A geometrical theorem for the static equilibrium of a common-point-force system has been proven by means of virtual-work principle: The equilibrium point of a common-point force system has a minimal weighted distance ...A geometrical theorem for the static equilibrium of a common-point-force system has been proven by means of virtual-work principle: The equilibrium point of a common-point force system has a minimal weighted distance summation to every fixed point arbitrarily given on each force line with a weighing factor proportional to corresponding force value. Especially the mechanical simulating technique for its inverse problem has been realized by means of pulley block. The conclusions for the inverse problem derived from mechanic method are in accordance with that given by the pure mathematical method, and the self-consistence of the theorem and its inverse problem has been demonstrated. Some application examples in engineering, economy and mathematics have been discussed, especially the possible application in the research of molecular structure, has also been predicted.展开更多
Our purpose is twofold: to present a prototypical example of the conditioning technique to obtain the best estimator of a parameter and to show that th</span><span style="font-family:Verdana;">is...Our purpose is twofold: to present a prototypical example of the conditioning technique to obtain the best estimator of a parameter and to show that th</span><span style="font-family:Verdana;">is technique resides in the structure of an inner product space. Th</span><span style="font-family:Verdana;">e technique uses conditioning </span></span><span style="font-family:Verdana;">of</span><span style="font-family:Verdana;"> an unbiased estimator </span><span style="font-family:Verdana;">on</span><span style="font-family:Verdana;"> a sufficient statistic. This procedure is founded upon the conditional variance formula, which leads to an inner product space and a geometric interpretation. The example clearly illustrates the dependence on the sampling methodology. These advantages show the power and centrality of this process.展开更多
文摘A geometrical theorem for the static equilibrium of a common-point-force system has been proven by means of virtual-work principle: The equilibrium point of a common-point force system has a minimal weighted distance summation to every fixed point arbitrarily given on each force line with a weighing factor proportional to corresponding force value. Especially the mechanical simulating technique for its inverse problem has been realized by means of pulley block. The conclusions for the inverse problem derived from mechanic method are in accordance with that given by the pure mathematical method, and the self-consistence of the theorem and its inverse problem has been demonstrated. Some application examples in engineering, economy and mathematics have been discussed, especially the possible application in the research of molecular structure, has also been predicted.
文摘Our purpose is twofold: to present a prototypical example of the conditioning technique to obtain the best estimator of a parameter and to show that th</span><span style="font-family:Verdana;">is technique resides in the structure of an inner product space. Th</span><span style="font-family:Verdana;">e technique uses conditioning </span></span><span style="font-family:Verdana;">of</span><span style="font-family:Verdana;"> an unbiased estimator </span><span style="font-family:Verdana;">on</span><span style="font-family:Verdana;"> a sufficient statistic. This procedure is founded upon the conditional variance formula, which leads to an inner product space and a geometric interpretation. The example clearly illustrates the dependence on the sampling methodology. These advantages show the power and centrality of this process.
