Let lK be a commutative field and (P, +) be a uniquely 2-divisible group (not necessarily abelian). We characterize all functions T: IK -+ P such that the Cauchy difference T(s+ t) T(t) T(s) depends only on the product st for all s, t E ~{. Further, we apply this result to describe solutions of the functional equation F(s + t) = K(st) 0 H(s) 0 G(t), where the unknown functions F, K, H, G map th...

abstract. certain facts about frames and generalized frames (g- frames) are extended for the g-frames for hilbert c*-modules. it is shown that g-frames for hilbert c*-modules share several useful properties with those for hilbert spaces. the paper also character- izes the operators which preserve the class of g-frames for hilbert c*-modules. moreover, a necessary and suffcient condition is ob- ...

3. Since Lp and Lr are subspaces of CX , their intersection is a vector space. It is clear that ‖ · ‖ is a norm (this follows directly from the fact that ‖ · ‖p and ‖ · ‖r are norms). Let 〈fn〉n=1 be a Cauchy sequence in Lp ∩ Lr. Since ‖fm − fn‖p ≤ ‖fm − fn‖ and ‖fm − fn‖r ≤ ‖fm − fn‖ for all m,n ∈ N, it is clear that 〈fn〉n=1 is a Cauchy sequence in both Lp and Lr. Let gp ∈ Lp and gr ∈ Lr be the...

The classical result on well-posedness of Cauchy problem for the linear ordinary differential system with the distributional right-hand side and smooth matrix of coefficients plays fundamental role in many applications of distribution theory to ordinary and partial differential equations. In the present paper we generalize this result to the case of system (0.1) x − A(t)x = f, where f is a dist...

The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := ∑k m=0 ( k m ) f(m)g(k −m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.

‎Let $X,Y$ be normed spaces with $L(X,Y)$ the space of continuous‎ ‎linear operators from $X$ into $Y$‎. ‎If ${T_{j}}$ is a sequence in $L(X,Y)$,‎ ‎the (bounded) multiplier space for the series $sum T_{j}$ is defined to be‎ [ ‎M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}%‎ ‎T_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $S:M^{infty}(sum T_{j})rightarrow Y$ associat...

*Correspondence: [email protected]; [email protected] 1Department of Mathematics, Maltepe University, Marmara Eğİtİm Köyü, TR 34857, Maltepe, İstanbul, Turkey Full list of author information is available at the end of the article Abstract An ideal I is a family of subsets of positive integersN which is closed under taking finite unions and subsets of its elements. A sequence (xn) of real...

Consider the Cauchy problem for an ordinary diierential equation _ x = g(t; x); x(0) = x; t 2 0; T]: (1:1) When g is continuous, the local existence of solutions is provided by Peano's theorem. Several existence and uniqueness results are known also in the case of a discontinuous right hand side 7]. We recall here the classical theorem of Carath eodory 8]: Theorem A. Let g : 0; T] IR n 7 ! IR n...

We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams binary signed digits. The main objective paper is the formalisation proof that numbers are closed with respect to limits. All proofs theorem and first application implemented in Minlog system extracted terms further translated into Haskel...