关于一个素数和一个素数的k次方和问题

设k≥2,Hk表示一个正整数n的集合,使对任意的正整数q,同余方程a+b2三n(modq)在模q的既约剩余系中有解a,b.Dk(N)表示n≤N,n∈Hk,但不能表成p1+p22=n的数的个数,其中p1,p2表示素数.则在GRH下,Dk(N)<<N1-1/k(h(k)+1)+ε,这里k=2,3;h(2)=2,h(3)=8.     

关于方程ax+by=cz的Terai-Jesmanowicz猜想

设m是正整数,证明了(A)如果b是奇素数,且a=m3-3m,b=3m2-1,c=m2+1,那么丢番图方程ax+by=cz(1)仅有正整数解(x,y,z)=(2,2,3);(B)如果b是奇素数,且a=m|m4-10m2+5|,b=5m4-10m2+1,c=m2+1,那么丢番图方程(1)仅有正整数解(x,y,z)=(2,2,5).     

关于方程ax+by=cz的Terai-Jesmanowicz猜想

设m是正整数,证明了:(A)如果b是奇素数,且a=m3-3m,b=3m2-1,c=m2+1,那么丢番图方程ax+by=cz(1)仅有正整数解(x,y,z)=(2,2,3);(B)如果b是奇素数,且a=m|m4-10m2+5|,b=5m4-10m2+1,c=m2+1,那么丢番图方程(1)仅有正整数解(x,y,z)=(2,2,5).     

余弦定理在四边形的一个推广

在△ABC中 ,设内角A ,B ,C的对边分别为a ,b,c,余弦定理cosB =a2 +c2 -b22ac ( 1 )是我们所熟悉的 .笔者在文 [1 ]中给出了余弦定理在四面体的推广 ,注意到文 [2 - 3]中给出了余弦定理在四边形的推广 ,本文试给出余弦定理在四边形的另一新颖推广 ,使得三角形的余弦定理成为该推广式极限情形的一个特例 .定理　记凸四边形ABCD的四边长依次为AB =a ,BC=b ,CD =c,DA =d ,两对角线长AC =p ,BD =q ,则cos(B+D) =(ac) 2 + (bd) 2 - (pq) 22abcd ( 2 )证明　如图 ,设两对角线交角为θ ,p ,q分别由p1 ,p2 与q1 ,q2 组成 .由余弦定理得p2 =…     

构造向量解题

文 [1 ]举例说明了平面向量在中学数学中的广泛应用 .作为文 [1 ]的补充 ,本文再举几例 ,说明构造向量 ,利用向量的内积在中学数学其它一些方面的应用 .1　求值例 1　设 a,b,c,x,y,z均为实数 ,且a2 b2 c2 =2 5,x2 y2 z2 =3 6,ax by cz =3 0 .求 a b cx y z的值 .解　由题设条件 ,考虑构造向量 p=(6a,6b) ,q=(5x,5y) .由 (p.q) 2 ≤ |p|2 |q|2 ,有 90 0 (ax by) 2 ≤ 90 0 (a2 b2 ) (x2 y2 ) ,即　 (3 0 - cz) 2 ≤ (2 5- c2 ) (3 6- z2 ) ,变形整理得　 (5z - 6c) 2≤ 0 ,∴　 5z =6c.同理　 5x =6a,　 5y =6b.∴…     

Adamovic猜测的证明

当q≥3p时,关于n的数列{S_n(p,g)}是单调增加的。当q≤(5/2)p且不存在a,b(a=2,3,……,b=1,2)使得p=2a 1,q=5a b时,数列{S_n(p,q)}是单调减少的。     

关于三项纯指数Diophantine方程ax+by=cz

设a、b、c是互素的正整数.本文证明了:当a b2l-1=c2,b≡5(mod 12),c是适合c≡-1(mod b2l)的奇素数,其中l是正整数时,方程ax by=cz仅有正整数解(x,y,z)=(1,2l-1,2).     

关于丢番图方程(na)~x+(nb)~y=(nc)~z（英文）

设(a,b,c)为本原的商高数组,满足a~2+b~2=c~2且2|b.1956年,Jesmanowicz猜想:对任给的正整数n,丢番图方程(na)~x+(nb)~y=(nc)~z仅有正整数解x=y=z=2.令P(n)表示n的所有不同素因子乘积.对商高数组(a,b,c)=(p~(2r)-4,4p~r,p~(2r)+4),其中p为大于3的素数且p■1(mod 8),本文证明在条件P(a)|n或者P(n)a下,Jesmanowicz猜想成立.     

一类3p~2阶4度1-正则图

设p为大于3的素数,群G=和H=(其中r(?)1(mod p~2),r~3≡1(mod p~2),3|(p-1))是两类3p~2阶非交换群.通过研究Cayley图的正规性,完成了对G和H的所有4度Cayley图的分类,并得到了一类新的4度1-正则图.     

对一个结论的再推敲

笔者在文 [1 ]中介绍了一个不完整的错误的结论——二次方程 f (x) =0 (其中 f(x) =ax2 +bx+c,a、b、c∈ R,a≠ 0 )在区间 (p,q)内至少有一个实根　 　 f (p) .f (q) &;lt;0　或　Δ≥ 0p &;lt;- b2 a0af (q) &;gt;0感谢周祥昌老师在文 [2 ]中指出了原稿的疏漏 ,并补充考虑 (确实应该考虑 )了下列两种直观图示 :图 1但从形到数的转化中 ,文 [2 ]却把上述两个图示依次表述为两个混合组 :f (p) =0af (q) &;gt;0 　或　 f(q) =0af (p) &;gt;0其实下列两个图示也分别适合这两个混合组 :图 2而此时与之对应的二次方程 f (x) =0在区间 (p,q)内却没有实数根 .再次校正推敲 ,我们得到完整的结论——二次方程 f (x) =0在区间 (p,q)内至少有一个实根 　 f(p) .f(q) &;lt;0　或　Δ≥ 0p &;lt;- b2 a0af (q) &;gt;0或　f (p) =0p &;lt;- b2 a     

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

A theorem on non-associative algebras and its application to differential equations

For a given field F, the set of F-algebras (resp. commutative F-algebras) of arity n≥2 and F-dimension m can be identified with the mⁿ⁺¹ (resp. m(^m+n−1 _n)) dimensional F-affine space S of structure coefficients. We show: If F is algebraically closed, then there exists an affine subvariety A of S with A≠S, which is defined over the prime field of F, such that all F-algebras corresponding to the points of S-A posses precisely n^m−1 idempotent elements ≠0 and fail to have nil potent elements ≠0. This implies for a system of ordinary differential equations $$\left( * \right)\dot X_i = D_i \left( {X_l ,..,X_m } \right),i = l,..,m,$$ with D_i(X_i,...,X_m)∈ℂ[X₁,...,X_m] homogeneous polynomials of degree n: If the coefficients of the polynomials D_i, i=1,...,m, are algebraically independent over the field of rationals, then (*) possesses precisely n^m−1 ray solutions and fails to have a critical point other than the origin.     

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y _n + py _n−k + q _n y _n−l = 0 for n∈ℤ⁺(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ₁(u _n,m + pu _n−k,m ) + q _n,m u _n−l,m = a ²Δ² ₂ u _{n +1, m−1} for (n,m) ∈ℤ⁺ (0) ×Ω, (2*) study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses. Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

一类双重退化抛物型不等式问题解的Liouville 型定理

该文对一类双重退化抛物型不等式问题建立了弱解的Liouville 型定理. 不同于通常的上下解方法, 这里采用更为简洁的适当选取试验函数与能量估计的方法证明整体解的不存在性.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).     

INTERPOLATION WITH LAGRANGE POLYNOMIALS A SIMPLE PROOF OF MARKOV INEQUALITY AND SOME OF ITS GENERALIZATIONS

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

On the diophantine equation x 2 − Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.     

On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over ℝ̄ + m

We investigate the regular convergence of the m-multiple series (*) $$\sum\limits_{j_1 = 0}^\infty {\sum\limits_{j_2 = 0}^\infty \cdots \sum\limits_{j_m = 0}^\infty {c_{j_1 ,j_2 } , \ldots j_m } }$$ of complex numbers, where m ≥ 2 is a fixed integer. We prove Fubini’s theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim’s sense can also be computed by successive summation. We introduce and investigate the regular convergence of the m-multiple integral (**) $$\int_0^\infty {\int_0^\infty { \cdots \int_0^\infty {f\left( {t_1 ,t_2 , \ldots ,t_m } \right)dt_1 } } } dt_2 \cdots dt_m ,$$ where f : ?? ₊ ^m → ? is a locally integrable function in Lebesgue’s sense over the closed nonnegative octant ?? ₊ ^m := [0,∞) ^m . Our main result is a generalized version of Fubini’s theorem on successive integration formulated in Theorem 4.1 as follows. If f ∈ L _loc ¹ (?? ₊ ^m ), the multiple integral (**) converges regularly, and m = p + q, where p and q are positive integers, then the finite limit $$\mathop {\lim }\limits_{v_{_{p + 1} } , \cdots ,v_m \to \infty } \int_{u_1 }^{v_1 } {\int_{u_2 }^{v_2 } { \cdots \int_0^{v_{p + 1} } { \cdots \int_0^{v_m } {f\left( {t_1 ,t_2 , \ldots t_m } \right)dt_1 dt_2 } \cdots dt_m = :J\left( {u_1 ,v_1 ;u_2 v_2 ; \ldots ;u_p ,v_p } \right)} , 0 \leqslant u_k \leqslant v_k < \infty } ,k = 1,2, \ldots p,}$$ exists uniformly in each of its variables, and the finite limit $$\mathop {\lim }\limits_{v_1 ,v_2 \cdots ,v_p \to \infty } J\left( {0,v_1 ;0,v_2 ; \ldots ;0,v_p } \right) = I$$ also exists, where I is the limit of the multiple integral (**) in Pringsheim’s sense. The main results of this paper were announced without proofs in the Comptes Rendus Sci. Paris (see [8] in the References).