Abstract: Toute solution saturée de ce système est donc définie dans l'intervalle I. Dans ce travail nous étudions les propriétés asymptotiques des solution du système (1.1) seulement dans le cas où les coefficients a = a(t), d = d(t) de ce système sont bornés pour t € I; au §2 nous admettons, de plus, que pour tout t e I les coefficients b = b(t), c = c(t) sont positifs,. au §3 nous les supposons négatifs* Dans l'étude des propriétés asymptotiques du système ( 1 . 1 ) nous appliquons la méthode topologique de Vatewski [3]• Dans ce travail nous admettons les définitions posées dans [1].

Abstract: 1 . P r e l i m i n a r e s Let M ¿ 0 be a s e t and C an a r b i t r a r y s e t of r e a l f u n c t i o n s d e f i n e d on M. We denote by r c the weakest t o p o logy on M such t h a t a l l f u n c t i o n s b e l o n g i n g t o C are c o n t i n u o u s . Por any s e t A c o n t a i n e d i n M we denote by ClA the s e t of f u n c t i o n s of the form » |A where or e C. We denote by C^ the s e t o f a l l r e a l f u n c t i o n s on A such t h a t f o r any p o i n t p of A t h e r e e x i s t s i n r c an open n e i g h bourhood U of p and a f u n c t i o n a e C, such t h a t n U = = a | A O U. I t i s easy to v e r i f y t h a t , f o r any s e t A c M, we have r c = t\"C|A = *\"C|A. I n p a r t i c u l a r r^, = r c > We denote

Abstract: All algebras in this paper are assumed to be commutative, unital, with the unit element denoted by e, and linear over the field C of complex numbers. A topological algebra is a topological linear space in which is defined a jointly continuous associative multiplication. We shall consider only the complete algebras, i.e. topological algebras which are complete as the topological linear spaces. By a superalgebra of a topological algebra A we mean any topological algebra B containing a subalgebra topologically isomorphic to A (which we identify with A and so write A c B) and having the same unit element as A. If K is any class of topological algebras, then ideal I of an algebra A e K is called K-non-removable if for any 3uperalgebra B d A, B e K, the ideal I is contained in a proper ideal of 3. Otherwise the ideal I is said K-removable. The latter means that there is an algebra B e K with A c B, and there are elements a^ e I, b^ e B t i = 1,...,n, such that

Abstract: This inequality has been extended to discrete harmonic functions by S.Verblunsky [8] and R.Duffin [1]. They have proved the existence of an absolute constant A(< 50) such that o every function f(p) defined on the integral lattice Z satisfying the equation A f ( p ) = f ( p + e 1 ) + f(p-e^) + f ( p + e 2 ) + f ( p e 2 ) 4f(p) = 0, e1 = (1,0), e 2 = (0,1) and positive in the disc D(o,R) satisfies the inequalities |f(e^) fi.o)|< | f(o). (j = 1,2).

Abstract: Introduction In the present paper-we consider the connection between r e l a t i o n s of tangency of arcs in general metric spaces. W.Wal i s z e w s k i in [5] gave the d a f i n i t i o n of tangency of se ts in a general ized metric space (E, 1 ) . Here E denotes a set and 1 i s a non-negative r e a l funct ion defined on the Cartesian product Eo* EQ, where E0 i s a family on non-empty subsets of the set E. Under some assumptions, the funct ion 1 induces a metric lQ(x, ,y) = l ( ( x j , [ y ] ) on the set E, which allows us to def ine the notions of a sphere and a b a l l in the space (Ε , 1 ) . According to t h i s d e f i n i t i o n a set Ae BQ i s (a, b) tangent of order k to the se t Be EQ at a point ρ of the space (E, 1) i f the pair (A, B) i s (a, b) concentrated at the point ρ in the considered space and we have

Abstract: The conjeoture (see [ 3 ] ) that every bounded subset A of the Euclide-an n-space En with pos i t i ve diameter 6'(A) i s the union of n+1 sets each of diameter less than ¿¡(A) has been considered by many authors ( in part icular by W.G.Boit i a n s k i j [1] and [ 2 ] , H.G. Eggleston [4 ] , B. Gale [ 5 ] , B. Griinbaum [6 ] , H. Hadwiger [ 7 ] , H. Lenz [ 8 ] ) and has been establ ished only in the case n < 3. In the present note, I consider another question of a s imi lar character. Let A be a subset of E*1 with diameter 1 and l e t a. be a r e a l number such that

Abstract: This r e l a t i o n w i l l be ca l l ed the r e l a t i o n of ( a , b ) tengency of s e t s of order k a t the point p . If (A,B) € T ^ ( a , b , k , p t h e n we sh&ll say tha t the set A i s l a , b ) tangent of order k at the point p to the set B. There a r i s e s the fol lowing ques t ion: i f the s e t s A,B 6jS0 are ( a ,b ) tangent of order k a t the point p of the space (£ | l ) » then when these s e t s are (a\", b') tangent of order k a t the point p of t h i s space? In the present paper we consider t h i s problem in a metr ic space (E,p) f o r r e c t i f i a b l e a r c s having the Archimedean proper ty a t the point p and f o r r e a l f unc t ions def ined by the e q u a l i t i e s and ^ l ( A ^ S ( p , r ) r ( r ) , B ^ S , ( p f r ) t { r ) ) -

Abstract: Désignons par E l ' e s p a c e des nombres r é e l s e t par l ' e s p a c e E ir S. Un ensemble A c E es t d i t r -ouvert l o r s q u ' i l es t l a somme d'une f a m i l l e d'ensembles étant du type Fget G^ e t d-ouverts (un ensemble e s t d i t d-ouvert l o r s que chacun de ses points e s t son point de d e n s i t é ) . La f a mi l le R de tous l e s ensembles r o u v e r t s es t une tropologie examinée dans le t r a v a i l [$]• Dans le t r a v a i l [3] l ' a u t e u r montre encore une autre topologie (c R) t e l l e ^ue toutes l e s fonct ions continues presque partout relat ivement à l a mesure de Lebesgue e t approximativement continues sont cont inues par rapport à c e t t e t o p o l o g i e . Cette topologie se compose de tous l e s ensembles presque ouverts (un ensemble A c E es t d i t presque ouvert l o r s q u ' i l e s t d-ouvert e t mi(A) = m(Int(A)) , où m désigne l a mesure de Lebesgue e t Int(A) désigne l ' i n t é r i e u r de l 'ensemble A. Dans c e t a r t i c l e j 'examine l e s topologies suivantes : R̂ = | a c E 2 : A e s t r o u v e r t } ^ R2 = R * R , où R x R désigne l a topologie ayant comme sa base l a f a m i l l e des ensembles de l a forme A x B, où A,B 6 R;

Abstract: I . Désignons par R l 'espace des nombres réels et par I l ' i n t e r v a l l e fermé [0 ,1] , D é f i n i t i o n 1. Une fonction f il—-R est dite F-continue lo r squ ' i l existe unje fonction continue g i i — R dont le graphe G(g) est contenu dans l a fermeture Cl(G(f)) du graphe de l a fonction f . R e m a r q u e 1. Soit f i l — R une fonction. S ' i l exis te une fonction continue g : I — R t e l l e que l'ensemble { x c l r f ( x j = g(x)| est dense dans I , l a fonction f est F-continue. D é m o n s t r a t i o n . En e f f e t , comme G(g) c Cl(G(f)4, l a fonction f est dono P-continue. Cependant i l existe une fonction F—continue f : I — R t e l l e que l'ensemble {x e I : f ( χ ) = g(x)} n» est pas dense dans I , quelle que soi t l a fonction continue g : I*-R.

Abstract: 1 . Introduction Predict ion problem, which i s r e c e i v i n g much a t t e n t i o n r e c e n t l y , has been viewed mostly in two d i r e c t i o n s . One i s .the c l a s s i c a l approach based on the independence of s t a t i s t i c s and t h e i r exact d i s t r i b u t i o n s . Such i s the case in Lawless [ 9 ] , Paulkenberry [8], Kaminsky, Luks and Nelson [7 J and a lso in Lingappaiah [10] , [i 1] . But, another method, i s the Bayes approach with poster ior d i s t r i b u t i o n s and su i tab le p r i o r s . Such works are seen i n Bancroft and Dunsmore [ l ] , Aitcheson and Dunsmore [2] and Dunsmore [3] , [4] . Our development here i s based on l a s t of these r e s u l t s Dunsmore [4] and Lingappaiah [ 1 1 ] . Our main motivation, here , i s about what can be done where more than a pair of samples are a v a i l a b l e . What we have done here i s to consider the poster ior d i s t r i b u t i o n at a c e r t a i n stage as the prior f o r the next s t a g e , on the l i n e s of Khan [12] and in so doing, we have developed the predict ive d i s t r i b u t i o n f o r an order s t a t i s t i c at the sth stage ((s+1)th sample) and a lso f o r the d i f f e r e n c e of two s t a t i s t i c s at t h i s s tage . We have discussed the variance in each of these two cases in r e l a t i o n to number of s t a g e s . Also , we have evaluated the probabi l i ty i n t e g r a l f o r both the s i t u a t i o n s and p a r t i c u l a r cases are considered f o r i l l u s t r a t i o n s .

Abstract: 1. P r e l i m i n a r i e s Let Vm be an m-dimensional Çiemannian mánifold immersed in an n-dimensional Riemannian maaifold V_, and l e t i u