THE SUBSTITUTION THEOREM FOR RIEMANN INTEGRALS

Bagby

doi:10.2307/44154126

THE SUBSTITUTION THEOREM FOR RIEMANN INTEGRALS

Real Analysis Exchange ◽

10.2307/44154126 ◽

2001 ◽

Vol 27 (1) ◽

pp. 309

Author(s):

Bagby

Keyword(s):

Substitution Theorem ◽

Riemann Integrals

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Upper and Lower Generalized Riemann Integrals

Real Analysis Exchange ◽

10.2307/44153691 ◽

1990 ◽

Vol 16 (1) ◽

pp. 215

Author(s):

Cater

Keyword(s):

Riemann Integrals

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A CONVERGENCE THEOREM FOR GENERALIZED RIEMANN INTEGRALS

Real Analysis Exchange ◽

10.2307/44152472 ◽

1994 ◽

Vol 20 (1) ◽

pp. 119 ◽

Cited By ~ 1

Author(s):

Bartle

Keyword(s):

Convergence Theorem ◽

Riemann Integrals

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ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

International Journal of Mathematics ◽

10.1142/s0129167x12500310 ◽

2012 ◽

Vol 23 (04) ◽

pp. 1250031 ◽

Cited By ~ 8

Author(s):

JOSÉ F. FERNANDO ◽

J. M. GAMBOA

Keyword(s):

Integral Domain ◽

Hilbert Problem ◽

Semialgebraic Sets ◽

Semialgebraic Set ◽

Irreducible Components ◽

Real Geometry ◽

Nash Manifold ◽

Substitution Theorem ◽

Satisfactory Theory ◽

17Th Hilbert Problem

In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

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