Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is ...Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.展开更多
Letf(x,y)=ax2+bxy+cy2,g(x,y)=Ax2+Bxy+Cy2,be two binary quadratic forms with real coefficients.A real number m is said to be represented by fif f(x,y)=m has a(rational)integer solution(x,y).We say f and g are equivalen...Letf(x,y)=ax2+bxy+cy2,g(x,y)=Ax2+Bxy+Cy2,be two binary quadratic forms with real coefficients.A real number m is said to be represented by fif f(x,y)=m has a(rational)integer solution(x,y).We say f and g are equivalent if there exists aninteger matrlx(r s t u)with determinant±1 such that f(x′,y′)=g(x,y),where展开更多
The normal form and modilied normal form for binary redundant representation are defined. A redundant binary algorithm to compute modular exponentiation for very large integers is proposed. It is shown that the propos...The normal form and modilied normal form for binary redundant representation are defined. A redundant binary algorithm to compute modular exponentiation for very large integers is proposed. It is shown that the proposed algorithm requires the minimum number of basic operations (modular multiplications) among all possible binary redundant representations.展开更多
For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be th...For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.展开更多
On the basis of analyzing some neural network storage capacity problems a network model comprising a new encoding and recalling scheme is presented. By using some logical operations which operate on the binary pattern...On the basis of analyzing some neural network storage capacity problems a network model comprising a new encoding and recalling scheme is presented. By using some logical operations which operate on the binary pattern strings during information processing procedure the model can reach a high storage capacity for a certain size of network framework.展开更多
基金Supported by Chinese National Basic Research Program(2007CB807902)
文摘Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.
文摘Letf(x,y)=ax2+bxy+cy2,g(x,y)=Ax2+Bxy+Cy2,be two binary quadratic forms with real coefficients.A real number m is said to be represented by fif f(x,y)=m has a(rational)integer solution(x,y).We say f and g are equivalent if there exists aninteger matrlx(r s t u)with determinant±1 such that f(x′,y′)=g(x,y),where
文摘The normal form and modilied normal form for binary redundant representation are defined. A redundant binary algorithm to compute modular exponentiation for very large integers is proposed. It is shown that the proposed algorithm requires the minimum number of basic operations (modular multiplications) among all possible binary redundant representations.
基金supported by National Natural Science Foundation of China (Grant No.11071121)
文摘For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.
文摘On the basis of analyzing some neural network storage capacity problems a network model comprising a new encoding and recalling scheme is presented. By using some logical operations which operate on the binary pattern strings during information processing procedure the model can reach a high storage capacity for a certain size of network framework.
