Tunnel numbers of small knots do not go down under connected sum

Let and be two knots in and , the tunnel numbers of them. In this paper, we show that if both and are small, then . Moreover we show that for any small knots .

     

On non-orientable surfaces in 4-space which are projected with at most one triple point

We show that if a non-orientable surface embedded in 4-space has a projection into 3-space with at most one triple point, then it is ambient isotopic to a connected sum of some unknotted projective planes and an embedded surface in 4-space with vanishing normal Euler number.

     

Band-sums are ribbon concordant to the connected sum

We show that an arbitrary band-connected sum of two or more knots are ribbon concordant to the connected sum of these knots. As an application we consider which knot can be a nontrivial band-connected sum.

     

Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers

We prove that a crossing change along a double point circle on a -knot is realized by ribbon-moves for a knotted torus obtained from the -knot by attaching a -handle. It follows that any -knots for which the crossing change is an unknotting operation, such as ribbon -knots and twist-spun knots, have trivial Khovanov-Jacobsson number.

     

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {X _i } _i=1^∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ _i=1 ⁿ X _i , and $\bar X_n = \tfrac{{S_n }} {n} $\bar X_n = \tfrac{{S_n }} {n} . And let N _n be the point process formed by the exceedances of random level $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n by X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].     

The 2-twist-spun trefoil has the triple point number four

The triple point number of an embedded surface in 4-space is the minimal number of the triple points on all the projection images into 3-space. We show that the 2-twist-spun trefoil has the triple point number four.

     

On the number of triple points of an immersed surface with boundary

Given a generic immersionf:S ¹→S ² of a circle into the sphere, we find the best possible lower estimation for the number of triple points of a generic immersionF: (M, S ¹)→(B ³,S ²) extendingf, whereM is an oriented surface with boundary ∂M=S ¹,B ³ is the 3-dimensional ball with boundaryS ². Supported by the Hungarian National Science and Research Foundation OTKA 2505 Supported by the Hungarian National Science and Research Foundation OTKA T4232 This article was processed by the author using the Springer-Verlag T_EX P Jour1g macro package 1991.     

On the triple points of singular maps

The number of triple points (mod 2) of a self-transverse immersion of a closed 2n-manifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9].? If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer- and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds. Received: October 12, 2001     

On the bilinear and cubic forms of some symplectic connected sums

Let ( M1 , M2 , N ) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M1 and M2 along their submanifold N to obtain M1#NM2 . In this paper, we consider the bilinear and cubic forms of H* (M1#NM2 , Z) when dim M1#NM2 = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M1#NM2 and M1■M2 .     

On the multidimensional distribution of the subset sum generator of pseudorandom numbers

We show that for a random choice of the parameters, the subset sum pseudorandom number generator produces a sequence of uniformly and independently distributed pseudorandom numbers. The result can be useful for both cryptographic and quasi-Monte Carlo applications and relies on bounds of exponential sums.

     

A degree sum condition for longest cycles in 3‐connected graphs

For a graph G, we denote by d_G(x) and κ(G) the degree of a vertex x in G and the connectivity of G, respectively. In this article, we show that if G is a 3‐connected graph of order n such that d_G(x) + d_G(y) + d_G(z) ≥ d for every independent set {x, y, z}, then G contains a cycle of length at least min{d ? κ(G), n}. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 277–283, 2007     

The intersection numbers of nearly Kirkman triple systems

In this paper,we investigate the intersection numbers of nearly Kirkman triple systems.J_N [v] is the set of all integers k such that there is a pair of NKTS(v)s with a common uncovered collection of 2-subset intersecting in k triples.It has been established that J_N[v]={0,1,...,v(v-2)/6-6,v(v-2)/6-4,v(v-2)/6} for any integers v ≡ 0(mod 6) and v≥66.For v≤60,there are 8 cases left undecided.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function forLaplace's equation in multiply connected circular domains arepresented. The method is constructive and relies on the useof a special function known as the Schottky–Klein primefunction associated with multiply connected circular domains.It is shown that all the important functions of potential theory,including the first-type Green's function, the modified Green'sfunctions and the harmonic measures of a domain, can be writtenin terms of this prime function. A broad range of representativeexamples are given to demonstrate the efficacy of the methodas well as a quantitative comparison, in special cases, withthe results obtained using other independent methods.     

Critical extreme points of the 2-edge connected spanning subgraph polytope

In this paper we study the extreme points of the polytope P(G), the linear relaxation of the 2-edge connected spanning subgraph polytope of a graph G. We introduce a partial ordering on the extreme points of P(G) and give necessary conditions for a non-integer extreme point of P(G) to be minimal with respect to that ordering. We show that, if is a non-integer minimal extreme point of P(G), then G and can be reduced, by means of some reduction operations, to a graph G' and an extreme point of P(G') where G' and satisfy some simple properties. As a consequence we obtain a characterization of the perfectly 2-edge connected graphs, the graphs for which the polytope P(G) is integral. Received: May, 2004 Part of this work has been done while the first author was visiting the Research Institute for Discrete Mathematics, University of Bonn, Germany.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down

In this paper, we show the following result: Let K _i be a knot in a closed orientable 3-manifold M _i such that (M _i ,K _i ) is not homeomorphic to (S ₂ ×S ₁, x ₀ ×S ₁), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K _i ) is less than the difference of one and twice of the tunnel number of K _i . Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive.