A Relation in the Stable Homotopy Groups of Spheres

Let p≥7 be an odd prime. Based on the Toda bracket α1βp-11, α1 β1, p, γs,the authors show that the relation α1βp-11h2,0 γs= βp/p-1γs holds. As a result, they can obtain α1βp1h2,0 γs = 0 ∈π*(S0) for 2≤s≤p- 2, even though α1h2,0γs and β1α1h2,0 γs are not trivial. They also prove that βp-11α1 h2,0 γ3 is nontrivial in π*(S0) and conjecture that βp-11α1 h2,0 γs is nontrivial in π*(S0) for 3≤s≤p- 2. Moreover, it is known thatβp/p-1γ3 = 0 ∈ Ext5,*BP*BP(BP*, BP*), but βp/p-1γ3 is nontrivial in π*(S0) and represents the element βp-11α1 h2,0 γ3.     

Mixing Homomorphisms,Recolorings, and Extending Circular Precolorings

This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the ‐colorings of a graph G can be obtained by successively recoloring a single vertex provided along the lines of Cereceda, van den Heuvel, and Johnson's result for k‐colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson‐type extension theorem for ‐precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.     

Homotopy groups of Hom complexes of graphs

The notion of ×-homotopy from [Anton Dochtermann, Hom complexes and homotopy theory in the category of graphs, European J. Combin., in press] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom_∗(G,H) with the homotopy groups of Hom_∗(G,HI). Here Hom_∗(G,H) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and HI is the graph of based paths in H. As a corollary it is shown that πi(Hom_∗(G,H))≅_×[G,ΩiH], where ΩH is the graph of based closed paths in H and _×[G,K] is the set of ×-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of [Eric Babson, Hélène Barcelo, Mark de Longueville, Reinhard Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (1) (2006) 31-44], where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.     

Absolute Lax 2-categories

We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ . Here we show that, when has a -topology, this structure can be embedded into a larger one, with the same objects (the points of ), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.Work supported by MIUR Research Projects.     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 John Wiley & Sons, Ltd.     

TANGENT UNIT-VECTOR FIELDS:NONABELIAN HOMOTOPY INVARIANTS AND THE DIRICHLET ENERGY

Let O be a closed geodesic polygon in S~2 . Maps from O into S~2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2,we compute the infimum Dirichlet energy E(H) for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H . The expression for E (H ) involves a topological invariant - the spelling length - associated with the (non-abelian) fundamental group of the n-times punctured two-sphere, π 1 (S 2 ? {s 1 , ··· , s n }, ?). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representa- tives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly con- formal and anticonformal representatives respectively), the expression for E (H ) reduces to a previous result involving the degrees of a set of regular values s 1 , ··· , s n in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π 1 (S 2 ? {s 1 , ··· , s n }, *). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit- vector fields in a rectangular prism.     

Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions

In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

非均匀圆柱壳中非线性波传播模型的同伦分析解法

圆柱形壳中非线性波传播问题的研究具有重要的理论和现实意义.本文用同伦分析方法,研究了描述非均匀圆柱壳中非线性波传播的一种新模型,得到了该模型的高精度近似孤波解和周期波解.这表明同伦分析方法是研究非线性波传播问题的一种十分有效的方法.     

On fixed point sets of distinguished collections for groups of parabolic characteristic

We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail.