Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygon...Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.展开更多
Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by...Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z1,...,z9)∈Z[z1,...,z9]whether the equation P(z1,...,z9)=0 has integral solutions with z9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,...,z11)=0(with integer coefficients)in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z1,...,z17)∈Z[z1,...,z17]whether P(z12,...,z172)=0 has integral solutions,and that there is a polynomial Q(z1,...,z20)∈Z[z1,...,z20]such that{Q(z12,...,z202):z1,...,z20∈Z}∩{0,1,2,...}coincides with the set of all primes.展开更多
The polygonal fuzzy numbers are employed to define a new fuzzy arithmetic. A novel ex-tension principle is also introduced for the increasing function σ:R→R. Thus it is convenient to con-struct a fuzzy neural networ...The polygonal fuzzy numbers are employed to define a new fuzzy arithmetic. A novel ex-tension principle is also introduced for the increasing function σ:R→R. Thus it is convenient to con-struct a fuzzy neural network model with succinct learning algorithms. Such a system possesses some universal approximation capabilities, that is, the corresponding three layer feedforward fuzzy neural networks can be universal approximators to the continuously increasing fuzzy functions.展开更多
文摘Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.
基金supported by National Natural Science Foundation of China(Grant No.11971222)。
文摘Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z1,...,z9)∈Z[z1,...,z9]whether the equation P(z1,...,z9)=0 has integral solutions with z9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,...,z11)=0(with integer coefficients)in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z1,...,z17)∈Z[z1,...,z17]whether P(z12,...,z172)=0 has integral solutions,and that there is a polynomial Q(z1,...,z20)∈Z[z1,...,z20]such that{Q(z12,...,z202):z1,...,z20∈Z}∩{0,1,2,...}coincides with the set of all primes.
基金The author would like to thank Professor H. Wang for helpful suggestions This work was supported by the National Natural Science Foundation of China( Grants Nos. 69974006 and 69974041) .
文摘The polygonal fuzzy numbers are employed to define a new fuzzy arithmetic. A novel ex-tension principle is also introduced for the increasing function σ:R→R. Thus it is convenient to con-struct a fuzzy neural network model with succinct learning algorithms. Such a system possesses some universal approximation capabilities, that is, the corresponding three layer feedforward fuzzy neural networks can be universal approximators to the continuously increasing fuzzy functions.
