Note on a Zero-Sum Problem
Let G be an additively written, finite abelian group, and exp(G ) its exponent. Let S=(a1 , ..., ak) be a sequence of elements in G; we say that S is a zero-sum sequence if i=1 ai=0. Let s(G ) be the samllest integer t such that every sequence of t elements in G contains a zero-sum subsequence of length exp(G ). This constant has been studied by serveral authors during last 20 years [1 9, 11]. Let Cn be the cyclic group of order n, and C n the direct product of k copies of Cn . In , Erdo s et al. proved that S(C n)=2n&1. A geometrical interpretation of s(C n) was given by Harborth in . In 1980, Kemntiz suggested the following

title: An addition theorem for finite cyclic groups
abstract: The following theorem is proved. Let 2 ⩽ k ⩽ [ n 4 ] + 1 , and let S be a sequence of 2n − k elements in Zn. Suppose that S does not contain any n-subsequence with 0-sum. Then, one can rearrange S to the type a, …,a, b, …, b, c1, …, c2n-k-u-v, where u ⩾ n − 2k + 3, v ⩾ n − 2k + 3, u + v ⩾ 2n − 2k + 2 and a − b generates Zn.

title: Covering a Finite Abelian Group by
Subset Sums
abstract: Let G be an abelian
group of order n. The
critical number c(G) of G is the smallest
s such that the subset sums
set Σ(S) covers all G for eachs ubset
S⊂G\{0} of cardinality |S|≥s. It has been recently proved that, if
p is the smallest prime
dividing n and
n/p is composite, then
c(G)=|G|/p+p−2, thus establishing a conjecture of
Diderrich.We characterize the critical sets with |S|=|G|/p+p−3 and Σ(S)=G, where p≥3 is the smallest prime dividing
n, n/p is composite and
n≥7p2+3p.We also extend a result of Diderrichan d Mann by proving
that, for n≥67, |S|≥n/3+2 and S=G
imply Σ(S)=G. Sets of cardinality 
$$
{\left| S \right|} \geqslant \frac{{n + 11}}

$$ for which
Σ(S) =G are also characterized when
n≥183, the smallest prime
p dividing
n is odd and
n/p is composite. Finally we
obtain a necessary and sufficient condition for the equality
Σ(G)=G
to hold when |S|≥n/(p+2)+p, where p≥5, n/p is composite and
n≥15p2.

title: Zero-sum problems and coverings by proper cosets
abstract: Let G be a finite Abelian group and D(G) its Davenport constant, which is defined as the maximal length of a minimal zero-sum sequence in G. We show that various problems on zero-sum sequences in G may be interpreted as certain covering problems. Using this approach we study the Davenport constant of groups of the form (Z/nZ)r, with n ≥ 2 and r ∈ N. For elementary p-groups G, we derive a result on the structure of minimal zero-sum sequences S having maximal length |S| = D(G).

title: Sequences not containing long zero-sum subsequences
abstract: Let G bea finite abelian group (written additively), and let D(G) denote the Davenport’s constant of G, i.e. the sma llest integerd such that every sequence of d elements (repetition allowed) inG contains a nonempty zero-sum subsequence. Let Sb a sequence of elements in G with |S| ≥ D(G). We say S is a normal sequence ifS contains no zero-sum subsequence of length larger than |S| − D(G) + 1. In this paper we obtain some results on the structure of normal sequences for arbitrary G. If G = Cn ⊕ Cn and n satisfies some well-investigated property, we determine all normal sequences. Applying these results, we obt ain correspondingly some results on the structure of the sequenceS in G of length|S| = |G| + D(G)− 2 andScontains no zero-sum subsequence of length |G|.

title: Weighted sums in finite cyclic groups
abstract: In this paper we prove the following result. Let p be a prime, and k a positive integer. Let { w 1 ,…, w k } be a sequence of k integers such that w 1 +⋯+w k ≡0 ( mod p 2 ) . Then, for every sequence a 1 ,…, a p 2 + k −1 of p 2 + k −1 elements in C p 2 , there are k distinct indices i 1 , i 2 ,…, i k such that w 1 a i 1 +⋯+ w k a i k =0. This confirms partly a conjecture by Caro.

title: A SRN/HMM system for signer-independent continuous sign language recognition
abstract: Sign language recognition is to provide an efficient and accurate mechanism to transcribe sign language into text or speech. State-of-the-art sign language recognition should be able to solve the signer-independent continuous problem for practical applications. A divide-and-conquer approach, which takes the problem of continuous Chinese Sign Language (CSL) recognition as subproblems of isolated CSL recognition, is presented for signer-independent continuous CSL recognition. In the proposed approach, the improved simple recurrent network (SRN) is used to segment the continuous CSL. The outputs of SRN are regarded as the states of hidden Markov models (HMM) in which the Lattice Viterbi algorithm is employed for searching for the best word sequence. Experimental results show that the SRN/HMM approach has a better performance than the standard HMM.