The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.     

A FORMALISM FOR SOME CLASS OF FORCING NOTIONS

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.     

The covering number and the uniformity of the ideal ℐf

Let f, g ∈ ^ω ω . We will denote by g ? f that for every k < ω, f (n ^k ) ≤ g (n ) except for finitely many n . The ideal ?_f on ^ω 2 is the collection of sets X such that, for some g ? f and τ ∈ ∏_{n <ω} ^{g (n )}2, every x ∈ X satisfies τ (n ) ? x for infinitely many n . In the present paper, we will prove the consistency of cov(?_f ) < ?? and non(?_f ) < ??. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere

Let u be a map from a Riemann surface M to a Riemannian manifold N and $\alpha >1$, the α energy functional is defined as

$E_{\alpha }(u)=\frac{1}{2}\underset{M}{\int }[{(1+|▽u{|}^2)}^{\alpha }?1]dV.$

We call $u_{\alpha }$ a sequence of Sacks–Uhlenbeck maps if $u_{\alpha }$ are critical points of $E_{\alpha }$ and

$\underset{\alpha >1}{\sup }?E_{\alpha }(u_{\alpha })<\infty .$

In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere $S^{K?1}$. The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3], [4]).     

Forcing operators on MTL‐algebras

We study the forcing operators on MTL‐algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t ‐norm based logic (MTL). At logical level, they provide the notion of the forcing value of an MTL‐formula. We characterize the forcing operators in terms of some MTL‐algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL‐formula (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forcing with Proper Classes

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FORCING BONDS OF A BENZENOID SYSTEM

ＦＯＲＣＩＮＧＢＯＮＤＳＯＦＡＢＥＮＺＥＮＯＩＤＳＹＳＴＥＭＺＨＡＮＧＦＵＪＩ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉａｍｅｎＵｎｉｖｅｒｓｉｔｙ，Ｘｉａｍｅｎ８５００４６，Ｃｈｉｎａ）ＬＩＸＵＥＬＩＡＮＧ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐ...     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.     

Forcing Linearity Numbers for Modules over PID's

Let V be a module over a principal ideal domain. Then V = M N where M is divisible and N has no nonzero divisible submodules. In this paper we determine the forcing linearity number for V when N is a direct sum of cyclic modules. As a consequence, the forcing linearity numbers of several classes of Abelian groups are obtained.     

Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

Consistently There Is No Non Trivial CCC Forcing Notion with the Sacks or Laver Property

Dedicated to the memory of Paul Erdős Received March 2, 2000 RID=" " ID=" " The research supported by the Israel Science Foundation (founded by the Israel Academy of Sciences). Publication number 723. We would like to thank Martin Goldstern for many improvements in this paper.     

带强迫项的高阶中立型时滞差分方程的振动定理

本文研究带有强迫项的高阶中立型时滞差分方程的振动性.我们建立了方程(E)的新的振动准则且给出了说明定理应用的例子.     

The Bounded Axiom A Forcing Axiom

We introduce the Bounded Axiom A Forcing Axiom (BAAFA). It turns out that it is equiconsistent with the existence of a regular ∑₂‐correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

孤立子型热源强迫与大气和海洋流场的奇异响应

运用非线性动力学分析方法,对一个广义的地球流体动力学正压准地转模式中,局地热源强迫对大气和海洋流场产生的响应结果,进行了解析研究.发现热源扰动和响应流场之间存在很好的对应关系:两者具有类似的分布模态,且孤立子型的响应流场要比孤立子型的热源强迫范围大得多,即"狭窄"的热源扰动可能导致"宽广"的流场响应;强迫热源的特殊结构有可能导致大气或海洋流场出现奇异响应,从而使大气或海洋环流发生异常(如大气阻塞现象);中、高纬地区的大气和海洋流场对热源强迫的响应要比低纬地区显著得多.上述结果与中、低纬大气试验和观测资料的研究结果相符,可部分解释地球流体中,局地热源异常可能导致的环流异常现象.     

Lower Bound Improvement and Forcing Rule for Quadratic Binary Programming

In this paper several equivalent formulations for the quadratic binary programming problem are presented. Based on these formulations we describe four different kinds of strategies for estimating lower bounds of the objective function, which can be integrated into a branch and bound algorithm for solving the quadratic binary programming problem. We also give a theoretical explanation for forcing rules used to branch the variables efficiently, and explore several properties related to obtained subproblems. From the viewpoint of the number of subproblems solved, new strategies for estimating lower bounds are better than those used before. A variant of a depth-first branch and bound algorithm is described and its numerical performance is presented.     

On a conjecture of Gentner and Rautenbach

Gentner and Rautenbach conjectured that the size of a minimum zero forcing set in a connected graph on $n$ vertices with maximum degree $3$ is at most $\frac{1}{3}n+2$. We disprove this conjecture by constructing a collection of connected graphs $\left\{G_n\right\}$ with maximum degree 3 of arbitrarily large order having zero forcing number at least $\frac{4}{9}\left|V\left(G_n\right)\right|$.     

热源强迫的边界层低涡解及其应用

考虑边界层低涡为受非绝热加热和摩擦强迫并满足热成风平衡的轴对称涡旋系统,采用Boussinesq近似,通过求解柱坐标系中涡旋模式的初值问题,分析了热源强迫对低涡流场结构的影响．结果表明:热源强迫对低涡的流场结构有重要影响,并且这种影响的具体表现形式与加热的径向分布有密切关系．对边界层涡旋解讨论的结果可以解释青藏高原低涡系统的某些重要结构特征．     

具有快速振荡外力的拟地转运动的平均原理

一类大尺度的地球物理流体流可以用拟地转方程来描述．有限、但是大时间区间和整个时间轴上在快速振荡外力下的拟地转运动的平均原理被得到了．其中包括比较估计,稳定性估计和拟地转运动及其平均运动之间的收敛性．进一步,几乎周期拟地转运动的存在性和吸引子的收敛性也被得到了．