Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

Bases and nonbases of square-free integers

A basis is a set A of nonnegative integers such that every sufficiently large integer n can be represented in the form n = a_i + a_j with a_i, a_i ∈ A. If A is a basis, but no proper subset of A is a basis, then A is a minimal basis. A nonbasis is a set of nonnegative integers that is not a basis, and a nonbasis A is maximal if every proper superset of A is a basis. In this paper, minimal bases consisting of square-free numbers are constructed, and also bases of square-free numbers no subset of which is minimal. Maximal nonbases of square-free numbers do not exist. However, nonbases of square-free numbers that are maximal with respect to the set of square-free numbers are constructed, and also nonbases of square-free numbers that are not contained in any nonbasis of square-free numbers maximal with respect to the square-free numbers.     

Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

We consider the Tikhonov-like dynamics where A is a maximal monotone operator on a Hilbert space and the parameter function ε(t) tends to 0 as t→∞ with . When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u(t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A⁻¹(0) provided that the function ε(t) has bounded variation, and provide a counterexample when this property fails.     

Nonexistence of stable currents

In this paper, we investigate nonexistence of stable integral currents in a compact hypersurface with positive Ricci curvature in a Euclidean spaceR ^m+1 and a vanishing theorem concerning the homology group is obtained.The project was supported by NNSFC, FECC and CPSF     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.     

Partitions of the natural numbers into infinitely oscillating bases and nonbases

The setA of nonnegative integers is a basis if every sufficiently large integerx can be written in the formx=a+a′ witha, a′∈A. IfA is not a basis, then it is a nonbasis. We construct a partition of the natural numbers into a basisA and a nonbasisB such that, as random elements are moved one at a time fromA toB, fromB toA, fromA toB, …, the setA oscillates from basis to nonbasis to basis … and the setB oscillates simultaneously from nonbasis to basis to nonbasis…     

Nonexistence of certain pseudogeometric graphs

Izo($r$) is the pseudogeometric graph for pG${}_r(s,t)$ which satisfies the Krein equality and of which the local graph also satisfies the Krein equality. We prove that Izo($r$) exists if and only if a spherical tight 4-design on $S^{{(2r+1)}^2?4}$ exists. Nonexistence of infinitely many spherical tight 4-designs is well known. It gives the nonexistence of Izo($r$) for infinitely many $r$. Especially, Izo(4) does not exist, which was the first open case.     

Nonexistence of stable threesome matchings

A counterexample shows stable matchings need not exist in societies where threesomes are to form. The stability theorem breaks down, in fact, for K-some formations for all K≥3, even when preferences are restricted to be separable.     

Nonexistence of block-transitive 6-designs

In this paper, we prove that there are no nontrivial block-transitive 6-designs for k ⩽ 100. This supports the long-standing conjecture of Cameron and Praeger that there are no nontrivial block-transitive 6-designs.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

Nonexistence of a code for

In this paper, we shall prove that there is no [3q⁴-q³-q²-3q-1,5,3q⁴-4q³-2q+1]_q code over the finite field for q11. Thus, we conclude the nonexistence of a [g_q(5,d),5,d]_q code for 3q⁴-4q³-2q+1d3q⁴-4q³-q.     

Nonexistence of o-injective partially ordered modules

It is shown that Ribenboim's proof of the nonexistence of nontrivial o-injective o-modules is incorrect, and new proof of this result is presented.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 693–696, June, 1970.     

Nonexistence of skew loops on ellipsoids

We prove that every closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve.

     

F-调和映照的不存在性定理

本文主要讨论一类F-调和映照的不存在性问题,从而得到相应的Liouville型定理．