Functional Analysis Framework Research Articles

Lower bounds for the approximation with variation-diminishing splines

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval [0,1] in terms of classical moduli of smoothness depending on the degree of the spline basis. For this purpose we use a functional analysis framework in order to characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates.

Functional logistic regression approach to detecting gene by longitudinal environmental exposure interaction in a case-control study.

Most complex human diseases are likely the consequence of the joint actions of genetic and environmental factors. Identification of gene-environment (G × E) interactions not only contributes to a better understanding of the disease mechanisms, but also improves disease risk prediction and targeted intervention. In contrast to the large number of genetic susceptibility loci discovered by genome-wide association studies, there have been very few successes in identifying G × E interactions, which may be partly due to limited statistical power and inaccurately measured exposures. Although existing statistical methods only consider interactions between genes and static environmental exposures, many environmental/lifestyle factors, such as air pollution and diet, change over time, and cannot be accurately captured at one measurement time point or by simply categorizing into static exposure categories. There is a dearth of statistical methods for detecting gene by time-varying environmental exposure interactions. Here, we propose a powerful functional logistic regression (FLR) approach to model the time-varying effect of longitudinal environmental exposure and its interaction with genetic factors on disease risk. Capitalizing on the powerful functional data analysis framework, our proposed FLR model is capable of accommodating longitudinal exposures measured at irregular time points and contaminated by measurement errors, commonly encountered in observational studies. We use extensive simulations to show that the proposed method can control the Type I error and is more powerful than alternative ad hoc methods. We demonstrate the utility of this new method using data from a case-control study of pancreatic cancer to identify the windows of vulnerability of lifetime body mass index on the risk of pancreatic cancer as well as genes that may modify this association.

A Multimodal Analysis of Image-text Relations in Picture Books

Image-text relations are expounded based on Systemic Functional Multimodal Analysis framework in this article. A though analysis of the logical relations between the visual and verbal elements in picture books is also given in terms of elaboration, extension, enhancement and projection. Furthermore, Divergence, a fundamental type of image-text relation in picture books, is thoroughly investigated. Generic framework of picture book is proposed for the exploration of image-text relations.

Three Dimensional Sound Field Reproduction using Multiple Circular Loudspeaker Arrays: Functional Analysis Guided Approach

Three dimensional sound field reproduction based on higher order Ambisonics requires the placement of loudspeakers on a sphere that surrounds the target reproduction region. The deployment of a spherical array is not trivial especially for implementation in real rooms where the placement flexibility is highly desirable. This paper proposes a design of multiple circular loudspeaker arrays for reproducing three dimensional sound fields originating from a limited region of interest. We apply a functional analysis framework to formulate the sound field reproduction problem in a closed form. Secondary source distributions and target sound fields are modeled as two Hilbert spaces and mapped by an integral operator and its adjoint operator, from which a self-adjoint operator is constructed and the singular value decomposition is applied to represent source distributions and sound fields with two sets of interrelated singular functions. We derive the solutions for a circular secondary source arrangement and propose the design of placing multiple circular loudspeaker arrays only over the limited region of interest. Such a design allows for non-spherical and non-uniform loudspeaker placement and thus provides a flexible array arrangement. The reproduction accuracy of the proposed method is verified through numerical simulations.

REGULARIZED 3D FUNCTIONAL REGRESSION FOR BRAIN IMAGE DATA VIA HAAR WAVELETS.

The primary motivation and application in this article come from brain imaging studies on cognitive impairment in elderly subjects with brain disorders. We propose a regularized Haar wavelet-based approach for the analysis of three-dimensional brain image data in the framework of functional data analysis, which automatically takes into account the spatial information among neighboring voxels. We conduct extensive simulation studies to evaluate the prediction performance of the proposed approach and its ability to identify related regions to the outcome of interest, with the underlying assumption that only few relatively small subregions are truly predictive of the outcome of interest. We then apply the proposed approach to searching for brain subregions that are associated with cognition using PET images of patients with Alzheimer's disease, patients with mild cognitive impairment, and normal controls.

Locally convex structures on higher local fields

Higher local fields can be described as locally convex vector spaces once an embedding of a local field into them has been fixed. This extends previous two-dimensional results of the author. We study these spaces in the framework of nonarchimedean functional analysis. In particular, we introduce new classes of bounded and compactoid submodules and establish a self-duality result once the dual space has been suitably topologized.

Arabidopsis transcriptome analysis reveals key roles of melatonin in plant defense systems.

Melatonin is a ubiquitous molecule and exists across kingdoms including plant species. Studies on melatonin in plants have mainly focused on its physiological influence on growth and development, and on its biosynthesis. Much less attention has been drawn to its affect on genome-wide gene expression. To comprehensively investigate the role(s) of melatonin at the genomics level, we utilized mRNA-seq technology to analyze Arabidopsis plants subjected to a 16-hour 100 pM (low) and 1 mM (high) melatonin treatment. The expression profiles were analyzed to identify differentially expressed genes. 100 pM melatonin treatment significantly affected the expression of only 81 genes with 51 down-regulated and 30 up-regulated. However, 1 mM melatonin significantly altered 1308 genes with 566 up-regulated and 742 down-regulated. Not all genes altered by low melatonin were affected by high melatonin, indicating different roles of melatonin in regulation of plant growth and development under low and high concentrations. Furthermore, a large number of genes altered by melatonin were involved in plant stress defense. Transcript levels for many stress receptors, kinases, and stress-associated calcium signals were up-regulated. The majority of transcription factors identified were also involved in plant stress defense. Additionally, most identified genes in ABA, ET, SA and JA pathways were up-regulated, while genes pertaining to auxin responses and signaling, peroxidases, and those associated with cell wall synthesis and modifications were mostly down-regulated. Our results indicate critical roles of melatonin in plant defense against various environmental stresses, and provide a framework for functional analysis of genes in melatonin-mediated signaling pathways.

Analysis of juggling data: Alignment, extraction, and modeling of juggling cycles

In this paper we present results from alignment, extraction, and statistical analysis of juggling trajectories using an elastic functional data analysis framework. This framework, specifically adapted for analyzing cyclostationary signals using an elastic Riemannian metric, was introduced recently by Kurtek et al. [2]. It relies on a special representation of curves called the square-root velocity function to pose the alignment problem as an optimization over the re-parametrization space. The cost function for alignment is a proper metric and is used to separate phase and amplitude components of juggling cycles. We present results of segmenting juggling trials into cycles, separating phase and amplitude components of cycles, and developing principal component analysis (PCA) based statistical models for these individual components.

When can the environmental profile and emissions reduction be optimised independently of the pollutant level?

Consider a model for optimal timing of a policy measure which changes the emission rate, e.g. trading off the cost of reduction against the time-additive aggregate of environmental damage, the disutility from the pollutant stock the infrastructure contributes to. Intuitively, the optimal timing for an infinitesimal pollution source should reasonably not depend on its historical contribution to the stock, as this is negligible. Dropping the size assumption, we show how to reduce the minimisation problem to one not depending on the history of , under linear evolution and suitable linearity or additivity conditions on the damage functional. We employ a functional analysis framework which allows for delay equations, non-Markovian driving noise, a choice between discrete and continuous time, and a menu of integral concepts covering stochastic calculi less frequently used in resource and environmental economics. Examples are given under the common (Markovian Itô) stochastic analysis framework.

Analytic imperfect bifurcation theorem and the Liouville‐Gel’fand equation on a perturbed annular domain

AbstractAn imperfect bifurcation is studied in a framework of functional analysis. We obtain a sufficient condition for an imperfect bifurcation and classify the types of imperfect bifurcations when an analyticity condition is imposed. It is almost impossible to ascertain the assumptions of existing imperfect bifurcation theorems when the degenerate solution is not constant. Our sufficient condition does not require the inverse of the linearized operator. As a nontrivial application of our imperfect bifurcation theorem, we are concerned with the Liouville‐Gel’fand equation on a two‐dimensional perturbed annular domain.

The extrapolation of numerical eigenvalues by finite elements for differential operators

This paper discusses the extrapolation of numerical eigenvalues by finite elements for differential operators and obtains the following new results: (a) By extending a theorem of eigenvalue error estimate, which was established by Osborn, a new expansion of eigenvalue error is obtained. Many achievements, which are about the asymptotic expansions of finite element methods of differential operator eigenvalue problems, are brought into the framework of functional analysis. (b) The Richardson extrapolation of nonconforming finite elements for multiple eigenvalues and splitting extrapolation of finite elements based on domain decomposition of non-selfadjoint differential operators for multiple eigenvalues are achieved. In addition, numerical examples are provided to support the theoretical analysis.

Varying coefficient model for modeling diffusion tensors along white matter tracts

Diffusion tensor imaging provides important information on tissue structure and orientation of fiber tracts in brain white matter in vivo. It results in diffusion tensors, which are 3×3 symmetric positive definite (SPD) matrices, along fiber bundles. This paper develops a functional data analysis framework to model diffusion tensors along fiber tracts as functional data in a Riemannian manifold with a set of covariates of interest, such as age and gender. We propose a statistical model with varying coefficient functions to characterize the dynamic association between functional SPD matrix-valued responses and covariates. We calculate weighted least squares estimators of the varying coefficient functions for the Log-Euclidean metric in the space of SPD matrices. We also develop a global test statistic to test specific hypotheses about these coefficient functions and construct their simultaneous confidence bands. Simulated data are further used to examine the finite sample performance of the estimated varying co-efficient functions. We apply our model to study potential gender differences and find a statistically significant aspect of the development of diffusion tensors along the right internal capsule tract in a clinical study of neurodevelopment.

Electrified thin films: Global existence of non-negative solutions

We consider an equation modeling the evolution of a viscous liquid thin film wetting a horizontal solid substrate destabilized by an electric field normal to the substrate. The effects of the electric field are modeled by a lower order non-local term. We introduce the good functional analysis framework to study this equation on a bounded domain and prove the existence of weak solutions defined globally in time for general initial data (with finite energy).

PROCESSING CONTINUOUS QUERIES ON SENSOR-BASED MULTIMEDIA DATA STREAMS BY MULTIMEDIA DEPENDENCY ANALYSIS AND ONTOLOGICAL FILTERING

We present a mathematical model of multimedia data streams and a framework for multimedia functional dependency analysis. The dual objectives are to effectively design multimedia data streams schema and to efficiently process continuous queries on sensor-based multimedia data streams. To further improve query processing, we introduce the concept of ontological filtering. A software tool to add multimedia functional dependencies to ontology is developed. Based upon multimedia functional dependency analysis and ontological filtering, query processing algorithms, illustrative examples and experimental results for sensor-based multimedia data streams continuous querying are presented to demonstrate the practical applications of our approach.

Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities

How can repulsive and attractive forces, acting on a conservative system, create stable travelling patterns or branching instabilities? We have proposed to study this question in the framework of the hyperbolic Keller–Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a field which is also called socio-biology. We continue earlier modelling work, where we have shown numerically that branching patterns arise for this system and we have analysed this instability by formal asymptotics for small diffusivity of the chemo-repellent.Here we are interested in the more general situation, where the diffusivities of both the chemo-attractant and the chemo-repellent are positive. To do so, we develop an appropriate functional analysis framework. We apply our method to two cases. Firstly we analyse steady states. Secondly we analyse travelling waves when neglecting the degradation coefficient of the chemo-repellent; the unique wave speed appears through a singularity cancellation which is the main theoretical difficulty. This shows that in different situations the cell density takes the shape of a plateau.The existence of steady states and travelling plateaus are a symptom of how rich the system is and why branching instabilities can occur. Numerical tests show that large plateaus may split into smaller ones, which remain stable.

Generalized random set framework for functional enrichment analysis using primary genomics datasets

Functional enrichment analysis using primary genomics datasets is an emerging approach to complement established methods for functional enrichment based on predefined lists of functionally related genes. Currently used methods depend on creating lists of 'significant' and 'non-significant' genes based on ad hoc significance cutoffs. This can lead to loss of statistical power and can introduce biases affecting the interpretation of experimental results. We developed and validated a new statistical framework, generalized random set (GRS) analysis, for comparing the genomic signatures in two datasets without the need for gene categorization. In our tests, GRS produced correct measures of statistical significance, and it showed dramatic improvement in the statistical power over other methods currently used in this setting. We also developed a procedure for identifying genes driving the concordance of the genomics profiles and demonstrated a dramatic improvement in functional coherence of genes identified in such analysis. GRS can be downloaded as part of the R package CLEAN from http://ClusterAnalysis.org/. An online implementation is available at http://GenomicsPortals.org/.

Dynamical systems on lattices with decaying interaction I: A functional analysis framework

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites.We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local.We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.

Discourse analysis versus philosophic reading of a literary text: The herne's egg, W.B. Yeats

This study explores a unique poetic narrative from two perspectives: a sociolinguistic point of view within the framework of functional discourse analysis and a literary critique through the prism of Indian philosophic ideas in the text. The research method is based on juxtaposing a social-semiotic interpretation and a literary commentary. Firstly, the article applies the Hallidayan model of the dimensions of discourse – field, tenor and mode – to an excerpt from Yeats's poetic drama, as the context of situation. Secondly, the article identifies specific Indian philosophic concepts in the text and establishes how they influence the structure of dramatic verse and the characterization of protagonists. The discussion reveals the extent of complementarity and tension between the two approaches and how they change the reading of the text when open to disparate disciplinary perspectives. Final results signify the potential and applicability of the analytical categories to research across diverse discourses.

Structural basis for the function of DEAH helicases

DEAH helicases participate in pre-messenger RNA splicing and ribosome biogenesis. The structure of yeast Prp43p-ADP reveals the homology of DEAH helicases to DNA helicases and the presence of an oligonucleotide-binding motif. A beta-hairpin from the second RecA domain is wedged between two carboxy-terminal domains and blocks access to the occluded RNA binding site formed by the RecA domains and a C-terminal domain. ATP binding and hydrolysis are likely to induce conformational changes in the hairpin that are important for RNA unwinding or ribonucleoprotein remodelling. The structure of Prp43p provides the framework for functional and genetic analysis of all DEAH helicases.