Uniform lower bound for the least common multiple of a polynomial sequence

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.Comment: 6 pages. To appear in Comptes Rendus Mathematiqu

The elementary symmetric functions of a reciprocal polynomial sequence

Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being an integer and $n=1$.Comment: 4 pages. To appear in Comptes Rendus Mathematiqu

Effective Discriminative Feature Selection with Non-trivial Solutions

Feature selection and feature transformation, the two main ways to reduce dimensionality, are often presented separately. In this paper, a feature selection method is proposed by combining the popular transformation based dimensionality reduction method Linear Discriminant Analysis (LDA) and sparsity regularization. We impose row sparsity on the transformation matrix of LDA through ${\ell}_{2,1}$-norm regularization to achieve feature selection, and the resultant formulation optimizes for selecting the most discriminative features and removing the redundant ones simultaneously. The formulation is extended to the ${\ell}_{2,p}$-norm regularized case: which is more likely to offer better sparsity when $0<p<1$. Thus the formulation is a better approximation to the feature selection problem. An efficient algorithm is developed to solve the ${\ell}_{2,p}$-norm based optimization problem and it is proved that the algorithm converges when $0<p\le 2$. Systematical experiments are conducted to understand the work of the proposed method. Promising experimental results on various types of real-world data sets demonstrate the effectiveness of our algorithm

Neural network-assisted decision-making for adaptive routing strategy in optical datacenter networks

To improve the blocking probability (BP) performance and enhance the resource utilization, a correct decision of routing strategy which is most adaptable to the network configuration and traffic dynamics is essential for adaptive routing in optical datacenter networks (DCNs). A neural network (NN)-assisted decision-making scheme is proposed to find the optimal routing strategy in optical DCNs by predicting the BP performance for various candidate routing strategies. The features of an optical DCN architecture (i.e., the rack number N, connection degree D, spectral slot number S and optical transceiver number M) and the traffic pattern (i.e., the ratio of requests of various capacities R, and the load of arriving request) are used as the input to the NN to estimate the optimal routing strategy. A case of two-strategy decision in the transparent optical multi-hop interconnected DCN is studied. Three metrics are defined for performance evaluation, which include (a) the ratio of the load range with wrong decision over the whole load range of interest (i.e., decision error E), (b) the maximum BP loss (BPL) and (c) the resource utilization loss (UL) caused by the wrong decision. Numerical results show that the ratio of error-free cases over tested cases always surpasses 83% and the average values of E, BPL and UL are less than 3.0%, 4.0% and 1.2%, respectively, which implies the high accuracy of the proposed scheme. The results validate the feasibility of the proposed scheme which facilitates the autonomous implementation of adaptive routing in optical DCNs

Numerical Investigation of Flow and Heat Transfer Characteristics in Plate with Multiple Incline Stage Holes

In this paper, the effects of impingement and film composite cooling on the design of combustion chamber cooling structure are simulated by numerical simulation. The focus of the investigation was on increased film cooling efficiency and enhanced heat transfer between the coolant and the hole wall. The five-stage shaped hole model and one cylindrical hole design have the same equivalent flow area. The flow and heat transfer characteristics of cylindrical hole and stage-shaped hole were numerically investigated under same blowing ratio, and compared at the same blowing ratio. The results showed the stage-shaped hole resulted in higher cooling effectiveness, especially in rear part, and the mechanisms of which were studied in details. The consequences of the phase parameters in the flow have very clearly dependedt on the internal shape of the corresponding hole. Stage-shaped holes formed impact inside the wall, tapped the coolant potential in cooling, and increased the heat transfer inside the solid wall. Further, stage-shaped hole resulted in unstabilized flow inside hole, gave an enhancement of lateral spreading ability, and brought a significant increase of the film lateral effectiveness. Further, the affection of area ratio and height ratio has been studied by five models. The results show that the increasing of area ratio leads to an increase in cooling efficiency, which also indicates the increasing of height ratio showed slight affection