On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections

It is shown that if a closed smooth orientable manifold M ⁿ , n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.     

The recurrence and the gradient-like structure of a flow

In this paper, some new properties of the nonwandering set, the partial order set, the chain recurrent set and the generalized recurrent set of a flow are proved. A new characterization of the gradient-like part of a flow is given. Some results of Conley are generalized.     

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Interaction information and Markov chain

The problem of multivariate information analysis is considered. First, the interaction information in each dimension is defined analogously according to McGill [4] and then applied to Markov chains. The property of interaction information zero deeply relates to a certain class of weakly dependent random variables. For homogeneous, recurrent Markov chains with m states, m ≥ n ≥3, the zero criterion of n-dimensional interaction information is achieved only by (n ? 2)-dependent Markov chains, which are generated by some nilpotent matrices. Further for Gaussian Markov chains, it gives the decomposition rule of the variables into mutually correlated subchains.     

A generalization of the implicit function theorem for mappings from Rn + 1 into Rn and its applications

Some regularity results about the local structure of the zero set of a mapping of n + 1 real variables with values in $\text{R}$ⁿ are presented. They all provide improved and generalized versions of the Morse lemma for plane curves as well as generalizations of the implicit function theorem. We describe their applications to one-parameter nonlinear problems, including an analysis of bifurcation phenomena at simple or multiple eigenvalues.     

3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.

     

A Note on Equilibrium Problems with Properly Quasimonotone Bifunctions

In this paper, we consider some well-known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on K x K,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.     

Computing the generalized eigenvalues of weakly symmetric tensors

Tensor is a hot topic in the past decade and eigenvalue problems of higher order tensors become more and more important in the numerical multilinear algebra. Several methods for finding the Z-eigenvalues and generalized eigenvalues of symmetric tensors have been given. However, the convergence of these methods when the tensor is not symmetric but weakly symmetric is not assured. In this paper, we give two convergent gradient projection methods for computing some generalized eigenvalues of weakly symmetric tensors. The gradient projection method with Armijo step-size rule (AGP) can be viewed as a modification of the GEAP method. The spectral gradient projection method which is born from the combination of the BB method with the gradient projection method is superior to the GEAP, AG and AGP methods. We also make comparisons among the four methods. Some competitive numerical results are reported at the end of this paper.     

Representations of epi-Lipschitzian sets

A closed subset M of a Banach space E is epi-Lipschitzian, i.e., can be represented locally as the epigraph of a Lipschitz function, if and only if it is the level set of some locally Lipschitz function f:E→R, for which Clarke’s generalized gradient does not contain 0 at points in the boundary of M, i.e., such that:

     

Bounds on the indices of the spectral-circle eigenvalues of a nonnegative matrix

Let P be a square, nonnegative matrix. A set of k classes of P is called a chain of length k if every class in the set has access to or from any other class in the set. A chain is called a λ-chain if λ is in the spectrum of each submatrix of P corresponding to a class in the chain. Our main result states that the index of a complex number λ lying on the spectral circle of P is bounded above by the length of the longest λ-chain. We illustrate, by examples, that this bound can be attained but could sometimes be arbitrarily poor.     

A variable selection method based on Tabu search for logistic regression models

A Tabu search method is proposed and analysed for selecting variables that are subsequently used in Logistic Regression Models. The aim is to find from among a set of m variables a smaller subset which enables the efficient classification of cases. Reducing dimensionality has some very well-known advantages that are summarized in literature. The specific problem consists in finding, for a small integer value of p, a subset of size p of the original set of variables that yields the greatest percentage of hits in Logistic Regression. The proposed Tabu search method performs a deep search in the solution space that alternates between a basic phase (that uses simple moves) and a diversification phase (to explore regions not previously visited). Testing shows that it obtains significantly better results than the Stepwise, Backward or Forward methods used by classic statistical packages. Some results of applying these methods are presented.     

Morse and semistable stratifications of K?hler spacesby \Bbb C*{\Bbb C}^{\ast}-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \Bbb C^*{\Bbb C}^{\ast} -K?hler manifold coincide if the moment map is proper and if the fixed points set X^{\Bbb C*}X^{{\Bbb C}^{\ast}} has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.     

On the sum-of-ranks winner when losers are removed

Suppose that m alternatives are linearly ranked from best to worst by each of a number of judges, and that alternative x is the unique winner on the sum-of-ranks basis. It is shown that it is possible to construct a situation (with an appropriate number of judges) such that the initial winner x will be a sum-of-ranks loser within every proper subset of the original set of alternatives that contains x and at least one other alternative, except that x is the winner in exactly one subset that contains x and one other alternative.     

Common fixed point results for two new classes of hybrid pairs in symmetric spaces

Some common fixed point theorems due to Abbas and Khan [M. Abbas, A.R. Khan, Common fixed points of generalized contractive hybrid pairs in symmetric spaces, Fixed Point Theor. Appl. 2009 (2009) 11, Article ID 869407, doi:10.1155/2009/869407], and Abbas and Rhoades [M. Abbas, B.E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings defined on symmetric spaces, Pan. Amer. Math. J. 18 (1) (2008) 55-62] are proved for two new classes of hybrid pair of mappings which contain occasionally weakly compatible hybrid pairs as a proper subclass. Consequently, some results proved by Hussain et al. [N. Hussain, M.A. Khamsi, A. Latif, Common fixed points for JH-operators and occasionally weakly biased pairs under relaxed conditions, Nonlinear Anal. 74 (2011) 2133-2140], Bhatt et al. [A. Bhatt, et al., Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions, Nonlinear Anal. 73 (2010) 176-182] and many others are extended to hybrid pair of mappings. Examples are also presented to support the concepts defined in the paper.     

Largest disk of stability of explicit Runge-Kutta methods

The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [–2m, 0] as a proper subset.     

Topological vector spaces problems in homology and homotopy theories

Three data are interesting here: domains of integration, integrands and integration itself. There is a lack of symmetry between polyhedral chains as domains of integration and differential forms as integrands. The non-symmetric situation disappears after considering the topological spaces of the de Rham differential forms and forms with compact supports and their strong duals, i.e., currents with compact supports and currents, respectively. This idea goes back to Schwartz distributions and Schwartz distributions with compact supports, in other terminology, generalized functions and generalized functions with compact supports.Some problems are raised, e.g., whether every quasi-complete barreled nuclear space E, whose strongly dual E^′ is nuclear, is strongly hereditary reflexive. This concerns the above mentioned de Rham spaces. Problems on R- and Q-homotopy, proper R- and Q-homotopy and proper R- and Q-homotopy at infinity are also considered as well as the coalgebra structure on currents and currents with compact supports.The classical theorem concerning derivation of additive functions with respect to volumes in points is generalized to a theorem on derivation of continuous m-forms with compact supports ωm of an oriented n-dimensional C¹-manifold Mn with respect to its m-dimensional oriented submanifolds Vm in compact regular oriented submanifolds Lk of Mn, 0?k<m?n.     

A minimization problem concerning subsets of a finite set

Let F be the set of subsets of a finite set S, and for H ? F, let H′ denote the elements of F which are contained in some element of H. Given integers m_l and m_l+1 does there exist a subset H of F consisting of exactly m_ll-element subsets of S and m_l+1 (l+1)-element subsets of S such that no two elements of H are related by set-wise inclusion, and if such sets H do exist what the smallest |(l?1)(H′)| can be, where |(l?1)(H′)| is the number of (l?1)-element subsets of S in H′? A generalization of this problem, which was posed by G. Katona, is solved in this paper with the help of the generalized Macaulay theorem [2].     

Morse Decompositions with Infinite Components for Multivalued Semiflows

In this paper we study the theory of Morse decompositions with an infinite number of components in the multivalued framework, proving that for a disjoint infinite family of weakly invariant sets (being all isolated but one) a Lyapunov function ordering them exists if and only if the multivalued semiflow is dynamically gradient. Moreover, these properties are equivalent to the existence of a Morse decomposition.This theorem is applied to a reaction-diffusion inclusion with an infinite number of equilibria.     

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.