三项式xn-x-a的二次不可约因式

设n是正整数,f(x)=xn-x-a,其中a是非零整数. 证明了当n>5时,如果f(x)有首项系数为1的二次整系数不可约因式g(x),则必有n≡2(mod6),a=-1,g(x)=x2-x+1或者n=7,a=±280,g(x)=x2t(±)x+5.     

On the Factorizations of （mg, mf）-graph

Let G be an (mg, mf)-graph, where g and f are integer-valued functions defined on V(G) and such that 0≤g(x)≤f(x) for each x ∈ V(G). It is proved that(1) If Z ≠ , both g and f may be not even, G has a (g, f)-factorization, where Z = {x ∈ V(G): mf(x)-dG(x)≤t(x) or dG(x)-mg(x)≤ t(x), t(x)= f(x)-g(x)>0}.(2) Let G be an m-regular graph with 2n vertices, m≥n. If (P1, P2,..., Pr) is a partition of m, P1 ≡ m (mod 2), Pi ≡ 0 (mod 2), i = 2,..., r, then the edge set E(G) of G can be parted into r parts E1 , E2,...,Er of E(G) such that G[Ei] is a Pi-factor of G.     

趣味数学几例

1.今年元旦是星期日,试问今年元旦后的第1984~(1984)天是星期几。解:∵1984~(1984)=(283×7+3)~(1984) =7m+3~(1984),m∈N。而 3~6≡1(mod7),3~(1984)=3~4×3~(6×330) 3~4≡4(mod7),∴1984~(1984)≡4 (mod7)。答:今年元旦后的第1984~(1984)天是丛期四。 2.若f(x+1)=|x-1|,求f(1984)。解:令 x+1=1984,则x-1=1982, ∴ f(1984)=1982。 3.已知 f(x)=3x+1,g(x)=2x-1,h(g〔f(x)〕)=f(x)。求h(1984)。解:∵ f(y)=3y+1, ∴ g〔f(y)〕=2(3y+1)-1=6y+1, 故h(6y+1)=3y+1。令6y+1=1984,     

K_m∨的最小直径定向(英文)

For a graph G,let D denote an orientation of G having minimum diameter. Define f(G)=diamD.In this paper,we concentrate on exploring the minimum diameter of K_m∨(m≥1,n≥1).Some special cases are known:f(K_m∨)=∞,2,3, where m=1 and n≥1,m=2 or m≥4 and n=1,m=3 and n=1,respectively. So we only consider the case when m≥2 and n≥2.The following results are obtained. (1) f(K_m∨)=3,where m=2,3,n≥2 and m=n=4.(2) f(K_m∨)=2, where m≥5 and m is odd,2≤n≤■-m.(3) f(K_m∨)=2,where m≥4 and m≡0(mod4),2≤n≤■-(m/2 1).(4) f(K_m∨)=2,where m≥6 and m≡2(mod4),2≤n≤■-m/2.(5) f(K_m∨)=3,where m≥4,n>■.     

关于矩阵方程A~m=λJ的一些结果(英文)

本文利用四个等阶的同余式得到命题2,从而概括了别人的一结果。 命题2 设Q(x)是g 循环阵A的Hall多项式 (1)假设g~m=0 (mod n),则A满足A~m=λJ当且仅当T_c(x)|Q(x),c=(g,n); (2)假设Q(x)=T_r(x) (mod x~n-1),则A满足A~m=λJ当且仅当rg~(m-1)=0 (mod n)和r≡0 (mod c)。它们的Hall-多项式如下: 在此基础上得到二组新解     

Universal sums of three quadratic polynomials

Let a,b,c,d,e and f be integers with a≥ c≥ e> 0,b>-a and b≡a(mod 2),d>-c and d≡c(mod 2),f>-e and f≡e(mod 2).Suppose that b≥d if a=c,and d≥f if c=e.When b(a-b),d(c-d) and f(e-f) are not all zero,we prove that if each n∈N={0,1,2,...} can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈N then the tuple(a,b,c,d,e,f) must be on our list of 473 candidates,and show that 56 of them meet our purpose.When b∈[0,a),d∈[0,c) and f∈[0,e),we investigate the universal tuples(a,b,c,d,e,f) over Z for which any n∈N can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈Z,and show that there are totally 12,082 such candidates some of which are proved to be universal tuples over Z.For example,we show that any n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈Z,and conjecture that each n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈N.     

关于Diophantine方程(a~n-1)((a+1)~n-1)=x~2

本文研究了指数Diophantine方程(an-1)((a+1)n-1)=x2的正整数解(n,x),其中a是大于1的正整数.运用初等数论方法证明了:当a≡2或3(mod4)时,该方程无解.     

Kloosterman sums with multiplicative coefficients

Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N~2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q~(1/4+ε/2)N~(2/1)(log(6N))~(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function.     

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 DIOPHANTINE EQUATION X4-Dy2=1(II)

For the Diophantine equation x^4 — Dy^2 = 1 (1) where D>0 and is not a perfect square, we prove the following theorems in this paper. Theorem 1. If D\[{\not \equiv }\]7 (mod 8)，D=p1p2...ps，s≥2，where pi（i = 1，…，s) are distincyt primes，p1≡1(mod 4) such that either 2p1=a^2+b^2，а≡\[ \pm \]3(mod 8)，b三\[ \pm \]3(mod 8) or there is a j(2≤j≤s), for which Legendre symbal \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\],and pi≡7(mod8) (i=2,..., s) or pi≡3(mod 8) (i=2,..., s), then (1) has no solutions in positive integer x,y. Theorem 2. If D=p1...ps,s≥2, where pi（i = 1，…，s) are distinct primes, and pi≡3(mod 4)（i = 1，…，s), then (1) has no solutions in positive integer x, y. Theorem 3. The equation (1) with D=2p1...ps has no solutions in positive integer x, y, if (1) p1≡(mod 4), pi≡7(mod 8) (i = 2, ???, s), snch that either 2p1 = a^2+b^2 a≡\[ \pm \]3(mod 8)，b≡\[ \pm \]3(mod 8)or there is a j (2≤j≤s)，for which \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\]; or (2) p1≡5(mod8),pi≡3(mod8) (i = 2,..., s)； or ⑶p1≡5(mod8)，pi≡7(mod 8) (i=2，…，s). Corollary of theorem 3. If D = 2pq, p≡5(mod 8), q≡3(mod 4), where p, q are distinct primes, then (1) has no solutions in positive integer x, y. Theorem 4. If D=2p1...ps, pi≡3(mod 4)(0 = 1,...,s), then (1) has no solutions In positive integer x, y.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].