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## Abstract

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

,{\rm{ }}{k_3} = 2{k_1}} \hfill \cr } } \right\}${ a 1 = a 1 , a 2 = 1 8 a 1 k 1 2 , a 3 = − 1 32 ( k 1 4 − 64 ) a 1 , a 4 = a 4 , a 5 = a 5 , a 6 = 1 8 a 5 k 1 2 , a 7 = − 1 32 a 5 ( k 1 4 − 64 ) , a 8 = a 8 , a 9 = a 9 , k 1 = k 1 , k 2 = − 1 8 k 1 3 , k 3 = − 1 32 k 1 ( 5 k 1 4 − 64 ) }$\left\{ {\matrix{ \matrix{ {a_1} = {a_1},{\rm{ }}{a_2} = {1 \over 8}{a_1}{k_1}^2,{\rm{ }}{a_3} = - {\kern 1pt} {1 \over {32}}({k_1}^4 - 64){a_1},{\rm{ }}{a_4} = {a_4},{\rm{ }} \hfill \cr {a_5} = {a_5},{\rm{ }}{a_6} = {1 \over 8}{a_5}{k_1

, hence 8 G Lp CI Np, provided p > 5. a2) p = 13 mod 16. If p = 48A; + 13, k = 0 ,1 ,2 , . . . , then l l p + 1 24 • = 1 mod p, 24 ^ hence 24 e Lp n Np, provided p > 13. If p = 48A; + 29, k = 0 , 1 , 2 . . . , then 2p - 16 = 3(32A; + 14), hence 32k + 14 is a quadratic nonresidue mod p. We Application of Nagel!'s estimate 653 have 48(43Ar + 26) = 1 mod p and (32k + 1 4 ) ( 4 5 * + 27) = 1 mod p. Therefore 32k + 14 G Lp n Np if k is odd or 48 e Lp ("I Np if k > 0 is even, b) p = 1 m o d 8. If p = 24k + 1 7 , k = 0 , 1 , 2 , . . . , then 7 p + 1 1 2

sequence the relation L) becomes * * *) F2j+1 + F2k+2j+2 ~ F2k+1 [F2j+A F2k+23+l ~ F2j+2 F2k+2j-l] = 0. Think of the left hand side of ***) as the function P of j. We shall prove that P{j) = 0. L e t M = ( ^ ) J + ! a n d I . = ( ^ ) f c . Then P(j) =-§Q, where Q is the expression [ u ^ - l ) [ ( 3 V 5 - 5 ) m + 2/?4 j] + (v/5 + 2) ( 2 ^ - 1 + /34fc. Notice that the second term is zero because (3 — and that ( 3 + y ^ \ 0 2 = ( 3 + %/5\ / l - y / 5 \ 2 = / S + V5 \ ( 3 - = l 58 Z. Cerin R) Vrn + 'Yl Uk+i Vk+i = < i=0 Vm = < so that v (32k1 = 0. Hence P

. 1 1 ) ^ ( z ) = 1 exp J o / L . / l + 2 g ( 1 - « J « - * j + ( 1 - 2 . t } e - 2 1 V f v „ - 2 i ® . 2 y ( 2 . 1 2 ) F 2 ( Z ) = £ exp where 0 < (p ^ 2 j t , however 32 K = 1 - e " 2 i V (s~ - h ) ( 1 - r 2 ) - ( 1 + r 2 ) y We have t h e f o l l o w i n g theorem. T h e o r e m . Denote by x t h e o n l y r o o t c f t h e e^ua4"! ->n 1 + P 1 ( 3 # 1 ) ( l + S ) f t ^ + " ^ ^ + ( 1 + r * - ^ [ r 2 - ( 1 + i J ) r - | i] = 0 i n t h e i n t e r v a l ( 0 , 1 ) and l e t ciQ d e u ^ e t h e e x p r e s s i o n - 28.5 - 6 Z. Pachulski (3.2) otn = o

+ 76 k 6 − 16 k 5 − 1226 k 4 + 5456 k 3 − 11348 k 2 + 11984 k − 5167 + 32 ( k1 ) ( k − 3 ) ( k − 2 ) 2 − ( k 2 − 10 k + 17 ) ( k 2 + 2 k − 7 ) , e ¯ = k 8 − 16 k 7 + 76 k 6 − 16 k 5 − 1226 k 4 + 5456 k 3 − 11348 k 2 + 11984 k − 5167 − 32 ( k1 ) ( k − 3 ) ( k − 2 ) 2 − ( k 2 − 10 k + 17 ) ( k 2 + 2 k − 7 ) $$\begin{array}{} \displaystyle e=k^8-16k^7+76k^6-16k^5-1226k^4+5456k^3-11348k^2+11984k-5167\\ \qquad \quad \,\,\,+32(k-1) (k-3) (k-2)^2 \sqrt{-(k^2-10 k+17) (k^2+2 k-7)} ,\\ \displaystyle \bar{e}=k^8-16k^7+76k^6-16k^5-1226k^4+5456k^3-11348k^2+11984k-5167 bounded from above by the valueq3k3k1 6+ 12 32k1 (4.2) and tends to M=m as k!1. Analogously,q3k3k2 = os2 23k21+sin2 23k 6 (21+32k)1 6 12q3k3k1 (4.3) and for k > 2 q3k3k6 6 q3k3k4; q3k3k4 6 q3k3k2 (4.4) because this triple of numbers forms the polynomial S3(T2(x);!;T2()) of the form (3.2) of the first type. In particular,qNN2 6 12qNN1; qNN1 6 + 12N21: (4.5) We estimate q3k3k1 . According to (3.18)Q3k3k1(x) = (2T23k1(x)1)=(2T23k1()1) (4.6) i.e. q3k3k1 6 3=(2T23k1 ())1)6 3: (4.7) In particular, qNN=3 6 3=(2T2N=3())1)6 3: (4.8) Expanding the numerator and the denominator - f r ^ t ) , r 2 ( t ) } Let Λ (9 ) «= = ( t ) ,  2 ( t ) } , Λ ( ν ) = ψ = [ V 1 ( t ) , ^ 2 ( t ) } According to (15) # k < t ) . * k ( t ) - + p ; ( t . s p 1 ( t ) f s p 2 l t ) ) + ^ - p * ( t , v 1 ( t ) > v 2 ( t ) ) + / 3 1 — ^ £ — V r (k = 1 , 2 ) We introduce the f o l l o w i n g n o t a t i o n A ( t ) = P ( t , 5 p . , ( t ) , y 2 ( t ) ) - F f t . V - j d O . V g C t ) ) (32) SPk(t) = x k + i y k X = ( X 1 f s 2 ) y = (y1 , y 2 ) (32') (k=1, 2) T k ( t ) = x k + i y k χ = ( x v x 2 ) y = (y . , ,y 2 ) Then re re r im im Ι A ( t ) = P ) .$$\eqalign{& {{dR} \over {d{T_{11}}}} = \cr & - {\kern 1pt} {\kern 1pt} {R \over {16{{({k_3} + 2 {\kern 1pt} {\kern 1pt} \sigma)}^2}}}\left({16{R^2}{k_2} {\kern 1pt} {\kern 1pt} {\Omega ^2}{\omega ^2} + ({R^2}{k_{2}}{\kern 1pt} - 8 {\kern 1pt} {\kern 1pt} {k_{1}}{\kern 1pt}){{({\kern 1pt} {k_3} + 2 {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma)}^2}} \right).}} By setting d R d T 11 = 0 ${{dR} \over {d{T_{11}}}} = 0$ , the nontrivial solution is obtained and given by (28) R = 8 k 1 k 3 2 + 32 k 1 k 3 σ + 32 k 1 σ 2 16 k 2 Ω 2 ω 2 + k 2 k 3 2 + 4 k 2 k 3 σ + 4 k

= 0,08 normal war. Der Quotient /± KCl : f ± KaCt in 0,08 n Lösung ist nur wenig von 1 verschieden, so daß innerhalb der Versuchsgenauig- c O keit - K • J?" gleich F . ECl: F . NaCI sein sollte. Für andere Alkaliionen c j i o CK gilt das entsprechende. Die Bestimmung der Alkaliionen in der Lösung geschah direkt, die der im Austauscher gebundenen Alkaliionen erst nach dem Aus- waschen mit 2 n HCl mit dem Flammenphotometer. Als Mittel aus fünf Messungen wurde erhalten. ,32, ^ • °K. =. 1,89; • °.K = 1,45. «JVa CLi CK CLi rK ^Na Die Messungen zeigen also, daß