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
Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However,...Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However, over time, the most inquisitive thinkers have questioned the validity of standard arithmetic in certain settings. It started in ancient Greece with the famous philosopher Zeno of Elea, who elaborated a number of paradoxes questioning popular knowledge. Millennia later, the famous German researcher Herman Helmholtz (1821-1894) [1] expressed reservations about applicability of conventional arithmetic with respect to physical phenomena. In the 20th and 21st century, mathematicians such as Yesenin-Volpin (1960) [2], Van Bendegem (1994) [3], Rosinger (2008) [4] and others articulated similar concerns. In validation, in the 20th century expressions such as 1 + 1 = 3 or 1 + 1 = 1 occurred to reflect important characteristics of economic, business, and social processes. We call these expressions synergy arithmetic. It is common notion that synergy arithmetic has no meaning mathematically. However in this paper we mathematically ground and explicate synergy arithmetic.展开更多
In this paper,some issues concerning the Chinese remaindering representation are discussed.A new converting method is described. An efficient refinement of the division algorithm of Chiu,Davida and Litow is given.
The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept calle...The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.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Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However, over time, the most inquisitive thinkers have questioned the validity of standard arithmetic in certain settings. It started in ancient Greece with the famous philosopher Zeno of Elea, who elaborated a number of paradoxes questioning popular knowledge. Millennia later, the famous German researcher Herman Helmholtz (1821-1894) [1] expressed reservations about applicability of conventional arithmetic with respect to physical phenomena. In the 20th and 21st century, mathematicians such as Yesenin-Volpin (1960) [2], Van Bendegem (1994) [3], Rosinger (2008) [4] and others articulated similar concerns. In validation, in the 20th century expressions such as 1 + 1 = 3 or 1 + 1 = 1 occurred to reflect important characteristics of economic, business, and social processes. We call these expressions synergy arithmetic. It is common notion that synergy arithmetic has no meaning mathematically. However in this paper we mathematically ground and explicate synergy arithmetic.
文摘In this paper,some issues concerning the Chinese remaindering representation are discussed.A new converting method is described. An efficient refinement of the division algorithm of Chiu,Davida and Litow is given.
基金Supported by National Natural Science Foundation of China(Grant No.11701549)。
文摘The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.
