关于Cn中积分表示的拓广问题

指出文［４］所得的结果只是Ｌｅｒａｙ公式和Ｃａｕｃｈｙ－Ｆａｎｔａｐｐｉｅ公式的特殊情况。     

The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

The simultaneous diophantine equations 5Y2 − 20 = X2 and 2Y2 + 1 = Z2

The three numbers 1, 5, 10 have the property that the product of any two numbers decreased by 1 is a perfect square. In this paper it is proved that there is no other positive integer N which shares this property with 1, 5, and 10.     

On the solution of the diophantine equation A3 + B3 + C3 + D3 = 0

This paper presents a complete and symmetric solution of the diophantine equation of the title in terms of four parameters α, β, γ, δ, subject to α + β + γ + δ = 0.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

行列式detVn-(x1,...,xn)的准确值及其应用

本文给出行列式Ｖ_n^-(x₁,...,x_n)的准确值，它是通常的Ｖａｎｄｅｒｍｏｄｅ行列式计算公式的推广，以及它在理论上的一些重要应用．     

The quadratic form x2−2py2

Using known properties of continued fractions, we give a very simple and elementary proof of the theorem of Epstein and Rédei on the impossibility in a certain case of representing ?1 by the quadratic form x² ? 2py². Two of our theorems, which concern the representation of a² and ?2a², serve to extend our method to an unknown case in which ?1 is not representable.     

On x3 + y3 = D

The simplest case of Fermat's last theorem, the impossibility of solving x³ + y³ = z³ in nonzero integers, has been proved. In other words, 1 is not expressible as a sum of two cubes of rational numbers. However, the slightly extended problem, in which integers D are expressible as a sum of two cubes of rational numbers, is unsolved. There is the conjecture (based on work of Birch, Swinnerton-Dyer, and Stephens) that x³ + y³ = D is solvable in the rational numbers for all square-free positive integers D ≡ 4 (mod 9). The condition that D should be square-free is necessary. As an example, it is shown near the end of this paper that x³ + y³ = 4 has no solutions in the rational numbers. The remainder of this paper is concerned with the proof published by the first author (Proc. Nat. Acad. Sci. USA., 1963) entitled “Remarks on a conjecture of C. L. Siegel.” This pointed out an error in a statement of Siegel that the diophantine equation ax³ + bx²y + cxy² + dy³ = n has a bounded number of integer solutions for fixed a, b, c, d, and, further, that the bound is independent of a, b, c, d, and n. However, x³ + y³ = n already has an unbounded number of solutions. The paper of S. Chowla itself contains an error or at least an omission. This can be rectified by quoting a theorem of E. Lutz.     

On Legendre's equation ax2 + by2 + cz2 = 0

Let a, b, c be nonzero integers having no prime factors ≡ 3 (mod 4), not all of the same sign, abc squarefree, and for which Legendre's equation ax² + by² + cz² = 0 is solvable in nonzero integers x, y, z. A property is proved yielding a congruence which must be satisfied by any solution x, y, z.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

广义函数x+λ和x-μ的乘积

For any complcx λ≠integral, and any natural number n, the product is calcuated and Hadmard''s finite part of obtained. The product is also obtained when λ+μ is a complex number of any other possible kind, and μ is a complex number ≠ integral. The products are also calculated. Any special example of these kinds of products has not yet been calculated.     

The diophantine equation x3 − 3xy2 − y3 = 1 and related equations

The diophantine equation of the title has been solved by Ljunggren, by indirect use of the p-adic method (use is made of intermediate algebraic extensions). It is generally accepted that an immediate application of the p-adic method for the aforementioned equation is impossible. In this paper, however, this view was overthrown by first solving x² + 3 = 4y³ and then x³ ? 3xy² ? y³ = 1 with direct application of the p-adic method, avoiding the use of intermediate algebraic extensions, fulfilling thus a desire of Professor Mordell. The method used in this paper has a general character, as it is shown in Appendix B, where three more examples are given.     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

Topological singularities in W S,P ( S N , S 1)

We are interested in the location of the singularities of maps u ∈ W ^s,p (S ^N , S ¹) when 1 ≤ sp and 1 < sp < 2. To this end, we consider the distributional Jacobian. We show that the range of this operator on W ^s,p (S ^N , S ¹) is the closure in W ^s−2,p ⋂ W ^−1,sp of the set of N − 2-currents defined as the integration on smooth oriented N − 2-dimensional boundaryless submanifolds.