On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

Morse Decompositions for Nonautonomous General Dynamical Systems

The Morse decomposition theory for nonautonomous general dynamical systems (set-valued dynamical systems) and differential inclusions is established. The stability of Morse decompositions of pullback attractors is also addressed.     

Infinite Time-Dependent Network Equilibria with a Multivalued Cost Function

In this paper, we consider the dynamic traffic network equilibria with possibly an infinite number of routes, a possibly multivalued cost function, and a not necessarily reflexive Banach space of flow trajectories. We investigate the existence of equilibria under a monotonicity assumption on the cost function, as well as an equivalent condition for equilibria under additional constraints. Finally, we give an iterative method for the computation of equilibria. Our results generalize and extend previous results in Refs. 1–2.Communicated by F. GiannessiThe authors are thankful to the referee and Professor F. Giannessi for valuable suggestions and comments leading the paper to its present form.All correspondence should be sent to the first author.     

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.     

On the Complexity of Multivalued Logic Functions over Some Infinite Basis

Under study is the complexity of the realization of k-valued logic functions (k ≥ 3) by logic circuits in the infinite basis consisting of the Post negation (i.e., the function (x + 1) mod k) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds of complexity, which differ from one another at most by 1 and have the form 3 log₃(d(f)+ 1)+O(1), where d(f) is the maximal number of the decrease of the value of f taken over all increasing chains of tuples of values of the variables. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.     

Sparse Components of Images and Optimal Atomic Decompositions

Recently, Field, Lewicki, Olshausen, and Sejnowski have reported efforts to identify the ``Sparse Components' of image data. Their empirical findings indicate that such components have elongated shapes and assume a wide range of positions, orientations, and scales. To date, sparse components analysis (SCA) has only been conducted on databases of small (e.g., 16 by 16) image patches and there seems limited prospect of dramatically increased resolving power. In this paper, we apply mathematical analysis to a specific formalization of SCA using synthetic image models, hoping to gain insight into what might emerge from a higher-resolution SCA based on n by n image patches for large n but a constant field of view. In our formalization, we study a class of objects \cal F in a functional space; they are to be represented by linear combinations of atoms from an overcomplete dictionary, and sparsity is measured by the ℓ <sup>p</sup> -norm of the coefficients in the linear combination. We focus on the class \cal F = \sc Star <sup>α</sup> of black and white images with the black region consisting of a star-shaped set with an α -smooth boundary. We aim to find an optimal dictionary, one achieving the optimal sparsity in an atomic decomposition uniformly over members of the class \sc Star <sup>α</sup> . We show that there is a well-defined optimal sparsity of representation of members of \sc Star <sup>α</sup> ; there are decompositions with finite ℓ <sup>p</sup> -norm for p > 2/(α+1) but not for p < 2/(α+1) . We show that the optimal degree of sparsity is nearly attained using atomic decompositions based on the wedgelet dictionary. Wedgelets provide a system of representation by elements in a dyadically organized collection, at all scales, locations, orientations, and positions. The atoms of our atomic decomposition contain both coarse-scale dyadic ``blobs,' which are simply wedgelets from our dictionary, and fine-scale ``needles,' which are differences of pairs of wedgelets. The fine-scale atoms used in the adaptive atomic decomposition are highly anisotropic and occupy a range of positions, scales, and locations. This agrees qualitatively with the visual appearance of empirically determined sparse components of natural images. The set has certain definite scaling properties; for example, the number of atoms of length l scales as 1/l , and, when the object has α -smooth boundaries, the number of atoms with anisotropy \approx A scales as \approx A ^α-1 . August 16, 1999. Date revised: April 24, 2000. Date accepted: April 4, 2000.     

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Abstract

Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver Γ_A of A in which each Γ_A-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τ_A-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α?γ and multiple coray-ray insertions of type α?γ.     

Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays

Differential equations with state-dependent delay can often be written as (t)=f(x_t) with a continuously differentiable map f from an open subset of the space C¹=C¹([-h,0], {}^n), {h>0}, into {}^n. In a previous paper we proved that under two mild additional conditions the set is a continuously differentiable n-codimensional submanifold of C ¹, on which the solutions define a continuous semiflow F with continuously differentiable solution operators F_t=F(t,·), t 0. Here we show that under slightly stronger conditions the semiflow F is continuously differentiable on the subset of its domain given by {t> h}. This yields, among others, Poincaré return maps on transversals to periodic orbits. All hypotheses hold for an example which is based on Newton's law and models automatic position control by echo.     

Halpern-Type Iterative Algorithm for an Infinite Family of Relatively Quasi-Nonexpansive Multivalued Mappings and Equilibrium Problem in Banach Spaces

In this paper, we introduce Halpern-type iterative algorithm for finding a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the set of solutions of an equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence to common elements of two set is established. Our results improved and extend the corresponding results announced by many others.     

混合单调半流与泛函微分方程的稳定性

本文首先提出混合单调半流的概念和泛函微分方程生成这种半流的条件。然后，利用半流的混合单调性，我们得到关于泛函微分方程的渐近稳定性和全局稳定性的新结果．     

Multivalued dynamic systems with weights

We consider m-valued transformations of a probability space (X, \(\mathcal{B}\), µ) endowed with a set of weights \(\left\{ {\alpha _j :X \to (0,1],\sum\limits_{j = 1}^m {\alpha _j } \equiv 1} \right\}\). For this case we introduce analogs of the basic notions of the ergodic theory, namely, the measure invariance, ergodicity, Koopman and Frobenius-Perron operators. We study the properties of these operators, prove ergodic theorems, and give some examples. We also propose a technique for reducing some problems of the fractal geometry to those of the functional analysis.     

Multivalued equations for granular avalanches

We study existence and uniqueness for the scalar equation of first order hyperbolic type. The RHS is in the form of a multifunction, which is physically motivated by an avalanche flow model. Therefore some kind of monotonicity to the RHS has to be assumed.     

渐近上半紧的多值随机半流的随机吸引子

该文研究多值随机半流的随机吸引子的存在性．首先证明在拉回渐近上半紧及吸收的条件下,关于极限集的一个抽象结果,然后证明了随机的吸引子的存在性．     

Lifting Modules with Indecomposable Decompositions

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.     

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

一个多值增算子的不动点定理

研究了一个多值增算子的不动点问题,获得了几个存在性定理,所获结果推广了已知的结论.