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Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.
      
Weighted Estimates for Singular Integral Operators Satisfying H?rmander's Conditions of Young Type
      
The identifiability of the dynamics problem is proved and necessary conditions of the optimality are given.
      
Sufficient conditions of the penalty function attaining a global minimum are obtained.
      
Degree conditions of induced matching extendable graphs
      
The degree conditions of IM-extendable graphs are researched in this paper.
      
The discrimination conditions of the existence of the nonzero steady states are given.
      
Global convergence of the algorithm is proved under the same conditions of usual trust region method.
      
This paper formulates a kind of dynamical macro-economic model based on Sidrauski's work, then presents the sufficient and necessary conditions of the stability of model at equilibrium states, and shows some results for special production functions.
      
The results support some conditions of a question posed by Lin Fanghua and Wang Changyou.
      
The necessary and sufficient conditions of the existence of a kind of Liapunov functional for the system are given.
      
The first order necessary conditions of optimality for problem (OH) is obtained by the conception of normal cone.
      
The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived.
      
For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied.
      
Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.
      
In these results, the staircase and quantitative-topological conditions of both functions involve strictly monotone transformation and mixing of functional values.
      
The spectral characteristics, effect factors, optimum conditions of the reaction and the influence of foreign substances were also investigated.
      
Well-defined core-shell structure and narrow particle size distribution could be achieved under starved conditions of monomer feeding.
      
It was found that the optimum conditions of the preparation are hydrothermal crystallization for 8 h at 180°C, followed by vacuum drying at 45°C and calcination at 500°C for 2 h.
      
The study showed that the water molecules between the HAP faces were under conditions of strong electrical field and high pressure, and hence formed 2-3 well-organized water layers on the crystal surfaces.
      
 

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