By Peng Wei, Sun Bingnan, Tang Jinchun

In line with analytical equations, a catenary aspect is gifted for thefinite point research of cable buildings. in comparison with frequently used aspect (3-node point, 5-node element), a application with the proposed aspect is of lesscomputer time and higher accuracy.

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**Example text**

As the proof will show, these properties, with the exception of for complex measures ; (b) is called is called finite additivity; (c) PROOF A e 9Jl so that Jl (A ) < oo, and take A1= A and A2= A3= · · ·= (a) Take 0 in 1 . 1 8(1). An+1= An+l= · · ·= 0 in 1 . 1 8(1). ) B1=A1, and put Bn=An-An-1 for n= 2, 3, 4, . . ::1 B;. Hence ) u + n E u n n Jl (An)= L Jl (B; ) and Jl (A)= L Jl (B;). i= 1 i= 1 oo (e) Now (d) follows, by the definition of the sum of an infinite series. Put en=A1-An. ) n-+ n-+ This implies (e).

Hint : If IX > 1, the integrands are dominated by IXj. If IX < 1, Fatou's lemma can be applied. , � f uniformly on X. (X) < oo " cannot be omitted. 4 1 , and hence prove the theorem without any reference t o integration. ). Prove that to each Jl(E) < b. 24(c) is also true when c = oo . JE I f I dJl < E whenever C H A PTER TWO POSITIVE BOREL MEASURES Vector Spaces complex vector space vectors addition scalar multiplication, x y, y x + x x z; = z) = (x (y y) + x + + y y + + + 0 zero vector origi n 0 = x x + xx + ( - x) = O.

If 1 < i < n, let H i be the union of those Wxi which lie in V; . By Urysohn's lemma, there are functions g i such that Hi � gi � V; . Define h1 = g 1 h2 = (1 - g 1 )g 2 (2) hn = (1 - g1)(1 - g 2) · · · (1 g n - )gn · Then h i � V; . It is easily verified, by induction, that (3) h1 + h 2 + · · + h n = 1 - ( 1 - g1)( 1 - g 2) ( 1 - g n)• Since K c H1 u · · u Hn , at least one g i(x) 1 at each point x e K ; hence (3) shows that ( 1 ) holds. c > > �. 1 3 Theorem E ••• e � � 1 � �m :::J - · = · l • •• IIII The Riesz Representation Theorem Let X be a local l y compact Hausdor ff space, and let A be a positive linear (X).

### A catenary element for the analysis of cable structures by Peng Wei, Sun Bingnan, Tang Jinchun

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