Self delta-equivalence of cobordant links

Self -equivalence is an equivalence relation for links, which is stronger than the link-homotopy defined by J. Milnor. It is known that cobordant links are link-homotopic and that they are not necessarily self -equivalent. In this paper, we will give a sufficient condition for cobordant links to be self -equivalent.

     

On the torus cobordant cohomology spheres

Let G be a compact Lie group. In 1960, P A Smith asked the following question: “Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?” A result due to Atiyah and Bott proves that the answer is ‘yes’ for ? _p and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on S ⁿ which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ?-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.     

Studying links via closed braids IV: composite links and split links

Summary The main result concerns changing an arbitrary closed braid representative of a split or composite link to one which is obviously recognizable as being split or composite. Exchange moves are introduced; they change the conjugacy class of a closed braid without changing its link type or its braid index. A closed braid representative of a composite (respectively split) link is composite (split) if there is a 2-sphere which realizes the connected sum decomposition (splitting) and meets the braid axis in 2 points. It is proved that exchange moves are the only obstruction to representing composite or split links by composite or split closed braids. A special version of these theorems holds for 3 and 4 braids, answering a question of H. Morton. As an immediate Corollary, it follows that braid index is additive (resp. additive minus 1) under disjoint union (resp. connected sum).Oblatum 29-XI-1988 & 25-I-1990Partially supported by NSF Grant #DMS-88-05672     

To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179-189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.     

Random groups do not split

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on simplicial trees. This implies that their Gromov boundaries, defined at density less than \frac12{\frac{1}{2}} , are Menger curves.     

On F-almost split sequences

Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End（T）. Then Horn（-, T） induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧（f（F）, T）. Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌（ΩCM^-dΩ^dDTrA^＊）^＊,where（-）^＊=Hom（g,T）.In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC.     

On split Lie triple systems

We begin the study of arbitrary split Lie triple systems by focussing on those with a coherent 0-root space. We show that any such triple systems T with a symmetric root system is of the form T = + Μ _j I _j with a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. Under certain conditions, it is shown that T is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. The key tool in this job is the notion of connection of roots in the framework of split Lie triple systems.     

On alternation numbers of links

We construct infinitely many hyperbolic links with x-distance far from the set of (possibly, splittable) alternating links in the concordance class of every link. A sensitive result is given for the concordance class of every (possibly, split) alternating link. Our proof uses an estimate of the τ-distance by an Alexander invariant and the topological imitation theory, both established earlier by the author.     

On geometric interpretations of split quaternions

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).     

On automorphisms of split metacyclic groups

Let D(m, n; k) be the semi-direct product of two finite cyclic groups and , where the action is given by yxy ⁻¹  =  x ^k . In particular, this includes the dihedral groups D _2m . We calculate the automorphism group Aut (D(m, n; k)).     

On split Lie triple systems II

In [4] it is studied that the structure of split Lie triple systems with a coherent 0-root space, that is, satisfying [T ₀, T ₀, T] = 0 and [T ₀, T _α , T ₀] ≠ 0 for any nonzero root α and where T ₀ denotes the 0-root space and T _α the α-root space, by showing that any of such triple systems T with a symmetric root system is of the form T = U + Σ _j I _j with U a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. It is also shown in [4] that under certain conditions, a split Lie triple system with a coherent 0-root space is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. In the present paper we extend these results to arbitrary split Lie triple systems with no restrictions on their 0-root spaces.     

On split common fixed point problems

Based on the convergence theorem recently proved by the second author, we modify the iterative scheme studied by Moudafi for quasi-nonexpansive operators to obtain strong convergence to a solution of the split common fixed point problem. It is noted that Moudafi's original scheme can conclude only weak convergence. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators, split common null point problems for maximal monotone operators, and Moudafi's split feasibility problem.     

On flat braidzel surfaces for links

Rudolph introduced the notion of braidzel surfaces as a generalization of pretzel surfaces, and Nakamura showed that any oriented link has a braidzel surface. In this paper, we introduce the notion of flat braidzel surfaces as a special kind of braidzel surfaces, and show that any oriented link has a flat braidzel surface. We also introduce and study a new integral invariant of links, named the flat braidzel genus, with respect to their flat braidzel surfaces. Moreover, we give a way to calculate the number of components, the distance from proper links, the Arf invariant, and a Seifert matrix of a given link through the flat braidzel notation.