This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to al...This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.展开更多
Presents the counting of the counts number of primitive elements of finite dimensional field extension GF(p nm )/GF(p n) using (1) the principle of inclusion exclusion, (2) the Mbius inversion, (3) the Euler ...Presents the counting of the counts number of primitive elements of finite dimensional field extension GF(p nm )/GF(p n) using (1) the principle of inclusion exclusion, (2) the Mbius inversion, (3) the Euler function, and the new identity obtained ∑t｜p nm-1 , t p nmq j -1(t)=p nm -∑jp nmq j +∑j 1<j 2p nmqj 1qj 2 -…+(-1) kp nmq 1…q k where, m>1, p is a prime, (·) is Euler function, and q 1,…,q k are the all distinct prime divisors of m.展开更多
The internal flow of a droplet in the nonlinear extensional flow field will exhibit more than two internal circulations with the variation of nonlinear intensity(E).In this paper,the effect of positions and sizes of i...The internal flow of a droplet in the nonlinear extensional flow field will exhibit more than two internal circulations with the variation of nonlinear intensity(E).In this paper,the effect of positions and sizes of internal circulations on internal mass transfer rate of a single spherical droplet in a nonlinear extensional flow field is studied and compared with that in a linear extensional flow field.The simulation results show that when E≥0,there are two symmetrical internal circulations in the droplet,which is the same with that in a linear extensional flow.The limit value of mass transfer rate Sh is 15,which is equal to that in a linear extensional flow,no matter how large E is.When E≤-3/7,the number of internal flow circulation of a droplet increase to four and the transfer rate Sh increases.When E=-1,the maximum internal transfer rate Sh equals 30 which is twice of that in a linear extensional flow.The generation of new flow circulations in droplets and the circulation positions will enhance mass transfer when E≤-3/7,which provides a new idea for enhancing the internal mass transfer rate of droplets.展开更多
The estimation of contemporary tectonic stress field and deformation in active fold-and-thrust belts are imperative in identifying active geodynamics and resulting faulting phenomenon. In this paper, we focus on conte...The estimation of contemporary tectonic stress field and deformation in active fold-and-thrust belts are imperative in identifying active geodynamics and resulting faulting phenomenon. In this paper, we focus on contemporary extensional tectonics in the overall compressive setting of the Himalayan orogen. Here we examine the regional tectonic stress field and upper crustal deformation in the Himalayan thrust wedge using a 2D finite element technique, incorporating elastic rheology under plain strain condition. The elastic models demonstrate that the extensional tectonic stress and related nor- mal faulting is extensively developed in the southern front of the Himalaya at shallow crustal level (&lt;10 km in depth). Our modelling shows a good consistency with the geological field evidences of active faulting, focal mechanism solutions of medium size earthquakes in the several sectors of the Himalaya. Results based on numerical simulation, tectonic analysis and taking geological and geophysical data into account, we interpret that the present-day extensional tectonic activity is not restricted in the southern Tibet but distributed in the different sectors of the Himalayan fold-and-thrust belt co-exist with compressional structures. Modelling results also indicate that the nature, distribution and orientation of the maximum compressive stress (?1) of the Himalaya are mainly controlled by the intra crustal Main Himalayan d&#233;collement (MHT). The significant amount of shear stress/strain concentration along the MHT in the western Nepal predict that the region is prone to moderate and great future earthquakes.展开更多
In brder to study the contradiction problem of multilayer multidimensional complex systems, the concepts of extension field and stable field of intersection and union of multilayer multidimensional extension set are g...In brder to study the contradiction problem of multilayer multidimensional complex systems, the concepts of extension field and stable field of intersection and union of multilayer multidimensional extension set are given. Then the related operations and properties are discussed. The results of study expand the concepts of intersection and union of extension set to a general situation, and provide the theoretical basis for production of the concepts of intersection and union of multilayer multidimensional matter element system extension set. In this way, it will be possible that matter element system theory is used to creative designs of complex systems.展开更多
The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> i...The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk</sub>, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZ<sub>k</sub> decomposes into a product of prime ideals of Zk</sub> as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each f<sub>θ</sub>(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05展开更多
文摘This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.
文摘Presents the counting of the counts number of primitive elements of finite dimensional field extension GF(p nm )/GF(p n) using (1) the principle of inclusion exclusion, (2) the Mbius inversion, (3) the Euler function, and the new identity obtained ∑t｜p nm-1 , t p nmq j -1(t)=p nm -∑jp nmq j +∑j 1<j 2p nmqj 1qj 2 -…+(-1) kp nmq 1…q k where, m>1, p is a prime, (·) is Euler function, and q 1,…,q k are the all distinct prime divisors of m.
基金supported by the National Key Research and Development Program of China(2021YFC2902502)the National Natural Science Foundation of China(22078320,22035007)+4 种基金the NSFC-EU project(31961133018)the Shandong Provincial Key Research and Development Program(2022CXGC020106)the Shandong Key Research and Development Program(International Cooperation Office)(2019GHZ018)the Shandong Province Postdoctoral Innovative Talents Support Plan(SDBX2020018)the External Cooperation Program of BIC,Chinese Academy of Sciences(122111KYSB20190032)。
文摘The internal flow of a droplet in the nonlinear extensional flow field will exhibit more than two internal circulations with the variation of nonlinear intensity(E).In this paper,the effect of positions and sizes of internal circulations on internal mass transfer rate of a single spherical droplet in a nonlinear extensional flow field is studied and compared with that in a linear extensional flow field.The simulation results show that when E≥0,there are two symmetrical internal circulations in the droplet,which is the same with that in a linear extensional flow.The limit value of mass transfer rate Sh is 15,which is equal to that in a linear extensional flow,no matter how large E is.When E≤-3/7,the number of internal flow circulation of a droplet increase to four and the transfer rate Sh increases.When E=-1,the maximum internal transfer rate Sh equals 30 which is twice of that in a linear extensional flow.The generation of new flow circulations in droplets and the circulation positions will enhance mass transfer when E≤-3/7,which provides a new idea for enhancing the internal mass transfer rate of droplets.
文摘The estimation of contemporary tectonic stress field and deformation in active fold-and-thrust belts are imperative in identifying active geodynamics and resulting faulting phenomenon. In this paper, we focus on contemporary extensional tectonics in the overall compressive setting of the Himalayan orogen. Here we examine the regional tectonic stress field and upper crustal deformation in the Himalayan thrust wedge using a 2D finite element technique, incorporating elastic rheology under plain strain condition. The elastic models demonstrate that the extensional tectonic stress and related nor- mal faulting is extensively developed in the southern front of the Himalaya at shallow crustal level (&lt;10 km in depth). Our modelling shows a good consistency with the geological field evidences of active faulting, focal mechanism solutions of medium size earthquakes in the several sectors of the Himalaya. Results based on numerical simulation, tectonic analysis and taking geological and geophysical data into account, we interpret that the present-day extensional tectonic activity is not restricted in the southern Tibet but distributed in the different sectors of the Himalayan fold-and-thrust belt co-exist with compressional structures. Modelling results also indicate that the nature, distribution and orientation of the maximum compressive stress (?1) of the Himalaya are mainly controlled by the intra crustal Main Himalayan d&#233;collement (MHT). The significant amount of shear stress/strain concentration along the MHT in the western Nepal predict that the region is prone to moderate and great future earthquakes.
基金Sponsored bythe National Natural Science Foundation of China(60375014) the Shandong Province Natural Science Foundation(Y2002G16)
文摘In brder to study the contradiction problem of multilayer multidimensional complex systems, the concepts of extension field and stable field of intersection and union of multilayer multidimensional extension set are given. Then the related operations and properties are discussed. The results of study expand the concepts of intersection and union of extension set to a general situation, and provide the theoretical basis for production of the concepts of intersection and union of multilayer multidimensional matter element system extension set. In this way, it will be possible that matter element system theory is used to creative designs of complex systems.
文摘The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk</sub> is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk</sub>, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZ<sub>k</sub> decomposes into a product of prime ideals of Zk</sub> as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each f<sub>θ</sub>(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05
