O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....holes and one 2 dimensional hole. Analogously, for the two dimensional sphere, S, # 0 (S) 1, # 1 (S) 0, # 2 (S) 1, # i (S) 0, i 2. The basic result in bounding the Betti numbers of semi algebraic sets defined by polynomial inequalities was proved independently by Oleinik and Petrovsky [14], Thom [18] and Milnor [13] They proved: Theorem 1 [14, 18, 13] Let S # R k be the set defined by the conjunction 2 of n inequalities, P 1 # 0, P n # 0, P i # R[X 1 , X k ] deg(P i ) # d, 1 # i # n. Then, # i # i (S) O(nd) k . The above bound is ....

....two dimensional sphere, S, # 0 (S) 1, # 1 (S) 0, # 2 (S) 1, # i (S) 0, i 2. The basic result in bounding the Betti numbers of semi algebraic sets defined by polynomial inequalities was proved independently by Oleinik and Petrovsky [14] Thom [18] and Milnor [13] They proved: Theorem 1 [14, 18, 13] Let S # R k be the set defined by the conjunction 2 of n inequalities, P 1 # 0, P n # 0, P i # R[X 1 , X k ] deg(P i ) # d, 1 # i # n. Then, # i # i (S) O(nd) k . The above bound is actually quite tight. One can see this by considering the following ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

....in computational geometry and robotics see [1, 10] Another problem that has received considerable attention from researchers interested in real algebraic geometry, is to bound the topological complexity of semi algebraic sets. A classical result in this area is due to Oleinik and Petrovsky [14], Thom [17] and Milnor [13] who independently proved that the sum of the Betti numbers, i (S) of a basic closed semi algebraic set S 2 R k de ned by P 1 0; P s 0; with deg(P i ) d; 1 i s; is bounded by O(sd) k : This bound was extended to arbitrary closed semi algebraic ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

....and one 2 dimensional hole. Analogously, for the two dimensional sphere, S, fi 0 (S) 1; fi 1 (S) 0; fi 2 (S) 1; fi i (S) 0; i 2: The basic result in bounding the Betti numbers of semi algebraic sets defined by polynomial inequalities was proved independently by Oleinik and Petrovsky [12], Thom [16] and Milnor [11] They proved: Theorem 1 [12, 16, 11] Let S ae R k be the set defined by the conjunction of n inequalities, P 1 0; Pn 0; P i 2 R[X 1 ; X k ] deg(P i ) d; 1 i n: Then, X i fi i (S) O(nd) k : The above bound is actually quite tight. One ....

.... sphere, S, fi 0 (S) 1; fi 1 (S) 0; fi 2 (S) 1; fi i (S) 0; i 2: The basic result in bounding the Betti numbers of semi algebraic sets defined by polynomial inequalities was proved independently by Oleinik and Petrovsky [12] Thom [16] and Milnor [11] They proved: Theorem 1 [12, 16, 11] Let S ae R k be the set defined by the conjunction of n inequalities, P 1 0; Pn 0; P i 2 R[X 1 ; X k ] deg(P i ) d; 1 i n: Then, X i fi i (S) O(nd) k : The above bound is actually quite tight. One can see this by considering the following example: Let P i = L 2 ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

....improving the results in [39] The third part of this work [12] deals with bounding the topological complexity of semi algebraic sets in terms of the number and the degrees of the 5 polynomials describing it. We extend and improve a classical and widely used result of Oleinik and Petrovsky [50] (1949) Thom [64] 1965) and Milnor [47] 1965) bounding the sum of the Betti numbers of semi algebraic sets. This result generalizes the Oleinik Petrovsky Thom Milnor bound in two different directions (see next subsection 1.2.3) It also generalizes our result in [10] on the number of connected ....

....not rely on the machinery of Morse Theory for stratified sets. 1.2.3 Bounding the Betti Numbers and Computing the Euler Characteristic of Semi algebraic Sets We prove a new bound on the sum of the Betti numbers of semi algebraic sets. This extends a well known bound due to Oleinik and Petrovsky [50], Thom [64] and Milnor [47] In separate papers they proved that the sum of the Betti numbers of a semi algebraic set S ae R k ; defined by P 1 0; P s 0; deg(P i ) d; 1 i s; is bounded by (O(sd) k : Given a closed semi algebraic set S ae R k defined as the intersection of a ....

[Article contains additional citation context not shown here]

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

....considered an outstanding open problem in algorithmic real algebraic geometry. Combinatorial and Topological Complexity of Semi algebraic Sets: ffl In [1] we prove a new bound on the sum of the Betti numbers of semi algebraic sets. This extends a well known bound due to Oleinik and Petrovsky [26], Thom [28] and Milnor [24] We proved that for any closed semi algebraic set S ae R k of real dimension is k 0 ; the the sum of the Betti numbers of S is bounded by n k 0 (O(d) k : Our method also gives a similar bound on the sum of the Betti numbers where the algebraic part of the ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

....computational geometry and robotics see [1, 9] Another problem that has received considerable attention from researchers interested in real algebraic geometry, is to bound the topological complexity of semialgebraic sets. The rst and classical result in this area is due to Oleinik and Petrovsky [13], Thom [16] and Milnor [12] who independently proved that the sum of the Betti numbers, i (S) of a basic closed semialgebraic set S 2 R k de ned by P 1 0; P s 0; with deg(P i ) d; 1 i s; is bounded by O(sd) k : This bound was extended to arbitrary closed semialgebraic ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

....Bounding the Betti Numbers and Computing the Euler Characteristic of Semi algebraic Sets Saugata Basu Abstract In this paper we give a new bound on the sum of the Betti numbers of semi algebraic sets. This extends a well known bound due to Oleinik and Petrovsky [19], Thom [23] and Milnor [18] In separate papers they proved that the sum of the Betti numbers of a semi algebraic set S ae R k ; defined by P 1 0; P s 0; deg(P i ) d; 1 i s; is bounded by (O(sd) k : Given a semialgebraic set S ae R k defined as the intersection of a real ....

.... the same bound applies to fi i (S) In the case where the set S is a basic closed semi algebraic set defined by, P 1 0; P s 0; with deg(P i ) d; there is a tighter bound of (O(sd) k on the sum of the Betti numbers of S: This was proved in separate papers by Oleinik and Petrovsky [19], Thom [23] and Milnor [18] Note that this includes the case when S is the real zeros of a set of polynomials. This bound plays an important role in algorithmic real algebraic geometry [17] and has been used recently in proving lower bounds in the algebraic computation tree model (see [24] ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

....Bounding the Betti Numbers and Computing the Euler Characteristic of Semi algebraic Sets Saugata Basu September 10, 1997 Abstract In this paper we give a new bound on the sum of the Betti numbers of closed semi algebraic sets. This extends a well known bound due to Oleinik and Petrovsky [28], Thom [35] and Milnor [27] In separate papers they proved that the sum of the Betti numbers of a semi algebraic set S ae R k ; defined by P 1 0; P s 0; deg(P i ) d; 1 i s; is bounded by (O(sd) k : Given a closed semi algebraic set S ae R k defined as the intersection of a ....

.... the same bound applies to fi i (S) In the case where the set S is a basic closed semi algebraic set defined by, P 1 0; P s 0; with deg(P i ) d; there is a tighter bound of (O(sd) k on the sum of the Betti numbers of S: This was proved in separate papers by Oleinik and Petrovsky [28], Thom [35] and Milnor [27] Note that this includes the case when S is the real zeros of a set of polynomials. This bound plays an important role in algorithmic real algebraic geometry [22] and has been used recently in proving lower bounds in the algebraic computation tree model (see [36] ....

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402, (1949).

No context found.

O. A. Oleinik, I. B. Petrovskii On the topology of real algebraic surfaces, Izv. Akad. Nauk SSSR 13, 389-402 (1949).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC