Homogeneous Map Research Articles

On the qualitative approximation of Lipschitzian functions

Problem of topological equivalence between a function and certain local approximations is studied. The study is carried out in a neighbourhood of a critical point with the concept of critical point of Clarke's theory. The function belongs to a particular class of non B-differentiable functions. The local approximations are positively homogeneous maps. Using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function. As a consequence we obtain sufficient conditions for the existence of local extremes for the initial map.

Stability and singularities of harmonic maps into 3-spheres

The behavior of harmonic maps from a domain in R 4 into S 3 is discussed. In this paper, we show that if u is a strictly stable stationary harmonic map : B 4→ S 3 , such that the singular set Sing( u) of u consists of {0}, then the degree deg( u,0) of u at 0 is zero; and that if u is a weakly stable harmonic map B 4→ S 3 with Sing( u)={0}, then deg( u,0)=0 or ±1. Furthermore, the characterization of homogeneous stable stationary harmonic maps from B 4 into S 3 is given.

Dimension change, coarse grained coding and pattern recognition in spatio-temporal nonlinear systems.

Several research programs employing spatio-temporal recurrent dynamics and changes in dimensionality have extended the dialog on neural computation and coding beyond classical frameworks such as feed forward and attractor neural networks and feature detectors. Some have emphasized spiking networks, while others emphasize oscillations and synchronization as the locus of computation and coding. In this paper, the formalism of locally connected homogeneous coupled map lattices is described. Its deployment in an extended version of the dynamical recognizer framework is described, and is compared with density coding, computational mechanics, and liquid state machine frameworks for neural computation. A population coding strategy based on coarse graining the continuous valued distribution of all sites in the lattice is developed and examined as a form of dimension reduction. Results on recognition of 3-D objects are reported. In order to better understand the dynamics supporting recognition, measures suggested by these other research programs and computational frameworks were examined. Dynamics trajectories from object recognition trials were examined for correlation with recognition rates and measures of the distance of the representation space statistics between the target objects and noise initial conditions, and the intrinsic separation between different objects in the set to be classified were performed. These results raise questions about the efficacy of density coding as an explanation for the results, and on the validity of recent criticisms that chaotic systems cannot satisfy separation requirements required for real time computation.

The Analysis of Spatial and Temporal Trends in Yield Map Data over Six Years

A quantitative analysis of yield data from four fields over 6 years was carried out to identify the spatial and temporal trends. The methodology was modified from previous work to separate the temporal effects into two parts; the inter-year offset and the temporal variance. The inter-year offset quantifies the overall differences in yield between 1 year and the next, whereas the temporal variance indicates the amount of change at a particular point over time. Results from these fields show that the significant spatial variability found within each individual yield map cancelled out over time, leaving a relatively homogenous spatial trend map. The implications of these findings are that each field should be managed according to the current years’ conditions.

Large-scale mapping of boreal forest in SIBERIA using ERS tandem coherence and JERS backscatter data

Siberia's boreal forests represent an economically and ecologically precious resource, a significant part of which is not monitored on a regular basis. Synthetic aperture radars (SARs), with their sensitivity to forest biomass, offer mapping capabilities that could provide valuable up-to-date information, for example about fire damage or logging activity. The European Commission SIBERIA project had the aim of mapping an area of approximately 1 million km 2 in Siberia using SAR data from two satellite sources: the tandem mission of the European Remote Sensing Satellites ERS-1/2 and the Japanese Earth Resource Satellite JERS-1. Mosaics of ERS tandem interferometric coherence and JERS backscattering coefficient show the wealth of information contained in these data but they also show large differences in radar response between neighbouring images. To create one homogeneous forest map, adaptive methods which are able to account for brightness changes due to environmental effects were required. In this paper an adaptive empirical model to determine growing stock volume classes using the ERS tandem coherence and the JERS backscatter data is described. For growing stock volume classes up to 80 m 3/ha, accuracies of over 80% are achieved for over a hundred ERS frames at a spatial resolution of 50 m.

Material homogeneity mapping using millimetre waves

A millimetre-wave bridge technique for non-destructive material homogeneity mapping is described. The concept of this technique is the local excitation of millimetre waves in testing material and the measurement of the transmitted (reflected) signal's amplitude and phase in different places. Some results of the homogeneity measurements for the dielectric substrates and metallic surfaces are also presented.

On the Euler characteristic of the link of a weighted homogeneous mapping

The paper is concerned with an effective formula for the Euler characteristic of the link of a weighted homogeneous mapping $F:{{\mathbb R}}^{n}\rightarrow {{\mathbb R}}^{k}$ with an isolated singularity. The formula is based on Szafraniec's method for ca

Approximations of nondifferentiable functions and local topological equivalence

A concept of local approximation of a function is introduced. This concept is defined via directional derivatives. In consequence, the local approximation is carried out by a positively homogeneous mapping. We obtain local approximations for functions that are not necessarily locally Lipschitzian nor continuous. This is the case of some large classes of functions such as stable functions or contingently epidifferentiable and directionally Lipschitzian functions. Using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function. This investigation is developed for contingently epidifferentiable functions around a noncritical point, and for noncontingently epidifferentiable functions under particular conditions.

Spectral theorem for convex monotone homogeneous maps, and ergodic control

We consider convex maps f : R n→ R n that are monotone (i.e., that preserve the product ordering of R n ), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of R n , whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

A remark about homogeneous polynomial maps

We consider homogeneous polynomial maps $F\colon {\mathbb R}^{n} \rightarrow {\mathbb R}^{n}$ of degree $p$. We classify the pairs $(p,n)$ for which there exists a surjective and non-proper such map and when the right inverse to $F$ exists but is unbounded.

Polynomial and Regular Maps into Grassmannians

We find a regular deformation retractionn,r (K) :I dem n,r (K) → Gn,r (K) from the manifold Idemn,r (K) of idempotent n × n matrices with rank r to the Grassmannian manifold Gn,r (K) over K the reals, complex numbers or quaternions. Then we derive an injection P C(VC, Idemn,r (K)) → RR(V, Gn,r (K)) from the sets of homotopy classes of complex-valued polynomial to such a set of real-valued regular maps, where VC denotes the Zariski closure in the affine space C n of a subset V ⊆ R n . Furthermore, we list complex-valued polynomial maps S 2 → S 2 of any Brouwer degree and deduce that the map � 2,1(C) :I dem2,1(C) → G2,1(C) yields an isomorphism PC(S 2 , S 2 ) ∼ −→ RR(S 2 , S 2 ) of cyclic infinite homotopy groups. Finally, we show that every nonzero even Brouwer degree of the spheres S n and S n cannot be realized by a real-valued (resp. complex-valued) homogeneous polynomial map provided that n is even.

The deformation multiplicity of a map germ with respect to a Boardman symbol

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

Continuous Extension of Order-Preserving Homogeneous Maps

Maps f defined on the interior of the standard non-negative cone K in RN which are both homogeneous of degree 1 and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson's part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous orderpreserving continuous extension to the whole cone. It follows that the extension must have at least one eigenvector in K - {0}. In the case where the cycle time X(f) of the original map does not exist, such eigenvectors must lie in ∂K - {0}.

SAR image data compression using a tree-structured wavelet transform

SAR image compression is very important in reducing the costs of data storage and transmission in relatively slow channels. The authors propose a compression scheme driven by texture analysis, homogeneity mapping and speckle noise reduction within the wavelet framework. The image compressibility and interpretability are improved by incorporating speckle reduction into the compression scheme. The authors begin with the classical set partitioning in hierarchical trees (SPIHT) wavelet compression scheme, and modify it to control the amount of speckle reduction, applying different encoding schemes to homogeneous and nonhomogeneous areas of the scene. The results compare favorably with the conventional SPIHT wavelet and the JPEG compression methods.

Complete form of X-linked congenital stationary night blindness: refined mapping and evidence of genetic homogeneity.

A number of distinct, partly non-overlapping genetic loci have been reported for the complete type of X-linked congenital stationary night blindness (CSNB1), suggesting genetic heterogeneity. In order to refine the localization of the CSNB1 gene and to demonstrate genetic homogeneity, linkage analysis was performed in two large CSNB1 families. Clinical features consistent with the diagnosis of CSNB1 were documented in five patients from a German seven-generation kindred by full ophthalmological examination including psychophysical and electroretinographical testing. Haplotype analysis in 30 members of the large German family was performed with 38 polymorphic markers predominantly covering the critical region. Linkage analyses defined a locus for CSNB1 with flanking markers DXS8042 and DXS228, refining the interval to 2.5 cM in Xp11.4. In addition, two-point linkage analysis was carried out using the MLINK computer program. In agreement with meiotic breakpoints, lod scores of 3.0 and greater were obtained for markers located to the proximal site of the former 5 cM CSNB consensus interval. A large Dutch CSNB1 family was re-evaluated with markers from the Xp11.4 region, and supports the CSNB1 minimal interval found in the German family. Together with previous results from three unrelated families from Sweden, Sardinia and Great Britain, our results provide evidence of genetic homogeneity in the disorder. Subsequent mutation analyses in CSNB1 patients revealed no pathogenic sequence alterations in DFFRX and CASK genes, but retain candidates for other diseases mapping to that region.

Development of a Homogeneous MAP Kinase Reporter Gene Screen for the Identification of Agonists and Antagonists at the CXCR1 Chemokine Receptor

Development of a Homogeneous MAP Kinase Reporter Gene Screen for the Identification of Agonists and Antagonists at the CXCR1 Chemokine Receptor

Graphical display of fMRI data: visualizing multidimensional space

Visualization of multidimensional data is an integral part of computational statistics and exploratory data analysis (EDA). We show how visualization of fMRI time-courses may be used to reveal the fMRI data structure. We consider fMRI time-courses (TCs) as points in multidimensional space. In simulated and in vivo data, we show that minimum spanning tree (MST)-based sequencing of multivariate time-courses, in combination with a homogeneity map visualization, allows for effective and useful graphical display of the groups of coactivated time-courses obtained by temporal clustering. This display may serve as a tool for investigation of brain connectivity. We also suggest a simple overall display of the entire fMRI data set.

CONTROLLING SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE SYSTEMS

We have effectively controled spatiotemporal chaos in homogeneous and heterogeneous coupled map lattice systems by the same phase space compression. Numerical simulation results show that there is a definite functional relationship between the control results and control parameters in a certain region of compressed phase space, when we control the spatiotemporal chaos to homogeneous stable state. Using this control equation and selecting different phase-space compression parameters to control spatiotemporal chaos in a coupled map lattice systems, we successfully obtain various desired stable patterns.

Infinitely Many Solutions for a Floquet-Type BVP with Superlinearity Indefinite in Sign

By the application of a technique developed by G. J. Butler we find infinitely many solutions of the BVP[formula]where q:[0,ω]→R is allowed to change sign, g:R→R is superlinear and g(x)x>0 for all x≠0, and Λ:R2→R2 is a continuous, positively homogeneous, and nondegenerate map. At first we apply the main result to obtain solutions with a prescribed large number of zeros when Λ is the rotation of a fixed angle λ; second, we find infinitely many subharmonic solutions of any order and, again, solutions with a prescribed large number of zeros for the periodic problem associated to the equation ẍ+cẋ+q(t)g(x)=0, with q and g as above and c∈R.

Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs

Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs