Non-additivity for triple point numbers on the connected sum of surface-knots

Any surface-knot in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, , is defined similarly to the crossing number of a classical knot. By definition, we have for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold.

     

Optimal filter and mollifier for piecewise smooth spectral data

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious Gibbs' oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal joint time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.

     

Links and cubic 3-polytopes

It is well known that a prime link diagram corresponds to a signed plane graph without cut vertices (Kauffman, 1989). In this paper, we present a new relation between prime links and cubic 3-polytopes. Let be the set of links such that each has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except -torus links and -pretzel links, can be obtained from by using some operation of untwining. Furthermore, we define the generalized cubic 3-polytope chains and then show that any nontrivial link can be obtained from by some untwining operations, where is the set of links corresponding to generalized cubic 3-polytope chains. These results are used to simplify the computation of the Kauffman brackets of links so that the computing can be done in a unified way for many infinite families of links.

     

Generalized Seifert surfaces and signatures of colored links

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in . This yields an integral valued function on the -dimensional torus, where is the number of colors of the link. The case corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this -variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a -dimensional interpretation and the Atiyah-Singer -signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.

     

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.

     

Viscosity solutions, almost everywhere solutions and explicit formulas

Consider the differential inclusion in . We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.

     

Morse theory for the Yang-Mills functional via equivariant homotopy theory

In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds with generic -invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337-356 and Råde in Compactness theorems for invariant connections, submitted for publication.     

Rational Gauss-Chebyshev quadrature formulas for complex poles outside

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside to arbitrary complex poles outside . The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside .

     

Complete nonorientable minimal surfaces in a ball of

The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.

     

The modal logic of forcing

A set theoretical assertion is forceable or possible, written , if holds in some forcing extension, and necessary, written , if holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory .

     

A uniqueness result concerning Schur ideals

A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of -dimensional complex space is said to be if it arises as the set of possible values arising from restricting contractive elements from some uniform algebra to a finite set in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.

     

Global existence of solutions to a hyperbolic-parabolic system

In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get -estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and a priori assumptions.

     

Quandle cohomology and state-sum invariants of knotted curves and surfaces

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).

A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in -space and knotted surfaces in -space.

Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

Robust a-posteriori estimator for advection-diffusion-reaction problems

We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem.

The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a factor, where is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature.

We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

     

Dual decompositions of 4-manifolds II: Linear invariants

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index . Part I (Trans. Amer. Math. Soc. 354 (2002), 1373-1392) gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles, this theory can be thought of as giving ``ambient' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type' invariants in the sense of Vassilaev.

     

A new approach to relatively nonexpansive mappings

In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets of a Banach space is a mapping defined from into such that for and . These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk's Fixed Point Theorem. In this work we show that, for certain proximinal pairs , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.

     

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.

     

Effectively dense Boolean algebras and their applications

A computably enumerable Boolean algebra is effectively dense if for each we can effectively determine an such that implies . We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals of subrecursive degree structures, where is the degree of a set of relatively small complexity, for instance a set in exponential time.