soxumis saxelmwifo universitetis S r o m e b i VII

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Theorem 3.1Let[ ai, bi]ОВ . Leti1distributed in the interval [ a , b ] for kОN. We setnn ( k)n= Х( j=1 j) ґ Х{ k}k= 1k>nkk( k )(n)n NxОbe uniformlyY U x a . Then ( Y ) Оis uniformly distributed inthe [ ai, bi],i1Theorem 3.2 Let f be a continuous (w.r.t. Tikhonov metric) functiononni1n N[ a , b]. Then the f is Riemann-integrable onii [ ai, bi].We have the following infinite-dimensional version of the Lebesguetheorem (see, [10], Lebesgue Theorem , p.359).Theorem 3.3 Let f be a bounded real-valued function oni1[ a , b]. Then f is Riemann integrable oniif is l -almost continuous (w.r.t. Tikhonov metric) onTheorem 3.4 Fori1i1i1 [ ai, bi]if and only if [ ai, bi].i1[ ai, bi]ОВ , let ( Yn)n О Nbe an increasingfamily its finite subsets. Then ( Y ) Оis uniformly distributed in thei1nn N [ ai, bi]if and only if for every continuous (w.r.t. Tikhonov metric)function f on [ ai, bi]the following equalityi1holds.limеf( y)тҐХ [ ai, bi]yОYni=1n®Ґ =Ґ#( Yn)( R) f ( x) dl( x)Хl ( [ a , b])i=1ii50
REFERENCES1. L. Kuipers and H. Niederreiter. Uniform distribution of sequences.– Pure and Applied Mathematics. Wiley-Interscience [John Wiley& Sons], New York-London-Sydney, 1974.2. G. H. Hardy, J. E. Littlewood. Some problems of diophantine approximation.– Acta Math. 41 no. 1 (1916)).3. R. Baker. "Lebesgue measure" on R . – Proc. Amer. Math. Soc.,113(4) (1991), 1023-1029.4. R. Baker. "Lebesgue measure" on R . II. – Proc. Amer. Math. Soc.,132(9), (2004), 2577-25915. G. Pantsulaia. Relations between Shy sets and Sets of np-MeasureZero in Solovay's Model. – Bull. Polish Acad.Sci., vol.52, No.1,(2004). 63-696. G. Pantsulaia. On generators of shy sets on Polish topological vectorspaces. – New York J. Math., 14 ( 2008), 235-2617. G. Pantsulaia. On ordinary and Standard Lebesgue Measures onҐЎ . – Bull. Polish Acad. Sci.73(3) (2009), 209-222.8. G. Pantsulaia. Riemann Integrability and Uniform Distribution inInfinite-Dimensional Rectangles. – Georg. Inter. J. Sci. Tech., NovaScience Publishers. Volume 2, Issue 3 (2010).9. G. Pantsulaia. On uniformly distributed sequences of an increasingfamily of finite sets in infinite-dimensional rectangles (Submittedfor consideration to publish).10. S. M. Nikolski. Course of mathematical analysis (in Russian), № 1,Moscow, 1983.51
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At t [ 0, ), N3(t ) , when t .
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N220N 10+ N210 N20 t2De N10 N201t1
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Let's really copy (3.5) - (3.7) as
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As to the international organizatio
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2 N0 t ln N2010 N N1020828(4.33
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5. M. Hananashvili. Information str
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Russia, though with delay, operates
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al organizations (the United Nation
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The third side initially does not s
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3. D = 2 8< 0N230N3(t )=2N10 N202
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2N10 2N30The sign N1(t)at big t def
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1 N10 N20t = Dln N30N 10N20
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whenceN210N 20 1e 2t- A= 0 ,e 2t =
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As in the right part of the last pa
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then, the antagonistic sides under
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1
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t0F .Thus, when,30The lemma is prov
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As for N1(t ), N1(t ) , at t ,
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1. Let, D x x, x2,,x ,2. The state
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Let h hbe the partial-time mesh o
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p 1By t F,where0 0B E R E , y y
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theorem 3.1 it is easily possible t
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ods used for numerical solution of
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x b b x b x 1 2 1 1 1 1u( x1) 1
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21k(1) ( i)z h h f x2andx2b21 N
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z(1) z(1)1 z(1)2b2 xb2b1 bxb(1)22b2
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