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A branch-and-price algorithm for solving the cutting strips problem
      
After giving a suitable model for the cutting strips problem, we present a branch-and-price algorithm for it by combining the column generation technique and the branch-and-bound method with LP relaxations.
      
The double points may be the double points of either branch, or the intersection points of the two branches.
      
When the thickness is d >amp;gt; 2 Rg (radius of gyration), the polymer can crystallize into spherulites; when Rg >amp;lt; d >amp;lt; 2 Rg, a dense-branch morphology and dendrites could be found; when d >amp;lt; Rg, an "islands" structure could be obtained.
      
In addition, considering the non-convex and non-concave nature of the sub-problem of combinational optimization, the branch-and-bound technique was adopted to obtain or approximate a global optimal solution.
      
To improve the efficiency of the branch-and-bound technique, some heuristic principles were proposed to cut those branches that may generate a global optimization solution with low probability.
      
It was lower in autumn (October) in root, branch, and leaf in the tree layer, and low in January in the understory.
      
In each plantation, fine roots of two species by sampling up to five fine root branch orders three times during the 2003 growing season from two soil depths (i.e., 0-10 and 10-20 cm) were obtained.
      
The results showed that average diameters of fine roots were significantly different among the five branch orders.
      
The first-order had the thinner roots and the fifth order had the thickest roots, the diameter increasing regularly with the ascending branch orders in both species.
      
Therefore, based on this study, it has been concluded that both diameter and branch order should be considered in the estimation of root lifespan and turnover.
      
Regression analysis of leaf biomass and total biomass of each branch against branch diameter (d), branch length (L), d3 and d2L was conducted with functions of linear, power and exponent.
      
These kinds of equations were also used to fit the relationship between total tree biomass, branch biomass, foliage biomass and tree diameter at breast height (D), tree height (H), D3 and D2H, respectively.
      
However, for foliage biomass and branch biomass, both parameters and equation forms showed some differences among species.
      
Correlations were highly significant (P>amp;lt;0.001) for foliage biomass, branch biomass and total biomass, among which the equation of the total biomass was the highest.
      
The above-ground biomass of Masson pine forest, slash pine forest and Chinese fir forest was 83.6, 72.1 and 59 t/hm2 respectively, and the stem biomass was more than the foliage biomass and the branch biomass.
      
The structural diversity of forests on Baihuashan Mountain, Beijing, was surveyed by a plotless method combined with branch and leaf coverage estimation in the different layers.
      
Numerical multilinear algebra (or called tensor computation), in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics.
      
First, the reverse solution models of position, velocity, and acceleration of parallelogram branch structure are deduced, and then, its dynamic model of a rigid body is set up by using the virtual work principle.
      
With the reverse solution module of the translation, the module with the exerted translation joint was obtained, which included the location, velocity and acceleration of the parallelogram carriage-branch.
      
 

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