Coherence, Homotopy and 2-Theories

2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories induces a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence.     

Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.     

Categorical non-abelian cohomology and the Schreier theory of groupoids

By regarding the classical non-abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non-abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids, cocycles are lax 2-functors, and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand, group extensions are fibrations of categories. Furthermore, n-simplices in the nerve of a 2-category are lax 2-functors.     

Algebraic classification of physical structures with zero. II. Topological aspects

Some topological analogs are obtained of the author’s previous results on the classification of physical structures. A topological-algebraic axiomatics is considered enabling us to replace the algebraic axiom corresponding to the main equation of a physical structure by a more natural axiom. A physical structure of rank different from (2, 2) is shown to be a pair of vector spaces with a nondegenerate bilinear form over a topological skew field. The obtained results are applied to the classification of physical structures of rank different from (2, 2). The structures of rank (2, 2) are also considered. To describe them, a topological group structure corresponding to the biform action is introduced on the set of measurements.     

Bifurcation of periodic solutions of a system close to a Lyapunov system

Bifurcation of 2π-periodic solutions (2π-ps) of a system of second-order differential equations close to a Lyapunov system is investigated. The case of principal resonance, when an eigenfrequency of the linear oscillations of the unperturbed system is close to the frequency of the perturbing impulse, is considered. It is shown that, at certain values of the problem parameters, bifurcation of the 2π-ps that are generated from an equilibrium position, occurs. A constructive method is proposed for finding the bifurcation curve, as well as 2π-ps on it. The examples considered are bifurcation of 2π-ps in the problem of the oscillations of a mathematical pendulum with a horizontally vibrating suspension point, and in the problem of the planar oscillations of an artificial satellite in a weakly elliptical orbit. The bifurcation curves for these examples are constructed and the corresponding 2π-ps are found.     

A novel 2D partial unwinding adaptive Fourier decomposition method with application to frequency domain system identification

This paper proposes a two‐dimensional (2D) partial unwinding adaptive Fourier decomposition method to identify 2D system functions. Starting from Coifman in 2000, one‐dimensional (1D) unwinding adaptive Fourier decomposition and later a type called unwinding AFD have been being studied. They are based on the Nevanlinna factorization and a maximal selection. This method provides fast‐converging rational approximations to 1D system functions. However, in the 2D case, there is no genuine unwinding decomposition. This paper proposes a 2D partial unwinding adaptive Fourier decomposition algorithm that is based on algebraic transforms reducing a 2D case to the 1D case. The proposed algorithm enables rational approximations of real coefficients to 2D system functions of real coefficients. Its fast convergence offers efficient system identification. Numerical experiments are provided, and the advantages of the proposed method are demonstrated.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case

A lattice graph with 2–3 reachability constraints is considered. The graph’s vertices are the points with integer nonnegative coordinates in the plane. Each vertex has two outgoing edges, one entering its immediate right neighbor and the other entering its immediate upper neighbor. The admissible paths for 2–3 reachability are those in which the numbers of edges in all but the last inclusion-maximal straight-line segments are divisible by 2 for horizontal segments and by 3 for vertical segments. A formula for the number of 2–3 paths from a vertex to a vertex is obtained. A random walk process on the 2–3 paths in the lattice graph is considered. It is proved that this process can locally be reduced to a Markov process on subgraphs determined by the type of the initial vertex. Formulas for the probabilities of vertex-to-vertex transitions along 2–3 paths are obtained.     

Pseudo-commutativity of KZ 2-monads

In this paper we prove that KZ 2-monads (also known as lax-idempotent 2-monads) are pseudo-commutative. The main examples of KZ 2-monads for us will be 2-monads whose algebras are V-categories with chosen colimits of a given class; this provides a large family of examples of pseudo-commutative 2-monads. In order to achieve this we characterise pseudo-commutativities on a 2-monad in terms of extra structure on its 2-category of algebras and pseudomorphisms. We also consider tensor products associated to pseudo-closed structures and show some results on preservation of colimits. To cover the general case of V-enriched categories and not only ordinary categories we are led to consider monads enriched in a 2-category, and some of the associated two-dimensional monad theory.     

2-nerves for bicategories

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N : NHom → [Δ^op,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories. The hospitality of Macquarie University and the support of the Australian Research Council are gratefully acknowledged by S. Lack. The support of the Australian Research Council is gratefully acknowledged by S. Paoli.     

On 2-Adjacency Between Links

The author discusses 2-adjacency of two-component links and study the relations between the signs of the crossings to realize 2-adjacency and the coefficients of the Conway polynomial of two related links. By discussing the coefficient of the lowest $m$ power in the Homfly polynomial, the author obtains some results and conditions on whether the trivial link is 2-adjacent to a nontrivial link, whether there are two links 2-adjacent to each other, etc. Finally, this paper shows that the Whitehead link is not 2-adjacent to the trivial link, and gives some examples to explain that for any given two-component link, there are infinitely many links 2-adjacent to it. In particular, there are infinitely many links 2-adjacent to it with the same Conway polynomial.     

二元语义Bonferroni集成算子及其在多属性群决策中的应用

Bonferroni平均算子的优点是在进行信息集成的过程中能够考虑到输入变量之间的相互影响。本文首先将其推广到语言环境中,提出了二元语义Bonferroni平均(2TLBA)算子、加权2TLBA(W2TLBA)算子和组合W2TLBA(C-W2TLBA)算子的概念,研究它们的相关性质。然后针对决策信息以语言变量给出的多属性群决策问题,提出一种基于W2TLBA算子和C-W2TLBA算子的群决策方法。最后应用实例说明了该方法是切实可行的。     

A model of a 2-player stopping game with priority and asynchronous observation

Various models of 2-player stopping games have been considered which assume that players simultaneously observe a sequence of objects. Nash equilibria for such games can be found by first solving the optimal stopping problems arising when one player remains and then defining by recursion the normal form of the game played at each stage when both players are still searching (a 2 × 2 matrix game). The model considered here assumes that Player 1 always observes an object before Player 2. If Player 1 accepts the object, then Player 2 does not see that object. If Player 1 rejects an object, then Player 2 observes it and may choose to accept or reject it. It is shown that such a game can be solved using recursion by solving appropriately defined subgames, which are played at each moment when both players are still searching. In these subgames Player 1 chooses a threshold, such that an object is accepted iff its value is above this threshold. The strategy of Player 2 in this subgame is a stopping rule to be used when Player 1 accepts this object, together with a threshold to be used when Player 1 rejects the object. Whenever the payoff of Player 1 does not depend on the value of the object taken by Player 2, such a game can be treated as two optimisation problems. Two examples are given to illustrate these approaches.     

On perfectly two-edge connected graphs

This paper studies the graphs for which the 2-edge connected spanning subgraph polytope is completely described by the trivial inequalities and the so-called cut inequalities. These graphs are called perfectly 2-edge connected. The class of perfectly 2-edge connected graphs contains for instance the class of series-parallel graphs. We introduce a new class of perfectly 2-edge connected graphs. We discuss some structural properties of graphs which are (minimally with respect to some reduction operations) nonperfectly 2-edge connected. Using this we give sufficient conditions for a graph to be perfectly 2-edge connected.     

一种多指标质量动态控制图及其应用

研究了将多指标在一个图上标示的多指标控制图。提出了三种质量控制限并作了比较。控制上限(UCL)取Hotelling的T2统计量的α/2上侧分位数。预控上限(UPCL)取Hotelling的T2统计量的λα/2上侧分位数。当第k个观测点的T2k值超过UPCL时,则接下去的抽样区间将缩短,以便及早发现失控状态。对超出控制限的情况下如何确定哪一个质量指标发生异常的问题,提出了一种可靠的分析方法。用这种方法解决了某卷烟厂的烟丝生产过程中的多指标质量控制和管理问题。     

Peakon Soliton Solutions Of K(2,-2,4) Equation

In this paper, the qualitative analysis methods of a dynamical system are used to investigate the peakon soliton solutions of K(2;-2; 4) equation: ut + a(u2)x + b[u-2(u4)xx]x = 0. The phase portrait bifurcation of the traveling wave system corresponding to the equation is given. The explicit expressions of the peakon soliton solutions are obtained by using the portraits. The graph of the solutions are given with the numerical simulation. This supplements the results obtained in [4].     

Two Necessary Conditions for λ 2 (G) = d 2 ( G )

In this article, two necessary conditions for a graph to have an equality of the second largest Laplacian eigenvalue λ 2 ( G ) and its lower bound d 2 ( G ) are presented. Some of the graphs who satisfy λ 2 ( G )= d 2 ( G ) are also described.     

A Regularization of Quantum Field Hamiltonians with the Aid of p–adic Numbers

Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L₂-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L₂-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L₂-spaces with respect to a p-adic Gaussian distribution.     

Off-day scheduling with hierarchical worker categories

A work force includes workers of m types. The worker categories are ordered, with type-1 workers the most highly qualified, type-2 the next, and so on. If the need arises, a type-k worker is able to substitute for a worker of any type j greater than k (k = 1, ..., m - 1). For 7-day-a-week operation, daily requirements are for at least Dk workers of type-k or better, of which at least dk must be precisely type-k. Formulas are given to find the smallest number and most economical mix of workers, assuming that each worker must have 2 off-days per week and a given fraction of weekends off. Algorithms are presented which generate a feasible schedule, and provide work stretches between 2 and 5 days, and consecutive weekdays off when on duty for 2 weekends in a row, without additional staff.