A note on theT-ideal generated bys 3[x 1,x 2,x 3]

The co-characters of theT-ideal generated by the standard identitys ₃[x ₁,x ₂,x ₃] are determined.     

[2] x [3] x N is not a circle order

The result stated in the title is proved in this note. Actually we show that S x N is not a circle order, where S={(1, 1), (1, 2), (1, 3), (2, 1), (2, 3)}. Furthermore this non-circle order is critical in the sense that (S-{x}) x N is a circle order for any x in S.     

关于holonomicA1-模k[x,p-1]的研究

本文研究了一阶Weyl代数A1上的Holonomic模k[x,p-1].利用与Bernstein-链对应的k[x,p-1]上的好链,证明了k[x,p-1]的重数为degp+1,且计算了k[x,p-1]上的一些元素的零化子.     

Dedekind subrings ofk[x 1,…,x n ] are rings of polynomials

The purpose of this note is to prove that a Dedekind domain R which contains a field k, and which is a subring ofk[x ₁,…,x _n ] is a ring of polynomials. This generalizes similar results of A. Evyatar and A. Zaks on principal ideal domains, and of P. M. Cohn for the casen=1. Our methods and proofs differ from those introduced previously. This research was partially supported by the National Science Foundation, Grant GP-23861.     

[1, 2]-sets in graphs

A subset S⊆V$S\subseteq V$ in a graph G=(V,E)$G=(V,E)$ is a [j,k]$[j,k]$-set if, for every vertex v∈V?S$v\in V?S$, j≤|N(v)∩S|≤k$j\le |N\left(v\right)\cap S|\le k$ for non-negative integers j$j$ and k$k$, that is, every vertex v∈V?S$v\in V?S$ is adjacent to at least j$j$ but not more than k$k$ vertices in S$S$. In this paper, we focus on small j$j$ and k$k$, and relate the concept of [j,k]$[j,k]$-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k$k$-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G$G$, the restrained domination number is equal to the domination number of G$G$.     

Homogeneous ideals inK[x,y, z]

Acta Mathematica Hungarica -     

The Jackson Inequality for the Best L2-Approximation of Functions on [0, 1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.     

Uniformly distributed functions inGF[q,x] andGF{q,x}

Summary Let A, B, C, ... denote polynomials over the finite field GF(q). It is shown that the sequence {B_i} is uniformly distributed modulo M if the sequence {B_i+k - B_i} is uniformly distributed modulo M for all integers k>0. A similar result holds for sequences defined by functional values. Also, a result of Weyl concerning uniform distribution modulo 1 is extended to polynomials over finite fields. Entrata in Redazione il 25 febbraio 1972.     

Zur Bestimmung vonE n [x n+2m ]

Ohne Zusammenfassung     

关于不定方程组x-1=3py^2，x^2＋x＋1=3z^2

设P为素数,利用同余及高次丢番图方程的一些结果证明了不定方程组x-1=3py^2,x^2＋x＋1=3z^2仅有正整数解（p,x,y,z）=（7,22,1,13）。     

Special isomorphisms of F[x 1,..., x n ] preserving GCD and their use

On the ring R = F[x ₁,..., x _n ] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x ₁,..., x _n ]]⁺ and the ring T: = F[[x ₁,..., x _n ]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x ₁,..., x _n ]] has a greatest common divisor.     

Einige Anwendungen der Function [x]

Ohne Zusammenfassung     

A variety generated by a finite algebra with 2x 0 subvarieties

As was indicated to me by Prof. A. Wroński, the following problem was suggested by Prof. B. Jónsson: is every subvariety of the variety of a finite algebra generated by a finite algebra? In this paper we solve this problem in the negative by constructing a finite algebra that generates a variety having 2^x ₀ subvarieties.     

On the bases of the spaces L1[0, 1] and C[0, 1]

New and simple proofs are given for the non-existence of unconditional bases in the spaces L₁[0, 1] and C[0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 241–249, March, 1972.The author wishes to thank P. L. Ul'yanov for his interest in the problem under consideration.