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de Souza P.N., Silva J.-N. — Berkeley Problems in Mathematics
de Souza P.N., Silva J.-N. — Berkeley Problems in Mathematics

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Название: Berkeley Problems in Mathematics

Авторы: de Souza P.N., Silva J.-N.

Аннотация:

In 1977, the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree.The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years since a) students are already deeply involved with the material and b) they will be prepared to take the exam within the first month of the graduate program rather than in the middle or end of the first year. The book is a compilation of approximately nine hundred problems which have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. Tags with the exact exam year provide the opportunity to rehearse complete examinations. The appendix includes instructions on accessing electronic versions of the exams as well as a syllabus, statistics of passing scores, and a Bibliography used throughout the solutions. This new edition contains approximately 120 new problems and 200 new solutions. It is an ideal means for students to strengthen their foundation in basic mathematics and to prepare for graduate studies.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 616

Добавлена в каталог: 09.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Algebraically independent      477
Argument principle      363 366
Arzela — Ascoli's Theorem      215 221—224 299
Baire category theorem      291
Bertrand's postulate      483
Binomial theorem      158 159
Bolzano — Weierstrass theorem      161 176 177 291 292 298
Casorati — Weierstrass theorem      334 342
Cauchy integral formula      326 333 345 347 348 356 378 403
Cauchy integral formula for derivatives      323 331 332 348
Cauchy root test      313
Cauchy sequence      184 249 298
Cauchy Theorem of      315 339 350 354 355 375 377 378 382 392—394 399 413 420 421 439
Cauchy — Riemann equations      330 334 336 337 341 372
Cauchy — Schwarz inequality      157 203 207 224 240 247 249 365 553 563
Cayley — Hamilton theorem      280 424 505 506
Cayley's theorem      454
Chain rule      252 262
Change of variables      259 264
Chinese remainder theorem      486
Comparison Test      314 315 395 396
Complexification      557
Contraction mapping principle      302 303
Critical point, nondegenerate      34
Deformation Theorem      375 381
Descartes' Rule of Signs      364
Dirichlet, integral of      393
Dirichlet, problem of      386
Dirichlet, test of      403
Eisenstein criterion      465 469 470 475
Euclidean algorithm      465 467 472 476 527
Euclidean Domain      476
Euclidean metric      245
Euler, theorem of      485
Euler, totient function      120 430 482 485 541
Extension theorem      285
Fermat little theorem      467 478 481 485 486
Fibonacci numbers      12 136 519
Fixed point theorem      180 300
Fubini's theorem      345
Function, concave      29
Function, convex      29
Function, harmonic      87
Function, orthonormal      93
Function, polynomial      117
Function, proper      33
Function, real analytic      4
Function, semicontinuous      10
Fundamental theorem of algebra      81 112 349—351 357 359
Fundamental theorem of calculus      170 248
Gauss lemma      469
Gauss theorem      263
Gauss — Lucas Theorem      359 361 363
Gaussian elimination      496
Gram — Schmidt procedure      510 546 553
Green's theorem      260 264 372
Group action, faithful      124
Group permutations, transitive      105
Group, center      97
Group, centralizer      99
Group, conjugacy class      99
Group, dihedral      105
Group, fixed point      105
Heine — Borel theorem      176 293 297
Hurwitz theorem      315
Identity Theorem      311 335 339 340 342
Implicit function theorem      252 273
Induction principle      158 171 180 181 183 184 187 190 198 441 449 477 492 494 497 498 508 516 518
Intermediate value property      9
Intermediate Value Theorem      161 162 174 201 205 236 275 326 363 364
Inverse function theorem      241—243 254—256 268 341
Jensen's inequality      239
Jordan canonical form      181 281 493 510 528 530 534—536 538—540 542—545
Jordan, Lemma of      167 396-398 404
l'Hopital's rule      196
Lagrange, multipliers      246
Lagrange, polynomials      531
Lagrange, theorem of      429 430 444 450 456 485
Laplace expansion      256 257
Laplacian, eigenfunction      41
Laurent expansion      339 342 356
Leibniz criterion      188 210
Liouville's theorem      329—331 333 334 336 342 349 373
Maclaurin expansion      166 185 189 206 315 332 337 343 345 369 370 565
Map, closed      32
Map, proper      32
Map, strict contraction      63
Matrix, axis      133
Matrix, column rank      125
Matrix, commutant of a      125 151
Matrix, finite order      138
Matrix, index      148
Matrix, order n      140
Matrix, orthogonal      97 152
Matrix, positive definite      148
Matrix, row rank      125
Matrix, skew-symmetric      149
Matrix, special orthogonal      138
Matrix, tridiagonal      126
Matrix, unitary      148
Matrix, Vandermonde      127 371 497 498 500 562 565
Maximum Modulus Principle      74 323—327 329—331 349 364 372 373
Mean Value for Integrals      227 232
Mean Value, Property      321
Mean Value, Sub-Property      74
Mean value, theorem      168 173 192 194 200 204 215 244 248 336
Method of undetermined coefficients      275
Morera's theorem      315 345 351 355
Nested set property      300
Norm      32
Nyquist diagram      362
Open mapping theorem      320 336
Parseval identity      229 231
Picard's theorem      266—268 273—275 277 284 287
Pigeonhole Principle      457 480
Poisson's formula      386
Rank theorem      254
Rank-nullity theorem      491 497 501 504 547
Rational canonical form      542
Rayleigh's theorem      288 523 524
Regular value      34
Residue theorem      348 374 375 377 379 381—385 387 389—392 397 398 400 405 406 409 410 414—416 419
Riemann Sums      185
Riemann — Lesbesgue Lemma      230 231
Ring, index      111
Ring, unit      98 119
Rolle's theorem      163 168 191 194 199
Rouche's theorem      323 349 357 358 360 362—364 368 369 381
Schwarz lemma      325—328
Schwarz reflection principle      340
Schwarz theorem      242
Second isomorphism theorem      438
Spectral theorem      497 567
Stokes' theorem      262
Stone — Weierstrass, Approximation Theorem      210 216 225 228 230 231 315 344
Structure theorem      431 449—451 458 474
Sylow's Theorems      428 442 452—455 478
Taylor's theorem      159 163 166 199 242 369
Triangle inequality      184 220 298 364
UFD      458 477
Weierstrass M-test      226 227
Weierstrass theorem      176 232
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