On the Sperner Capacity of the Cyclic Triangle

Using a single trick it is shown that the Sperner capacity of the cyclic triangle equals log 2.     

The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle

Shannon introduced the concept of zero-error capacity of a discrete memoryless channel. The channel determines an undirected graph on the symbol alphabet, where adjacency means that symbols cannot be confused at the receiver. The zero-error or Shannon capacity is an invariant of this graph. Gargano, Körner, and Vaccaro have recently extended the concept of Shannon capacity to directed graphs. Their generalization of Shannon capacity is called Sperner capacity. We resolve a problem posed by these authors by giving the first example (the two orientations of the triangle) of a graph where the Sperner capacity depends on the orientations of the edges. Sperner capacity seems to be achieved by nonlinear codes, whereas Shannon capacity seems to be attainable by linear codes. In particular, linear codes do not achieve Sperner capacity for the cyclic triangle. We use Fourier analysis or linear programming to obtain the best upper bounds for linear codes. The bounds for unrestricted codes are obtained from rank arguments, eigenvalue interlacing inequalities and polynomial algebra. The statement of the cyclic q-gon problem is very simple: what is the maximum size N _q(n) of a subset S _n of {0, 1, \(\ldots\) , q?1} ⁿ with the property that for every pair of distinct vectors x = (x _i), y = (y _i) \(\in \) S _n, we have x _j ?y _j ≡ 1(mod q) for some j? For q = 3 (the cyclic triangle), we show N ₃(n)?2 ⁿ . If however S _n is a subgroup, then we give a simple proof that \(\left| {S_n } \right| \leqslant \sqrt 3 ^n \) .     

On the efficiency and capacity of computers

We address the problems of estimating the computer efficiency and the computer capacity. We define the computer efficiency and capacity and suggest a method for their estimation, based on the analysis of processor instructions and kinds of accessible memory. Obtained results can be of some interest for practical applications.     

斜对角无穷维Hamilton算子的点谱和特征函数系辛结构的非退化性

本文研究斜对角无穷维Hamilton算子$H=\begin{pmatrix}0&B\\C&0\end{pmatrix}$的点谱和特征函数系辛结构的非退化性, 给出斜对角无穷维Hamilton算子$H$的特征函数系具有非退化辛结构的充分必要条件. 基于此, 进一步刻画了斜对角无穷维Hamilton算子$H$的点谱分别包含于实轴、虚轴以及其它区域的充分必要条件. 最后, 以板弯曲问题和弦振动问题中导出的斜对角无穷维Hamilton算子为例, 验证了所得结论的正确性.     

Path Capacity Dimensioning in a Multiprotocol Label Switched Network: Analysis of Optimal and Suboptimal Solutions

In recent years, the problem of capacity allocation for a label switched patch (LSP) in a multiprotocol label switched (MPLS) network has received great attention due to its relevance in the context of traffic control. In this paper, the problem of capacity allocation is formulated as an optimal control problem and its solution is obtained by assuming the knowledge of the bandwidth requests on the entire control interval. A suboptimal solution is also given which has the advantage of requiring limited information about future bandwidth requests. The analysis of the suboptimal solution is explored both analytically and numerically by using simulated and real data. This study demonstrates that the suboptimal solution, also with limited knowledge of the future, yields a good approximation of the optimal one and requires little additional cost.