Generic-Type Results for Chaotic Dynamics in Equations with Periodic Forcing Terms

We prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces. The approach is based on global variational methods.     

Global search perspectives for multiobjective optimization

Extending the notion of global search to multiobjective optimization is far than straightforward, mainly for the reason that one almost always has to deal with infinite Pareto optima and correspondingly infinite optimal values. Adopting Stephen Smale’s global analysis framework, we highlight the geometrical features of the set of Pareto optima and we are led to consistent notions of global convergence. We formulate then a multiobjective version of a celebrated result by Stephens and Baritompa, about the necessity of generating everywhere dense sample sequences, and describe a globally convergent algorithm in case the Lipschitz constant of the determinant of the Jacobian is known.     

Infinite lexicographic products

We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.     

Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of the Jacobi elliptic functions. We find explicit expression for these polynomials in terms of elliptic hypergeometric functions. We show that the obtained polynomials are orthogonal on the unit circle with respect to a dense point measure. We also construct corresponding explicit systems of polynomials orthogonal on the interval of the real axis with respect to a dense point measure. They can be considered as an elliptic generalization of the Askey-Wilson polynomials of a special type.      

Topological aspects of the Medvedev lattice

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.     

Dense subtrees in complete Boolean algebras

We characterize complete Boolean algebras with dense subtrees. The main results show that a complete Boolean algebra contains a dense tree if its generic filter collapses the algebra's density to its distributivity number and the reverse holds for homogeneous algebras. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dense computability structures

We examine computability structures on a metric space and the relationships between maximal, separable and dense computability structures. We prove that in a computable metric space which has the effective covering property and compact closed balls for a given computable sequence which is a metric basis there exists a unique maximal computability structure which contains that sequence. Furthermore, we prove that each maximal computability structure on a convex subspace of Euclidean space is dense. We also examine subspaces of Euclidean space on which each dense maximal computability structure is separable and prove that spheres, boundaries of simplices and conics are such spaces.     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

SMOOTHNESS PROPERTIES OF REAL-VALUED FUNCTIONS BASED ON THEIR OSCILLATION

Abstract

We consider the following two selection principles for topological spaces:

Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;

Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.

We show that for separable metric space X one of these principles holds for the space C_p (X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhász.     

Unbounded Hermitian operators and relative reproducing kernel Hilbert space

We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

Counting Homomorphisms to Sparse Graphs

We define nowhere dense and somewhere dense classes by means of counting of homomorphisms from test graphs. This seems to be bridging the gap between existential and counting theorems (for graph homomorphisms) and it has application to complexity of Boolean queries.     

ODC: Frame for definition of Dense codes

Natural language compression has made great progress in the last two decades. The main step in this evolution was the introduction of word-based compression by Moffat. Another improvement came with so-called Dense codes, which proved to be very fast in compression and decompression while keeping a good compression ratio and direct search capability. Many variants of the Dense codes have been described, each of them using its own definition. In this paper, we present a generalized concept of dense coding called Open Dense Code (ODC), which aims to be a frame for the definition of many other dense code schemas. ODC underlines common features of the dense code schemas but at the same time allows one to express the divergences of each of them. Using the frame of ODC, we present two new word-based statistical compression algorithms based on the dense coding idea: Two Byte Dense Code (TBDC) and Self-Tuning Dense Code (STDC). Our algorithms improve the compression ratio and are considerate to smaller files, which are very often omitted by other compressors.     

ACO classification of thermogram symmetry features for breast cancer diagnosis

Breast cancer is the most commonly diagnosed form of cancer in women. While mammography is the standard modality for detecting breast cancer, it has been shown that medical thermography, which uses cameras with sensitivities in the thermal infrared, is also well suited for the task, especially in dense tissue and for small tumors. In this paper, we present an approach of analysing breast thermograms that first extracts a series of image features describing bilateral (a)symmetry between left and right breast regions, and then uses these features in a subsequent classification step. For the classification, we employ an ant colony optimisation based pattern recognition algorithm that is shown to provide a concise rule base with good classification performance.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

Interfacial Contact Model in a Dense Network of Elastic Materials

Functional Analysis and Its Applications - We consider a dense network of elastic materials modeled by a dense network of elastic disks. More specifically, we consider a dense network of elastic...     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

The Distance-2-Sets of the Slim Dense Near Hexagons

We determine all distance-2-sets and all valuations of the 11 slim dense near hexagons. We consider this as a necessary step in order to classify all slim dense near octagons. These near octagons will be classified in a forthcoming paper. Received November 29, 2004