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2007 621.315.3 .. . . - .: 2007. - 60 .

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11. .......................................................................................................... . Rietveld . Powder Match. . .

. .................................................................................................................................................................................... X F (x), X, F (x) = P ( < x) , X = 0, 1, P ( = 1) = p, P ( = 0) = 1 - p.

= p, V = p(1 - p) = p - p2.

n = i = 0, 1,..., n i=m P ( = m) = Cn pm(1 - p)n-m = pn, V = np(1 - p) = np - np2.

[0, 1, 2,..., ) am P ( = m) = exp(-a) m! = a; V = a.

[a, b] [a, b] 0 t < a P ( < t) = (t - a)/(b - a) a t b p(t) = [a,b](t) b - a 1 t > b a + b (b - a) =, V =.

2 = - a [0, c] c = b-a c (c) =, V =.

2 (-, ), F = N(a, 2) 1 (t - a) p(t) = exp(- ) = a; V = 1 p(t) =, (1 + ((t - a)/)2) n (-, ).

1 1 < t 2 2 < t = P = p(t1, t2,..., tn)dt1dt2...dtn.

......

n n < tn n 1 2 p(t1, t2,..., tn) = (2)- det(A) exp(- (AT, T )), t1 - a1 a t2 - a2 a A > 0, T = V =......

tn - an an n = 1 0 - r 2 1 1 1 1. A = 2. A = 2 1 - r2 - r 2 2 r [0, ) 1 - exp(-t) t P ( < t) = 1 p(t) = exp(-t)[0,)(t); =, V = ln2 = P ( < T1/2) = T1/2 2.

n n = i, i N(0, 1). [0, ) i= = n, V = 2n.

2.

n (0, 1) x 1 (0, 1) N(0, 1) a = u1 s, b = u2 s lnm u1 = 21 - 1, u2 = 22 - 1, s = -2, m = u2 + u2, m < 1.

1 m (0, 1) a = -ln() p(t) = exp(-t) n (0, 1) xi, i = 1, 2,..., n m1 v1 y m2 v xi - myi = v2 + m v xi k k yi = (xi + xi+1) v xi, i = 1, 2,..., n yi Y = A X, A n T t1, t2,..., tm, P ( < t) atrue P ( < t, atrue) T P atrue atrue T a atrue atrue = a P atrue atrue atrue = aT = (t1, t2,..., tm) (0, a) a (0, a) a m = 2 ti/m i= = 2 (m a/2)/m = a a a V = 4 m /m2 = 12 3m a a = max ti, i = 1, 2,..., m.



(x)m 0 x a a P ( < x) = 1 x > a 0 x < xm- p(x) = (0, a) am a m ma (max ti) = xmdx = am 0 m + a m a m m (max ti)2 = um+1du = aam 0 m + m V (max ti) = a(m + 2)(m + 1)m + = max ti, i = 1, 2,..., m.

m a m + 1 (m + 1)2 m V ( max ti) = a2 = m m (m + 2)(m + 1)2 m(m + 2)a m m m m s2 = (ti - t)m a s2 = m2 = 2.

m s2 = (ti - a)2 - (t a)2 s2 = m m-m m sm- B2 + V B, V a = f(Mi) Mi i = f(Mi) Mi i T = (t1,..., tm) P ( < t, a) p(t, a) a L(a) m Lc(a) = p(ti, a) i= m Ld(a) = P (ti, a) i= m lc(a) = lnLc(a) = ln(p(ti, a)) i=m ld(a) = lnLd(a) = ln(P (ti, a)) i= m E(a) = p(ti, a)ln(p(ti, a)).

i= a = ARG MAXL(a) L(a) a dL = da A = (ai), i = 1,..., n L = 0, i = 1,..., n ai n = i ti=tlnL(p) = ln(Cn ) + t0ln(p) + (n - t0)ln(1 - p) l t0 n - t0 t - = 0 p = p p 1 - p n t1,..., tm k ti i=p = m n tat l lnL(a) = ln( exp(-a)). = 0 t0 ln(a) - 1 = 0. = tt0! a t1,..., tm k ti i= = m = [0, a] m 1 p(t, a) = [0,a](t). L(a) = ( )m [0,a](ti). = max (ti) a a i= > tmax < tmax 0,(ti) = m l m lnL() = nln() - ti. - ti = 0.

i=m m ln2 ti i= = ; T1/2 = m ti m i= m (ti - a)lnL(a, ) = -mln() - - m/2 ln(2) i=m m l ti - a l m (ti - a) = 0. - + = 0.

a 2 i=1 i=m m = ti/m; 2 = (ti - )m i=1 i= = ARG MAXE(a) E E(a) a = 0.

a (a) a = ad((a)) L(a) (a) = L(a)da aL(a)da = L(a)da am/m!exp(-a) T t a aat/t!e-ada at+1e-ada ate-ada 0 0 = = = (t + 1) = t + at/t!e-ada ate-ada ate-ada T = (t1,..., tm) (0, a) = max (ti) m m t P (max ti < t) = P (ti < t) = b i=a a t t am (max ti) = t(( )m) dt = m ( )mdt = 0 a 0 a m + a m+ln(p(a)) I1 = ( )a p a I1 = ( )p m Im(a) = mI1(a) V Im(a)-(ti - a)2 1 m I1(a) = ( ) =. Im = 4 2 V = m m0 1 a 1 m I1(a) = ( - 1)2 = (m0 - a)2 = =. Im = a a2 a2 a a a V = m m a limm = a P (limm = a) = C V a m ( - a)/ m N(0, V a) m T = t1,..., tn p(t, a) a a a = a0 a = a1 T T f(t1, t2,..., tm) a f G(f) g(f) [f1, f2] f(T ) ti, i = 1, 2,..., m a = a 1 - f T, a [f1, f2] a = a m = 1 f t p(t, a) m 2 f a ak, k = 1, 2,...

H : a = a0, n 1 ti - af(t1,..., tn) = n i=g(f, a) = N(0, ) n H : a = a0, n f(t1,..., tn) = n - 1fracb - a0s, b = ti n i=n s = (ti - b)n i=g(f, a) = S(n - 1) = n - 1.

H : a = an ti - af(t1,..., tn) = ( ) i=g(f, a) = 2 = n n a H : = sn n ti - b f(t1,..., tn) = ( )2, b = ti s0 n i=1 i=g(f, a) = n- H0 : a = aH1 : a = a H H1 HL(a H0) lnL(a H0) r = ; r = ;

L(a H1) lnL(a H1) I = (1 - 1, 1 + 2) r > 1 + 2 H0 r < 1 - 1 Hr I r y(x), x [x1, x2] y(x) = m(x) + e(x), e(x) x (P (e(x) < t(x))) x e(xj) = 0, V e(xj) = v(xj), (e(xj)e(xi)) = cij = 0 i = j c(xi, xj) = V y(x) = V m(x) + V e(x) = v(x). v(x) x y(x) t t y(x) y(x) y(x) + m(x) m(x) m(x) m(x) P m(x) m(x)- > m(x, P ) P n m(x, P ) = ax + b, P = (a, b);

m(x, P ) = ax2 + bx + c, P = (a, b, c); x n- m(x, P ) = aixi, P = (a0, a1,..., an-1);

i=n- m(x, P ) = aifi(x), P = (a0, a1,..., an-1).

i= ai fi(x) m(x) x P m(x, P ) P x x - C ln2t A T Aexp(-( )2), Ae-,, Asin(W t).

W 1 + ((x - C)2/(W )) exp, sin, m(x) = exp(-x ) m(x) = 1+xA m(x) = Aexp(-(x-C) ) m(x) = ) 2 2W 1+(x-C)2/W A C W A, W (x-C) m(x) = Aexp(-(Lx+W )2 ) L = L = n (x - C)aixiexp(- ), 2W i= n + 2 ai, C, W, i = 0, 1,.., n C, W n n aixi n i= u(x) Ok(x)u(Pn(x)), Qk(x), Pn(x) k, n k + n + k, n u(x) u(x) x - C Au( ), W 1 C Qk(x) = A, Pn(x) = x +, W W k = 0, n = 1 u(x) Qk(x), Pn(x) Au(Wx-C ) x+Wy(xj) j = 1,..., m f(x, P ) P - n d(x) x P y(x) P Ptrue = P P Ptrue P Ptrue = P y(xj) j = 1,..., m f(x, P ) P n d(x) P m S(P )2 = w(xj)(y(xj) - f(xj, P ))2, w(x) =.

d(x) j=f(x, P ) n S(P )= 0, i = 1,..., n pi f w(x)(y(x) - f(x, P )) = pi x n f(x, P ) = pigi(x) i=n w(x)gi(x) gk(x)pk = w(x)y(x)gi(x) x x k= AP = B, aik = w(x)gi(x)gk(x)pk, bi = w(x)y(x)gi(x) x x bi A aiipi = bi pii = aii = aii P = A-1B A P P f(x, P ) d(x) P = P A- y(xj) (N(f(x, P ), d(x))) N(P, A-1) S(P ) m-n y(x) f(x, Pmin) Pmin S(Pmin)2 m-n y(x) S(Pmin)2 m-n d(x) S(Pmin)2 m-n d(x) 2+ m-n y(xj) - f(xj, P ) S(Pmin)0.(m-n) 2+ m-n d(x) y(x) f(x, Pmin) P m S(P )2 = w(xj)(y(xj) - f(xj, P ))j=S2 m f = w(xj)(y(xj) - f(xj, P )) = 0, pi j=1 pi P f n f(x, P0) f(x, P ) f(x, P0) + pi pi i=B = AP, m m f0 f0 fB = w(xj)(y(xj) - f0), A = w(xj), pi pi pj j=1 j=P pi f0 = f(xj, P0) P = A-1B P P1 = P0 + P.

PPk+1 = Pk + A(k)-1B(k), k A(k) B(k) A B P = Pk Px e(x) A-f(xi) = 1, 1, 1, 1,..., 1, 101, 1,..., 1, n = c(x) = C x = 1, 2,..., 100 c(x) n fi i= f(x) c(x) = = n f(x) n (f, c) = |f(xi) - c(xi)|, i= c(x) = f(x) = 101.

w(x) Pm S(P )2 = w(xj, P0)(y(xj) - f(xj, P ))2 P, j=1/d(x) y(xj) - f(xj, P0) k d(x) w(x, P0) = 1/|y(xj) - f(xj, P0)|1/d(x) h(j) const;

w(x, P0) = c(j)/d(x) y(xj) - f(xj, P0) 1 + h(j) = ; c(j) =.

(h(j)/const)2 + d(x) = = P Pi, i = 1, 2,..

Px P P PPk+1 = kG(k)B(k), k G G Pk+1 = kB(k).

B P Pk+1 = (RA)-1(k)B(k), 0 < 1 k R A B R(A) = (A + I)- P P P P S(Pk+1)2 - S(Pk) S(P )2 m-n S2+ S A Pk+1 = Pk - kMG(Pk) G(Pk) M M D(P ) G P P G(P0) = k Pk P0 M, D P Pk+1 - P0 = Pk+1 - P0 - kMG(Pk) = Pk+1 - P0 - kM(G(Pk) - G(P0)) = Pk+1 - P0 - kMD(Pk)(Pk - P0) = (I - kMD(Ptk))(Pk - P0) Ptk = P0 + t(Pk - P0), 0 t 1, I (I - kMD)(Pk - P0) Pk - P0 I - kMD I - kMD = maxi |1 - kik| = qk, ik M D qk 1 - kmin = -(1 - kmax) k = n = mini ik x = maxi ik x+n M D Pk+1 - P0 qk Pk - P0, qk 1.





k+ Pk+1 - P0 qi P1 - P0, qi 1.

i= n > 0, x - n Pk+1 - P0 ( )k+1 P1 - P0, x + n P M = D D = A- k = 1 = F (f(P ), g(Q)) P g Q qi Q P Q Q = Q0 Q Q P Q Q P pi = qi F pi m F gB = w(xj)(y(xj) - F (f, g)), g qi j=f A pi P (A + A)B.

A + A B P m S(P )2 = w(xj)(y(xj) - f(xj, P ))j=g(P ) 0.

pil pi piu; pj = kjpi + bj. i = 1, 2,..., n j > i.

pj = kjpi+bj i i kj bj j P pi pil pi piu pi = pil pi < pil piu pi > piu pi P [pil, piu], i = 1, 2,..., n, m S(P )2 = w(xj)(y(xj) - f(xj, P ))2 + g(P ).

j= g(P ) g(P ) g(P ) g(P ) f(t) g() = f(t)e-itdt f(t) = g()eitd 2 f(t)) F (f (t) = iF R(x) y(x) = R(x) f(t) = R(t - x)f(t)dt y R f F = F F y(x) R(x) f(t) -1(F F R y) f(t) = F s(k), k = 0,..., m s(v), v = 0,..., m s(k), k = 0,..., m m s(v) = s(k)e-qkv, v = 0,..., m m + k=m s(k) = s(v)eqkv, k = 0,..., m;

v=2i q = m+ s(v + m - 1) = s(v) m s(m/2 + 1) = s(k) F (s(k + 1) - s(k)) = (eqkv - 1)F s(0) = s(m) = s(k) = 1, k = 0,..., m;

m s(v) = e-qkv = 1 + e-qv + e-q2v +... + e-qmv, m + k=1 v = s(v) = 1 1-e-qv(m+1) = m+1 1-e-qv s(k) = eqkn, k = 0,..., m m 1 1 - e-q(v-n)(m+1) s(v) = eqkne-qkv = 1 + e-q(v-n +... + e-qm(v-n) = m + 1 (m + 1)(1 - e-q(v-n) k=1 v = n s(v) = s(v) s(k) m/ 2/(m + 1) s(k) sin(wit) cos(wit) wi w w 2/(m + 1) s(v) v1, v2,... vm-v, vm-v,...

1 v1, v2,...

si, Ri i = 0, 1,.., 2n + Ri i = n + yj n yj = sj+iRn+i, j = 0, 1, 2,..., 2n + i=-n 2n+2 n yv = ( sk+iRn+i)e-qkv 2n + k=0 i=-n sk+ie-q(k+i)vRn+ie-q(n+i)veq(n)v eq(n)v sk+i Rn+i DF y = DF R DF s eq(n)v 2n + y(a, b) = C ((b - x)/a)f(x)dx C (x, a, b) exp(ixw) a, b dn n(x) = (-1)n exp(-x2/2), n = 0, 1, 2,...

dxn f(x) n(x) h(xk), k = 0, 1, 2,.., m m/2 n(x) f(k) F s(l), k K, l L f(k) s(l) n fk Pl(k) f(k - 1), f(k), f(k + 1) f-1, f0, f1 P1(k) = ak + b, k = -1, 0, 3 0 b fk = 0 2 a fkk f0 = b b = (f-1 + f0 + f1) f(k - 2), f(k - 1), f(k), f(k + 1), f(k + 2) f-2, f-1, f0, f1, f2 P2(k) = a(k2 - 2) + bk + c, k = -2, -1, 0, 1, 5 0 0 c fk 0 10 0 b = fkk 0 0 14 a fk(k2 - 2) 1 2 1 f0 = -2a + c c = fk a = fk(k2 - 2) k=-2 k=-5 f0 = (-3f-2 + 12f-1 + 17f0 + 12f1 - 3f2) F (f(t), f (t), f (t)dt + r(f(t), y(t)) t1,t f(t) F f(t) r f(t) y(t) r F (x, x + dx) f (x + dx) - f (x) c (x) =.

(1 + f (x)2)dx dx c(x) = lim c (x) = dxf (x + dx) - f (x) 1 f (x) lim =.

dx 1 + f (x)2 1 + f (x) x1, xx2 xf (x)(x1, x2) = c(x)2dx =.

x1 x1 1 + f (x)df(x) = (f(x + dx) - f(x))/dx.

dx d(x) = f (x).

x(x1, x2) = d(x)2dx.

x b f (x)f (x) (a, b) = dx.

a 1 + f (x)f (x) (f ) (f ) f f b f (x) (x) =.(a, b) = (x)2dx b a 1 + f (x) exp(-ikx) k = 0, 1,...,, (0, 2) (0, 2) = k4/(1 + k2)bdx = 2k4/(1 + k2)b.

c k4-2b, exp(-ikx) b = k.

b f /(1 + f 2)1.f(x, a, p, w) = a m((x - p)/w).

((a2m 2/w4)/(1 + (a2m 2/w2)b a2b-2/w2b- b = 1, f(x) f(x) b = f(t) y(t) (t1, t2) F (f(t)) = f (t)m-1 m f (i)2 + q w(i)(f(i) - y(i))i=2 i= f(1) = y(1), f(m) = y(m) q = w(i) i w(i) y(i) y (i) y(i) f(i) w(i) 1 | f0(i) - y(i) | c(i) w(i) = 1/(1+ | f0(i) - y(i) |) c c = w(i) w(i)q f(i + 1) - 2f(i) + f(i - 1) - w(i)f(i) = -w(i)y(i), i = 2,..., m - 1, g(1) = f(1), g(m) = f(m) f(i) = c(i)f(i + 1) + h(i), c(1) = 0, h(1) = y(0), i = 1,..., m - 1, c(i), h(i), i > 1 f(i) f(i + 1) - 2g(i) + c(i - 1)f(i) + h(i - 1) - w(i)f(i) = -w(i)y(i).

c(i), h(i) c(i) = 1/(2 + w(i) - c(i - 1)), h(i) = c(i)(w(i)f(i) + h(i - 1)), c(1) = 0, h(1) = y(1) f(i) y(i) f(i) m-1 m f (i)2e(i) + w(i)(y(i) - f(i))i=2 i= f(1) = y(1), f(m) = y(m), f (1) = 0, f (m) = 0.

F (f (t)) = f (t)2e(t) e(t) w(t) e(i) = 1/(1 + y (i)2) w(i) = 1/(i)w(i) = e(i + 1)d (i + 1) - 2e(i)d (i) + e(i - 1)d (i - 1) + w(i)f(i) = w(i)y(i).

d(i) = f (i) f(1) = y(1); f(m) = y(m); f (1) = 0; f (m) = 0;

P (i) = Ai P (i + 1) + B(i) P (i) f(i), f (i)) Ai 22 B(i) Ai(1, 1) = ui(1, 1); Ai(1, 2) = ui(1, 2);

Ai(2, 1) = ui(2, 1); Ai(2, 2) = ui(2, 2);

A1(1, 1) = A1(1, 2) = A1(2, 1) = A1(2, 2) = 0;

B1(i) = ui(1, 1)B1(i - 1) - ui(1, 2)( w(i)f(i) - e(i - 1)B2(i - 1))/e(i + 1);

B2(i) = ui(2, 1)B1(i - 1) - ui(2, 2)( w(i)f(i) - e(i - 1)B2(i - 1))/e(i + 1);

B1(1) = f(1); B2(1) = 0;

ui(k, j) 2 - Ai-1(1, 1) 1 - Ai-1(1, 2).

-(e(i - 1)Ai-1(2, 1) + w(i))/e(i + 1) (2e(i) - e(i - 1)Ai-1(2, 2)/e(i + 1) f(i), f (i)) r (x) f(x) = max(0, - ) 1 + r (x) n s(x) = fi(x) + b(x) + e(x) i= fi b(x) e(x) x x = 1, 2,..., m fi mi(x) mb(x) fi(x, A, P, W, C) = A mi((x - P )/(C x + W )) A P W C s(x) / 1 + s(x) xj s(xj) > err(xj) err(x) x s(x) s(x) s(x) A, P, W, C z = (x - P )/(C x + W ) fi = mi((x - P )/(C x + W )) A fi mi(z) = -A /(C x + W ) P x fi mi(z) = -A z/(C x + W ) W x fi mi(z) = -A z x/(C x + W ) C x s(X) n s(X) = fi(X) + b(X) + e(X) i= fi i b e X fi m(X), xk - Pik Aim(Mi Zi), Zi =, Cikxk + Wik Ai i xk k X Pik, Cik, Wik k Mi X 1 M =.

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