Weighted sums in finite cyclic groups
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: 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: 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: On the number of zero sum subsequences
abstract: For a sequence S of elements from an additive abelian group G, let f(S) denote the number of subsequences of S the sum of whose terms is zero. In this paper we characterize all sequences S in G with f(S) > 2 |S| 2 , where |S| denotes the number of terms of S.

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: A Chinese sign language recognition system based on SOFM/SRN/HMM
abstract: In sign language recognition (SLR), the major challenges now are developing methods that solve signer-independent continuous sign problems. In this paper, SOFM/HMM is first presented for modeling signer-independent isolated signs. The proposed method uses the self-organizing feature maps (SOFM) as different signers' feature extractor for continuous hidden Markov models (HMM) so as to transform input signs into significant and low-dimensional representations that can be well modeled by the emission probabilities of HMM. Based on these isolated sign models, a SOFM/SRN/HMM model is then proposed for signer-independent continuous SLR. This model applies the improved simple recurrent network (SRN) to segment continuous sign language in terms of transformed SOFM representations, and the outputs of SRN are taken as the HMM states in which the lattice Viterbi algorithm is employed to search the best matched word sequence. Experimental results demonstrate that the proposed system has better performance compared with conventional HMM system and obtains a word recognition rate of 82.9% over a 5113-sign vocabulary and an accuracy of 86.3% for signer-independent continuous SLR.