On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)^1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55.     

On Waring's Problem for Six Cubes and Higher Degrees

It is proved that the equation $$n = x_{\,1}^3 + x_{\,2}^3 + x_{\,3}^3 + x_{\,4}^3 + x_{\,5}^3 + x_{\,6}^3 + u^4 + v^9$$ has nonnegative integral solutions if $n \equiv 1\left( {\bmod 5} \right)$ is even and sufficiently large. Bibliography: 8 titles.     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.     

On the Number of Integral Ideals in Two Different Quadratic Number Fields

Let K be an algebraic number field of finite degree over the rational field Q,and a K(n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of a K(n),Σn≤x a K(n)~l, l = 1, 2, 3, ···.This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum Σn≤x aK_1(n~j)~laK_2(n~j)~l, j = 1, 2, l = 2, 3, ···,where K_1 and K_2 are two different quadratic fields.     

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

二次函数域理想类数问题的一个注记

应用F_q[t]上的Pell方程这一初等方法重新证明一个已知的结果:实二次函数域F_q(t)(D~(1/2))理想类数为1时,D只能为P或QR,其中P,Q,R是F_q[t]中的首一不可约多项式且Q,R次数为奇数.     

On Real Quadratic Number Fields and Simultaneous Diophantine Approximation

 Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from . Specifically, for all such pairs we have . (Received 10 August 1998; in revised form 23 November 1998)     

On a Class of Number Fields of Eisenstein Type

In this paper the author determine the factorization of p and the number of p-th subfields of the splitting field K of the irreducible binomial xp ?m(m is p-th powerfree and p | m) and obtain that all p-th subfields of K have power integral bases and the same discriminant.     

On the On-line Number of Snacks Problem

In the number of snacks problem (NSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as _: C_A()/C_OPT(), where denotes a sequence of numbers of customers, C _OPT() is the cost of satisfying by an optimal off-line algorithm, and C _A() is the cost of satisfying by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm(ENHA), with competitive ratio 1+p(M-m)/(M+m), where M and m are two extreme numbers of customers over the total period of the game, and p is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP.     

Fractals and Number Systems in Real Quadratic Number Fields

In this paper we compute the box counting dimension of sets, that are related to number systems in real quadratic number fields. The sets under discussion are so-called graph-directed self affine sets. Contrary to the case of self similar sets, for self affine sets there does not exist a general theory for the determination of the box counting dimension. Thus we are forced to construct the covers, needed for its calculation, directly. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Torsion of Elliptic Curves Over Quartic Number Fields

We determine which groups Z/MZZ/NZ occur infinitely often astorsion groups E(K)_tors when K varies over all quartic numberfields and E varies over all elliptic curves over K.     

On A Minimization Problem for Vector Fields in L1

This paper treats the problem of minimizing the norm of vectorfields in L¹ with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R²is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rⁿ is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form ₁ = –Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R² is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique.     

On the Number of Solutions to Carrier's Problem

Let N _ɛ denote the maximum number of spikes that a solution to Carrier's problem can have, where ɛ is a small positive parameter. We show that N _ɛ is asymptotically equal to [ K /ɛ], where   K = 0.4725⋯  , and the square brackets represent the greatest integer less than or equal to the quantity inside. If n (ɛ) stands for the number of solutions to this problem, then it is also shown that 4 N _ɛ− 3 ≤ n (ɛ) ≤ 4 N _ɛ. Our approach is based on the shooting method used by Ou and Wong ( Stud. Appl. Math. 111 (2003)) and on the construction of an envelope function for the minimum values of the solutions as ɛ approaches zero.     

关于有限域上一类方程零点个数的估计

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.     

On Exponential Sums Involving the Ideal Counting Function in Quadratic Number Fields

 Let χ be a Dirichlet character modulo k > 1, and F _χ(n) the arithmetical function which is generated by the product of the Riemann zeta-function and the Dirichlet L-function corresponding to χ in . In this paper we study the asymptotic behaviour of the exponential sums involving the arithmetical function F _χ(n). In particular, we study summation formulas for these exponential sums and mean square formulas for the error term. Received April 17, 2001; in revised form April 2, 2002     

关于有限域上一类方程解的个数

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N(a1x12 + ···+anxn2 = bx1 ···xs) denote the number of solutions(x1,...,xn) of the equation a1x12 +···+ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F*q,b ∈ F*q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.