Showing papers on "Disjoint sets published in 1974"

Local Contributions to Global Deformations of Surfaces

Abstract: In [23], Schiffer and Spencer prove that all small deformations of complex structure on a compact Riemann surface may be realized by altering the complex structure only within an arbitrarily small neighborhood of a point on the surface. It seems interesting in general to consider whether it is possible to construct deformations of an algebraic variety or complex manifold from deformations of neighborhoods of certain subvarieties. Further motivation for trying to understand the role of subvarieties in deformations is suggested by the instability under deformation of the Neron-Severi group, i.e., the group of divisors modulo numerical equivalence. As an example, one may consider the family of affine surfaces Vt: x 2 + y 2 + z 2 = t z (t is a parameter; V o has a nodal singularity at the origin). This family admits a resolution {Xt} --, {V~}, with {X,} a smooth family of non-singular surfaces, and each X~ is a minimal resolution of V t ([4] or [5]). The exceptional curve E in X o is a IP 1 with self-intersection 2 which does not appear in any Xt, for t#:0. One may ask whether every smooth surface X with such a curve in it admits a one-parameter family of deformations arising from this local model. Furthermore, if X contains several disjoint such curves, does each one independently contribute one dimension to the moduli of X? The Hartogs ' theorem of [14] says that one cannot simply plumb in the local deformation, leaving the structure of X unchanged outside a small neighborhood of E. Moreover, old examples of Segre [26] show that the nodes on certain hypersurfaces V in IP 3 are not "independent", i.e., there aren't enough deformations of the resolution X of V to allow for a one-dimensional contribution from each node. Theorem (3.7) of this paper says that the regularity of the Kuranishi variety of X is sufficient for the deformations of X to realize independently

Triple points and surgery of immersed surfaces

Abstract: For a sufficiently general immersion of a smooth or polyhedral closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper involves elementary notions of modification of surfaces by surgery. Let f: M2 2E3 be an immersion in general position of a closed surface M2 into Euclidean 3-space so that there are finitely many N(f) triple points of /. The purpose of this note is to give a direct and elementary proof of the fact that N(f) x(M) (mod 2). The techniques used in this paper are related to the notion of surgery for surfaces, by which a pair of discs is replaced by a cylinder having the same boundary. The approach used in this paper was reported on at the American Mathematical Society Annual Meeting in Las Vegas, January 1972. With the added assumption of differentiability, it is possible to give a new proof of this same result using normal characteristic classes and singularities of projections, as in [l]. For an immersion /: M2 -. E3, set Gr(f) = lx in E3 f| -1(x) consists of precisely r points of M2}. The condition that f is in general position implies that if x e Gr(f) and f1(x) = tp1, P2,..., P,i then there are disjoint disc neighborhoods D(p1), D(p2),.9., D(pr) of these points in M and a homeomorphism w: B D3 of a ball neighborhood B of x to the unit ball in E3 so that (w of) (D(pi)) is the intersection of D3 with the plane orthogonal to the ith coordinate vector. If the immersion f is from a particular category, for example, differentiable or piecewise-linear, then we may assume that the homeomorphism w is in the same category, and in fact all the constructions which will be described can be altered in fairly standard ways so Presented to the Society January 18, 1972; received by the editors September 10, 1973. AMS(MOS) subject classfications(1970). Primary 57C35, 57D40, 57D65; Secondary 55A20.

A representation of closed, orientable 3-manifolds as 3-fold branched coverings of $S^3$

Abstract: In 1920, J. W. Alexander proved that, if M3 is a closed orientable three-dimensional manifold, then there exists a covering M3→S3 that branches over a link [same Bull. 26 (1919/20), 370–372; Jbuch 47, 529]. In the paper under review, the author proves a precision, piquant reformulation of Alexander's result: M3 is a closed orientable three-manifold, then there is a threefold irregular covering M3→S3 that branches over a knot; exactly two points of M3 cover each point of the singular set (the branching knot), one point with index of branching one; the other, with index of branching two. H. M. Hilden has independently proved the same theorem [ibid. 80 (1974), 1243–1244]. Suppose that g is the genus of M3, let both Xg and Xg′ denote a handlebody of genus g, and let φ:∂Xg→∂Xg′ be a homeomorphism for which Xg∪φXg′ is a Heegard splitting of M3. Let B and B′ both denote three-cells, and let A be a collection of g+2 disjoint arcs properly imbedded in B; let A′ be a similar collection of arcs in B′. Hilden constructs two irregular three-fold coverings p:Xg→B and p′:Xg′→B′; the covering p branches over A and the covering p′, over A′. The homeomorphism φ:∂Xg→∂Xg′ (or a homeomorphism isotopic to φ) projects to a homeomorphism γ:∂B→∂B′ such that γ(A∩∂B)=A′∩∂B′ and such that A∪γ|(A∩∂B)A′ is a knot in B∪γB′, the three-sphere. The branched covering we are seeking is p∪p′:Xg∪φXg′→B∪γB′. The author proves the theorem differently. Let L denote two unliked trivial knots, K1 and K2, in S3, and let Σ3 denote the symmetric group on {0,1,2}. The assignment of a meridian of Ki to the transposition (0i) (i=1,2) induces a representation π1(S3−L)→Σ3 and, thereby, a three-fold irregular covering, p:Σ3→S3, branched over L. The manifold Σ3 is S3, and p−1(Ki) contains exactly two curves, one with branching index one, the other, K˜i, with branching index two (i=1,2). Furthermore, the curves of p−1(Ki) are unknotted and unlinked. Now surgery on an appropriate μ-link L in Σ3 produces the manifold M3 [W. B. R. Lickorish, Ann. of Math. (2) 82 (1965), 414–420]. We can assume that each component of L cuts K˜1∪K˜2 in exactly two points, and we can find a second-regular neighborhood Vj for each component kj of L such that p(Vj) is a three-cell and such that p(Vj)∩L consists of two disjoint arcs (j=1,⋯,μ). Appropriate surgery on the solid tori V1,⋯,Vμ in Σ3 induces surgery on the corresponding three-cells p(V1),⋯,p(Vμ), and one obtains a three-fold, irregular covering M3→S3, branched over a link. Then, applying tools he developed in a previous paper [Rev. Mat. Hisp-Amer. (4) 32 (1972), 33–51], the author modifies the covering so that branching occurs over a knot.

On covering the points of a graph with point disjoint paths

Abstract: The minimum number of point disjoint paths which cover all the points of a graph defines a covering number denoted by ζ. The relation of ζ to some other well-known graphical invariants is discussed, and ζ is evaluated for a variety of special classes of graphs. A simple algorithm is developed for determining ζ in the case of a tree, and it is shown that this tree algorithm can be generalized to yield ζ for any connected graph. Degree conditions are also derived which yield simple upper bounds for ζ.

Enumeration of posets generated by disjoint unions and ordinal sums

Abstract: Let fn be the number of n-element posets which can be built up from a given collection of finite posets using the operations of disjoint union and ordinal sum. A curious functional equation is obtained for the generating function Z;fnxn. Using a result of Bender, an asymptotic estimate can sometimes be given for fn The analogous problem for labeled posets is also considered. Let P and Q be partially ordered sets (or posets). Regard P and Q as being relations on two disjoint sets T and T?, respectively. The disjoint union P + Q is defined to be the partial ordering on T U T' satisfying: (1) If x eT, y ET,and x

On Subcreative Sets and S-Reducibility

Abstract: : Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete. (Author)

Injective Banach spaces of typeC(T)

Abstract: A Banach spaceX is aPλ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aPλ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setTAsuch thatC(TA) is aPλ−1-space.

Edge-disjoint spanning trees, dominators, and depth-first search.

Abstract: This paper presents an algorithm for finding two edge-disjoint spanning trees rooted at a fixed vertex of a directed graph. The algorithm uses depth-first search, an efficient method for computing disjoint set unions, and an efficient method for computing dominators. It requires O(V log V + E) time and O(V + E) space to analyze a graph with V vertices and E edges.

On the Complexity of Strictly Nonblocking Concentration Networks

Abstract: A concentration network is a contact switching network that provides a number of potential users (connected to its inputs) with access to a smaller number of equivalent resources (connected to its outputs). Its basic property is that any sufficiently small subset of the inputs can be simultaneously connected by disjoint paths to distinct outputs, although the particular outputs to which they are to be connected cannot, in general, be specified arbitrarily. We show that a strictly nonblocking concentration network must have at least 3n \log_{3} n - O(n) contacts where n is the number of connections to be established simultaneously.

Ultrafilters and almost disjoint sets

Abstract: Let μκ denote the space of uniform ultrafilters on κ, and let λ be a cardinal. U ϵμκ is said to be a λ-point of μκ if U is a boundary point of λ pairwise disjoint open subsets of μκ. We prove that if κ is a successor cardinal, 2 κ = κ + , and Kurepa's hypothesis for κ holds, then each U ϵ μκ is a 2 κ -point of μκ.

Disjoint Representation of Tree Realizable Sequences

Abstract: A necessary and sufficient condition is obtained for the edge disjoint tree realization of two degree sequences each of which is tree realizable. The condition is that the sum sequence be graphical.

Properties of boolean functions with a tree decomposition

Abstract: Boolean functions that have a multiple disjoint decomposition scheme in the form of a tree are considered. Properties of such functions are given for the case that the functions are increasing, unate, and/or have no vacuous variables. The functions with a binary decomposition scheme are of special interest. The modulus of sensitivity is defined, and evaluated for some classes of functions. The modulus of sensitivity is interesting from the point of view of semantic information processing. It is found that the sensitivity for the class of functions with a given disjoint binary decomposition scheme is much smaller than for the unrestricted class of boolean functions. This indicates that these functions are potentially useful in pattern recognition of discrete data.

connectivity and mappings onto a chainable indecomposable continuum

Abstract: A continuum (i.e., a compact connected nondegenerate metric space) M is said to be X connected if any two of its points can be joined by a hereditarily decomposable continuum in M. Here we prove that a plane continuum is X connected if and only if it cannot be mapped continuously onto Knaster's chainable indecomposable continuum with one endpoint. Recent results involving aposyndesis and decompositions to a simple closed curve are extended to X connected continua. Throughout this paper D will denote Knaster's chainable indecomposable continuum with one endpoint (see [7, p. 332] or [9, Example 19 p. 205]), I will denote the unit interval, and h will denote the function of I onto itself defined by h(t) = 2t for t '2. D can be represented as an inverse limit of unit intervals, indexed by the positive integers, where the bonding map between successive terms is always h. In [10] J. W. Rogers, Jr. proved that every indecomposable continuum can be. mapped continuously onto D. Recently D. P. Bellamy [1] generalized this theorem by showing that D is a continuous image of each indecomposable compact connected nondegenerate Hausdorff space. Our principal tool (presented in the foll6wing theorem) is derived from Bellamy's proof. Theorem 1. Suppose that M is a continuum and GnIn_= is a sequence of nonempty open sets in M such that (1) the closures of G1 and G2 are disjoint, (2) for each n, G2n+1 U G2n+2 C G2n-1. and (3) for each n, there is a separation An U Bn of M -'G2n such that G2n+1 C An and G 2n+2 C BnThen M can be mapped continuously onto D. Proof. Following Bellamy [19 Theorem (proof)], we let f, be a Urysohn Presented to the Society, August 23, 1973 under the title X connected continua; received by the editors August 6, 1973. AMS (MOS) subject classifications (1970). Primary 54C05, 54F20, 54F25, 54F60; Secondary 54F 15, 54A05.

Polynomial invariants of boundary links

Abstract: An m-link is a (smooth, polygonal) embedding l : mS 1 →S 3 of the disjoint (1) union of m circles S 1 1 +... S 1 m into S 3 .

Decomposition theorems for $3$-convex subsets of the plane.

Abstract: Let S be a 3-convex subset of the plane. If (cl S ~ £>) S int (cl S) or if (cl S ~ S) g bdry (cl S), then S is expressible as a union of four or fewer convex sets. Otherwise, S is a union of six or fewer convex sets. In each case, the bound is best possible. 1* Introduction. Let S be a subset of Rd. Then S is said to be 3-convex iff for every three distinct points in S, at least one of the segments determined by these points lies in S. Valentine [2] has proved that for S a closed, 3-convex subset of the plane, S is expressible as a union of three or fewer closed convex sets. We are interested in obtaining a similar decomposition without requiring the set S to be closed. The following definitions and results obtained by Valentine will be useful. For S S Rd> a point x in S is a point of local convexity of S iff there is some neighborhood U of x such that, if y, zeSCiU, then [y, A S S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S. Let S be a closed, connected, 3-convex subset of the plane, and let Q denote the closure of the set of isolated lnc points of S. Valentine has proved that for S not convex, then card Q ^ 1, Q lies in the convex kernel of S, and Q § bdry (conv Q). An edge of bdry (conv Q) is a closed segment (or ray) in bdry (conv Q) whose endpoints are in Q. We define a leaf of S in the following manner: In case card Q ^ 3, let L be the line determined by an edge of bdry (conv Q), Llf L2 the corresponding open half spaces. Then L supports conv Q, and we may assume conv Q § cl (LJ. We define W — cl (L2 f] S) to be a leaf of S. For 2 ^ card Q ^ 1, constructions used by Valentine may be employed to decompose S into two closed convex sets, and we define each of these convex sets to be a leaf of S. By Valentine's results, every point of S is either in conv Q or in some leaf W of S (or both), and every leaf W is convex. Moreover, Valentine obtains his decomposition of S by showing that for any collection {sj of disjoint edges of bdry (conv Q), with {Wt} the corresponding collection of leaves, conv Q U (U Wt) is closed and convex. Finally, we will use the following familiar definitions: For x, y in S, we say x see y via S iff the corresponding segment [x, y] lies in S. A subset T of S is visually independent via S iff for every