. | a | 2 + | b | 2 + + | h | 2 |a|^{2}+|b|^{2}+\cdots+|h|^{2}
  12. | a | 2 |a|^{2}
  13. | b | 2 |b|^{2}
  14. ( | a | 2 , | b | 2 , , | h | 2 ) (|a|^{2},|b|^{2},\ldots,|h|^{2})
  15. a | 000 + b | 001 + c | 010 + d | 011 + e | 100 + f | 101 + g | 110 + h | 111 a\,|000\rangle+b\,|001\rangle+c\,|010\rangle+d\,|011\rangle+e\,|100\rangle+f\,% |101\rangle+g\,|110\rangle+h\,|111\rangle
  16. | 010 = ( 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 ) |010\rangle=\left(0,0,1,0,0,0,0,0\right)
  17. | 0 = ( 1 , 0 ) |0\rangle=\left(1,0\right)
  18. | 1 = ( 0 , 1 ) |1\rangle=\left(0,1\right)
  19. | + = 1 2 ( 1 , 1 ) |+\rangle=\tfrac{1}{\sqrt{2}}\left(1,1\right)
  20. | - = 1 2 ( 1 , - 1 ) |-\rangle=\tfrac{1}{\sqrt{2}}\left(1,-1\right)
  21. | 000 |000\rangle

Quantum_electrodynamics.html

  1. P = | 𝐯 + 𝐰 | 2 P=|\mathbf{v}+\mathbf{w}|^{2}
  2. P = | 𝐯 𝐰 | 2 . P=|\mathbf{v}\,\mathbf{w}|^{2}.
  3. P ( A to B ) D F ( x B - x A ) , E ( C to D ) S F ( x D - x C ) P(\mbox{A to B}~{})\rightarrow D_{F}(x_{B}-x_{A}),\quad E(\mbox{C to D}~{})% \rightarrow S_{F}(x_{D}-x_{C})
  4. x A x_{A}
  5. γ μ \gamma^{\mu}
  6. ψ \psi
  7. ψ ¯ ψ γ 0 \bar{\psi}\equiv\psi^{\dagger}\gamma^{0}
  8. D μ μ + i e A μ + i e B μ D_{\mu}\equiv\partial_{\mu}+ieA_{\mu}+ieB_{\mu}\,\!
  9. F μ ν = μ A ν - ν A μ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,\!
  10. = i ψ ¯ γ μ μ ψ - e ψ ¯ γ μ ( A μ + B μ ) ψ - m ψ ¯ ψ - 1 4 F μ ν F μ ν . \mathcal{L}=i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-e\bar{\psi}\gamma_{\mu}(% A^{\mu}+B^{\mu})\psi-m\bar{\psi}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.\,
  11. μ ( ( μ ψ ) ) = μ ( i ψ ¯ γ μ ) , \partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\psi)}% \right)=\partial_{\mu}\left(i\bar{\psi}\gamma^{\mu}\right),\,
  12. ψ = - e ψ ¯ γ μ ( A μ + B μ ) - m ψ ¯ . \frac{\partial\mathcal{L}}{\partial\psi}=-e\bar{\psi}\gamma_{\mu}(A^{\mu}+B^{% \mu})-m\bar{\psi}.\,
  13. i μ ψ ¯ γ μ + e ψ ¯ γ μ ( A μ + B μ ) + m ψ ¯ = 0 i\partial_{\mu}\bar{\psi}\gamma^{\mu}+e\bar{\psi}\gamma_{\mu}(A^{\mu}+B^{\mu})% +m\bar{\psi}=0\,
  14. i γ μ μ ψ - e γ μ ( A μ + B μ ) ψ - m ψ = 0. i\gamma^{\mu}\partial_{\mu}\psi-e\gamma_{\mu}(A^{\mu}+B^{\mu})\psi-m\psi=0.\,
  15. ν ( ( ν A μ ) ) - A μ = 0 . \partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}A_{\mu})% }\right)-\frac{\partial\mathcal{L}}{\partial A_{\mu}}=0\,.
  16. ν ( ( ν A μ ) ) = ν ( μ A ν - ν A μ ) , \partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}A_{\mu})% }\right)=\partial_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right% ),\,
  17. A μ = - e ψ ¯ γ μ ψ \frac{\partial\mathcal{L}}{\partial A_{\mu}}=-e\bar{\psi}\gamma^{\mu}\psi\,
  18. μ A μ = 0 \partial_{\mu}A^{\mu}=0
  19. A μ = e ψ ¯ γ μ ψ , \Box A^{\mu}=e\bar{\psi}\gamma^{\mu}\psi\,,
  20. | i |i\rangle
  21. f | \langle f|
  22. M f i = f | U | i . M_{fi}=\langle f|U|i\rangle.
  23. V = e d 3 x ψ ¯ γ μ ψ A μ V=e\int d^{3}x\bar{\psi}\gamma^{\mu}\psi A_{\mu}
  24. U = T exp [ - i t 0 t d t V ( t ) ] U=T\exp\left[-\frac{i}{\hbar}\int_{t_{0}}^{t}dt^{\prime}V(t^{\prime})\right]
  25. d 4 p / ( 2 π ) 4 \int d^{4}p/(2\pi)^{4}
  26. M f i = ( i e ) 2 u ¯ ( p , s ) ϵ / ( k , λ ) * p / + k / + m e ( p + k ) 2 - m e 2 ϵ / ( k , λ ) u ( p , s ) + ( i e ) 2 u ¯ ( p , s ) ϵ / ( k , λ ) p / - k / + m e ( p - k ) 2 - m e 2 ϵ / ( k , λ ) * u ( p , s ) M_{fi}=(ie)^{2}\overline{u}(\vec{p}\,^{\prime},s^{\prime})\epsilon\!\!\!/\,^{% \prime}(\vec{k}\,^{\prime},\lambda^{\prime})^{*}{p\!\!\!/+k\!\!\!/+m_{e}\over(% p+k)^{2}-m^{2}_{e}}\epsilon\!\!\!/(\vec{k},\lambda)u(\vec{p},s)+(ie)^{2}% \overline{u}(\vec{p}\,^{\prime},s^{\prime})\epsilon\!\!\!/(\vec{k},\lambda){p% \!\!\!/-k\!\!\!/^{\prime}+m_{e}\over(p-k^{\prime})^{2}-m^{2}_{e}}\epsilon\!\!% \!/\,^{\prime}(\vec{k}\,^{\prime},\lambda^{\prime})^{*}u(\vec{p},s)
  27. Π \Pi\,
  28. Σ \Sigma\,
  29. Γ \Gamma\,

Quantum_entanglement.html

  1. p s y m p_{sym}
  2. A A
  3. B B
  4. H A H B . H_{A}\otimes H_{B}.
  5. | ψ A \scriptstyle|\psi\rangle_{A}
  6. | ϕ B \scriptstyle|\phi\rangle_{B}
  7. | ψ A | ϕ B . |\psi\rangle_{A}\otimes|\phi\rangle_{B}.
  8. { | i A } \scriptstyle\{|i\rangle_{A}\}
  9. { | j B } \scriptstyle\{|j\rangle_{B}\}
  10. | ψ A B = i , j c i j | i A | j B |\psi\rangle_{AB}=\sum_{i,j}c_{ij}|i\rangle_{A}\otimes|j\rangle_{B}
  11. c i A , c j B c^{A}_{i},c^{B}_{j}
  12. c i j = c i A c j B , \scriptstyle c_{ij}=c^{A}_{i}c^{B}_{j},
  13. | ψ A = i c i A | i A \scriptstyle|\psi\rangle_{A}=\sum_{i}c^{A}_{i}|i\rangle_{A}
  14. | ϕ B = j c j B | j B . \scriptstyle|\phi\rangle_{B}=\sum_{j}c^{B}_{j}|j\rangle_{B}.
  15. c i A , c j B c^{A}_{i},c^{B}_{j}
  16. c i j c i A c j B . \scriptstyle c_{ij}\neq c^{A}_{i}c^{B}_{j}.
  17. { | 0 A , | 1 A } \scriptstyle\{|0\rangle_{A},|1\rangle_{A}\}
  18. { | 0 B , | 1 B } \scriptstyle\{|0\rangle_{B},|1\rangle_{B}\}
  19. 1 2 ( | 0 A | 1 B - | 1 A | 0 B ) . \tfrac{1}{\sqrt{2}}\left(|0\rangle_{A}\otimes|1\rangle_{B}-|1\rangle_{A}% \otimes|0\rangle_{B}\right).
  20. A A
  21. B B
  22. A A
  23. B B
  24. { | 0 , | 1 } \scriptstyle\{|0\rangle,|1\rangle\}
  25. A A
  26. | 0 A | 1 B \scriptstyle|0\rangle_{A}|1\rangle_{B}
  27. | 1 A | 0 B \scriptstyle|1\rangle_{A}|0\rangle_{B}
  28. B B
  29. A A
  30. A A
  31. B B
  32. ρ = i w i | α i α i | , \rho=\sum_{i}w_{i}|\alpha_{i}\rangle\langle\alpha_{i}|,
  33. ρ ρ
  34. | α i |\alpha_{i}\rangle
  35. | 𝐳 + |\mathbf{z}+\rangle
  36. 𝐳 \mathbf{z}
  37. | 𝐲 - |\mathbf{y}-\rangle
  38. 𝐲 \mathbf{y}
  39. ρ = i w i [ j c ¯ i j ( | α i j | β i j ) ] [ k c i k ( α i k | β i k | ) ] \rho=\sum_{i}w_{i}\left[\sum_{j}\bar{c}_{ij}(|\alpha_{ij}\rangle\otimes|\beta_% {ij}\rangle)\right]\otimes\left[\sum_{k}c_{ik}(\langle\alpha_{ik}|\otimes% \langle\beta_{ik}|)\right]
  40. j | c i j | 2 = 1 \sum_{j}|c_{ij}|^{2}=1
  41. ρ = i w i ρ i A ρ i B , \rho=\sum_{i}w_{i}\rho_{i}^{A}\otimes\rho_{i}^{B},
  42. ρ i A \rho_{i}^{A}
  43. ρ i B \rho_{i}^{B}
  44. A A
  45. B B
  46. ρ i A \rho_{i}^{A}
  47. ρ i B \rho_{i}^{B}
  48. 2 × 2 2×2
  49. 2 × 3 2×3
  50. A A
  51. B B
  52. | Ψ H A H B . |\Psi\rangle\in H_{A}\otimes H_{B}.
  53. A A
  54. ρ T = | Ψ Ψ | \rho_{T}=|\Psi\rangle\;\langle\Psi|
  55. A A
  56. B B
  57. ρ A = def j j | B ( | Ψ Ψ | ) | j B = Tr B ρ T . \rho_{A}\ \stackrel{\mathrm{def}}{=}\ \sum_{j}\langle j|_{B}\left(|\Psi\rangle% \langle\Psi|\right)|j\rangle_{B}=\hbox{Tr}_{B}\;\rho_{T}.
  58. ρ ρ
  59. A A
  60. B B
  61. A A
  62. A A
  63. 1 2 ( | 0 A | 1 B - | 1 A | 0 B ) , \tfrac{1}{\sqrt{2}}\left(|0\rangle_{A}\otimes|1\rangle_{B}-|1\rangle_{A}% \otimes|0\rangle_{B}\right),
  64. ρ A = 1 2 ( | 0 A 0 | A + | 1 A 1 | A ) \rho_{A}=\tfrac{1}{2}\left(|0\rangle_{A}\langle 0|_{A}+|1\rangle_{A}\langle 1|% _{A}\right)
  65. A A
  66. | ψ A | ϕ B |\psi\rangle_{A}\otimes|\phi\rangle_{B}
  67. ρ A = | ψ A ψ | A . \rho_{A}=|\psi\rangle_{A}\langle\psi|_{A}.
  68. H H
  69. p 1 , , p n p_{1},\cdots,p_{n}
  70. H ( p 1 , , p n ) = - i p i log 2 p i . H(p_{1},\cdots,p_{n})=-\sum_{i}p_{i}\log_{2}p_{i}.
  71. ρ ρ
  72. S ( ρ ) = - Tr ( ρ log 2 ρ ) . S(\rho)=-\hbox{Tr}\left(\rho\log_{2}{\rho}\right).
  73. ρ ρ
  74. λ 1 , , λ n \lambda_{1},\cdots,\lambda_{n}
  75. log 2 ( λ 1 ) , , log 2 ( λ n ) \log_{2}(\lambda_{1}),\cdots,\log_{2}(\lambda_{n})
  76. S ( ρ ) = - Tr ( ρ log 2 ρ ) = - i λ i log 2 λ i S(\rho)=-\hbox{Tr}\left(\rho\log_{2}{\rho}\right)=-\sum_{i}\lambda_{i}\log_{2}% \lambda_{i}
  77. lim p 0 p log p = 0 , \lim_{p\to 0}p\log p=0,
  78. 0 l o g ( 0 ) = 0 0log(0)=0
  79. ρ ρ
  80. ρ = λ d P λ , \rho=\int\lambda dP_{\lambda},
  81. ρ log 2 ρ = λ log 2 λ d P λ . \rho\log_{2}\rho=\int\lambda\log_{2}\lambda dP_{\lambda}.
  82. l o g ( 2 ) log(2)
  83. 2 × 2 2×2
  84. ρ ρ
  85. [ 1 n 1 n ] . \begin{bmatrix}\frac{1}{n}&&\\ &\ddots&\\ &&\frac{1}{n}\end{bmatrix}.
  86. k = 1 k=1
  87. U U
  88. S ( ρ ) = S ( U ρ U * ) . S(\rho)=S\left(U\rho U^{*}\right).
  89. U U
  90. U ( t ) = exp ( - i H t ) , U(t)=\exp\left(\frac{-iHt}{\hbar}\right),
  91. H H
  92. | Φ ± = 1 2 ( | 0 A | 0 B ± | 1 A | 1 B ) |\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}\pm|1% \rangle_{A}\otimes|1\rangle_{B})
  93. | Ψ ± = 1 2 ( | 0 A | 1 B ± | 1 A | 0 B ) |\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|1\rangle_{B}\pm|1% \rangle_{A}\otimes|0\rangle_{B})
  94. | GHZ = | 0 M + | 1 M 2 , |\mathrm{GHZ}\rangle=\frac{|0\rangle^{\otimes M}+|1\rangle^{\otimes M}}{\sqrt{% 2}},
  95. | Φ + |\Phi^{+}\rangle
  96. M = 2 M=2
  97. M = 3 M=3
  98. | ψ NOON = | N a | 0 b + | 0 a | N b 2 , |\psi\text{NOON}\rangle=\frac{|N\rangle_{a}|0\rangle_{b}+|{0}\rangle_{a}|{N}% \rangle_{b}}{\sqrt{2}},\,
  99. | Φ + |\Phi^{+}\rangle

Quantum_field_theory.html

  1. x ^ \hat{x}
  2. p ^ \hat{p}
  3. \mathcal{L}
  4. x μ [ ( ϕ / x μ ) ] - ϕ = 0 , \frac{\partial}{\partial x^{\mu}}\left[\frac{\partial\mathcal{L}}{\partial(% \partial\phi/\partial x^{\mu})}\right]-\frac{\partial\mathcal{L}}{\partial\phi% }=0,
  5. ( ϕ , ϕ ) = - ρ ( t , 𝐱 ) ϕ ( t , 𝐱 ) - 1 8 π G | ϕ | 2 , \mathcal{L}(\phi,\nabla\phi)=-\rho(t,\mathbf{x})\,\phi(t,\mathbf{x})-\frac{1}{% 8\pi G}|\nabla\phi|^{2},
  6. 4 π G ρ ( t , 𝐱 ) = 2 ϕ . 4\pi G\rho(t,\mathbf{x})=\nabla^{2}\phi.
  7. ψ ( x , t ) ψ(x,t)
  8. - 2 2 m 2 x 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) = i t ψ ( x , t ) . -\frac{{\hbar}^{2}}{2m}\frac{{\partial}^{2}}{\partial x^{2}}\psi(x,t)+V(x)\psi% (x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t).
  9. m m
  10. V ( x ) V(x)
  11. V ( x ) V(x)
  12. N N
  13. N N
  14. | ϕ 1 ϕ N = j N j ! N ! p S N | ϕ p ( 1 ) | ϕ p ( N ) , |\phi_{1}\cdots\phi_{N}\rangle=\sqrt{\frac{\prod_{j}N_{j}!}{N!}}\sum_{p\in S_{% N}}|\phi_{p(1)}\rangle\otimes\cdots\otimes|\phi_{p(N)}\rangle,
  15. | ϕ i |\phi_{i}\rangle
  16. j j
  17. p p
  18. N N
  19. N ! N!
  20. N N
  21. j N j ! N ! \sqrt{\frac{\prod_{j}N_{j}!}{N!}}
  22. | ϕ 1 , | ϕ 2 , | ϕ 3 , |\phi_{1}\rangle,|\phi_{2}\rangle,|\phi_{3}\rangle,
  23. | ϕ 1 |\phi_{1}\rangle
  24. | ϕ 2 |\phi_{2}\rangle
  25. 1 3 [ | ϕ 1 | ϕ 2 | ϕ 2 + | ϕ 2 | ϕ 1 | ϕ 2 + | ϕ 2 | ϕ 2 | ϕ 1 ] . \frac{1}{\sqrt{3}}\left[|\phi_{1}\rangle|\phi_{2}\rangle|\phi_{2}\rangle+|\phi% _{2}\rangle|\phi_{1}\rangle|\phi_{2}\rangle+|\phi_{2}\rangle|\phi_{2}\rangle|% \phi_{1}\rangle\right].
  26. | ϕ 1 , | ϕ 2 , |\phi_{1}\rangle,|\phi_{2}\rangle,
  27. | 1 , 2 , 0 , 0 , 0 , . |1,2,0,0,0,\dots\rangle.
  28. | 0 |0\rangle
  29. j j
  30. | ϕ 1 , | ϕ 2 , , | ϕ j , |\phi_{1}\rangle,|\phi_{2}\rangle,\dots,|\phi_{j}\rangle,\dots
  31. | N 1 , N 2 , N 3 , , N j , . |N_{1},N_{2},N_{3},\dots,N_{j},\dots\rangle.
  32. a 2 a_{2}
  33. a 2 a_{2}^{\dagger}
  34. a 2 | N 1 , N 2 , N 3 , = N 2 N 1 , ( N 2 - 1 ) , N 3 , , a_{2}|N_{1},N_{2},N_{3},\dots\rangle=\sqrt{N_{2}}\mid N_{1},(N_{2}-1),N_{3},% \dots\rangle,
  35. a 2 | N 1 , N 2 , N 3 , = N 2 + 1 N 1 , ( N 2 + 1 ) , N 3 , . a_{2}^{\dagger}|N_{1},N_{2},N_{3},\dots\rangle=\sqrt{N_{2}+1}\mid N_{1},(N_{2}% +1),N_{3},\dots\rangle.
  36. [ a i , a j ] = 0 , [ a i , a j ] = 0 , [ a i , a j ] = δ i j , \left[a_{i},a_{j}\right]=0\quad,\quad\left[a_{i}^{\dagger},a_{j}^{\dagger}% \right]=0\quad,\quad\left[a_{i},a_{j}^{\dagger}\right]=\delta_{ij},
  37. δ \delta
  38. a k a_{k}
  39. a k a_{k}^{\dagger}
  40. N k N_{k}
  41. a k a k | , N k , = N k | , N k , . a_{k}^{\dagger}\,a_{k}|\dots,N_{k},\dots\rangle=N_{k}|\dots,N_{k},\dots\rangle.
  42. a k a k a_{k}^{\dagger}a_{k}
  43. E k E_{k}
  44. N k N_{k}
  45. E k N k E_{k}N_{k}
  46. k k
  47. E tot = k E k N k E_{\mathrm{tot}}=\sum_{k}E_{k}N_{k}
  48. N k N_{k}
  49. a k a k a_{k}^{\dagger}a_{k}
  50. H = k E k a k a k . H=\sum_{k}E_{k}\,a^{\dagger}_{k}\,a_{k}.
  51. c c^{\dagger}
  52. c j | N 1 , N 2 , , N j = 0 , = 0 c_{j}|N_{1},N_{2},\dots,N_{j}=0,\dots\rangle=0
  53. c j | N 1 , N 2 , , N j = 1 , = ( - 1 ) ( N 1 + + N j - 1 ) | N 1 , N 2 , , N j = 0 , c_{j}|N_{1},N_{2},\dots,N_{j}=1,\dots\rangle=(-1)^{(N_{1}+\cdots+N_{j-1})}|N_{% 1},N_{2},\dots,N_{j}=0,\dots\rangle
  54. c j | N 1 , N 2 , , N j = 0 , = ( - 1 ) ( N 1 + + N j - 1 ) | N 1 , N 2 , , N j = 1 , c_{j}^{\dagger}|N_{1},N_{2},\dots,N_{j}=0,\dots\rangle=(-1)^{(N_{1}+\cdots+N_{% j-1})}|N_{1},N_{2},\dots,N_{j}=1,\dots\rangle
  55. c j | N 1 , N 2 , , N j = 1 , = 0. c_{j}^{\dagger}|N_{1},N_{2},\dots,N_{j}=1,\dots\rangle=0.
  56. { c i , c j } = 0 , { c i , c j } = 0 , { c i , c j } = δ i j . \left\{c_{i},c_{j}\right\}=0\quad,\quad\left\{c_{i}^{\dagger},c_{j}^{\dagger}% \right\}=0\quad,\quad\left\{c_{i},c_{j}^{\dagger}\right\}=\delta_{ij}.
  57. ϕ ( 𝐫 ) \phi(\mathbf{r})
  58. ϕ ( 𝐫 ) = def j e i 𝐤 j 𝐫 a j . \phi(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \sum_{j}e^{i\mathbf{k}_{j}\cdot% \mathbf{r}}a_{j}.
  59. [ ϕ ( 𝐫 ) , ϕ ( 𝐫 ) ] = 0 , [ ϕ ( 𝐫 ) , ϕ ( 𝐫 ) ] = 0 , [ ϕ ( 𝐫 ) , ϕ ( 𝐫 ) ] = δ 3 ( 𝐫 - 𝐫 ) \left[\phi(\mathbf{r}),\phi(\mathbf{r^{\prime}})\right]=0\quad,\quad\left[\phi% ^{\dagger}(\mathbf{r}),\phi^{\dagger}(\mathbf{r^{\prime}})\right]=0\quad,\quad% \left[\phi(\mathbf{r}),\phi^{\dagger}(\mathbf{r^{\prime}})\right]=\delta^{3}(% \mathbf{r}-\mathbf{r^{\prime}})
  60. δ ( x ) \delta(x)
  61. H = - 2 2 m i i 2 + i < j U ( | 𝐫 i - 𝐫 j | ) H=-\frac{\hbar^{2}}{2m}\sum_{i}\nabla_{i}^{2}+\sum_{i<j}U(|\mathbf{r}_{i}-% \mathbf{r}_{j}|)
  62. H = - 2 2 m d 3 r ϕ ( 𝐫 ) 2 ϕ ( 𝐫 ) + 1 2 d 3 r d 3 r ϕ ( 𝐫 ) ϕ ( 𝐫 ) U ( | 𝐫 - 𝐫 | ) ϕ ( 𝐫 ) ϕ ( 𝐫 ) . H=-\frac{\hbar^{2}}{2m}\int d^{3}\!r\ \phi^{\dagger}(\mathbf{r})\nabla^{2}\phi% (\mathbf{r})+\frac{1}{2}\int\!d^{3}\!r\int\!d^{3}\!r^{\prime}\;\phi^{\dagger}(% \mathbf{r})\phi^{\dagger}(\mathbf{r}^{\prime})U(|\mathbf{r}-\mathbf{r}^{\prime% }|)\phi(\mathbf{r^{\prime}})\phi(\mathbf{r}).
  63. ϕ \phi
  64. N ^ \hat{N}
  65. N ^ \hat{N}
  66. a k a_{k}
  67. a k a_{k}^{\dagger}
  68. N ^ \hat{N}
  69. H I = k , q V q ( a q + a - q ) c k + q c k , H_{I}=\sum_{k,q}V_{q}(a_{q}+a_{-q}^{\dagger})c_{k+q}^{\dagger}c_{k},
  70. a k a_{k}^{\dagger}
  71. a k a_{k}
  72. c k c_{k}^{\dagger}
  73. c k c_{k}
  74. V q V_{q}
  75. k + q k+q
  76. Λ ( x i ) \Lambda(x_{i})
  77. x i x j Λ = x j x i Λ . \partial_{x_{i}x_{j}}\Lambda=\partial_{x_{j}x_{i}}\Lambda.
  78. Λ ( x i ) \Lambda(x_{i})

Quantum_Hall_effect.html

  1. σ = I channel V Hall = ν e 2 h , \sigma=\frac{I\text{channel}}{V\text{Hall}}=\nu\;\frac{e^{2}}{h},
  2. I channel I\text{channel}
  3. V Hall V\text{Hall}
  4. E n = ω c ( n + 1 / 2 ) , \textstyle E_{n}=\hbar\omega_{c}(n+1/2),
  5. N = g s B A / ϕ 0 , \textstyle N=g_{s}BA/\phi_{0},

Quantum_harmonic_oscillator.html

  1. H ^ = p ^ 2 2 m + 1 2 m ω 2 x ^ 2 , \hat{H}=\frac{{\hat{p}}^{2}}{2m}+\frac{1}{2}m\omega^{2}{\hat{x}}^{2}\,,
  2. m m
  3. ω ω
  4. x x
  5. p ^ = - i x . \hat{p}=-i\hbar{\partial\over\partial x}\,.
  6. H ^ | ψ = E | ψ , \hat{H}\left|\psi\right\rangle=E\left|\psi\right\rangle\,,
  7. E E
  8. | ψ |ψ⟩
  9. x | ψ = ψ ( x ) ⟨x|ψ⟩=ψ(x)
  10. ψ n ( x ) = 1 2 n n ! ( m ω π ) 1 / 4 e - m ω x 2 2 H n ( m ω x ) , n = 0 , 1 , 2 , . \psi_{n}(x)=\frac{1}{\sqrt{2^{n}\,n!}}\cdot\left(\frac{m\omega}{\pi\hbar}% \right)^{1/4}\cdot e^{-\frac{m\omega x^{2}}{2\hbar}}\cdot H_{n}\left(\sqrt{% \frac{m\omega}{\hbar}}x\right),\qquad n=0,1,2,\ldots.
  11. H n ( x ) = ( - 1 ) n e x 2 d n d x n ( e - x 2 ) . H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}\left(e^{-x^{2}}\right).
  12. E n = ω ( n + 1 2 ) = ( 2 n + 1 ) 2 ω . E_{n}=\hbar\omega\left(n+{1\over 2}\right)=(2n+1){\hbar\over 2}\omega.
  13. ħ ω ħω
  14. n = 0 n=0
  15. ħ ω / 2 ħω/2
  16. a a
  17. a = m ω 2 ( x ^ + i m ω p ^ ) a = m ω 2 ( x ^ - i m ω p ^ ) \begin{aligned}\displaystyle a&\displaystyle=\sqrt{m\omega\over 2\hbar}\left(% \hat{x}+{i\over m\omega}\hat{p}\right)\\ \displaystyle a^{\dagger}&\displaystyle=\sqrt{m\omega\over 2\hbar}\left(\hat{x% }-{i\over m\omega}\hat{p}\right)\end{aligned}
  18. x ^ = 2 m ω ( a + a ) \hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})
  19. p ^ = i m ω 2 ( a - a ) . \hat{p}=i\sqrt{\frac{m\omega\hbar}{2}}(a^{\dagger}-a)~{}.
  20. a a
  21. a | n = n + 1 | n + 1 a^{\dagger}|n\rangle=\sqrt{n+1}\,|n+1\rangle
  22. a | n = n n - 1 . a|n\rangle=\sqrt{n}\mid n-1\rangle.
  23. a a
  24. N N
  25. N = a a N=a^{\dagger}a
  26. N | n = n | n . N\left|n\right\rangle=n\left|n\right\rangle.
  27. [ a , a ] = 1 , [ N , a ] = a , [ N , a ] = - a , [a,a^{\dagger}]=1,\qquad[N,a^{\dagger}]=a^{\dagger},\qquad[N,a]=-a,
  28. H = ( N + 1 2 ) ω , H=\left(N+\frac{1}{2}\right)\hbar\omega,
  29. N N
  30. N a | n = ( a N + [ N , a ] ) | n = ( a N + a ) | n = ( n + 1 ) a | n , \begin{aligned}\displaystyle Na^{\dagger}|n\rangle&\displaystyle=\left(a^{% \dagger}N+[N,a^{\dagger}]\right)|n\rangle\\ &\displaystyle=\left(a^{\dagger}N+a^{\dagger}\right)|n\rangle\\ &\displaystyle=(n+1)a^{\dagger}|n\rangle,\end{aligned}
  31. N a | n = ( n - 1 ) a n . Na|n\rangle=(n-1)a\mid n\rangle.
  32. a a
  33. | n |n⟩
  34. | n 1 |n–1⟩
  35. | n |n⟩
  36. | n + 1 |n+1⟩
  37. a a
  38. a a
  39. a a
  40. ħ ω ħω
  41. E = E=−∞
  42. n = n N n = n a a n = ( a n ) a n 0 , n=\langle n\mid N\mid n\rangle=\langle n\mid a^{\dagger}a\mid n\rangle=\left(a% \mid n\rangle\right)^{\dagger}a\mid n\rangle\geqslant 0,
  43. a | 0 = 0. a\left|0\right\rangle=0.
  44. H | 0 = ω 2 | 0 H\left|0\right\rangle=\frac{\hbar\omega}{2}\left|0\right\rangle
  45. { | 0 , | 1 , | 2 , , | n , } , \left\{\left|0\right\rangle,\left|1\right\rangle,\left|2\right\rangle,\ldots,% \left|n\right\rangle,\ldots\right\},
  46. H | n = ω ( n + 1 2 ) | n , H\left|n\right\rangle=\hbar\omega\left(n+\frac{1}{2}\right)\left|n\right\rangle,
  47. | n = ( a ) n n ! | 0 . |n\rangle=\frac{(a^{\dagger})^{n}}{\sqrt{n!}}|0\rangle.
  48. n a a | n = n | ( [ a , a ] + a a ) n = n | ( N + 1 ) | n = n + 1 a n = n + 1 n + 1 | n = a n n - 1 = ( a ) 2 n ( n - 1 ) n - 2 = = ( a ) n n ! | 0 . \begin{aligned}\displaystyle\langle n\mid aa^{\dagger}|n\rangle&\displaystyle=% \langle n|\left([a,a^{\dagger}]+a^{\dagger}a\right)\mid n\rangle=\langle n|(N+% 1)|n\rangle=n+1\\ \displaystyle\Rightarrow a^{\dagger}\mid n\rangle&\displaystyle=\sqrt{n+1}\mid n% +1\rangle\\ \displaystyle\Rightarrow|n\rangle&\displaystyle=\frac{a^{\dagger}}{\sqrt{n}}% \mid n-1\rangle=\frac{(a^{\dagger})^{2}}{\sqrt{n(n-1)}}\mid n-2\rangle=\cdots=% \frac{(a^{\dagger})^{n}}{\sqrt{n!}}|0\rangle.\end{aligned}
  49. x a 0 = 0 ( x + m ω d d x ) x 0 = 0 x 0 = ( m ω π ) 1 / 4 exp ( - m ω 2 x 2 ) = ψ 0 , \begin{aligned}&\displaystyle\left\langle x\mid a\mid 0\right\rangle=0~{}~{}~{% }~{}~{}~{}~{}~{}~{}~{}\Longrightarrow\\ &\displaystyle\left(x+\frac{\hbar}{m\omega}\frac{d}{dx}\right)\left\langle x% \mid 0\right\rangle=0~{}~{}~{}~{}~{}~{}\Longrightarrow\\ &\displaystyle\left\langle x\mid 0\right\rangle=\left(\frac{m\omega}{\pi\hbar}% \right)^{1/4}\exp\left(-\frac{m\omega}{2\hbar}x^{2}\right)=\psi_{0}~{},\end{aligned}
  50. x a 0 = ψ 1 , \langle x\mid a^{\dagger}\mid 0\rangle=\psi_{1}~{},
  51. ħ ω ħω
  52. ħ / ( m ω ) \sqrt{ħ}{/(}{mω}{)}
  53. H = - 1 2 d 2 d x 2 + 1 2 x 2 , H=-\tfrac{1}{2}{d^{2}\over dx^{2}}+\tfrac{1}{2}x^{2},
  54. ψ n ( x ) x n = 1 2 n n ! π - 1 / 4 exp ( - x 2 / 2 ) H n ( x ) , \psi_{n}(x)\equiv\left\langle x\mid n\right\rangle={1\over\sqrt{2^{n}n!}}~{}% \pi^{-1/4}\exp(-x^{2}/2)H_{n}(x),
  55. E n = n + 1 2 , E_{n}=n+\tfrac{1}{2},
  56. x exp ( - i t H ) y K ( x , y ; t ) = 1 2 π i sin t exp ( i 2 sin t ( ( x 2 + y 2 ) cos t - 2 x y ) ) , \langle x\mid\exp(-itH)\mid y\rangle\equiv K(x,y;t)=\frac{1}{\sqrt{2\pi i\sin t% }}\exp\left(\frac{i}{2\sin t}\left((x^{2}+y^{2})\cos t-2xy\right)\right)~{},
  57. K ( x , y ; 0 ) = δ ( x y ) K(x,y;0)=δ(x−y)
  58. ψ ( x , 0 ) ψ(x,0)
  59. ψ ( x , t ) = d y K ( x , y ; t ) ψ ( y , 0 ) . \psi(x,t)=\int dy~{}K(x,y;t)\psi(y,0)~{}.
  60. F n ( u ) = ( - 1 ) n π L n ( 4 u ω ) e - 2 u / ω , F_{n}(u)=\frac{(-1)^{n}}{\pi\hbar}L_{n}\left(4\frac{u}{\hbar\omega}\right)e^{-% 2u/\hbar\omega}~{},
  61. u = 1 2 m ω 2 x 2 + p 2 2 m u=\frac{1}{2}m\omega^{2}x^{2}+\frac{p^{2}}{2m}
  62. [ x i , p j ] = i δ i , j [ x i , x j ] = 0 [ p i , p j ] = 0 \begin{aligned}\displaystyle{[}x_{i},p_{j}{]}&\displaystyle=i\hbar\delta_{i,j}% \\ \displaystyle{[}x_{i},x_{j}{]}&\displaystyle=0\\ \displaystyle{[}p_{i},p_{j}{]}&\displaystyle=0\end{aligned}
  63. H = i = 1 N ( p i 2 2 m + 1 2 m ω 2 x i 2 ) . H=\sum_{i=1}^{N}\left({p_{i}^{2}\over 2m}+{1\over 2}m\omega^{2}x_{i}^{2}\right).
  64. r 2 r^{2}
  65. 𝐱 | ψ { n } = i = 1 N x i ψ n i \langle\mathbf{x}|\psi_{\{n\}}\rangle=\prod_{i=1}^{N}\langle x_{i}\mid\psi_{n_% {i}}\rangle
  66. a i = m ω 2 ( x i + i m ω p i ) , a i = m ω 2 ( x i - i m ω p i ) . \begin{aligned}\displaystyle a_{i}&\displaystyle=\sqrt{m\omega\over 2\hbar}% \left(x_{i}+{i\over m\omega}p_{i}\right),\\ \displaystyle a^{\dagger}_{i}&\displaystyle=\sqrt{m\omega\over 2\hbar}\left(x_% {i}-{i\over m\omega}p_{i}\right).\end{aligned}
  67. H = ω i = 1 N ( a i a i + 1 2 ) . H=\hbar\omega\,\sum_{i=1}^{N}\left(a_{i}^{\dagger}\,a_{i}+\frac{1}{2}\right).
  68. U a i U = j = 1 N a j U j i for all U U ( N ) , U\,a_{i}^{\dagger}\,U^{\dagger}=\sum_{j=1}^{N}a_{j}^{\dagger}\,U_{ji}\quad\,% \text{for all}\quad U\in U(N),
  69. U j i U_{ji}
  70. E = ω [ ( n 1 + + n N ) + N 2 ] . E=\hbar\omega\left[(n_{1}+\cdots+n_{N})+{N\over 2}\right].
  71. n i = 0 , 1 , 2 , ( the energy level in dimension i ) . n_{i}=0,1,2,\dots\quad(\,\text{the energy level in dimension }i).
  72. g n = n 1 = 0 n n - n 1 + 1 = ( n + 1 ) ( n + 2 ) 2 g_{n}=\sum_{n_{1}=0}^{n}n-n_{1}+1=\frac{(n+1)(n+2)}{2}
  73. g n = ( N + n - 1 n ) g_{n}={\left({{N+n-1}\atop{n}}\right)}
  74. g n = p ( N - , n ) g_{n}=p(N_{-},n)
  75. k = 0 k n k = n \sum_{k=0}^{\infty}kn_{k}=n
  76. k = 0 n k = N \sum_{k=0}^{\infty}n_{k}=N
  77. V ( r ) = 1 2 μ ω 2 r 2 , V(r)={1\over 2}\mu\omega^{2}r^{2},
  78. μ μ
  79. m m
  80. μ μ
  81. m m
  82. ψ k l m ( r , θ , ϕ ) = N k l r l e - ν r 2 L k ( l + 1 2 ) ( 2 ν r 2 ) Y l m ( θ , ϕ ) \psi_{klm}(r,\theta,\phi)=N_{kl}r^{l}e^{-\nu r^{2}}L_{k}^{(l+{1\over 2})}(2\nu r% ^{2})Y_{lm}(\theta,\phi)
  83. N k l = 2 ν 3 π 2 k + 2 l + 3 k ! ν l ( 2 k + 2 l + 1 ) ! ! N_{kl}=\sqrt{\sqrt{\frac{2\nu^{3}}{\pi}}\frac{2^{k+2l+3}\;k!\;\nu^{l}}{(2k+2l+% 1)!!}}~{}~{}
  84. ν μ ω 2 \nu\equiv{\mu\omega\over 2\hbar}~{}
  85. L k ( l + 1 2 ) ( 2 ν r 2 ) {L_{k}}^{(l+{1\over 2})}(2\nu r^{2})
  86. k k
  87. Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi)\,
  88. ħ ħ
  89. h 2 π . \hbar\equiv\frac{h}{2\pi}~{}.
  90. E = ω ( 2 k + l + 3 2 ) . E=\hbar\omega\left(2k+l+\frac{3}{2}\right)~{}.
  91. n 2 k + l . n\equiv 2k+l~{}.
  92. k k
  93. n n
  94. = 0 , 2 , , n 2 , n ℓ=0,2,...,n−2,n
  95. n n
  96. = 1 , 3 , , n 2 , n ℓ=1,3,...,n− 2,n
  97. m m
  98. m −ℓ≤m≤ℓ
  99. n n
  100. m m
  101. n n
  102. l = , n - 2 , n ( 2 l + 1 ) = ( n + 1 ) ( n + 2 ) 2 , \sum_{l=\ldots,n-2,n}(2l+1)={(n+1)(n+2)\over 2}~{},
  103. n n
  104. S U ( 3 ) SU(3)
  105. i i
  106. 𝐇 = i = 1 N p i 2 2 m + 1 2 m ω 2 { i j } ( n n ) ( x i - x j ) 2 , \mathbf{H}=\sum_{i=1}^{N}{p_{i}^{2}\over 2m}+{1\over 2}m\omega^{2}\sum_{\{ij\}% (nn)}(x_{i}-x_{j})^{2}~{},
  107. m m
  108. N N
  109. x x
  110. N N
  111. Π Π
  112. p p
  113. Q k = 1 N l e i k a l x l Q_{k}={1\over\sqrt{N}}\sum_{l}e^{ikal}x_{l}
  114. Π k = 1 N l e - i k a l p l . \Pi_{k}={1\over\sqrt{N}}\sum_{l}e^{-ikal}p_{l}~{}.
  115. [ x l , p m ] \displaystyle\left[x_{l},p_{m}\right]
  116. l x l x l + m \displaystyle\sum_{l}x_{l}x_{l+m}
  117. 1 2 m ω 2 j ( x j - x j + 1 ) 2 = 1 2 m ω 2 k Q k Q - k ( 2 - e i k a - e - i k a ) = 1 2 m k ω k 2 Q k Q - k , {1\over 2}m\omega^{2}\sum_{j}(x_{j}-x_{j+1})^{2}={1\over 2}m\omega^{2}\sum_{k}% Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={1\over 2}m\sum_{k}{\omega_{k}}^{2}Q_{k}Q_{-k}% ~{},
  118. ω k = 2 ω 2 ( 1 - cos ( k a ) ) . \omega_{k}=\sqrt{2\omega^{2}(1-\cos(ka))}~{}.
  119. 𝐇 = 1 2 m k ( Π k Π - k + m 2 ω k 2 Q k Q - k ) . \mathbf{H}={1\over{2m}}\sum_{k}\left({\Pi_{k}\Pi_{-k}}+m^{2}\omega_{k}^{2}Q_{k% }Q_{-k}\right)~{}.
  120. Q Q
  121. Π Π
  122. N N
  123. ( N + 1 ) (N+1)
  124. k = k n = 2 n π N a for n = 0 , ± 1 , ± 2 , , ± N 2 . k=k_{n}={2n\pi\over Na}\quad\hbox{for}\ n=0,\pm 1,\pm 2,\ldots,\pm{N\over 2}.
  125. n n
  126. a a
  127. E n = ( 1 2 + n ) ω k n = 0 , 1 , 2 , 3 , E_{n}=\left({1\over 2}+n\right)\hbar\omega_{k}\quad\quad\quad n=0,1,2,3,\ldots
  128. ω , 2 ω , 3 ω , \hbar\omega,\,2\hbar\omega,\,3\hbar\omega,\,\ldots
  129. ħ ω ħω
  130. ω = k μ \omega=\sqrt{\frac{k}{\mu}}
  131. μ = m < s u b > 1 m 2 / ( m 1 + m 2 ) μ=m<sub>1m_{2}/(m_{1}+m_{2})

Quantum_information.html

  1. ρ \rho
  2. S ( ρ ) = - Tr ( ρ ln ρ ) . S(\rho)=-\operatorname{Tr}(\rho\ln\rho).\,

Quantum_key_distribution.html

  1. p p
  2. p p
  3. Z 0 , Z π 8 , Z π 4 Z_{0},Z_{\frac{\pi}{8}},Z_{\frac{\pi}{4}}
  4. Z 0 , Z π 8 , Z - π 8 Z_{0},Z_{\frac{\pi}{8}},Z_{-\frac{\pi}{8}}
  5. Z θ Z_{\theta}
  6. { | , | } \{|\uparrow\rangle,\;|\downarrow\rangle\}
  7. θ \theta
  8. S S
  9. | S | = - 2 2 |S|=-2\sqrt{2}
  10. n n
  11. P d = 1 - ( 3 4 ) n P_{d}=1-\left(\frac{3}{4}\right)^{n}
  12. P d = 0.999999999 P_{d}=0.999999999
  13. n = 72 n=72
  14. t 2 t^{2}
  15. t t
  16. t t
  17. t 3 / 2 t^{3/2}
  18. t t

Quantum_mechanics.html

  1. E = h ν E=h\nu
  2. - e 2 / ( 4 π ϵ 0 r ) \scriptstyle-e^{2}/(4\pi\ \epsilon_{{}_{0}}\ r)
  3. V ( x ) = { 0 , x < 0 , V 0 , x 0. V(x)=\begin{cases}0,&x<0,\\ V_{0},&x\geq 0.\end{cases}
  4. ψ 1 ( x ) = 1 k 1 ( A e i k 1 x + A e - i k 1 x ) x < 0 \psi_{1}(x)=\frac{1}{\sqrt{k_{1}}}\left(A_{\rightarrow}e^{ik_{1}x}+A_{% \leftarrow}e^{-ik_{1}x}\right)\quad x<0
  5. ψ 2 ( x ) = 1 k 2 ( B e i k 2 x + B e - i k 2 x ) x > 0 \psi_{2}(x)=\frac{1}{\sqrt{k_{2}}}\left(B_{\rightarrow}e^{ik_{2}x}+B_{% \leftarrow}e^{-ik_{2}x}\right)\quad x>0
  6. k 1 = 2 m E / 2 k_{1}=\sqrt{2mE/\hbar^{2}}
  7. k 2 = 2 m ( E - V 0 ) / 2 k_{2}=\sqrt{2m(E-V_{0})/\hbar^{2}}
  8. x x
  9. - 2 2 m d 2 ψ d x 2 = E ψ . -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}=E\psi.
  10. p ^ x = - i d d x \hat{p}_{x}=-i\hbar\frac{d}{dx}
  11. 1 2 m p ^ x 2 = E , \frac{1}{2m}\hat{p}_{x}^{2}=E,
  12. ψ \psi
  13. E E
  14. ψ ( x ) = A e i k x + B e - i k x E = 2 k 2 2 m \psi(x)=Ae^{ikx}+Be^{-ikx}\qquad\qquad E=\frac{\hbar^{2}k^{2}}{2m}
  15. ψ ( x ) = C sin k x + D cos k x . \psi(x)=C\sin kx+D\cos kx.\!
  16. ψ ( 0 ) = 0 = C sin 0 + D cos 0 = D \psi(0)=0=C\sin 0+D\cos 0=D\!
  17. ψ ( L ) = 0 = C sin k L . \psi(L)=0=C\sin kL.\!
  18. k = n π L n = 1 , 2 , 3 , . k=\frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.
  19. E = 2 π 2 n 2 2 m L 2 = n 2 h 2 8 m L 2 . E=\frac{\hbar^{2}\pi^{2}n^{2}}{2mL^{2}}=\frac{n^{2}h^{2}}{8mL^{2}}.
  20. V ( x ) = 1 2 m ω 2 x 2 V(x)=\frac{1}{2}m\omega^{2}x^{2}
  21. ψ n ( x ) = 1 2 n n ! ( m ω π ) 1 / 4 e - m ω x 2 2 H n ( m ω x ) , n = 0 , 1 , 2 , . \psi_{n}(x)=\sqrt{\frac{1}{2^{n}\,n!}}\cdot\left(\frac{m\omega}{\pi\hbar}% \right)^{1/4}\cdot e^{-\frac{m\omega x^{2}}{2\hbar}}\cdot H_{n}\left(\sqrt{% \frac{m\omega}{\hbar}}x\right),\qquad n=0,1,2,\ldots.
  22. H n ( x ) = ( - 1 ) n e x 2 d n d x n ( e - x 2 ) H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}\left(e^{-x^{2}}\right)
  23. E n = ω ( n + 1 2 ) E_{n}=\hbar\omega\left(n+{1\over 2}\right)
  24. I I II
  25. X I I I XIII

Quantum_teleportation.html

  1. | ϕ |\phi\rangle
  2. | ϕ |\phi\rangle
  3. | ϕ |\phi\rangle
  4. | ψ |\psi\rangle
  5. | ψ C = α | 0 C + β | 1 C . |\psi\rangle_{C}=\alpha|0\rangle_{C}+\beta|1\rangle_{C}.
  6. | Φ + A B = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) |\Phi^{+}\rangle_{AB}=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1% \rangle_{A}\otimes|1\rangle_{B})
  7. | Φ - A B = 1 2 ( | 0 A | 0 B - | 1 A | 1 B ) |\Phi^{-}\rangle_{AB}=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}-|1% \rangle_{A}\otimes|1\rangle_{B})
  8. | Ψ + A B = 1 2 ( | 0 A | 1 B + | 1 A | 0 B ) |\Psi^{+}\rangle_{AB}=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|1\rangle_{B}+|1% \rangle_{A}\otimes|0\rangle_{B})
  9. | Ψ - A B = 1 2 ( | 0 A | 1 B - | 1 A | 0 B ) |\Psi^{-}\rangle_{AB}=\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|1\rangle_{B}-|1% \rangle_{A}\otimes|0\rangle_{B})
  10. | Φ + A B . |\Phi^{+}\rangle_{AB}.
  11. | Φ + A B | ψ C = 1 2 ( | 0 A | 0 B + | 1 A | 1 B ) ( α | 0 C + β | 1 C ) . |\Phi^{+}\rangle_{AB}\otimes|\psi\rangle_{C}=\frac{1}{\sqrt{2}}(|0\rangle_{A}% \otimes|0\rangle_{B}+|1\rangle_{A}\otimes|1\rangle_{B})\otimes(\alpha|0\rangle% _{C}+\beta|1\rangle_{C}).
  12. | 0 | 0 = 1 2 ( | Φ + + | Φ - ) , |0\rangle\otimes|0\rangle=\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle+|\Phi^{-}\rangle),
  13. | 0 | 1 = 1 2 ( | Ψ + + | Ψ - ) , |0\rangle\otimes|1\rangle=\frac{1}{\sqrt{2}}(|\Psi^{+}\rangle+|\Psi^{-}\rangle),
  14. | 1 | 0 = 1 2 ( | Ψ + - | Ψ - ) , |1\rangle\otimes|0\rangle=\frac{1}{\sqrt{2}}(|\Psi^{+}\rangle-|\Psi^{-}\rangle),
  15. | 1 | 1 = 1 2 ( | Φ + - | Φ - ) . |1\rangle\otimes|1\rangle=\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle-|\Phi^{-}\rangle).
  16. | Φ + A B | \displaystyle|\Phi^{+}\rangle_{AB}\ \otimes\ |
  17. | Φ + A C , | Φ - A C , | Ψ + A C , | Ψ - A C |\Phi^{+}\rangle_{AC},|\Phi^{-}\rangle_{AC},|\Psi^{+}\rangle_{AC},|\Psi^{-}% \rangle_{AC}
  18. | Φ + A C ( α | 0 B + β | 1 B ) |\Phi^{+}\rangle_{AC}\otimes(\alpha|0\rangle_{B}+\beta|1\rangle_{B})
  19. | Φ - A C ( α | 0 B - β | 1 B ) |\Phi^{-}\rangle_{AC}\otimes(\alpha|0\rangle_{B}-\beta|1\rangle_{B})
  20. | Ψ + A C ( β | 0 B + α | 1 B ) |\Psi^{+}\rangle_{AC}\otimes(\beta|0\rangle_{B}+\alpha|1\rangle_{B})
  21. | Ψ - A C ( β | 0 B - α | 1 B ) |\Psi^{-}\rangle_{AC}\otimes(\beta|0\rangle_{B}-\alpha|1\rangle_{B})
  22. α | 0 B + β | 1 B \alpha|0\rangle_{B}+\beta|1\rangle_{B}
  23. | Φ + A C |\Phi^{+}\rangle_{AC}
  24. | Φ - A C |\Phi^{-}\rangle_{AC}
  25. σ 3 = [ 1 0 0 - 1 ] \sigma_{3}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  26. | Ψ + A C |\Psi^{+}\rangle_{AC}
  27. σ 1 = [ 0 1 1 0 ] \sigma_{1}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  28. - σ 3 σ 1 = i σ 2 = [ 0 - 1 1 0 ] . -\sigma_{3}\sigma_{1}=i\sigma_{2}=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}.
  29. | ψ B = α | 0 B + β | 1 B |\psi\rangle_{B}=\alpha|0\rangle_{B}+\beta|1\rangle_{B}
  30. G = ( H I ) C N G=(H\otimes I)\;C_{N}
  31. C N C_{N}
  32. N 3 N^{3}
  33. N 2 N^{2}
  34. N 2 N^{2}
  35. ρ ω . \rho\otimes\omega.
  36. ( Tr 12 Φ ) ( ρ ω ) = ρ , (\operatorname{Tr}_{12}\circ\Phi)(\rho\otimes\omega)=\rho\,,
  37. \circ
  38. Φ ( ρ ω ) | I O = ρ | O \langle\Phi(\rho\otimes\omega)|I\otimes O\rangle=\langle\rho|O\rangle
  39. I O I\otimes O
  40. 12 3 12\otimes 3
  41. ρ ω \rho\otimes\omega
  42. 1 23 1\otimes 23
  43. F i = M i 2 . {F_{i}}={M_{i}^{2}}.
  44. ( M i I ) ( ρ ω ) ( M i I ) . (M_{i}\otimes I)(\rho\otimes\omega)(M_{i}\otimes I).
  45. ( M i I ) (M_{i}\otimes I)
  46. 12 3 12\otimes 3
  47. ρ ω \rho\otimes\omega
  48. 1 23 1\otimes 23
  49. ( I d Ψ i ) ( M i I ) ( ρ ω ) ( M i I ) . (Id\otimes\Psi_{i})(M_{i}\otimes I)(\rho\otimes\omega)(M_{i}\otimes I).
  50. 1 2 1\otimes 2
  51. Φ ( ρ ω ) = i ( I d Ψ i ) ( M i I ) ( ρ ω ) ( M i I ) \Phi(\rho\otimes\omega)=\sum_{i}(Id\otimes\Psi_{i})(M_{i}\otimes I)(\rho% \otimes\omega)(M_{i}\otimes I)
  52. Φ ( ρ ω ) , I O = ρ , O \langle\Phi(\rho\otimes\omega),I\otimes O\rangle=\langle\rho,O\rangle
  53. i ( I d Ψ i ) ( M i I ) ( ρ ω ) ( M i I ) , I O \sum_{i}\langle(Id\otimes\Psi_{i})(M_{i}\otimes I)(\rho\otimes\omega)(M_{i}% \otimes I),\;I\otimes O\rangle
  54. = i ( M i I ) ( ρ ω ) ( M i I ) , I Ψ i * ( O ) =\sum_{i}\langle(M_{i}\otimes I)(\rho\otimes\omega)(M_{i}\otimes I),\;I\otimes% \Psi_{i}^{*}(O)\rangle
  55. i Tr ( ρ ω ) ( F i Ψ i * ( O ) ) . \sum_{i}\operatorname{Tr}\;(\rho\otimes\omega)(F_{i}\otimes\Psi_{i}^{*}(O)).
  56. i Tr ( ρ ω ) ( F i Ψ i * ( O ) ) = Tr ρ O . \sum_{i}\operatorname{Tr}\;(\rho\otimes\omega)(F_{i}\otimes\Psi_{i}^{*}(O))=% \operatorname{Tr}\;\rho\cdot O.

Quark.html

  1. J J
  2. ψ ψ
  3. [ | V ud | | V us | | V ub | | V cd | | V cs | | V cb | | V td | | V ts | | V tb | ] [ 0.974 0.225 0.003 0.225 0.973 0.041 0.009 0.040 0.999 ] , \begin{bmatrix}|V_{\mathrm{ud}}|&|V_{\mathrm{us}}|&|V_{\mathrm{ub}}|\\ |V_{\mathrm{cd}}|&|V_{\mathrm{cs}}|&|V_{\mathrm{cb}}|\\ |V_{\mathrm{td}}|&|V_{\mathrm{ts}}|&|V_{\mathrm{tb}}|\end{bmatrix}\approx% \begin{bmatrix}0.974&0.225&0.003\\ 0.225&0.973&0.041\\ 0.009&0.040&0.999\end{bmatrix},

Quartile.html

  1. { Q 1 = 15 Q 2 = 40 Q 3 = 43 \begin{cases}Q_{1}=15\\ Q_{2}=40\\ Q_{3}=43\end{cases}
  2. { Q 1 = 25.5 Q 2 = 40 Q 3 = 42.5 \begin{cases}Q_{1}=25.5\\ Q_{2}=40\\ Q_{3}=42.5\end{cases}
  3. { Q 1 = 20.25 Q 2 = 40 Q 3 = 42.75 \begin{cases}Q_{1}=20.25\\ Q_{2}=40\\ Q_{3}=42.75\end{cases}
  4. { Q 1 = 15 Q 2 = 37.5 Q 3 = 40 \begin{cases}Q_{1}=15\\ Q_{2}=37.5\\ Q_{3}=40\end{cases}
  5. { Q 1 = 15 Q 2 = 37.5 Q 3 = 40 \begin{cases}Q_{1}=15\\ Q_{2}=37.5\\ Q_{3}=40\end{cases}
  6. { Q 1 = 15.0 Q 2 = 37.5 Q 3 = 40.0 \begin{cases}Q_{1}=15.0\\ Q_{2}=37.5\\ Q_{3}=40.0\end{cases}
  7. Lower fence = Q 1 - 1.5 ( IQR ) \,\text{Lower fence}=Q_{1}-1.5(\mathrm{IQR})\,
  8. Upper fence = Q 3 + 1.5 ( IQR ) , \,\text{Upper fence}=Q_{3}+1.5(\mathrm{IQR}),\,

Quasigroup.html

  1. ( x , y , z ) ( x , - y , z ) (x,y,z)\mapsto(x,-y,z)
  2. ( x 1 , x 2 , x 3 ) ( y 1 , y 2 , y 3 ) = ( y 3 / x 2 , y 1 \ x 3 , x 1 y 2 ) = ( x 2 / / y 3 , x 3 \ \ y 1 , x 1 y 2 ) (x_{1},x_{2},x_{3})\cdot(y_{1},y_{2},y_{3})=(y_{3}/x_{2},y_{1}\backslash x_{3}% ,x_{1}y_{2})=(x_{2}//y_{3},x_{3}\backslash\backslash y_{1},x_{1}y_{2})
  3. L ( x ) y = x y R ( x ) y = y x \begin{aligned}\displaystyle L(x)y&\displaystyle=xy\\ \displaystyle R(x)y&\displaystyle=yx\\ \end{aligned}
  4. L ( x ) - 1 y = x \ y R ( x ) - 1 y = y / x \begin{aligned}\displaystyle L(x)^{-1}y&\displaystyle=x\backslash y\\ \displaystyle R(x)^{-1}y&\displaystyle=y/x\end{aligned}
  5. L ( x ) L ( x ) - 1 = 1 corresponding to x ( x \ y ) = y L ( x ) - 1 L ( x ) = 1 corresponding to x \ ( x y ) = y R ( x ) R ( x ) - 1 = 1 corresponding to ( y / x ) x = y R ( x ) - 1 R ( x ) = 1 corresponding to ( y x ) / x = y \begin{aligned}\displaystyle L(x)L(x)^{-1}&\displaystyle=1&\displaystyle\,% \text{corresponding to}\qquad x(x\backslash y)&\displaystyle=y\\ \displaystyle L(x)^{-1}L(x)&\displaystyle=1&\displaystyle\,\text{corresponding% to}\qquad x\backslash(xy)&\displaystyle=y\\ \displaystyle R(x)R(x)^{-1}&\displaystyle=1&\displaystyle\,\text{corresponding% to}\qquad(y/x)x&\displaystyle=y\\ \displaystyle R(x)^{-1}R(x)&\displaystyle=1&\displaystyle\,\text{corresponding% to}\qquad(yx)/x&\displaystyle=y\end{aligned}
  6. x λ = e / x x λ x = e x^{\lambda}=e/x\qquad x^{\lambda}x=e
  7. x ρ = x \ e x x ρ = e x^{\rho}=x\backslash e\qquad xx^{\rho}=e
  8. x λ = x ρ x^{\lambda}=x^{\rho}
  9. x - 1 x^{-1}
  10. x λ ( x y ) = y x^{\lambda}(xy)=y
  11. x x
  12. y y
  13. L ( x ) - 1 = L ( x λ ) L(x)^{-1}=L(x^{\lambda})
  14. x \ y = x λ y x\backslash y=x^{\lambda}y
  15. ( y x ) x ρ = y (yx)x^{\rho}=y
  16. x x
  17. y y
  18. R ( x ) - 1 = R ( x ρ ) R(x)^{-1}=R(x^{\rho})
  19. y / x = y x ρ y/x=yx^{\rho}
  20. ( x y ) λ = y λ x λ (xy)^{\lambda}=y^{\lambda}x^{\lambda}
  21. ( x y ) ρ = y ρ x ρ (xy)^{\rho}=y^{\rho}x^{\rho}
  22. ( x y ) z = e (xy)z=e
  23. x ( y z ) = e x(yz)=e
  24. ( x y ) λ x = y λ (xy)^{\lambda}x=y^{\lambda}
  25. x ( y x ) ρ = y ρ x(yx)^{\rho}=y^{\rho}
  26. α ( x ) β ( y ) = γ ( x y ) \alpha(x)\beta(y)=\gamma(xy)\,
  27. f ( x 1 , , x n ) = g ( x 1 , , x i - 1 , h ( x i , , x j ) , x j + 1 , , x n ) , f(x_{1},\dots,x_{n})=g(x_{1},\dots,x_{i-1},\,h(x_{i},\dots,x_{j}),\,x_{j+1},% \dots,x_{n}),

Quasispecies_model.html

  1. S S
  2. n i n_{i}
  3. A i A_{i}
  4. q i j q_{ij}
  5. w i j = A j q i j w_{ij}=A_{j}q_{ij}
  6. i q i j = 1 \sum_{i}q_{ij}=1\,
  7. n i n^{\prime}_{i}
  8. n i = j w i j n j n^{\prime}_{i}=\sum_{j}w_{ij}n_{j}\,
  9. D i D_{i}
  10. w i j = A j q i j - D i δ i j w_{ij}=A_{j}q_{ij}-D_{i}\delta_{ij}\,
  11. δ i j \delta_{ij}
  12. n > 0 n>0
  13. n t h n^{th}
  14. n i n^{\prime}_{i}
  15. n i n_{i}
  16. n i n_{i}
  17. n ˙ i = n i - n i \dot{n}_{i}=n^{\prime}_{i}-n_{i}
  18. n ˙ i = j w i j n j - n i \dot{n}_{i}=\sum_{j}w_{ij}n_{j}-n_{i}\,
  19. x i x_{i}
  20. x i = def n i j n j x_{i}\ \stackrel{\mathrm{def}}{=}\ \frac{n_{i}}{\sum_{j}n_{j}}
  21. x i = def n i j n j x^{\prime}_{i}\ \stackrel{\mathrm{def}}{=}\ \frac{n^{\prime}_{i}}{\sum_{j}n^{% \prime}_{j}}
  22. x i = j w i j x j i j w i j x j x^{\prime}_{i}=\frac{\sum_{j}w_{ij}x_{j}}{\sum_{ij}w_{ij}x_{j}}
  23. x ˙ i = j w i j x j - x i i j w i j x j . \dot{x}_{i}=\sum_{j}w_{ij}x_{j}-x_{i}\sum_{ij}w_{ij}x_{j}.
  24. 1 - k 1-k
  25. k k
  26. 0 k 1 0\leq k\leq 1
  27. 𝐖 = [ 1 0 0 0 0 1 - k k k 0 k 1 - k k 0 k k 1 - k ] \mathbf{W}=\begin{bmatrix}1&0&0&0\\ 0&1-k&k&k\\ 0&k&1-k&k\\ 0&k&k&1-k\end{bmatrix}
  28. 𝐖 = [ 1 - 2 k 0 0 0 0 1 - 2 k 0 0 0 0 1 0 0 0 0 1 + k ] \mathbf{W^{\prime}}=\begin{bmatrix}1-2k&0&0&0\\ 0&1-2k&0&0\\ 0&0&1&0\\ 0&0&0&1+k\end{bmatrix}
  29. 1 + k 1+k
  30. ( 1 + k ) n (1+k)^{n}

Quaternion.html

  1. \mathbb{H}
  2. i < s u p > 2 = j 2 = k 2 = i j k = 1 i<sup>2=j^{2}=k^{2}=ijk=−1
  3. i 2 = j 2 = k 2 = i j k = - 1 i^{2}=j^{2}=k^{2}=ijk=-1
  4. - k = i j k k = i j ( k 2 ) = i j ( - 1 ) , k = i j . \begin{aligned}\displaystyle-k&\displaystyle=ijkk=ij(k^{2})=ij(-1),\\ \displaystyle k&\displaystyle=ij.\end{aligned}
  5. i j = k , j i = - k , j k = i , k j = - i , k i = j , i k = - j , \begin{aligned}\displaystyle ij&\displaystyle=k,&\displaystyle\qquad ji&% \displaystyle=-k,\\ \displaystyle jk&\displaystyle=i,&\displaystyle kj&\displaystyle=-i,\\ \displaystyle ki&\displaystyle=j,&\displaystyle ik&\displaystyle=-j,\end{aligned}
  6. i i
  7. i −i
  8. a 1 a 2 + a 1 b 2 i + a 1 c 2 j + a 1 d 2 k a_{1}a_{2}+a_{1}b_{2}i+a_{1}c_{2}j+a_{1}d_{2}k
  9. + b 1 a 2 i + b 1 b 2 i 2 + b 1 c 2 i j + b 1 d 2 i k {}+b_{1}a_{2}i+b_{1}b_{2}i^{2}+b_{1}c_{2}ij+b_{1}d_{2}ik
  10. + c 1 a 2 j + c 1 b 2 j i + c 1 c 2 j 2 + c 1 d 2 j k {}+c_{1}a_{2}j+c_{1}b_{2}ji+c_{1}c_{2}j^{2}+c_{1}d_{2}jk
  11. + d 1 a 2 k + d 1 b 2 k i + d 1 c 2 k j + d 1 d 2 k 2 . {}+d_{1}a_{2}k+d_{1}b_{2}ki+d_{1}c_{2}kj+d_{1}d_{2}k^{2}.
  12. a 1 a 2 - b 1 b 2 - c 1 c 2 - d 1 d 2 a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}
  13. + ( a 1 b 2 + b 1 a 2 + c 1 d 2 - d 1 c 2 ) i {}+(a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2})i
  14. + ( a 1 c 2 - b 1 d 2 + c 1 a 2 + d 1 b 2 ) j {}+(a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2})j
  15. + ( a 1 d 2 + b 1 c 2 - c 1 b 2 + d 1 a 2 ) k . {}+(a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2})k.
  16. 𝐇 = { ( a , b , c , d ) a , b , c , d 𝐑 } . \mathbf{H}=\{(a,b,c,d)\mid a,b,c,d\in\mathbf{R}\}.
  17. 1 \displaystyle 1
  18. ( a 1 , b 1 , c 1 , d 1 ) + ( a 2 , b 2 , c 2 , d 2 ) = ( a 1 + a 2 , b 1 + b 2 , c 1 + c 2 , d 1 + d 2 ) . (a_{1},\ b_{1},\ c_{1},\ d_{1})+(a_{2},\ b_{2},\ c_{2},\ d_{2})=(a_{1}+a_{2},% \ b_{1}+b_{2},\ c_{1}+c_{2},\ d_{1}+d_{2}).
  19. ( a 1 , b 1 , c 1 , d 1 ) ( a 2 , b 2 , c 2 , d 2 ) = = ( a 1 a 2 - b 1 b 2 - c 1 c 2 - d 1 d 2 , a 1 b 2 + b 1 a 2 + c 1 d 2 - d 1 c 2 , a 1 c 2 - b 1 d 2 + c 1 a 2 + d 1 b 2 , a 1 d 2 + b 1 c 2 - c 1 b 2 + d 1 a 2 ) . \begin{aligned}\displaystyle(a_{1},\ b_{1},\ c_{1},\ d_{1})&\displaystyle(a_{2% },\ b_{2},\ c_{2},\ d_{2})=\\ &\displaystyle=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2},\\ &\displaystyle{}\qquad a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2},\\ &\displaystyle{}\qquad a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2},\\ &\displaystyle{}\qquad a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2}).\end{aligned}
  20. q = ( r , v ) , q 𝐇 , r 𝐑 , v 𝐑 3 q=(r,\ \vec{v}),\ q\in\mathbf{H},\ r\in\mathbf{R},\ \vec{v}\in\mathbf{R}^{3}
  21. ( r 1 , v 1 ) + ( r 2 , v 2 ) = ( r 1 + r 2 , v 1 + v 2 ) (r_{1},\ \vec{v}_{1})+(r_{2},\ \vec{v}_{2})=(r_{1}+r_{2},\ \vec{v}_{1}+\vec{v}% _{2})
  22. ( r 1 , v 1 ) ( r 2 , v 2 ) = ( r 1 r 2 - v 1 v 2 , r 1 v 2 + r 2 v 1 + v 1 × v 2 ) (r_{1},\ \vec{v}_{1})(r_{2},\ \vec{v}_{2})=(r_{1}r_{2}-\vec{v}_{1}\cdot\vec{v}% _{2},r_{1}\vec{v}_{2}+r_{2}\vec{v}_{1}+\vec{v}_{1}\times\vec{v}_{2})
  23. q = a + b i + c j + d k q=a+bi+cj+dk
  24. q * = a - b i - c j - d k q^{*}=a-bi-cj-dk
  25. q ¯ \overline{q}
  26. q ~ \tilde{q}
  27. q * = - 1 2 ( q + i q i + j q j + k q k ) . q^{*}=-\frac{1}{2}(q+iqi+jqj+kqk).
  28. q = q q * = q * q = a 2 + b 2 + c 2 + d 2 \lVert q\rVert=\sqrt{qq^{*}}=\sqrt{q^{*}q}=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}
  29. α q = | α | q . \lVert\alpha q\rVert=|\alpha|\lVert q\rVert.
  30. p q = p q . \lVert pq\rVert=\lVert p\rVert\lVert q\rVert.
  31. det ( a + i b i d + c i d - c a - i b ) = a 2 + b 2 + c 2 + d 2 , \det\Bigl(\begin{array}[]{cc}a+ib&id+c\\ id-c&a-ib\end{array}\Bigr)=a^{2}+b^{2}+c^{2}+d^{2},
  32. d ( p , q ) = p - q . d(p,q)=\lVert p-q\rVert.
  33. ( p + a p 1 + q + a q 1 ) - ( p + q ) = a p 1 + q 1 . \lVert(p+ap_{1}+q+aq_{1})-(p+q)\rVert=a\lVert p_{1}+q_{1}\rVert.
  34. 𝐔 q = q q . \mathbf{U}q=\frac{q}{\lVert q\rVert}.
  35. q q
  36. q / q 2 q^{\ast}/\left\|q\right\|^{2}
  37. q - 1 = q * q 2 . q^{-1}=\frac{q^{*}}{\lVert q\rVert^{2}}.
  38. p q \frac{p}{q}
  39. p q = b 1 b 2 + c 1 c 2 + d 1 d 2 . p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}.
  40. p q = 1 2 ( p * q + q * p ) = 1 2 ( p q * + q p * ) . p\cdot q=\textstyle\frac{1}{2}(p^{*}q+q^{*}p)=\textstyle\frac{1}{2}(pq^{*}+qp^% {*}).
  41. p × q = ( c 1 d 2 - d 1 c 2 ) i + ( d 1 b 2 - b 1 d 2 ) j + ( b 1 c 2 - c 1 b 2 ) k . p\times q=(c_{1}d_{2}-d_{1}c_{2})i+(d_{1}b_{2}-b_{1}d_{2})j+(b_{1}c_{2}-c_{1}b% _{2})k.
  42. p × q = 1 2 ( p q - q * p * ) . p\times q=\textstyle\frac{1}{2}(pq-q^{*}p^{*}).
  43. [ p , q ] = 2 p × q . [p,q]=2p\times q.
  44. p = p s + p v , p=p_{s}+\vec{p}_{v},
  45. q = q s + q v , q=q_{s}+\vec{q}_{v},
  46. p v \vec{p}_{v}
  47. q v \vec{q}_{v}
  48. p q = ( p q ) s + ( p q ) v = ( p s q s - p v q v ) + ( p s q v + p v q s + p v × q v ) . pq=(pq)_{s}+(\vec{pq})_{v}=(p_{s}q_{s}-\vec{p}_{v}\cdot\vec{q}_{v})+(p_{s}\vec% {q}_{v}+\vec{p}_{v}q_{s}+\vec{p}_{v}\times\vec{q}_{v}).
  49. [ a + b i c + d i - c + d i a - b i ] . \begin{bmatrix}a+bi&c+di\\ -c+di&a-bi\end{bmatrix}.
  50. [ a b c d - b a - d c - c d a - b - d - c b a ] = a [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] + b [ 0 1 0 0 - 1 0 0 0 0 0 0 - 1 0 0 1 0 ] + c [ 0 0 1 0 0 0 0 1 - 1 0 0 0 0 - 1 0 0 ] + d [ 0 0 0 1 0 0 - 1 0 0 1 0 0 - 1 0 0 0 ] . \begin{bmatrix}a&b&c&d\\ -b&a&-d&c\\ -c&d&a&-b\\ -d&-c&b&a\end{bmatrix}=a\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}+b\begin{bmatrix}0&1&0&0\\ -1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\end{bmatrix}+c\begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ -1&0&0&0\\ 0&-1&0&0\end{bmatrix}+d\begin{bmatrix}0&0&0&1\\ 0&0&-1&0\\ 0&1&0&0\\ -1&0&0&0\end{bmatrix}.
  51. ( a + b i ) 1 + ( c + d i ) j . (a+bi)1+(c+di)j.
  52. ( a + b i , c + d i ) ( a , b , c , d ) . (a+bi,\ c+di)\leftrightarrow(a,b,c,d).
  53. a 2 - b 2 - c 2 - d 2 = - 1 , a^{2}-b^{2}-c^{2}-d^{2}=-1,
  54. 2 a b = 0 , 2ab=0,
  55. 2 a c = 0 , 2ac=0,
  56. 2 a d = 0. 2ad=0.
  57. a + b - 1 a + b q . a+b\sqrt{-1}\mapsto a+bq.
  58. q = q s + q v . q=q_{s}+\vec{q}_{v}.
  59. q = q s + q v 𝐔 q v . q=q_{s}+\lVert\vec{q}_{v}\rVert\cdot\mathbf{U}\vec{q}_{v}.
  60. q s + q 𝐔 q q_{s}+\lVert q\rVert\cdot\mathbf{U}q
  61. 𝐔 q v \mathbf{U}\vec{q}_{v}
  62. a + b - 1 a + b 𝐔 q v . a+b\sqrt{-1}\mapsto a+b\mathbf{U}\vec{q}_{v}.
  63. q q
  64. q s + q v i q_{s}+\lVert\vec{q}_{v}\rVert i
  65. q = a + b i + c j + d k = a + 𝐯 q=a+bi+cj+dk=a+\mathbf{v}
  66. exp ( q ) = n = 0 q n n ! = e a ( cos 𝐯 + 𝐯 𝐯 sin 𝐯 ) \exp(q)=\sum_{n=0}^{\infty}\frac{q^{n}}{n!}=e^{a}\left(\cos\|\mathbf{v}\|+% \frac{\mathbf{v}}{\|\mathbf{v}\|}\sin\|\mathbf{v}\|\right)
  67. ln ( q ) = ln q + 𝐯 𝐯 arccos a q \ln(q)=\ln\|q\|+\frac{\mathbf{v}}{\|\mathbf{v}\|}\arccos\frac{a}{\|q\|}
  68. q = q e n ^ θ = q ( cos ( θ ) + n ^ sin ( θ ) ) , q=\|q\|e^{\hat{n}\theta}=\|q\|\left(\cos(\theta)+\hat{n}\sin(\theta)\right),
  69. θ \theta
  70. n ^ \hat{n}
  71. a = q cos ( θ ) a=\|q\|\cos(\theta)
  72. 𝐯 = n ^ 𝐯 = n ^ q sin ( θ ) . \mathbf{v}=\hat{n}\|\mathbf{v}\|=\hat{n}\|q\|\sin(\theta).
  73. e n ^ θ e^{\hat{n}\theta}
  74. α \alpha
  75. q α = q α e n ^ α θ = q α ( cos ( α θ ) + n ^ sin ( α θ ) ) . q^{\alpha}=\|q\|^{\alpha}e^{\hat{n}\alpha\theta}=\|q\|^{\alpha}\left(\cos(% \alpha\theta)+\hat{n}\sin(\alpha\theta)\right).
  76. σ 1 2 = σ 2 2 = σ 3 2 = 1 , \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=1,
  77. σ i σ j = - σ j σ i ( j i ) . \sigma_{i}\sigma_{j}=-\sigma_{j}\sigma_{i}\qquad(j\neq i).
  78. r = - w r w . r^{\prime}=-w\,r\,w.
  79. r ′′ = σ 2 σ 1 r σ 1 σ 2 r^{\prime\prime}=\sigma_{2}\sigma_{1}\,r\,\sigma_{1}\sigma_{2}
  80. r ′′ = - 𝐤 r 𝐤 . r^{\prime\prime}=-\mathbf{k}\,r\,\mathbf{k}.
  81. 𝐤 = σ 2 σ 1 , 𝐢 = σ 3 σ 2 , 𝐣 = σ 1 σ 3 \mathbf{k}=\sigma_{2}\sigma_{1},\mathbf{i}=\sigma_{3}\sigma_{2},\mathbf{j}=% \sigma_{1}\sigma_{3}
  82. 𝐢 2 = 𝐣 2 = 𝐤 2 = 𝐢𝐣𝐤 = - 1. \mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\mathbf{j}\mathbf{k}=-1.

Qubit.html

  1. | 0 |0\rangle
  2. | 1 |1\rangle
  3. | 0 |0\rangle
  4. | 1 |1\rangle
  5. | ψ = α | 0 + β | 1 , |\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\,
  6. | 0 |0\rangle
  7. | α | 2 |\alpha|^{2}
  8. | 1 |1\rangle
  9. | β | 2 |\beta|^{2}
  10. | α | 2 + | β | 2 = 1 |\alpha|^{2}+|\beta|^{2}=1\,
  11. | 0 |0\rangle
  12. | 1 |1\rangle
  13. | 0 + i | 1 2 {|0\rangle+i|1\rangle}\over{\sqrt{2}}
  14. | α | 2 + | β | 2 = 1 |\alpha|^{2}+|\beta|^{2}=1\,
  15. ϕ \phi
  16. θ \theta
  17. | 0 |0\rangle
  18. | α | 2 |\alpha|^{2}
  19. | 1 |1\rangle
  20. | β | 2 |\beta|^{2}
  21. | 0 |0\rangle
  22. 1 2 ( | 00 + | 11 ) . \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).
  23. | 00 |00\rangle
  24. | 11 |11\rangle
  25. | 1 / 2 | 2 = 1 / 2 |1/\sqrt{2}|^{2}=1/2
  26. | 0 |0\rangle
  27. | 1 |1\rangle
  28. | 0 |0\rangle
  29. | 00 |00\rangle
  30. | 0 |0\rangle
  31. | 0 |0\rangle
  32. | 1 |1\rangle

Queueing_theory.html

  1. | E - L | { 0 , 1 } |E-L|\in\{0,1\}
  2. μ 1 P 1 = λ 0 P 0 \mu_{1}P_{1}=\lambda_{0}P_{0}
  3. λ 0 P 0 + μ 2 P 2 = ( λ 1 + μ 1 ) P 1 \lambda_{0}P_{0}+\mu_{2}P_{2}=(\lambda_{1}+\mu_{1})P_{1}
  4. λ n - 1 P n - 1 + μ n + 1 P n + 1 = ( λ n + μ n ) P n \lambda_{n-1}P_{n-1}+\mu_{n+1}P_{n+1}=(\lambda_{n}+\mu_{n})P_{n}
  5. P 1 = λ 0 μ 1 P 0 P 2 = λ 1 μ 2 P 1 + 1 μ 2 ( μ 1 P 1 - λ 0 P 0 ) = λ 1 μ 2 P 1 = λ 1 λ 0 μ 2 μ 1 P 0 P_{1}=\frac{\lambda_{0}}{\mu_{1}}P_{0}\;\;\;P_{2}=\frac{\lambda_{1}}{\mu_{2}}P% _{1}+\frac{1}{\mu_{2}}(\mu_{1}P_{1}-\lambda_{0}P_{0})=\frac{\lambda_{1}}{\mu_{% 2}}P_{1}=\frac{\lambda_{1}\lambda_{0}}{\mu_{2}\mu_{1}}P_{0}
  6. P n = λ n - 1 λ n - 2 λ 0 μ n μ n - 1 μ 1 P 0 = P 0 i = 0 n - 1 λ i μ i + 1 P_{n}=\frac{\lambda_{n-1}\lambda_{n-2}\cdots\lambda_{0}}{\mu_{n}\mu_{n-1}% \cdots\mu_{1}}P_{0}=P_{0}\prod_{i=0}^{n-1}\frac{\lambda_{i}}{\mu_{i+1}}
  7. n = 0 P n = P 0 + P 0 n = 1 i = 0 n - 1 λ i μ i + 1 = 1 \sum_{n=0}^{\infty}P_{n}=P_{0}+P_{0}\sum_{n=1}^{\infty}\prod_{i=0}^{n-1}\frac{% \lambda_{i}}{\mu_{i+1}}=1
  8. P 0 = 1 1 + n = 1 i = 0 n - 1 λ i μ i + 1 P_{0}=\frac{1}{1+\sum_{n=1}^{\infty}\prod_{i=0}^{n-1}\frac{\lambda_{i}}{\mu_{i% +1}}}

Quintessence_(physics).html

  1. ρ \rho
  2. V ( Q ) V(Q)
  3. w q = p q / ρ q = 1 2 Q ˙ 2 - V ( Q ) 1 2 Q ˙ 2 + V ( Q ) w_{q}=p_{q}/\rho_{q}=\frac{\frac{1}{2}\dot{Q}^{2}-V(Q)}{\frac{1}{2}\dot{Q}^{2}% +V(Q)}

Quotient_group.html

  1. G = 𝐙 n 2 * G=\mathbf{Z}^{*}_{n^{2}}
  2. 𝐙 n * \mathbf{Z}^{*}_{n}

R.html

  1. \mathbb{R}

Radar.html

  1. P r = P t G t A r σ F 4 ( 4 π ) 2 R t 2 R r 2 P_{r}={{P_{t}G_{t}A_{r}\sigma F^{4}}\over{{(4\pi)}^{2}R_{t}^{2}R_{r}^{2}}}
  2. G r λ 2 4 π {G_{r}\lambda^{2}}\over{4\pi}
  3. λ \lambda
  4. P r = P t G t A r σ F 4 ( 4 π ) 2 R 4 . P_{r}={{P_{t}G_{t}A_{r}\sigma F^{4}}\over{{(4\pi)}^{2}R^{4}}}.
  5. F D F_{D}
  6. F T F_{T}
  7. V R V_{R}
  8. C C
  9. F D = 2 × F T × ( V R C ) F_{D}=2\times F_{T}\times\left(\frac{V_{R}}{C}\right)
  10. F D = F T × ( V R C ) F_{D}=F_{T}\times\left(\frac{V_{R}}{C}\right)
  11. 2 \sqrt{2}

Radian.html

  1. r a d {}^{rad}
  2. c {}^{c}
  3. 1 60 \frac{1}{60}
  4. angle in degrees = angle in radians 180 π \,\text{angle in degrees}=\,\text{angle in radians}\cdot\frac{180^{\circ}}{\pi}
  5. 1 rad = 1 180 π 57.2958 1\,\text{ rad}=1\cdot\frac{180^{\circ}}{\pi}\approx 57.2958^{\circ}
  6. 2.5 rad = 2.5 180 π 143.2394 2.5\,\text{ rad}=2.5\cdot\frac{180^{\circ}}{\pi}\approx 143.2394^{\circ}
  7. π 3 rad = π 3 180 π = 60 \frac{\pi}{3}\,\text{ rad}=\frac{\pi}{3}\cdot\frac{180^{\circ}}{\pi}=60^{\circ}
  8. angle in radians = angle in degrees π 180 \,\text{angle in radians}=\,\text{angle in degrees}\cdot\frac{\pi}{180^{\circ}}
  9. 1 = 1 π 180 0.0175 rad 1^{\circ}=1\cdot\frac{\pi}{180^{\circ}}\approx 0.0175\,\text{ rad}
  10. 23 = 23 π 180 0.4014 rad 23^{\circ}=23\cdot\frac{\pi}{180^{\circ}}\approx 0.4014\,\text{ rad}
  11. 2 π r 2\pi r
  12. r r
  13. 360 2 π r 360^{\circ}\iff 2\pi r
  14. 360 360^{\circ}
  15. 2 π r r rad \frac{2\pi r}{r}\,\text{ rad}
  16. = 2 π rad =2\pi\,\text{ rad}
  17. 2 π rad = 360 2\pi\,\text{ rad}=360^{\circ}
  18. 1 rad = 360 2 π \Rrightarrow 1\,\text{ rad}=\frac{360^{\circ}}{2\pi}
  19. 1 rad = 180 π \Rrightarrow 1\,\text{ rad}=\frac{180^{\circ}}{\pi}
  20. 2 π 2\pi
  21. 200 / π 200/\pi
  22. π / 200 \pi/200
  23. 1.2 rad = 1.2 200 g π 76.3944 g 1.2\,\text{ rad}=1.2\cdot\frac{200\text{g}}{\pi}\approx 76.3944\text{g}
  24. 50 g = 50 π 200 g 0.7854 rad 50\text{g}=50\cdot\frac{\pi}{200\text{g}}\approx 0.7854\,\text{ rad}
  25. 1 24 \frac{1}{24}
  26. 1 12 \frac{1}{12}
  27. 1 10 \frac{1}{10}
  28. 1 8 \frac{1}{8}
  29. 1 6 \frac{1}{6}
  30. 1 5 \frac{1}{5}
  31. 1 4 \frac{1}{4}
  32. 1 3 \frac{1}{3}
  33. 2 5 \frac{2}{5}
  34. 1 2 \frac{1}{2}
  35. 3 4 \frac{3}{4}
  36. π 12 \frac{\pi}{12}
  37. π 6 \frac{\pi}{6}
  38. π 5 \frac{\pi}{5}
  39. π 4 \frac{\pi}{4}
  40. π 3 \frac{\pi}{3}
  41. 2 π 5 2\frac{\pi}{5}
  42. π 2 \frac{\pi}{2}
  43. 2 π 3 2\frac{\pi}{3}
  44. 4 π 5 4\frac{\pi}{5}
  45. π \pi
  46. 3 π 2 3\frac{\pi}{2}
  47. π \pi
  48. 2 3 \frac{2}{3}
  49. 1 3 \frac{1}{3}
  50. 2 3 \frac{2}{3}
  51. 1 3 \frac{1}{3}
  52. lim h 0 sin h h = 1 , \lim_{h\rightarrow 0}\frac{\sin h}{h}=1,
  53. d d x sin x = cos x \frac{d}{dx}\sin x=\cos x
  54. d 2 d x 2 sin x = - sin x . \frac{d^{2}}{dx^{2}}\sin x=-\sin x.
  55. d 2 y d x 2 = - y \frac{d^{2}y}{dx^{2}}=-y
  56. d x 1 + x 2 \int\frac{dx}{1+x^{2}}
  57. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + . \sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots.
  58. sin x deg = sin y rad = π 180 x - ( π 180 ) 3 x 3 3 ! + ( π 180 ) 5 x 5 5 ! - ( π 180 ) 7 x 7 7 ! + . \sin x_{\mathrm{deg}}=\sin y_{\mathrm{rad}}=\frac{\pi}{180}x-\left(\frac{\pi}{% 180}\right)^{3}\ \frac{x^{3}}{3!}+\left(\frac{\pi}{180}\right)^{5}\ \frac{x^{5% }}{5!}-\left(\frac{\pi}{180}\right)^{7}\ \frac{x^{7}}{7!}+\cdots.
  59. 1 / 6283 {1}/{6283}
  60. 1 / 6400 {1}/{6400}
  61. 17 / 8 1{7}/{8}
  62. 1 / 2000 π {1}/{2000π}
  63. 1 / 6300 {1}/{6300}
  64. 1 / 6000 {1}/{6000}
  65. π / 648 , 000 {π}/{648,000}