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Problems in Modern Mathematics Book

Problems in Modern Mathematics
Problems in Modern Mathematics, Ulam, famous for his solution to the difficulties of initiating fusion in the hydrogen bomb, devised the well-known Monte-Carlo method. Here he presents challenges in the areas of set theory, algebra, metric and topological spaces, and topological groups., Problems in Modern Mathematics has a rating of 5 stars
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Problems in Modern Mathematics, Ulam, famous for his solution to the difficulties of initiating fusion in the hydrogen bomb, devised the well-known Monte-Carlo method. Here he presents challenges in the areas of set theory, algebra, metric and topological spaces, and topological groups., Problems in Modern Mathematics
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  • Problems in Modern Mathematics
  • Written by author Stanislaw M. Ulam
  • Published by Dover Publications, June 2004
  • Ulam, famous for his solution to the difficulties of initiating fusion in the hydrogen bomb, devised the well-known Monte-Carlo method. Here he presents challenges in the areas of set theory, algebra, metric and topological spaces, and topological groups.
  • Ulam, famous for his solution to the difficulties of initiating fusion in the hydrogen bomb, devised the well-known Monte-Carlo method. Here he presents challenges in the areas of set theory, algebra, metric and topological spaces, and topological groups.
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Chapter ISet Theory
1.Introductory remarks1
2.The operation of direct product3
3.Product-isomorphisms and some generalizations5
4.Generalized projective sets9
4a.Relations between products of different orders11
5.Projective algebras12
6.Generalized logic13
7.Some problems on infinite sets15
8.Measure in abstract sets16
9.Nonmeasurable projective sets17
10.Infinite games23
11.Situations involving many quantifiers24
12.Some problems of P. Erdos25
Chapter IIAlgebraic Problems
1.An inductive lemma in combinatorial analysis29
2.A problem on matrices arising in the theory of automata30
3.A fundamental transformation in the "theory of equations"30
4.A problem on Peano mappings32
5.The determination of a mathematical structure from a given set of endomorphisms32
6.A problem on continued fractions32
7.Some questions about groups33
8.Semi-groups35
8a.Topological semi-groups35
9.A problem in the game of bridge36
10.A problem on arithmetic functions36
Chapter IIIMetric Spaces
1.Invariant properties of trajectories observed from moving coordinate systems37
2.Problems on convex bodies38
3.Some problems on isometry39
4.Systems of vectors39
5.Other problems on metrics40
Chapter IVTopological Spaces
1.A problem on measure43
2.Approximation of homeomorphisms of E[superscript n]43
2a.On the approximability of transformations in three dimensions by compositions of cylindrical mappings44
3.A problem on the invariance of dimension45
4.Homeomorphisms of the sphere46
5.Some topological invariants46
6.Quasi-fixed points48
7.Connectedness questions50
8.Two problems about the disk50
9.Approximation of continua by polyhedra51
10.The symmetric product51
11.A method of proof based on Baire category of sets53
12.Quasi-homeomorphisms54
13.Some problems of Borsuk55
Chapter VTopological Groups
1.Metrization questions57
2.Universal groups58
3.Basis problems59
4.Conditionally convergent sequences61
Chapter VISome Questions in Analysis
1.Stability63
2.Conjugate functions69
3.Ergodic phenomena70
4.The Frobenius transform73
5.Functions of two variables75
6.Measure-preserving transformations76
7.Relative measure77
8.Vitali-Lebesque and Laplace-Liapounoff theorems78
9.A problem in the calculus of variations79
10.A problem on formal integration80
11.Geometrical properties of the set of all solutions of certain equations80
Chapter VIIPhysical Systems
1.Generating functions and multiplicative systems83
1a.Examples of mathematical problems suggested by biological schemata85
2.Infinities in physics89
3.Motion of infinite systems, randomly distributed91
4.Infinite systems in equilibrium96
5.Random Cantor sets97
6.Dynamical flow in phase space104
7.Some problems on electromagnetic fields107
8.Nonlinear problems109
Chapter VIIIComputing Machines as a Heuristic Aid
1.Introduction115
2.Some combinatorial examples116
3.Some experiments on finite games118
4.Lucky numbers120
5.Remarks on computations in mathematical physics121
6.Examples from electromagnetism122
7.The Schrodinger equation123
8.Monte Carlo methods125
9.Hydrodynamical problems128
10.Synergesis135
Bibliography145


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