In some long term studies, a series of dependent and possibly truncated lifetime data may be observed. Suppose that the lifetimes have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the lifetimes and its kernel estimate fn is the integrated square error (ISE). In this paper, we derive a central limit theorem for t...

In this paper we study the duality of Bessel and g-Bessel sequences in Hilbert spaces. We show that a Bessel sequence is an inner summand of a frame and the sum of any Bessel sequence with Bessel bound less than one with a Parseval frame is a frame. Next we develop this results to the g-frame situation.

Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper’s algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is inreased.

We present various comments on a question about systems over rings posed in a recent note by Sharma, proving that a ring R is pole-assignable if and only if, for every reachable system (F,G), G contains a rank-one summand of the state space. We also provide a generalization to deal with dynamic feedback.

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).

We provide a new algorithm for indefinite nested summation which is applicable to summands involving unspecified sequences x(n). More than that, we show how to extend Karr’s algorithm to a general summation framework by which additional types of summand expressions can be handled. Our treatment of unspecified sequences can be seen as a first illustrative application of this approach.

In this paper, we prove the following result conjectured by Z.-W. Sun: (2n− 1) ( 3n n ) ∣∣∣∣ n ∑ k=0 ( 6k 3k )( 3k k )( 6(n− k) 3(n− k) )( 3(n− k) n− k ) by showing that the left-hand side divides each summand on the right-hand side.

in this paper we study the duality of bessel and g-bessel sequences in hilbertspaces. we show that a bessel sequence is an inner summand of a frame and the sum of anybessel sequence with bessel bound less than one with a parseval frame is a frame. next wedevelop this results to the g-frame situation.

In this article, the effect of different yield functions on prediction of forming limit curve (FLC) for aluminum sheet is studied. Due to importance of FLC in sheet metal forming, concentration on effective parameters must be considered exactly in order to have better theoretical prediction comparing experimental results. Yield function is one of the factors which are improved by adding new coe...

It is shown how the performance of Zeilberger’s algorithm and its q-version for proving (q-)hypergeometric summation identities can be dramatically improved by a frequently missed optimization on the programming level and by applying certain kinds of substitutions to the summand. These methods lead to computer proofs of identities for which all existing programs have failed so far.