Arithmetic of dimension theory

Abstract

Fiber optic sensors (FOS) for vibration monitoring of smart structures have certain advantages over conventional strain gage based sensors due to electromagnetic environment insensitivity and high response bandwidth. During the present study, a spatially integrating fiber optic sensor was used for vibration monitoring. It is based on the concept that the optimal placement of the sensing element can be sought by using a priori knowledge of the mode shapes of the structure. The applicability ofthis approach to polarimetric optical fibers is described. The acquired integrating FOS signal was used to sense and control the vibration of an eiectrorheoiogicai adaptive structure subjected to a random external excitation.

Extension theory of infinite symmetric products

“…Those talks were the starting point to the author's understanding of paper [28] which eventually led to the ideas developed in this paper. Also, the author would like to thank the referee for excellent work in detecting numerous typos and simplifying several proofs.…”

“…If X = ∞ n=1 X n , X n is a closed subset of X and K ∈ AE(X n ) for all n, then K ∈ AE(X) provided X is normal and K is an absolute neighborhood extensor of X (K ∈ ANE(X)). Theorem 2.3 (see [31] and [28]…”

“…Thus, in this paper we show that the relation SP(K) ≤ SP(L) is of purely algebraic nature. We analyze it by generalizing the connectivity index of Shchepin [28] to the concept of homological dimension of CW complexes. To analyze the relation ext-dim(X) ≤ SP(L) we introduce the concept of cohomology groups H * (X; L) of X with coefficients in a CW complex L (see Section 4).…”

“…Similarly, one can introduce the homotopy connectivity index hcin(K). We start with the concept of the total function space which is related to Shchepin's [28] concept of the total cohomology of a space. Definition 8.3.…”

“…Let us move to algebraic concepts associated with extension theory by employing the connectivity index introduced by E. Shchepin [28]. (1) cin(K), (2) the supremum of all n ≥ 0 so that K ∈ AE(S n ), (3) the supremum of all n ≥ 0 so that K is homotopy k-connected for all k < n. Proof.…”