Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

Graph Minors. XX. Wagner''s conjecture

In a previous paper (J. Combin. Theory 48 (1990) 255) we showed that for any infinite set of (finite) graphs drawn in a fixed surface, one of the graphs is isomorphic to a minor of another. In this paper we extend that result in two ways:

• we generalize from graphs to hypergraphs drawn in a fixed surface, in which each edge has two or three ends, and

• the edges of our hypergraphs are labelled from a well-quasi-order, and the minor relation is required to respect this order.

This result is another step in the proof of Wagner's conjecture, that for any infinite set of graphs, one is isomorphic to a minor of another.     

Duals Invert

Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229–260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229–260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733–742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25–47, 2008) asserting that, for objects A and X in a cartesian bicategory , if A is Frobenius then the category Map(X,A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25–47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184–190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory.     

网格状有向图上的部分合作对策

本文通过在有向图上每个状态结点处定义合作函数,运用Berge C的关于图匕对策中策略的概念,在网格状有向图上考察部分合作动态对策.局中人在对策进程中将采取部分合作而不是完全合作,部分合作的主要特征是每个局中人的行为是合作行动与单独行动的组合.本文合作函数的设定允许局中人加入某个联盟之后再脱离该联盟,同时给出了有向图上部分合作对策的值、最优路径的算法及示例.     

Duals of Frame Sequences

Frames provide unconditional basis-like, but generally nonunique, representations of vectors in a Hilbert space H. The redundancy of frame expansions allows the flexibility of choosing different dual sequences to employ in frame representations. In particular, oblique duals, Type I duals, and Type II duals have been introduced in the literature because of the special properties that they possess. A Type I dual is a dual such that the range of its synthesis operator is contained in the range of the synthesis operator of the original frame sequence, and a Type II dual is a dual such that the range of its analysis operator is contained in the range of the analysis operator of the original frame sequence. This paper proves that all Type I and Type II duals are oblique duals, but not conversely, and characterizes the existence of oblique and Type II duals in terms of direct sum decompositions of H, as well as characterizing when the Type I, Type II, and oblique duals will be unique. These results are also applied to the case of shift-generated sequences that are frames for shift-invariant subspaces of L ²(? ^d ).     

Models with Network Duals

It is pointed out that a number of practical problems can be formulated as linear programmes whose duals are network flow models. The interpretation of these networks in relation to the original problem can be highly illuminating. If, as is frequently the case, the objective of the original problem is monetary, the network dual concerns the optimal pattern of re-allocation of money. It is suggested that the conceptual simplicity of a network makes this an attractive way to view problems as well as indicating efficient computational procedures.     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here. Received 8 July 1997; in revised form 17 November 1997     

On the Duals of Segre Varieties

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M.M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?     

Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A new class of alternative dual frames is introduced in the setting of finite frames for ? ^d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order $\mathcal{O}(N^{-r})$ for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.     

Duals and envelopes of some Hardy-Lorentz spaces

For we describe the dual spaces and Banach envelopes of the spaces for finite values of and for , the closure of the polynomials in . In addition, we determine the -Banach envelopes for the spaces in the cases and .

     

Volterra-like Banach Algebras and their Second Duals

 This paper presents and studies a class of algebras which includes the usual Volterra algebra. Roughly speaking, they relate to the Volterra algebra in the way a general locally compact group relates to ℝ. We show that they can be viewed as quotients of some semigroup algebras introduced by Baker and Baker [1]. Their sets of nilpotent elements are dense. We investigate the second duals of these algebras and find that most of the properties found in [7] for the biduals of the group algebras L ¹(G) for compact G are retained here.     

Ribbon tile invariants

Let be a finite set of tiles, and a set of regions tileable by . We introduce a tile counting group as a group of all linear relations for the number of times each tile can occur in a tiling of a region . We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group.

The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group.

The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.

     

Grothendieck Spaces and Duals of Injective Tensor Products

Let E and F be Fréchet spaces. We prove that if E isreflexive, then the strong bidual is a topological subspace of . We also prove that if, moreover, E is Montel and F has the Grothendieckproperty, then has the Grothendieck property whenever either E or has the approximation property. A similar result is obtainedfor the property of containing no complemented copy of c₀.