Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an...Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.展开更多
In this paper,a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed.The classes of this complete congruence which are intervals are discussed and their least and...In this paper,a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed.The classes of this complete congruence which are intervals are discussed and their least and greatest elements are presented clearly.展开更多
Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their p...Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their proof for the soundness part:p = q implies p = q is true in every inductive model. The aim of this paper is to give acorrect characterization of inductive soundness from an algebraic view by introducingstrong inductive models.展开更多
Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of...Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign （or a root of unity） ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign （root of unity） ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.展开更多
基金Supported by the National Natural Science Foundation of China(No.61772031)the Special Energy Saving Foundation of Changsha,Hunan Province in 2017
文摘Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.
基金Foundation item： the National Natural Science Foundation of China （No. 10571061） the Science Foundation of Guangdong Province （No. 0501332） and the Science Foundation of the Department of Education of Yunnan Province （No. 07C10110）.
文摘In this paper,a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed.The classes of this complete congruence which are intervals are discussed and their least and greatest elements are presented clearly.
文摘Kapur and Musser studied the theoretical basis for proof by consistency and ob-tained an inductive completeness result: p q if and only if p = q is true in everyinductive model. However, there is a loophole in their proof for the soundness part:p = q implies p = q is true in every inductive model. The aim of this paper is to give acorrect characterization of inductive soundness from an algebraic view by introducingstrong inductive models.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 10990011, 11001145, 61170289） and the Ph. D. Programs Foundation of Ministry of Education of China （No. 20090002120013）.
文摘Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign （or a root of unity） ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign （root of unity） ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.
