In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values o...In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values of the same dimension withinεrange.For rational vectors of a fixed dimension m,they can form a field that is an mth order extension Q(α)of the rational field Q whereαhas its minimal polynomial of degree m over Q.Then,the arithmetics,such as addition,subtraction,multiplication,and division,of real vectors can be defined by using that of their approximated rational vectors withinεrange.We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences(signals)of finite length.With these newly defined concepts for real vectors,linear processing,such as linear filtering,ARMA modeling,and least squares fitting,can be implemented to real vectorvalued signals with real vector-valued coefficients,which will broaden the existing linear processing to scalar-valued signals.展开更多
The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century...The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.展开更多
Through denoting each expert as an agent and viewing a multiple criteria decision-making as a synthesis problem of aggregating experts＇ ratings, a multi-agent blind model （MABM） is developed for regional eco-enviro...Through denoting each expert as an agent and viewing a multiple criteria decision-making as a synthesis problem of aggregating experts＇ ratings, a multi-agent blind model （MABM） is developed for regional eco-environmental quality assessment. In this model, the ratings of the evaluated object under an index, given by expert group, are first utilized to construct a series of blind numbers. In general, each index will correspond to different blind numbers. On the basis of aggregating index weights, the rank score in the form of a blind number is obtained for the evaluated object. Then, by means of calculating expected value of the above blind number, its rank score is further converted into a crisp value. By way of comparing the expected value with classification standards, eco-environmental quality of the evaluated sample could he identified successfully in the end. As a case, the MABM is used to evaluate the eco-environmental quality of Chaohu Lake basin. Study result shows that the MABM is a useful model for regional eco-environmental quality assessment.展开更多
文摘In this paper,we introduce a new concept,namelyε-arithmetics,for real vectors of any fixed dimension.The basic idea is to use vectors of rational values(called rational vectors)to approximate vectors of real values of the same dimension withinεrange.For rational vectors of a fixed dimension m,they can form a field that is an mth order extension Q(α)of the rational field Q whereαhas its minimal polynomial of degree m over Q.Then,the arithmetics,such as addition,subtraction,multiplication,and division,of real vectors can be defined by using that of their approximated rational vectors withinεrange.We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences(signals)of finite length.With these newly defined concepts for real vectors,linear processing,such as linear filtering,ARMA modeling,and least squares fitting,can be implemented to real vectorvalued signals with real vector-valued coefficients,which will broaden the existing linear processing to scalar-valued signals.
文摘The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.
基金Under the auspices of the Natural Science Foundation of Anhui Province (No. 050450303 )
文摘Through denoting each expert as an agent and viewing a multiple criteria decision-making as a synthesis problem of aggregating experts＇ ratings, a multi-agent blind model （MABM） is developed for regional eco-environmental quality assessment. In this model, the ratings of the evaluated object under an index, given by expert group, are first utilized to construct a series of blind numbers. In general, each index will correspond to different blind numbers. On the basis of aggregating index weights, the rank score in the form of a blind number is obtained for the evaluated object. Then, by means of calculating expected value of the above blind number, its rank score is further converted into a crisp value. By way of comparing the expected value with classification standards, eco-environmental quality of the evaluated sample could he identified successfully in the end. As a case, the MABM is used to evaluate the eco-environmental quality of Chaohu Lake basin. Study result shows that the MABM is a useful model for regional eco-environmental quality assessment.
