1 y fi x X j 1 2 a regio R o euclidea mspac int par o euclidea wspace Sup pos tha eac f unctio 0 1 n i o clas C i R ql 1 critical point o th ma 11 i a poin i R a whic th matri x firs derivative 2 i l mj l n i o les s tha maximu rank Th ID: 26852
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map y* = fi{%\ x\ , X j = , a R f = , n) C R critical point a R (i = l , m;j = l, , n) rank a # x. A critical value a» = a If m^n, the set of critical values of the map is of m-dimensional measure q; if� mn, the set of critical values of the map is of n-dimensional measure zero providing that q^m + q a (m/q)- first » = a m C a a m. The measure of the critical values of differentiable maps of euclidean spaces \ C q A function not constant on a connected set of critical points, 1 The behaviour of a function on its critical set, m = 3 m = 6 a a q a A 5 a. A d(A). Ai, } a A a; L ; a) s-dimensional outer measure A L = c c + c s, a A s-null L = c a (s+p)-null (p L a L = \A\,| A | A. a x� r0. A = | | d = , r) x x to u u f x = , r; g = , m - r. Xo / \j\ x ^(u), i , m, C u y u = , (3.3) \y+ f = F*+ = , n - r. = , m;j = l, , n). Functional Dimension Mass, Theory A | 0 =\j\ we may consider 3.3) near of near xo without changing either critical values or ranks of critical points. m^n. The critical values of the map constitute an m-null set if m Sn. first A a N A f(NA), NA A N N R. X\a NA a N. y y[ - y[ = £ - x\), j = , n, � X2X\. a ra-cube C(y) a A, ô{f[NC(y)]}^2m ô{f[NC(y)] }S (2m, f | Xi NA, X2 N , i = � a0 a a A, y S{f[NC(y)]} a,]} - ' C(y) C(y) NA 7 a a Ci, } NA a a Eineindeutige Abbildung und Messbarkeit, a a T,\Ck\£\N\ l. k {f(NCk)}(NA) 8f(NC Ô = ) k \N\ + 1 f(NA) ra-null first B r, r m. f(B) ra-null a N B f(NB) a XQ B. a K u D r K. U\ a D u^ a K. y F(u d d d d y2 Ji = u a = ' , r yi = + - = , n r, f U\ a y (u , C(y) (u , u pa 2 a strip C(y) u , uu , u 5 a D. Ui u% S y = l, I ^ i^ 1 \y* - yi , d = , I I ^ 2rUy/2 + A = , n - U | Fl | u K fUi'm D, | ul = , r; h , n r\ i = l, - - , m). ö F(S). Ô V W=(n F W V^r^l. a� = * a/(V + y WÇ p 2V(4:V) 2 4V(4:V) y d y Ô D. 2 2 yy = t) y F[DC(y) a mih rj | C(y) C{y) D y Ck} D + k F(D)F(DCk) a a vHk\ F(D) N K. N a x f(NB) F(D) Given a positive integer q and a set A in the space of the variables x. There exists a sequence A Ai, A of sets with the following properties: A Ao is denumerable, Ak (k = ) is bounded, if g(x) is any function of class Ccritical set includes A and if x\ and xi are points of Ak, then = k = . x X2 ~ Xi \ Let A be the set of critical points of rank zero of the map Then f (A) is an s-null set if s^m/q. A (A (A k) (k a Ak (k fixedB=A� a0 a x B. C(y) y centered at a G 1 y 8{f[C(y)]} (6.3) | f'XxO - P\x) | , , - j = , Xi BC{y). Snty/2 iixi BC(y). (6.3) j j + 2 / (\B\ + iy' *{f[BC(y)]}° B\ + l), 7 G finite. = - ) x%. C(y), y x B. aCk B ^ + k Ck (k = {f(BC} (B). [f(BC ] ; T,S[f(BC ej: + è | C | + ^ 6 fc qs^m | Ck\ =7 (k = � mn. Let A be the set of critical points of rank r of the map Thenf(A) is an n-null set if q^(m� mn. r s 0 a N Af(NA) a XQ A. a N UQ. u , u a (nt-space , u (y , y , u r (u , a 3.3) r, , u a (3.3)u , u (n B 3.3) r N. F(B) , y (n F(B) F(B) If� mn the critical values of the map constitute an n-null set providing that q^m r = , n r, ^ (m q W(x , x a / x , x\ C 0 1 a ^ a y W(x\ , x x = , n; m� n, C y