ON VARMA''s LACUNARY INTERPOLATION BY SPLINES OHEN TIANPING

In articles [1], [2], A. K. Varma investigated some lacunary interpolations by Spline. In this article we point out a mistake in the proof of Theorem 2 in [1] and give a new proof. Moreover, the results obtained in [1], [2] are improved.     

The Compact Characterhations of the Operator H_fH_g on the Bergman Space

ＴｈｅＣｏｍｐａｃｔＣｈａｒａｃｔｅｒｈａｔｉｏｎｓｏｆｔｈｅＯｐｅｒａｔｏｒＨ＿ｆＨ＿ｇｏｎｔｈｅＢｅｒｇｍａｎＳｐａｃｅＧｕｏＫｕｎｙｕ（郭坤宇)（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ?..     

An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts. Oblatum 3-IX-1996 & 4-VIII-1997     

Über die Moduln algebraischer Flächen

In this paper we want to extend results of Kodaira [4], Horikawa [1], and Wahl [14] concerning infinitesimal families of surfaces with generic singularities over the complex numbers to fields of arbitrary characteristic. The main obstruction in getting complete analogy will be the nonexistence of a proof of Kodaira's Vanishing Theoreme which we substitute by partial results. Characteristic 2 provides for extra difficulties but in 6. we state a result which combines this paper with an interesting statement of Roberts in [12], Besides this we look at a common obstructionspace for various deformation functors. The main technique is to compare on the surface the intrinsically defined Jacobi- and conductor ideal sheaves. In contrast to [1] and [4] we work as Wahl [14] completely in an algebraic category.     

Explicit maximal and minimal curves over finite fields of odd characteristics

In this work we present explicit classes of maximal and minimal Artin–Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1].     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

关于环Z[m~（1/2）]的商环元素个数的一个新的证明

文献[1]热运用环论的方法证明了环Z[m~（1/2）]热的商环Z[m~（1/2）]/（a＋bm~（1/2））的元素个数是｜a2-b2m｜.我们将用主理想整环上的模的理论给出一种简洁的证明.     

Garsia and Milne's bijective proof of the inclusion-exclusion principle

Although the following proof is implicit in Garsia and Milne's paper [1], it is so elegant that we felt that it should be presented by itself for the benefit of the general mathematical public. The idea behind the proof was further exploited by Remmel [2] and Wilf [3].     

Proof of a theorem of Lusin

In [1, 2], Lusin published a theorem (with proof) asserting that a very simple set constructed by him is not Borel. Lunina [3] discovered an error in Lusin's proof. It is proved that Lusin's theorem is nonetheless valid.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 61–66, January, 1978.The author is grateful to V. A. Uspenskii for his interest in this work and to M. A. Lunina for valuable remarks.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

On Strong Uniform Distribution, III

We construct infinite dimensional chains that are ¹ good for almost sure convergence, which settles a question raised in this journal [7] and earlier in [6] by R. Nair. In [7] it was stated that the construction proposed in [4] was invalid. We complete the construction proposed in [4], where it is true that a piece of proof was forgotten. The technic remains the same and the completion of the proof rather natural.     

On Strong Uniform Distribution,III

We construct infinite dimensional chains that are ¹ good for almost sure convergence, which settles a question raised in this journal [7] and earlier in [6] by R. Nair. In [7] it was stated that the construction proposed in [4] was invalid. We complete the construction proposed in [4], where it is true that a piece of proof was forgotten. The technic remains the same and the completion of the proof rather natural.     

One more proof of the Borodin-Okounkov formula for Toeplitz determinants

Recently, Borodin and Okounkov [2] established a remarkable identity for Toeplitz determinants. Two other proofs of this identity were subsequently found by Basor and Widom [1], who also extended the formula to the block case. We here give one more proof, also for the block case. This proof is based on a formula for the inverse of a finite block Toeplitz matrix obtained in the late seventies by Silbermann and the author.     

Orderings of higher level on noetherian rings

The noncommutative algebraic geometry has found fruitful applications in quantum geometry. Similar applications are expected to be found for its younger sister the noncommutative real algebraic geometry

One of the basic results in real algebraic geometry is the Positivestellensatz. The original results of Dubois and Risler (see section 3.3 of [13]) have been extended in many directions. We refer to [14], [1], [2], [3] for commutative rings and [9], [4] for associative rings. The aim of this paper is to prove the higher level Posit ivstellensatz for noncommutative Noetherian rings. Our proof depends on the intersection theorem for orderings of higher level on skew fields ([11], Theorem 3.13). The general case of orderings of higher level on associative rings remains open.     

Countable Products of Čech-scattered supercomplete spaces

We prove by using well-founded trees that a countable product of supercomplete spaces, scattered with respect to ech-complete subsets, is supercomplete. This result extends results given in [1], [5], [6], [19], [26] and its proof improves that given in [19].     

关于广义逆矩阵AT，S^（1,2）的刻画推广

本文通过一类秩等方程给出了AT,S^(1,2)、AT,S^(2)的一种刻画及一类秩等方程有解的充分必要条件，推广了文献[1]、[4]的结论，并改进了[4]关于矩阵A的Drazin逆Af的一类刻画的证明。     

Karl M. Peterson: The earliest derivation of the Mainardi-Codazzi equations and the fundamental theorem of surface theory

In 1853 Karl M. Peterson, a young Latvian student at the University of Dorpat [Tartu], submitted a candidates' dissertation [a requirement for obtaining a degree] containing a derivation of the Gauss-Mainardi-Codazzi equations and a proof of the Fundamental Theorem of Surface Theory. This predates the publications of equivalent equations derived by G. Mainardi [1856] and D. Codazzi [1860 and 1868–1869], as well as O. Bonnet's proof of the Fundamental Theorem. Although Peterson later [1865–1881] played a major role in the formation of a school of differential geometry at Moscow University, this early work remained virtually unknown until its translation [into Russian] and its publication in 1952. The contents of the dissertation and the probable sources of Peterson's ideas and techniques are discussed.     

Concentration through large advection

In this paper we extend the elegant results of Chen, Lam and Lou [6, Section 2], where a concentration phenomenon was established as the advection blows up, to a general class of adventive–diffusive generalized logistic equations of degenerate type. Our improvements are really sharp as we allow the carrying capacity of the species to vanish in some subdomain with non-empty interior. The main technical devices used in the derivation of the concentration phenomenon are Proposition 3.2 of Cano-Casanova and López-Gómez [5], Theorem 2.4 of Amann and López-Gómez [1] and the classical Harnack inequality. By the relevance of these results in spatial ecology, complete technical details seem imperative, because the proof of Theorem 2.2 of [6] contains some gaps originated by an “optimistic” use of Proposition 3.2 of [5]. Some of the general assumptions of [6] are substantially relaxed.     

Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof. The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.