The Isolated D. R. E. Degrees are Dense in the R. E. Degrees

In the present paper we prove that the isolated differences of r. e. degrees are dense in the r. e. degrees. Mathematics Subject Classification: 03D25.     

Discontinuity of Cappings in the Recursively Enumerable Degrees and Strongly Nonbranching Degrees

We construct an r. e. degree a which possesses a greatest a-minimal pair b₀, b₁, i.e., r. e. degrees b₀ and b₁ such that b₀, b₁ < a, b₀ ∩ b₁ = a, and, for any other pair c₀, c₁ with these properties, c₀ ≤ b_i and c₁ ≤ b_1-i for some i ≤ 1. By extending this result, we show that there are strongly nonbranching degrees which are not strongly noncappable. Finally, by introducing a new genericity concept for r. e. sets, we prove a jump theorem for the strongly nonbranching and strongly noncappable r. e. degrees. Mathematics Subject Classification : 03D25.     

On the Density of the Pseudo-Isolated Degrees

A d.c.e. (2-computably enumerable) degree d is pseudo-isolatedif d itself is non-isolated (in the sense that no computablyenumerable (c.e.) degree below d can bound the c.e. degreesbelow d) and there is a d.c.e. degree b < d bounding allc.e. degrees below d. We prove in this paper that the pseudo-isolateddegrees are densely distributed in the c.e. degrees. 2000 MathematicsSubject Classification 03D25, 03D28.     

Non‐isolated quasi‐degrees

We show that non‐isolated from below 2‐c.e. Q ‐degrees are dense in the structure of c.e. Q ‐degrees. We construct a 2‐c.e. Q ‐degree, which can't be isolated from below not only by c.e. Q ‐degrees, but by any Q ‐degree. We also prove that below any c.e. Q ‐degree there is a 2‐c.e. Q ‐degree, which is non‐isolated from below and from above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the number of conjugacy classes of zeros of characters and the length of solvable groups

Abstract

We apply an orbit theorem to a few questions about character degrees. We investigate the relation of the number of conjugacy classes where characters vanish and the length of the solvable groups. As another application, we give a bound for the size of defect groups of blocks of solvable groups.

Communicated by J. Zhang     

On the bounded quasi‐degrees of c.e. sets

We study the degree structure of bQ‐reducibility and we prove that for any noncomputable c.e. incomplete bQ‐degree a, there exists a nonspeedable bQ‐degree incomparable with it. The structure $\mathcal {D}_{\mbox{bs}}$ of the $\mbox{bs}$‐degrees is not elementary equivalent neither to the structure of the $\mbox{be}$‐degrees nor to the structure of the $\mbox{e}$‐degrees. If c.e. degrees a and b form a minimal pair in the c.e. bQ‐degrees, then a and b form a minimal pair in the bQ‐degrees. Also, for every simple set S there is a noncomputable nonspeedable set A which is bQ‐incomparable with S and bQ‐degrees of S and A does not form a minimal pair.     

考虑多救援点的突发事件应急救援人员派遣模型

针对具有多救援点的突发事件应急救援人员派遣问题,给出了一种应急救援人员派遣模型。首先,依据救援人员关于救援任务的能力指标评价值计算出不同出救点的救援人员对救援点中救援任务的胜任度;其次,依据救援人员到达救援点的应急救援时间计算出应急救援时间满意度;然后,将救援人员对救援点的胜任度与应急救援时间满意度进行集结,获得应急救援人员与各救援点的综合匹配度;进一步地,以综合匹配度最大为目标,构建应急救援人员派遣优化模型,并通过模型求解获得最优的应急救援人员派遣方案;最后,通过一个算例说明了所构建的应急救援人员派遣模型具有可用性。     

Derivations and generating degrees in the ring of arithmetical functions

In this paper we study a family of derivations in the ring of arithmetical functions of several variables over an integral domain, and compute the generating degrees of the ring of arithmetical functions over the kernel of these derivations.     

Density of the cototal enumeration degrees

We prove that the cototal enumeration degrees are exactly the enumeration degrees of sets with good approximations, as introduced by Lachlan and Shore [17]. Good approximations have been used as a tool to prove density results in the enumerations degrees, and indeed, we prove that the cototal enumerations degrees are dense.     

基于时间度的动态直觉模糊多属性妥协决策

针对当前动态直觉模糊多属性决策方法存在的不足,提出一种基于时间度的动态直觉模糊妥协决策方法。引入时间度准则,基于逼近理想解法融合主客观两类赋权法,获得兼顾主观偏好和样本客观信息的时序权重,克服现有时序权重主观赋值的随意性,同时运用直觉模糊熵(IFE)确定不同时序状态下各属性权重;根据动态直觉模糊加权几何算子(DIFWG)集结不同时序直觉模糊决策矩阵,构造动态直觉模糊综合决策矩阵,并利用VIKOR法,提供兼顾群体效用最大化与个体后悔最小化的各方案妥协折中排序,得到与理想解最近的妥协方案;以分布式创新企业合作伙伴选择为例,验证该方法在实际决策过程中的可行性和有效性。     

The low<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> and low<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> r. e. degrees are not elementarily equivalent

Jockusch, Li and Yang showed that the Lown and Low1 r.e. degrees are not elementarily equivalent for n > 1. We answer a question they raise by using the results of Nies, Shore and Slaman to show that the Lown and Lowm r.e. degrees are not elementarily equivalent for n > m > 1.     

Banach流形上映射度

本文讨论一种无限维流形上拓扑不变量，即具有可定向Ｆｒｅｄｈｏｌｍ结构的实Ｂａｎａｃｈ流形上　　Ｃｒ　　映射度．它是通常有限维流形上光滑映射度的一种自然推广．     

Degrees of presentability of structures. I

Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of Σ-definability are the main object of investigation. It is shown that the semilattice of degrees of Σ-definability of countable structures agrees well with semilattices of T-and e-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations. Supported by RFBR grant Nos. 05-0100481 and 06-0104002, by the Council for Grants (under RF President) for State Support of Young Candidates of Science and Their Supervisors via project MK-1239.2005.1, and via INTAS project YSF 04-83-3310. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 763–788, November–December, 2007.     

The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We say G satisfies the one-prime hypothesis if whenever a and b are distinct degrees in cd(G), then the greatest common divisor of a and b is either 1 or a prime. We show that if G is a solvable group satisfying the one-prime hypothesis, then |cd(G)|≤9. We also construct a solvable group G satisfying the one-prime hypothesis with |cd(G)|=9 which shows that the bound found in this paper is the best possible bound. Presented by D. Passman Mathematics Subject Classification (2000) 20C15.     

Modulo computably enumerable degrees by cupping partners

Cupping partners of an element in an upper semilattice with a greatest element 1 are those joining the element to 1. We define a congruence relation on such an upper semilattice by considering the elements having the same cupping partners as equivalent. It is interesting that this congruence relation induces a non-dense quotient structure of computably enumerable Turing degrees. Another main interesting phenomenon in this article is that on the computably enumerable degrees, this relation is different from that modulo the noncuppable ideal, though they define a same equivalent class for the computable Turing degree.     

The sum of the squares of degrees: Sharp asymptotics

Let f(n,m) be the maximum of the sum of the squares of degrees of a graph with n vertices and m edges. Summarizing earlier research, we present a concise, asymptotically sharp upper bound on f(n,m), better than the bound of de Caen for almost all n and m.     

Bipartite graphs with the maximum sum of squares of degrees

In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.     

Positions of characters in finite groups and the Taketa inequality

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.     

不可约特征标次数为等差数的有限可解群

设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2^e+1,2e+1,2^(e+1)},并给出了d>1时群的结构.