Recurrent sets for endomorphisms of topological groups

The concept of n-normed spaces is a generalization of the concept of normed spaces. Some characteristics of n-normed spaces have been discussed by many researchers. These spaces are usually observed using a set of n linear independent vectors. In this paper, we will construct quotient spaces of an n-normed space with respect to n linear independent vectors. We define a norm in each quotient space by using the n-norm that we have. Norms of these quotient spaces will be a new viewpoint in observing characteristics of n-normed spaces.

The concept of moduli space was introduced by Riemann in his study of the conformal (or equivalently, complex) structures on a Riemann surface. Let us consider the simplest non-trivial case, namely, that of a Riemann surface of genus 1 or the torus T 2. The set of all complex structures \(\mathcal{C}({T}^{2})\) on the torus is an infinite-dimensional space acted on by the infinite-dimensional group Diff(T 2). The quotient space $$\mathcal{M}({T}^{2}) := \mathcal{C}({T}^{2})/\mathrm{Diff}({T}^{2})$$ is the moduli space of complex structures on T 2. Since T 2 with a given complex structure defines an elliptic curve, \(\mathcal{M}({T}^{2})\) is, in fact, the moduli space of elliptic curves. It is well known that a point ω = ω1 + iω2 in the upper half-plane H (ω2 > 0) determines a complex structure and is called the modulus of the corresponding elliptic curve. The modular group SL(2, Z) acts on H by modular transformations and we can identify \(\mathcal{M}({T}^{2})\) with H ∕ SL(2, Z). This is the reason for calling \(\mathcal{M}({T}^{2})\) the space of moduli of elliptic curves or simply the moduli space. The topology and geometry of the moduli space has rich structure. The natural boundary ω2 = 0 of the upper half plane corresponds to singular structures. Several important aspects of this classical example are also found in the moduli spaces of other geometric structures. Typically, there is an infinite-dimensional group acting on an infinite-dimensional space of geometric structures with quotient a “nice space” (for example, a finite-dimensional manifold with singularities). For a general discussion of moduli spaces arising in various applications see, for example, [193].

Before resuming the general study of the geometry of numbers, it is convenient to introduce here the concept of the quotient space of an n-dimensional space by a lattice. This concept plays an important role in the discussion of inhomogeneous problems in Chapter XI: but we shall also need it in Chapter VIII as it gives the most natural interpretation of Minkowski’S theorem about the successive minima of a convex body with respect to a lattice.

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group is trivial.

Let G be an arbitrary topological group. We prove that the cellularity of G is equal to the cellularity of the quotient space G / K {G/K} for every compact subgroup K of G.

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

This paper reviews some of the recent progresses in understanding the dynamical and statistical properties of anyons in fractional quantum Hall (FQH) systems. These are strongly interacting topological systems for electrons confined to a two-dimensional manifold, and anyons are fundamental particles emerging from the truncation of the Hilbert space as a result of both the kinetic and interaction energies. We introduce the concept of the conformal Hilbert spaces (CHS) from such Hilbert space truncation, and the hierarchy of these Hilbert spaces allows us to understand the internal structures of the anyons that can strongly affect their dynamics. The bulk-edge correspondence and the conformal mapping of the CHS also enable us to derive a rigorous bosonization scheme for anyons in these two-dimensional systems, and to capture the statistical interaction between anyons in the form of microscopic interaction Hamiltonians. Examples about the fractionalization of Laughlin quasiholes and a new family of bosonic FQH phases as the dual descriptions of the well-known fermionic FQH phases are given, as applications to the proposed theoretical constructions in this work.

Both this paper and Chain recurrence and attraction in noncompact spaces, [Ergodic Theory Dynamical Systems (to appear)] are concerned with the question of extending certain results obtained by C. Conley for dynamical systems on compact spaces to systems on arbitrary metric spaces.The basic result is the analogue of Conley's theorem that characterizes the chain recurrent set of / in terms of the attractors of / and their basins of attraction.The point of view taken in the above-mentioned paper was that the given metric was of primary importance rather than the topology that it generated.The purpose of this note is to give results that depend on the topology induced by a metric rather than on the particular choice of the metric.

In this position paper we describe the concept of geographic relevance and its potential for mobile location-based services employing the mobile Internet. We argue that existing LBS have a too limited concept of location and its application for filtering geographic content. We propose an approach for geographic relevance that extends LBS and location-aware web applications and aims at better supporting mobile users' decision-making based on geographic information. After a short description of an ongoing project we discuss the different roles of location and the different conceptions of space that can be involved in assessing and representing geographic relevance. Finally we provide a few concluding statements that aim at stimulating a cross-disciplinary discussion about location and its importance for relevance.

Based on a simplified model, the methods and rules for generating multi-cavity chaotic map are presented. On this basis, a new rhombic cavity hyperchaotic map is constructed. Some typical dynamic properties of the new system are investigated, including phase trace, Lyapunov exponent spectrum (LEs) and bifurcations. The results show that the new chaotic map has rich dynamical behaviors, including complicated phase space trajectory, hyperchaotic behavior, large Lyapunov exponent. By changing the system parameters, the rhombic cavity with adjustable size and number are obtained. The rhombic cavity hyperchaotic map is improved to the multi-directional mode which has more complex topological structure than the original system. The complexity of the improved system is high at the entire parameter space. To verify the validity of the model and the feasibility of the systems, the digital circuits of the rhombic cavity hyperchaotic maps are implemented based on DSP technique. This lays a foundation for the application of these hyperchaotic maps in chaotic secure communication.

A higher order method of interpolated cell mapping

This paper addresses the problem of controlling the predator-prey like model of the interaction among CD4/sup +/ T-cell, CD8/sup +/ T-cell and HIV-1 by an external drug agency. By exploring the dynamic properties of the system, the original system is first regrouped into two subsystems, then a nonlinear global controller is presented by designing two controllers over two complimentary zones: a local controller on a finite region and a global boundary controller over its compliment. The local controller is developed to guarantee the nonnegative properties and avoid control singularity problem within the neighborhood of origin /spl Omega/. The complimentary controller is designed via backstepping for both the subsystems respectively over the complimentary region. The closed-loop system is globally stable at nominal values, the resulting controller is singularity free and guarantees the nonnegative properties. Simulation results are demonstrated to show the effectiveness of the proposed methods.

In this paper, we address the problem of controlling the synaptic plasticity to constraint bursting activity in epileptic seizures by a direct drug injection. With a good understanding of dynamical changes in seizures onset and the mechanisms that cause these changes, we design a control based on a model of interaction between synaptic strength and the intracellular calcium concentration [Ca <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+</sup> ] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> . By exploring the dynamic properties of the system, the original system is first regrouped into two subsystems, the first one is a self-stable system; and the second one is a second order system in strict feedback form. Then, nonlinear control for the second subsystem is presented through backstepping design. The closed-loop system is shown to be globally stable. Simulation studies are conducted to show the effectiveness of the control for stopping the bursting activity in brain with epilepsy.

For a class of third-order nonlinear systems, a one-dimensional model reflecting dynamic properties of the original system is proposed. Properties of this map, called the squeezed spiral map, are investigated. Dynamic phenomena encountered in the original system are reproduced by the model map with very good accuracy. >

The common tags given by multiple users to a particular document are often semantically relevant to the document and each tag represents a specific topic. In this paper, we attempt to emulate human tagging behavior to recommend tags by considering the concepts contained in documents. Specifically, we represent each document using a few most relevant concepts contained in the document, where the concept space is derived from Wikipedia. Tags are then recommended based on the tag concept model derived from the annotated documents of each tag. Evaluated on a Delicious dataset of more than 53K documents, the proposed technique achieved comparable tag recommendation accuracy as the state-of-the-art, while yielding an order of magnitude speed-up.