### Table 1. the lower bounds of previous works and this paper

"... In PAGE 2: ... For n = 2r 1, sizer 1(Parityn) 2n log2(n + 1) 1 + sizemono(Majorityn) 6n log2(n + 1) O(1): Theorem 2. For n = 2r 1, sizer(Invn) 4n log2(n + 1) + sizemono(Majorityn) 8n log2(n + 1) O(1): Now we explain the previously known lower bounds shown in Table1 , and how we obtain our improvements focusing on Parityn. Table 1.... ..."

### Table 1. Summary of previous works and the results of this paper.

1998

Cited by 9

### Table 1. Summary of previous works and the results of this paper.

### Table 1: results of previous work The following are notations used in this paper

"... In PAGE 3: ... [1] extended the method of Reed-Solomon codes and solves the problem for M replicated sites, where M 3. Table1 summarizes the characteristics of a number of previous work for this problem. In Table 1, N is the number of pages in the le, a1 is the assumption that an uncorrupted copy resides in a known site, and a2 is the assumption that N... In PAGE 3: ... Table 1 summarizes the characteristics of a number of previous work for this problem. In Table1 , N is the number of pages in the le, a1 is the assumption that an uncorrupted copy resides in a known site, and a2 is the assumption that N... ..."

### Table 2.1 summarizes the requirements described in the previous sections. In the remainder of the paper we will refer to these requirements via their numbers where appropriate.

1999

Cited by 22

### Table 5. Single-epoch solution using the real-time stochastic model derived using residuals from previous epochs, as proposed in this paper

### Table 1: The function test bed. The limits given below have been introduced in previous research and are used throughout this paper. Function Limits Name

1996

"... In PAGE 3: ... In the present context, an algorithm converges, if it nds at least one solution ~ xs; k~xs? ~xok with a prespeci ed probability p and an arbitrary neighborhood of the optimum ~xo. Table1 summarizes some of the widely-used test functions. The rst ve test func- tions have been proposed by De Jong (1975).... ..."

Cited by 51

### Table 1: The function test bed. The limits given below have been introduced in previous research and are used throughout this paper. Function Limits Name

"... In PAGE 8: ...1.1 Function Test Bed Table1 summarizes some of the widely-used test functions. The rst ve test functions have been proposed by De Jong (1975).... ..."

### Table 2 shows the WER using the MLP-based feature on each single MLP proposed in this paper, and the MLPs used in our previous papers, the HATS and the PLP-MLP3, where three-layer MLPs or concatenated three-layer MLPs are used. The previous structures, HATS and PLP-MLP3, have WER around 35.6% in this test set. Each of the new MLP structures from this paper is significantly better than all the old versions by WER reductions more than 1% absolution. All the three MLPs proposed in this paper have similar performance.

2005

"... In PAGE 4: ... Table2 : The word error rate using MLP-based feature from each MLP in this paper and the previous MLP structures. We know based on our previous experience that MLPs complementary to each other can give large improvements through combination.... ..."

Cited by 2

### Table 2. QoSMT problem with an arbitrary number of rates. Approximation ratios of previously known algorithms and of the algorithms given in this paper.

2003

Cited by 5