On the cofinality of the splitting number

The splitting number $\mathfrak{s}$ can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and Shelah (1989).     

Splitting of exact sequences of Fréchet spaces in the absence of continuous norms

In the present paper we study the splitting of a short exact sequence

     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

The additivity of porosity ideals

We show that several -ideals related to porous sets have additivity and cofinality . This answers a question addressed by Miroslav Repický.

     

Splitting Superhereditary Preradicals

Abstract

We study the structure of rings R with the property that, for every right R-module M and an ideal I of R, the annihilator of I in M is a direct summand of M.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

Splitting criterion for reflexive sheaves

We give a criterion for a reflexive sheaf to split into a direct sum of line bundles.

     

Van Der Waerden's Construction of a Splitting Field

In this article, we first present a unified technique in the discussion of the additivity of multiplicative maps on rings with idempotents. We next apply the obtained result to discuss the additivity of surjective elementary maps in rings with idempotents.     

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that \boldsymbol\aleph_1 $"> and find some topological characterizations of perfectly meager and universally meager sets.

     

On the Linear Stability of Splitting Methods

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2×2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By conveniently selecting the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.     

On Shattering,Splitting and Reaping Partitions

In this article we investigate the dual-shattering cardinal ?, the dual-splitting cardinal ?? and the dual-reaping cardinal ??, which are dualizations of the well-known cardinals ?? (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and ?? (the reaping number). Using some properties of the ideal ?? of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(??) = cov(??) = ?. With this result we can show that ? > ω₁ is consistent with ZFC and as a corollary we get the relative consistency of ? > ?? t, where t is the tower number. Concerning ?? we show that cov(M) ? ?? ?? (where M is the ideal of the meager sets). For the dual-reaping cardinal ?? we get p ?? ? ?? ? ?? (where ?? is the pseudo-intersection number) and for a modified dual-reaping number ??′ we get ??′ ? ?? (where ?? is the dominating number). As a consistency result we get ?? < cov(??).     

Z／（p^e）上分裂环的结构

本文研究了环ｚ／（ｐｅ）上多项式的性质和分裂环的结构．主要分析了分裂环中元素的极小多项式，零化理想的结构，和分裂环子环性质．     

Splitting lemma at infinity and a strongly resonant problem with periodic nonlinearity

In this paper we study the following problem:with periodic nonlinearity g, where and λ₂ is the second eigenvalue of −Δ, on H ¹ ₀(B). We proved that the problem has infinitely many solutions under some additional conditions on g and h. The method we used is a new variational reduction method. Mathematics Subject Classi cation (2000) 35J20, 35J70     

对称正定矩阵多级分裂预处理子

本文基于对称正定矩阵的多级分裂,讨论了这类方程组的多级分裂预处理子的构造,证明了预处理子的合理性.数值试验显示我们的预处理子是有效的.     

Explicit Volume-Preserving Splitting Methods for Linear and Quadratic Divergence-Free Vector Fields

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.     

Takens—Bogdanov分岐点的分裂迭代算法

本文构造分裂迭代算法用于计算Takens-Bogdanov分岐点,该方法将减少计算的工作量和占用的内存,可以调节的速度线性收敛,并且可以求得Takens-Bogdanov分岐点处fx^及fx^0的广义零特征向量,数值计算说明了算法的有效性。     

Splitting and composition methods for explicit time dependence in separable dynamical systems

We consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.